0`
0`The class of abelian groups,{} \spadignore{i.e.} additive monoids where each element has an additive inverse. \blankline
42`Abelian monoid ring elements (not necessarily of finite support) of this ring are of the form formal SUM (r_i * e_i) where the r_i are coefficents and the e_i,{} elements of the ordered abelian monoid,{} are thought of as exponents or monomials. The monomials commute with each other,{} and with the coefficients (which themselves may or may not be commutative). See \spadtype{FiniteAbelianMonoidRing} for the case of finite supp
42`--ort. A useful common model for polynomials and power series. Conceptually at least,{} only the non-zero terms are ever operated on.
117`The class of multiplicative monoids,{} \spadignore{i.e.} semigroups with an additive identity element. \blankline
162`the class of all additive (commutative) semigroups,{} \spadignore{i.e.} a set with a commutative and associative operation \spadop{+}. \blankline
209`The notion of aggregate serves to model any data structure aggregate,{} designating any collection of objects,{} with heterogenous or homogeneous members,{} with a finite or infinite number of members,{} explicitly or implicitly represented. An aggregate can in principle represent everything from a string of characters to abstract sets such as "the set of \spad{x} satisfying relation {\em r(x)}" An attribute \spadatt{finiteAggregate} is used to assert that a doma
209`--in has a finite number of elements.
246`The category of associative algebras (modules which are themselves rings). \blankline
304`Model for algebraically closed fields.
357`Model for algebraically closed function spaces.
442`An approximation to the real numbers.
486`Category for the inverse hyperbolic trigonometric functions.
545`Category for the inverse trigonometric functions.
608`An association list is a list of key entry pairs which may be viewed as a table. It is a poor mans version of a table: searching for a key is a linear operation.
697`A bag aggregate is an aggregate for which one can insert and extract objects,{} and where the order in which objects are inserted determines the order of extraction. Examples of bags are stacks,{} queues,{} and dequeues.
747`\spadtype{BasicType} is the basic category for describing a collection of elements with \spadop{=} (equality).
789`A \spadtype{BiModule} is both a left and right module with respect to potentially different rings. \blankline
844`A bialgebra is a coalgebra which at the same time is an algebra such that the comultiplication is also an algebra homomorphism. MxM: Module(\spad{R}) should be replaced by a more restricted category,{} but it is not clear at this point which one.
918`A binary-recursive aggregate has 0,{} 1 or 2 children and serves as a model for a binary tree or a doubly-linked aggregate structure
980`\spadtype{BinaryTreeCategory(S)} is the category of binary trees: a tree which is either empty or else is a \spadfun{node} consisting of a value and a \spadfun{left} and \spadfun{right},{} both binary trees.
1043`The bit aggregate category models aggregates representing large quantities of Boolean data.
1086`A cachable set is a set whose elements keep an integer as part of their structure.
1130`This is an \spadtype{AbelianMonoid} with the cancellation property,{} \spadignore{i.e.} \spad{ a+b = a+c => b=c }. This is formalised by the partial subtraction operator,{} which satisfies the axioms listed below: \blankline
1187`The class of all domains which have canonical representaion,{} that is if mathematicaly equal elements have the same data structures.
1229`Rings of Characteristic Non Zero
1282`Rings of Characteristic Zero.
1331`A coalgebra A over a ring is an \spad{R}-module with a coassociative comultiplication from A to the tensor product of A with itself and which possesses a counit.
1405`A is coercible to \spad{B} means any element of A can automatically be converted into an element of \spad{B} by the interpreter.
1455`A collection is a homogeneous aggregate which can built from list of members. The operation used to build the aggregate is generically named \spadfun{construct}. However,{} each collection provides its own special function with the same name as the data type,{} except with an initial lower case letter,{} \spadignore{e.g.} \spadfun{list} for \spadtype{List},{} \spadfun{flexibleArray} for \spadtype{FlexibleArray},{} and so on.
1503`Category for the usual combinatorial functions.
1563`CombinatorialOpsCategory is the category obtaining by adjoining summations and products to the usual combinatorial operations.
1620`The category of commutative rings with unity,{} \spadignore{i.e.} rings where \spadop{*} is commutative,{} and which have a multiplicative identity. element.
1668`The class of all commutative semigroups in multiplicative notation. In other words domains \spad{D} with \spad{"*": (D,{}D) -> D} which is commutative. Typically applied to rings.
1713`The class of set equipped with possibly unnatural linear order (needed for technical reasons).
1755`This category represents the extension of a ring by a square root of \spad{-1}.
1821`A is convertible to \spad{B} means any element of A can be converted into an element of \spad{B},{} but not automatically by the interpreter.
1874`A dequeue is a doubly ended stack,{} that is,{} a bag where first items inserted are the first items extracted,{} at either the front or the back end of the data structure.
1928`This category is a collection of operations common to both categories \spadtype{Dictionary} and \spadtype{MultiDictionary}
1993`A dictionary is an aggregate in which entries can be inserted,{} searched for and removed. Duplicates are thrown away on insertion. This category models the usual notion of dictionary which involves large amounts of data where copying is impractical. Principal operations are thus destructive (non-copying) ones.
2048`Differential extensions of a ring \spad{R}. Given a differentiation on \spad{R},{} extend it to a differentiation on \%.
2108`\spadtype{DifferentialPolynomialCategory} is a category constructor specifying basic functions in an ordinary differential polynomial ring with a given ordered set of differential indeterminates. In addition,{} it implements defaults for the basic functions. The functions \spadfun{order} and \spadfun{weight} are extended from the set of derivatives of differ
2108`--ential indeterminates to the set of differential polynomials. Other operations provided on differential polynomials are \spadfun{leader},{} \spadfun{initial},{} \spadfun{separant},{} \spadfun{differentialVariables},{} and \spadfun{isobaric?}. Furthermore,{} if the ground ring is a differential ring,{} then evaluation (substitution of differential indeterminates by elements of the ground ring or by differential polynomials) is provided by \spadfun{eval}. A convenient way of referencing derivati
2108`--ves is provided by the functions \spadfun{makeVariable}. \blankline To construct a domain using this constructor,{} one needs to provide a ground ring \spad{R},{} an ordered set \spad{S} of differential indeterminates,{} a ranking \spad{V} on the set of derivatives of the differential indeterminates,{} and a set \spad{E} of exponents in bijection with the set of differential monomials in the given differential indeterminates. \blankline
2253`An ordinary differential ring,{} that is,{} a ring with an operation \spadfun{differentiate}. \blankline
2302`\spadtype{DifferentialVariableCategory} constructs the set of derivatives of a given set of (ordinary) differential indeterminates. If \spad{x},{}...,{}\spad{y} is an ordered set of differential indeterminates,{} and the prime notation is used for differentiation,{} then the set of derivatives (including zero-th order) of the differential indeterminates is \spad{x},{}\spad{x'},{}\spad{x''},{}...,{} \spad{y},{}\spad{y'},{}\spad{
2302`--y''},{}... (Note: in the interpreter,{} the \spad{n}-th derivative of \spad{y} is displayed as \spad{y} with a subscript \spad{n}.) This set is viewed as a set of algebraic indeterminates,{} totally ordered in a way compatible with differentiation and the given order on the differential indeterminates. Such a total order is called a ranking of the differential indeterminates. \blankline A domain in this category is needed to construct a differential polynomial domain. Differential polynomials 
2302`--are ordered by a ranking on the derivatives,{} and by an order (extending the ranking) on on the set of differential monomials. One may thus associate a domain in this category with a ranking of the differential indeterminates,{} just as one associates a domain in the category \spadtype{OrderedAbelianMonoidSup} with an ordering of the set of monomials in a set of algebraic indeterminates. The ranking is specified through the binary relation \spadfun{<}. For example,{} one may define one deriva
2302`--tive to be less than another by lexicographically comparing first the \spadfun{order},{} then the given order of the differential indeterminates appearing in the derivatives. This is the default implementation. \blankline The notion of weight generalizes that of degree. A polynomial domain may be made into a graded ring if a weight function is given on the set of indeterminates,{} Very often,{} a grading is the first step in ordering the set of monomials. For differential polynomial domains,{}
2302`-- this constructor provides a function \spadfun{weight},{} which allows the assignment of a non-negative number to each derivative of a differential indeterminate. For example,{} one may define the weight of a derivative to be simply its \spadfun{order} (this is the default assignment). This weight function can then be extended to the set of all differential polynomials,{} providing a graded ring structure.
2376`attribute to indicate an aggregate of finite size
2460`A division ring (sometimes called a skew field),{} \spadignore{i.e.} a not necessarily commutative ring where all non-zero elements have multiplicative inverses.
2505`A doubly-linked aggregate serves as a model for a doubly-linked list,{} that is,{} a list which can has links to both next and previous nodes and thus can be efficiently traversed in both directions.
2565`Category for the elementary functions.
2625`An eltable aggregate is one which can be viewed as a function. For example,{} the list \axiom{[1,{}7,{}4]} can applied to 0,{}1,{} and 2 respectively will return the integers 1,{}7,{} and 4; thus this list may be viewed as mapping 0 to 1,{} 1 to 7 and 2 to 4. In general,{} an aggregate can map members of a domain {\em Dom} to an image domain {\em Im}.
2698`An eltable over domains \spad{D} and \spad{I} is a structure which can be viewed as a function from \spad{D} to \spad{I}. Examples of eltable structures range from data structures,{} \spadignore{e.g.} those of type \spadtype{List},{} to algebraic structures,{} \spadignore{e.g.} \spadtype{Polynomial}.
2762`Entire Rings (non-commutative Integral Domains),{} \spadignore{i.e.} a ring not necessarily commutative which has no zero divisors. \blankline
2806`A constructive euclidean domain,{} \spadignore{i.e.} one can divide producing a quotient and a remainder where the remainder is either zero or is smaller (\spadfun{euclideanSize}) than the divisor. \blankline Conditional attributes: \indented{2}{multiplicativeValuation\tab{25}\spad{Size(a*b)=Size(a)*Size(b)}} \indented{2}{additiveValuation\tab{25}\spad{Size(a*b)=Size(a)+Size(b)}}
2854`This category provides \spadfun{eval} operations. A domain may belong to this category if it is possible to make ``evaluation\spad{''} substitutions.
2909`An expression space is a set which is closed under certain operators.
2953`An extensible aggregate is one which allows insertion and deletion of entries. These aggregates are models of lists and streams which are represented by linked structures so as to make insertion,{} deletion,{} and concatenation efficient. However,{} access to elements of these extensible aggregates is generally slow since access is made from the end. See \spadtype{FlexibleArray} for an exception.
3017`ExtensionField {\em F} is the category of fields which extend the field \spad{F}
3068`FieldOfPrimeCharacteristic is the category of fields of prime characteristic,{} \spadignore{e.g.} finite fields,{} algebraic closures of fields of prime characteristic,{} transcendental extensions of of fields of prime characteristic.
3124`The category of commutative fields,{} \spadignore{i.e.} commutative rings where all non-zero elements have multiplicative inverses. The \spadfun{factor} operation while trivial is useful to have defined. \blankline
3161`This category provides an interface to operate on files in the computer\spad{'s} file system. The precise method of naming files is determined by the Name parameter. The type of the contents of the file is determined by \spad{S}.
3238`This category provides an interface to names in the file system.
3286`This category is similar to AbelianMonoidRing,{} except that the sum is assumed to be finite. It is a useful model for polynomials,{} but is somewhat more general.
3370`FiniteAlgebraicExtensionField {\em F} is the category of fields which are finite algebraic extensions of the field {\em F}. If {\em F} is finite then any finite algebraic extension of {\em F} is finite,{} too. Let {\em K} be a finite algebraic extension of the finite field {\em F}. The exponentiation of elements of {\em K} defines a \spad{Z}-module structure on the multiplicative group of {\em K}. The additive group of {\em K} becomes
3370`-- a module over the ring of polynomials over {\em F} via the operation \spadfun{linearAssociatedExp}(a:K,{}f:SparseUnivariatePolynomial \spad{F}) which is linear over {\em F},{} \spadignore{i.e.} for elements {\em a} from {\em K},{} {\em c,{}d} from {\em F} and {\em f,{}g} univariate polynomials over {\em F} we have \spadfun{linearAssociatedExp}(a,{}cf+dg) equals {\em c} times \spadfun{linearAssociatedExp}(a,{}\spad{f}) plus {\em d} times \spadfun{linearAssociatedExp}(a,{}\spad{g}). Therefore \
3370`--spadfun{linearAssociatedExp} is defined completely by its action on monomials from {\em F[X]}: \spadfun{linearAssociatedExp}(a,{}monomial(1,{}\spad{k})\spad{\$}SUP(\spad{F})) is defined to be \spadfun{Frobenius}(a,{}\spad{k}) which is {\em a^(q^k)} where {\em q=size()\$F}. The operations order and discreteLog associated with the multiplicative exponentiation have additive analogues associated to the operation \spadfun{linearAssociatedExp}. These are the functions \spadfun{linearAssociatedOrder
3370`--} and \spadfun{linearAssociatedLog},{} respectively.
3438`This category describes finite rational divisors on a curve,{} that is finite formal sums SUM(\spad{n} * \spad{P}) where the \spad{n}\spad{'s} are integers and the \spad{P}\spad{'s} are finite rational points on the curve.
3629`FiniteFieldCategory is the category of finite fields
3682`A finite linear aggregate is a linear aggregate of finite length. The finite property of the aggregate adds several exports to the list of exports from \spadtype{LinearAggregate} such as \spadfun{reverse},{} \spadfun{sort},{} and so on.
3742`A FiniteRankAlgebra is an algebra over a commutative ring \spad{R} which is a free \spad{R}-module of finite rank.
3848`A FiniteRankNonAssociativeAlgebra is a non associative algebra over a commutative ring \spad{R} which is a free \spad{R}-module of finite rank.
3931`A finite-set aggregate models the notion of a finite set,{} that is,{} a collection of elements characterized by membership,{} but not by order or multiplicity. See \spadtype{Set} for an example.
3995`The category of domains composed of a finite set of elements. We include the functions \spadfun{lookup} and \spadfun{index} to give a bijection between the finite set and an initial segment of positive integers. \blankline
4034`This category is intended as a model for floating point systems. A floating point system is a model for the real numbers. In fact,{} it is an approximation in the sense that not all real numbers are exactly representable by floating point numbers. A floating point system is characterized by the following: \blankline \indented{2}{1: \spadfunFrom{base}{FloatingPointSystem} of the \spadfunFrom{exponent}{FloatingPointSystem}.} \indented{9}{(actual implemena
4034`--tions are usually binary or decimal)} \indented{2}{2: \spadfunFrom{precision}{FloatingPointSystem} of the \spadfunFrom{mantissa}{FloatingPointSystem} (arbitrary or fixed)} \indented{2}{3: rounding error for operations} \blankline Because a Float is an approximation to the real numbers,{} even though it is defined to be a join of a Field and OrderedRing,{} some of the attributes do not hold. In particular associative(\spad{"+"}) does not hold. Algorithms defined over a field need special consid
4034`--erations when the field is a floating point system.
4083`\axiomType{FortranFunctionCategory} is the category of arguments to NAG Library routines which return (sets of) function values.
4139`A category of domains which model machine arithmetic used by machines in the AXIOM-NAG link.
4196`\axiomType{FortranMatrixCategory} provides support for producing Functions and Subroutines when the input to these is an FriCAS object of type \axiomType{Matrix} or in domains involving \axiomType{FortranCode}.
4247`\axiomType{FortranMatrixFunctionCategory} provides support for producing Functions and Subroutines representing matrices of expressions.
4308`\axiomType{FortranProgramCategory} provides various models of FORTRAN subprograms. These can be transformed into actual FORTRAN code.
4364`\axiomType{FortranVectorCategory} provides support for producing Functions and Subroutines when the input to these is an FriCAS object of type \axiomType{Vector} or in domains involving \axiomType{FortranCode}.
4415`\axiomType{FortranVectorFunctionCategory} is the catagory of arguments to NAG Library routines which return the values of vectors of functions.
4476`A \spadtype{FramedAlgebra} is a \spadtype{FiniteRankAlgebra} together with a fixed \spad{R}-module basis.
4578`FramedNonAssociativeAlgebra(\spad{R}) is a \spadtype{FiniteRankNonAssociativeAlgebra} (\spadignore{i.e.} a non associative algebra over \spad{R} which is a free \spad{R}-module of finite rank) over a commutative ring \spad{R} together with a fixed \spad{R}-module basis.
4657`A free abelian monoid on a set \spad{S} is the monoid of finite sums of the form \spad{reduce(+,{}[\spad{ni} * \spad{si}])} where the \spad{si}\spad{'s} are in \spad{S},{} and the \spad{ni}\spad{'s} are in a given abelian monoid. The operation is commutative.
4757`The category of free Lie algebras. It is used by domains of non-commutative algebra: \spadtype{LiePolynomial} and \spadtype{XPBWPolynomial}. \newline Author: Michel Petitot (petitot@lifl.\spad{fr})
4839`A domain of this category implements formal linear combinations of elements from a domain \spad{Basis} with coefficients in a domain \spad{R}. The domain \spad{Basis} needs only to belong to the category \spadtype{SetCategory} and \spad{R} to the category \spadtype{Ring}. Thus the coefficient ring may be non-commutative. See the \spadtype{XDistributedPolynomial} constructor for examples of domains built with the \spadtype{FreeModule
4839`--Category} category constructor. Author: Michel Petitot (petitot@lifl.\spad{fr}) \blankline Note (Franz Lehner,{} June 2009): Since \spad{leadingTerm} makes no sense for unordered base sets,{} and at the time of this writing this domain was never used for such,{} it was changed to \spad{OrderedSet}. \spad{FreeModule} originally was not of FreeModuleCategory. Some functions (like \spad{support},{} \spad{coefficients},{} \spad{monomials},{} ...) from here could be moved to \spad{IndexedDirectProd
4839`--uctCategory} but at the moment there is no need for this.
4909`This category provides a selection of evaluation operations depending on what the argument type \spad{R} provides.
4974`\spad{S} is \spadtype{FullyLinearlyExplicitRingOver R} means that \spad{S} is a \spadtype{LinearlyExplicitRingOver R} and,{} in addition,{} if \spad{R} is a \spadtype{LinearlyExplicitRingOver Integer},{} then so is \spad{S}
5044`A set \spad{S} is PatternMatchable over \spad{R} if \spad{S} can lift the pattern-matching functions of \spad{S} over the integers and float to itself (necessary for matching in towers).
5106`\indented{2}{A is fully retractable to \spad{B} means that A is retractable to \spad{B},{} and,{}} \indented{2}{in addition,{} if \spad{B} is retractable to the integers or rational} \indented{2}{numbers then so is A.} \indented{2}{In particular,{} what we are asserting is that there are no integers} \indented{2}{(rationals) in A which don\spad{'t} retract into \spad{B}.} Date Created: March 1990 Date Last Updated: 9 April 1991
5165`This category is a model for the function field of a plane algebraic curve.
5334`A space of formal functions with arguments in an arbitrary ordered set.
5389`This category describes domains where \spadfun{\spad{gcd}} can be computed but where there is no guarantee of the existence of \spadfun{factor} operation for factorisation into irreducibles. However,{} if such a \spadfun{factor} operation exist,{} factorization will be unique up to order and units.
5431`GradedAlgebra(\spad{R},{}\spad{E}) denotes ``E-graded \spad{R}-algebra\spad{''}. A graded algebra is a graded module together with a degree preserving \spad{R}-linear map,{} called the {\em product}. \blankline The name ``product\spad{''} is written out in full so inner and outer products with the same mapping type can be distinguished by name.
5510`GradedModule(\spad{R},{}\spad{E}) denotes ``E-graded \spad{R}-module\spad{''},{} \spadignore{i.e.} collection of \spad{R}-modules indexed by an abelian monoid \spad{E}. An element \spad{g} of \spad{G[s]} for some specific \spad{s} in \spad{E} is said to be an element of \spad{G} with {\em degree} \spad{s}. Sums are defined in each module \spad{G[s]} so two elements of \spad{G} have a sum if they have the same degree. \blankl
5510`--ine Morphisms can be defined and composed by degree to give the mathematical category of graded modules.
5588`The class of multiplicative groups,{} \spadignore{i.e.} monoids with multiplicative inverses. \blankline
5625`A homogeneous aggregate is an aggregate of elements all of the same type. In the current system,{} all aggregates are homogeneous. Two attributes characterize classes of aggregates. Aggregates from domains with attribute \spadatt{finiteAggregate} have a finite number of members. Of course,{} such a domain may have an infinite number of elements,{} like,{} for example \spadtype{List}. Those domains with attribute \spadatt{shallowlyMutable} allo
5625`--w an element to be modified or updated without changing its overall value.
5684`A Hopf algebra is a bialgebra with antipode.
5763`Category for the hyperbolic trigonometric functions.
5822`An indexed aggregate is a many-to-one mapping of indices to entries. For example,{} a one-dimensional-array is an indexed aggregate where the index is an integer. Also,{} a table is an indexed aggregate where the indices and entries may have any type.
5899`This category represents the direct product of some set with respect to an ordered indexing set.
5985`This category provides \spadfun{eval} operations. A domain may belong to this category if it is possible to make ``evaluation\spad{''} substitutions. The difference between this and \spadtype{Evalable} is that the operations in this category specify the substitution as a pair of arguments rather than as an equation.
6053`An \spad{IntegerNumberSystem} is a model for the integers.
6102`The category of commutative integral domains,{} \spadignore{i.e.} commutative rings with no zero divisors. \blankline Conditional attributes: \indented{2}{canonicalUnitNormal\tab{20}the canonical field is the same for all associates} \indented{2}{canonicalsClosed\tab{20}the product of two canonicals is itself canonical}
6149`\indented{1}{+ Author: Mike Dewar} + Date Created: November 1996 + Date Last Updated: + Basic Functions: + Related Constructors: + Also See: + AMS Classifications: + Keywords: + References: + Description: + This category implements of interval arithmetic and transcendental + functions over intervals.
6230`A keyed dictionary is a dictionary of key-entry pairs for which there is a unique entry for each key.
6311`LazyStreamAggregate is the category of streams with lazy evaluation. It is understood that the function 'empty?' will cause lazy evaluation if necessary to determine if there are entries. Functions which call 'empty?',{} \spadignore{e.g.} 'first' and 'rest',{} will also cause lazy evaluation if necessary.
6371`The category of all left algebras over an arbitrary ring.
6420`The category of left modules over an \spad{rng} (ring not necessarily with unit). This is an abelian group which supports left multiplation by elements of the \spad{rng}. \blankline
6470`The category of Lie Algebras. It is used by the following domains of non-commutative algebra: \axiomType{LiePolynomial} and \axiomType{XPBWPolynomial}. \newline Author : Michel Petitot (petitot@lifl.\spad{fr}).
6531`A linear aggregate is an aggregate whose elements are indexed by integers. Examples of linear aggregates are strings,{} lists,{} and arrays. Most of the exported operations for linear aggregates are non-destructive but are not always efficient for a particular aggregate. For example,{} \spadfun{concat} of two lists needs only to copy its first argument,{} whereas \spadfun{concat} of two arrays needs to copy both arguments. Most of the operations ex
6531`--ported here apply to infinite objects (\spadignore{e.g.} streams) as well to finite ones. For finite linear aggregates,{} see \spadtype{FiniteLinearAggregate}.
6585`\spad{LinearOrdinaryDifferentialOperatorCategory} is the category of differential operators with coefficients in a ring A with a given derivation. Multiplication of operators corresponds to functional composition: \indented{4}{\spad{(L1 * L2).(f) = L1 L2 f}}
6668`An extension ring with an explicit linear dependence test.
6732`Category for the transcendental Liouvillian functions.
6791`A list aggregate is a model for a linked list data structure. A linked list is a versatile data structure. Insertion and deletion are efficient and searching is a linear operation.
6843``Logic' provides the basic operations for lattices,{} \spadignore{e.g.} boolean algebra.
6880`\spadtype{MatrixCategory} is a general matrix category which allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and colums returned as objects of type Col. A domain belonging to this category will be shallowly mutable. The index of the 'first' row may be obtained by calling the function \spadfun{minRowIndex}. T
6880`--he index of the 'first' column may be obtained by calling the function \spadfun{minColIndex}. The index of the first element of a Row is the same as the index of the first column in a matrix and vice versa.
6994`The category of modules over a commutative ring. \blankline
7051`\indented{1}{MonadWithUnit is the class of multiplicative monads with unit,{}} \indented{1}{\spadignore{i.e.} sets with a binary operation and a unit element.} Axioms \indented{3}{leftIdentity("*":(\%,{}\%)\spad{->}\%,{}1)\space{3}\tab{30} 1*x=x} \indented{3}{rightIdentity("*":(\%,{}\%)\spad{->}\%,{}1)\space{2}\tab{30} x*1=x} Common Additional Axioms \indented{3}{unitsKnown
7051`---if "recip" says "failed",{} that PROVES input wasn\spad{'t} a unit}
7098`Monad is the class of all multiplicative monads,{} \spadignore{i.e.} sets with a binary operation.
7135`A \spadtype{MonogenicAlgebra} is an algebra of finite rank which can be generated by a single element.
7240`This is the category of linear operator rings with one generator. The generator is not named by the category but can always be constructed as \spad{monomial(1,{}1)}. \blankline For convenience,{} call the generator \spad{G}. Then each value is equal to \indented{4}{\spad{sum(a(i)*G^i,{} i = 0..n)}} for some unique \spad{n} and \spad{a(i)} in \spad{R}. \blankline Note that multiplication is not necessarily commutative. In fact,{} if \spad{a} i
7240`--s in \spad{R},{} it is quite normal to have \spad{a*G \~= G*a}.
7300`\spadtype{MonoidRingCategory}(\spad{R},{}\spad{M}) defines the algebra of all maps from the monoid \spad{M} to the commutative ring \spad{R} with finite support.
7366`The class of multiplicative monoids,{} \spadignore{i.e.} semigroups with a multiplicative identity element. \blankline
7405`A multi-dictionary is a dictionary which may contain duplicates. As for any dictionary,{} its size is assumed large so that copying (non-destructive) operations are generally to be avoided.
7466`A multi-set aggregate is a set which keeps track of the multiplicity of its elements.
7531`\spadtype{MultivariateTaylorSeriesCategory} is the most general multivariate Taylor series category.
7621`NonAssociativeAlgebra is the category of non associative algebras (modules which are themselves non associative rngs). Axioms \indented{3}{\spad{r*}(a*b) = (r*a)\spad{*b} = a*(\spad{r*b})}
7692`A NonAssociativeRing is a non associative \spad{rng} which has a unit,{} the multiplication is not necessarily commutative or associative.
7744`NonAssociativeRng is a basic ring-type structure,{} not necessarily commutative or associative,{} and not necessarily with unit. Axioms \indented{2}{\spad{x*}(\spad{y+z}) = x*y + \spad{x*z}} \indented{2}{(x+y)\spad{*z} = \spad{x*z} + \spad{y*z}} Common Additional Axioms \indented{2}{noZeroDivisors\space{2}ab = 0 \spad{=>} \spad{a=0} or \spad{b=0}}
7793`The category of normalized triangular sets. A triangular set \spad{ts} is said normalized if for every algebraic variable \spad{v} of \spad{ts} the polynomial \spad{select(ts,{}v)} is normalized \spad{w}.\spad{r}.\spad{t}. every polynomial in \spad{collectUnder(ts,{}v)}. A polynomial \spad{p} is said normalized \spad{w}.\spad{r}.\spad{t}. a non-c
7793`--onstant polynomial \spad{q} if \spad{p} is constant or \spad{degree(p,{}mdeg(q)) = 0} and \spad{init(p)} is normalized \spad{w}.\spad{r}.\spad{t}. \spad{q}. One of the important features of normalized triangular sets is that they are regular sets.\newline References : \indented{1}{[1] \spad{D}. LAZARD "A new method for solving algebraic systems of} \indented{5}{positive dimension" Discr. App. Math. 33:147-160,{}1991} \indented{1}{[2] \spad{P}. AUBRY,{} \spad{D}. LAZARD and \spad{M}. MORENO MAZ
7793`--A "On the Theories} \indented{5}{of Triangular Sets" Journal of Symbol. Comp. (to appear)} \indented{1}{[3] \spad{M}. MORENO MAZA and \spad{R}. RIOBOO "Computations of \spad{gcd} over} \indented{5}{algebraic towers of simple extensions" In proceedings of \spad{AAECC11}} \indented{5}{Paris,{} 1995.} \indented{1}{[4] \spad{M}. MORENO MAZA "Calculs de pgcd au-dessus des tours} \indented{5}{d'extensions simples et resolution des systemes d'equations} \indented{5}{algebriques" These,{} Universite \
7793`--spad{P}.etM. Curie,{} Paris,{} 1997.}
7951`OctonionCategory gives the categorial frame for the octonions,{} and eight-dimensional non-associative algebra,{} doubling the the quaternions in the same way as doubling the Complex numbers to get the quaternions.
8014`One-dimensional-array aggregates serves as models for one-dimensional arrays. Categorically,{} these aggregates are finite linear aggregates with the \spadatt{shallowlyMutable} property,{} that is,{} any component of the array may be changed without affecting the identity of the overall array. Array data structures are typically represented by a fixed area in storage and therefore cannot efficiently grow or shrink on demand as can list
8014`-- structures (see however \spadtype{FlexibleArray} for a data structure which is a cross between a list and an array). Iteration over,{} and access to,{} elements of arrays is extremely fast (and often can be optimized to open-code). Insertion and deletion however is generally slow since an entirely new data structure must be created for the result.
8081`\spadtype{OpenMath} provides operations for exporting an object in OpenMath format.
8118`Ordered sets which are also abelian groups,{} such that the addition preserves the ordering.
8171`This domain is an OrderedAbelianMonoid with a \spadfun{sup} operation added. The purpose of the \spadfun{sup} operator in this domain is to act as a supremum with respect to the partial order imposed by \spadop{-},{} rather than with respect to the total \spad{>} order (since that is "max"). \blankline
8227`Ordered sets which are also abelian monoids,{} such that the addition preserves the ordering.
8279`Ordered sets which are also abelian semigroups,{} such that the addition preserves the ordering. \indented{2}{\spad{ x < y => x+z < y+z}}
8334`Ordered sets which are also abelian cancellation monoids,{} such that the addition preserves the ordering.
8399`Ordered finite sets.
8445`The category of ordered commutative integral domains,{} where ordering and the arithmetic operations are compatible \blankline
8500`Ordered sets which are also monoids,{} such that multiplication preserves the ordering. \blankline
8546`to become an in order iterator
8616`Ordered sets which are also rings,{} that is,{} domains where the ring operations are compatible with the ordering. \blankline
8661`Semigroups with compatible ordering.
8711`The class of totally ordered sets,{} that is,{} sets such that for each pair of elements \spad{(a,{}b)} exactly one of the following relations holds \spad{a<b or a=b or b<a} and the relation is transitive,{} \spadignore{i.e.} \spad{a<b and b<c => a<c}. This order should be the natural order on given structure.
8754`This is the catefory of stream-based representations of \indented{2}{the \spad{p}-adic integers.}
8818`A partial differential ring with differentiations indexed by a parameter type \spad{S}. \blankline
8888`This is the description of any package which provides partial functions on a domain belonging to TranscendentalFunctionCategory.
8985`A set \spad{R} is PatternMatchable over \spad{S} if elements of \spad{R} can be matched to patterns over \spad{S}.
9048`An object \spad{S} is Patternable over an object \spad{R} if \spad{S} can lift the conversions from \spad{R} into \spadtype{Pattern(Integer)} and \spadtype{Pattern(Float)} to itself.
9098`PermutationCategory provides a categorial environment \indented{1}{for subgroups of bijections of a set (\spadignore{i.e.} permutations)}
9165`PlottablePlaneCurveCategory is the category of curves in the plane which may be plotted via the graphics facilities. Functions are provided for obtaining lists of lists of points,{} representing the branches of the curve,{} and for determining the ranges of the \spad{x}-coordinates and \spad{y}-coordinates of the points on the curve.
9226`PlottableSpaceCurveCategory is the category of curves in 3-space which may be plotted via the graphics facilities. Functions are provided for obtaining lists of lists of points,{} representing the branches of the curve,{} and for determining the ranges of the \spad{x-},{} \spad{y-},{} and \spad{z}-coordinates of the points on the curve.
9287`PointCategory is the category of points in space which may be plotted via the graphics facilities. Functions are provided for defining points and handling elements of points.
9339`The category for general multi-variate polynomials over a ring \spad{R},{} in variables from VarSet,{} with exponents from the \spadtype{OrderedAbelianMonoidSup}.
9442`This is the category of domains that know "enough" about themselves in order to factor univariate polynomials over themselves. This will be used in future releases for supporting factorization over finitely generated coefficient fields,{} it is not yet available in the current release of axiom.
9506`A category for finite subsets of a polynomial ring. Such a set is only regarded as a set of polynomials and not identified to the ideal it generates. So two distinct sets may generate the same the ideal. Furthermore,{} for \spad{R} being an integral domain,{} a set of polynomials may be viewed as a representation of the ideal it generates in the
9506`-- polynomial ring \spad{(R)^(-1) P},{} or the set of its zeros (described for instance by the radical of the previous ideal,{} or a split of the associated affine variety) and so on. So this category provides operations about those different notions.
9665`\spadtype{PowerSeriesCategory} is the most general power series category with exponents in an ordered abelian monoid.
9768`Category for the functions defined by integrals.
9827`The category of constructive principal ideal domains,{} \spadignore{i.e.} where a single generator can be constructively found for any ideal given by a finite set of generators. Note that this constructive definition only implies that finitely generated ideals are principal. It is not clear what we would mean by an infinitely generated ideal.
9877`A priority queue is a bag of items from an ordered set where the item extracted is always the maximum element.
9945`\spadtype{QuaternionCategory} describes the category of quaternions and implements functions that are not representation specific.
10015`A queue is a bag where the first item inserted is the first item extracted.
10068`QuotientField(\spad{S}) is the category of fractions of an Integral Domain \spad{S}.
10138`The \spad{RadicalCategory} is a model for the rational numbers.
10186`\axiomType{RealClosedField} provides common acces functions for all real closed fields.
10235`The category of real numeric domains,{} \spadignore{i.e.} convertible to floats.
10278`The real number system category is intended as a model for the real numbers. The real numbers form an ordered normed field. Note that we have purposely not included \spadtype{DifferentialRing} or the elementary functions (see \spadtype{TranscendentalFunctionCategory}) in the definition.
10324`\axiomType{RealRootCharacterizationCategory} provides common acces functions for all real root codings.
10474`\spadtype{RectangularMatrixCategory} is a category of matrices of fixed dimensions. The dimensions of the matrix will be parameters of the domain. Domains in this category will be \spad{R}-modules and will be non-mutable.
10650`A recursive aggregate over a type \spad{S} is a model for a a directed graph containing values of type \spad{S}. Recursively,{} a recursive aggregate is a {\em node} consisting of a \spadfun{value} from \spad{S} and 0 or more \spadfun{children} which are recursive aggregates. A node with no children is called a \spadfun{leaf} node. A recursive aggregate may be cyclic for which some operations as noted may go into an infinite loop.
10707`A category for general multi-variate polynomials with coefficients in a ring,{} variables in an ordered set,{} and exponents from an ordered abelian monoid,{} with a \axiomOp{sup} operation. When not constant,{} such a polynomial is viewed as a univariate polynomial in its main variable \spad{w}. \spad{r}. \spad{t}. to the total ordering on the elements in the ordered set,{} so that some operatio
10707`--ns usually defined for univariate polynomials make sense here.
10814`The category of regular triangular sets,{} introduced under the name regular chains in [1] (and other papers). In [3] it is proved that regular triangular sets and towers of simple extensions of a field are equivalent notions. In the following definitions,{} all polynomials and ideals are taken from the polynomial ring \spad{k[x1,{}...,{}xn]} where
10814`-- \spad{k} is the fraction field of \spad{R}. The triangular set \spad{[t1,{}...,{}tm]} is regular iff for every \spad{i} the initial of \spad{ti+1} is invertible in the tower of simple extensions associated with \spad{[t1,{}...,{}\spad{ti}]}. A family \spad{[T1,{}...,{}Ts]} of regular triangular sets is a split of Kalkbrener of a given ideal \spad{I} iff the radical of \spad{I} is equal to the intersection of the radical ideals generated by the saturated ideals of the \spad{[T1,{}...,{}\spad{T
10814`--i}]}. A family \spad{[T1,{}...,{}Ts]} of regular triangular sets is a split of Kalkbrener of a given triangular set \spad{T} iff it is a split of Kalkbrener of the saturated ideal of \spad{T}. Let \spad{K} be an algebraic closure of \spad{k}. Assume that \spad{V} is finite with cardinality \spad{n} and let \spad{A} be the affine space \spad{K^n}. For a regular triangular set \spad{T} let denote by \spad{W(T)} the set of regular zeros of \spad{T}. A family \spad{[T1,{}...,{}Ts]} of regular tria
10814`--ngular sets is a split of Lazard of a given subset \spad{S} of \spad{A} iff the union of the \spad{W(\spad{Ti})} contains \spad{S} and is contained in the closure of \spad{S} (\spad{w}.\spad{r}.\spad{t}. Zariski topology). A family \spad{[T1,{}...,{}Ts]} of regular triangular sets is a split of Lazard of a given triangular set \spad{T} if it is a split of Lazard of \spad{W(T)}. Note that if \spad{[T1,{}...,{}Ts]} is a split of Lazard of \spad{T} then it is also a split of Kalkbrener of \spad{T
10814`--}. The converse is \spad{false}. This category provides operations related to both kinds of splits,{} the former being related to ideals decomposition whereas the latter deals with varieties decomposition. See the example illustrating the \spadtype{RegularTriangularSet} constructor for more explanations about decompositions by means of regular triangular sets. \newline References : \indented{1}{[1] \spad{M}. KALKBRENER "Three contributions to elimination theory"} \indented{5}{\spad{Phd} Thesis
10814`--,{} University of Linz,{} Austria,{} 1991.} \indented{1}{[2] \spad{M}. KALKBRENER "Algorithmic properties of polynomial rings"} \indented{5}{Journal of Symbol. Comp. 1998} \indented{1}{[3] \spad{P}. AUBRY,{} \spad{D}. LAZARD and \spad{M}. MORENO MAZA "On the Theories} \indented{5}{of Triangular Sets" Journal of Symbol. Comp. (to appear)} \indented{1}{[4] \spad{M}. MORENO MAZA "A new algorithm for computing triangular} \indented{5}{decomposition of algebraic varieties" NAG Tech. Rep. 4/98.}
10970`A is retractable to \spad{B} means that some elementsif A can be converted into elements of \spad{B} and any element of \spad{B} can be converted into an element of A.
11024`The category of right modules over an \spad{rng} (ring not necessarily with unit). This is an abelian group which supports right multiplation by elements of the \spad{rng}. \blankline
11075`The category of rings with unity,{} always associative,{} but not necessarily commutative.
11110`The category of associative rings,{} not necessarily commutative,{} and not necessarily with a 1. This is a combination of an abelian group and a semigroup,{} with multiplication distributing over addition. \blankline
11143`This category allows the manipulation of Lisp values while keeping the grunge fairly localized.
11276`PointCategory is the category of points and vectors in space. Points may be used to represent shapes in a scenegraph such as: the verticies of a cube or points along a line. Shapes such as these may be defined explicitly or may be plotted. We implement SetCategory to allow us to curry transforms from one point to another.
11323`This category provides operations on ranges,{} or {\em segments} as they are called.
11378`This category provides an interface for expanding segments to a stream of elements.
11473`the class of all multiplicative semigroups,{} \spadignore{i.e.} a set with an associative operation \spadop{*}. \blankline
11515`The category of associative semirings,{} not necessarily commutative,{} with a 1.
11555`The category of associative semirings,{} not necessarily commutative,{} and not necessarily with a 1.
11593`A set category lists a collection of set-theoretic operations useful for both finite sets and multisets. Note however that finite sets are distinct from multisets. Although the operations defined for set categories are common to both,{} the relationship between the two cannot be described by inclusion or inheritance.
11652`\spadtype{SetCategory} is the basic category for describing a collection of elements with \spadop{=} (equality) and \spadfun{coerce} to output form. \blankline Conditional Attributes: \indented{3}{canonical\tab{15}data structure equality is the same as \spadop{=}}
11696`Category for the other special functions.
11752`The category of square-free and normalized triangular sets. Thus,{} up to the primitivity axiom of [1],{} these sets are Lazard triangular sets.\newline References : \indented{1}{[1] \spad{D}. LAZARD "A new method for solving algebraic systems of} \indented{5}{positive dimension" Discr. App. Math. 33:147-160,{}1991}
11921`The category of square-free regular triangular sets. A regular triangular set \spad{ts} is square-free if the \spad{gcd} of any polynomial \spad{p} in \spad{ts} and \spad{differentiate(p,{}mvar(p))} \spad{w}.\spad{r}.\spad{t}. \axiomOpFrom{collectUnder}{TriangularSetCategory}(\spad{ts},{}\axiomOpFrom{mvar}{RecursivePolynomialCategory}(\sp
11921`--ad{p})) has degree zero \spad{w}.\spad{r}.\spad{t}. \spad{mvar(p)}. Thus any square-free regular set defines a tower of square-free simple extensions.\newline References : \indented{1}{[1] \spad{D}. LAZARD "A new method for solving algebraic systems of} \indented{5}{positive dimension" Discr. App. Math. 33:147-160,{}1991} \indented{1}{[2] \spad{M}. KALKBRENER "Algorithmic properties of polynomial rings"} \indented{5}{Habilitation Thesis,{} ETZH,{} Zurich,{} 1995.} \indented{1}{[3] \spad{M}. MO
11921`--RENO MAZA "A new algorithm for computing triangular} \indented{5}{decomposition of algebraic varieties" NAG Tech. Rep. 4/98.}
12087`\spadtype{SquareMatrixCategory} is a general square matrix category which allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and colums returned as objects of type Col.
12247`A stack is a bag where the last item inserted is the first item extracted.
12300`A class of objects which can be 'stepped through'. Repeated applications of \spadfun{nextItem} is guaranteed never to return duplicate items and only return "failed" after exhausting all elements of the domain. This assumes that the sequence starts with \spad{init()}. For infinite domains,{} repeated application of \spadfun{nextItem} is not required to reach all possible domain elements starting from any initial element. \blankline Conditional attributes: \ind
12300`--ented{2}{infinite\tab{15}repeated \spad{nextItem}\spad{'s} are never "failed".}
12342`A stream aggregate is a linear aggregate which possibly has an infinite number of elements. A basic domain constructor which builds stream aggregates is \spadtype{Stream}. From streams,{} a number of infinite structures such power series can be built. A stream aggregate may also be infinite since it may be cyclic. For example,{} see \spadtype{DecimalExpansion}.
12396`A string aggregate is a category for strings,{} that is,{} one dimensional arrays of characters.
12443`A category for string-like objects
12491`A table aggregate is a model of a table,{} \spadignore{i.e.} a discrete many-to-one mapping from keys to entries.
12571`Category of tensor powers of modules over commutative rings.
12676`Category of tensor products of modules over commutative rings.
12775`Universal property of tensor products.
12923`The category ThreeSpaceCategory is used for creating three dimensional objects using functions for defining points,{} curves,{} polygons,{} constructs and the subspaces containing them.
12981`Category for the transcendental elementary functions.
13045`The category of triangular sets of multivariate polynomials with coefficients in an integral domain. Let \axiom{\spad{R}} be an integral domain and \axiom{\spad{V}} a finite ordered set of variables,{} say \axiom{\spad{X1} < \spad{X2} < ... < \spad{Xn}}. A set \axiom{\spad{S}} of polynomials in \axiom{\spad{R}[\spad{X1},{}\spad{X2},{}...,{}\spad{Xn}]
13045`--} is triangular if no elements of \axiom{\spad{S}} lies in \axiom{\spad{R}},{} and if two distinct elements of \axiom{\spad{S}} have distinct main variables. Note that the empty set is a triangular set. A triangular set is not necessarily a (lexicographical) Groebner basis and the notion of reduction related to triangular sets is based on the recursive view of polynomials. We recall this notion here and refer to [1] for more details. A polynomial \axiom{\spad{P}} is reduced \spad{w}.\spad{r}.\
13045`--spad{t} a non-constant polynomial \axiom{\spad{Q}} if the degree of \axiom{\spad{P}} in the main variable of \axiom{\spad{Q}} is less than the main degree of \axiom{\spad{Q}}. A polynomial \axiom{\spad{P}} is reduced \spad{w}.\spad{r}.\spad{t} a triangular set \axiom{\spad{T}} if it is reduced \spad{w}.\spad{r}.\spad{t}. every polynomial of \axiom{\spad{T}}. \newline References : \indented{1}{[1] \spad{P}. AUBRY,{} \spad{D}. LAZARD and \spad{M}. MORENO MAZA "On the Theories} \indented{5}{of Tr
13045`--iangular Sets" Journal of Symbol. Comp. (to appear)}
13199`Category for the trigonometric functions.
13262`\indented{1}{TwoDimensionalArrayCategory is a general array category which} allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and columns returned as objects of type Col. The index of the 'first' row may be obtained by calling the function 'minRowIndex'. The index of the 'first' column may be obt
13262`--ained by calling the function 'minColIndex'. The index of the first element of a 'Row' is the same as the index of the first column in an array and vice versa.
13390`The fundamental Type.
13425`A unary-recursive aggregate is a one where nodes may have either 0 or 1 children. This aggregate models,{} though not precisely,{} a linked list possibly with a single cycle. A node with one children models a non-empty list,{} with the \spadfun{value} of the list designating the head,{} or \spadfun{first},{} of the list,{} and the child designating the tail,{} or \spadfun{rest},{} of the list. A node with no child then designates the empty 
13425`--list. Since these aggregates are recursive aggregates,{} they may be cyclic.
13487`A constructive unique factorization domain,{} \spadignore{i.e.} where we can constructively factor members into a product of a finite number of irreducible elements.
13542`\spadtype{UnivariateLaurentSeriesCategory} is the category of Laurent series in one variable.
13616`This is a category of univariate Laurent series constructed from univariate Taylor series. A Laurent series is represented by a pair \spad{[n,{}f(x)]},{} where \spad{n} is an arbitrary integer and \spad{f(x)} is a Taylor series. This pair represents the Laurent series \spad{x^n * f(x)}.
13748`The category of univariate polynomials over a ring \spad{R}. No particular model is assumed - implementations can be either sparse or dense.
13816`\spadtype{UnivariatePowerSeriesCategory} is the most general univariate power series category with exponents in an ordered abelian monoid. Note: this category exports a substitution function if it is possible to multiply exponents. Note: this category exports a derivative operation if it is possible to multiply coefficients by exponents.
13915`\spadtype{UnivariatePuiseuxSeriesCategory} is the category of Puiseux series in one variable.
13990`This is a category of univariate Puiseux series constructed from univariate Laurent series. A Puiseux series is represented by a pair \spad{[r,{}f(x)]},{} where \spad{r} is a positive rational number and \spad{f(x)} is a Laurent series. This pair represents the Puiseux series \spad{f(x^r)}.
14123`This is the category of univariate skew polynomials over an Ore coefficient ring. The multiplication is given by \spad{x a = \sigma(a) x + \delta a}. This category is an evolution of the types \indented{2}{MonogenicLinearOperator,{} OppositeMonogenicLinearOperator,{} and} \indented{2}{NonCommutativeOperatorDivision} developped by Jean Della Dora and Stephen \spad{M}. Watt.
14196`\spadtype{UnivariateTaylorSeriesCategory} is the category of Taylor series in one variable.
14269`\spadtype{VectorCategory} represents the type of vector like objects,{} \spadignore{i.e.} finite sequences indexed by some finite segment of the integers. The operations available on vectors depend on the structure of the underlying components. Many operations from the component domain are defined for vectors componentwise. It can by assumed that extraction or updating components can be done in constant time.
14324`Vector Spaces (not necessarily finite dimensional) over a field.
14376`This is the category of algebras over non-commutative rings. It is used by constructors of non-commutative algebras such as: \indented{4}{\spadtype{XPolynomialRing}.} \indented{4}{\spadtype{XFreeAlgebra}} Author: Michel Petitot (petitot@lifl.\spad{fr})
14422`This category specifies opeations for polynomials and formal series with non-commutative variables.
14487`The Category of polynomial rings with non-commutative variables. The coefficient ring may be non-commutative too. However coefficients commute with vaiables.
14556`The class of all euclidean domains such that \spad{euclideanSize(a*b)=euclideanSize(a)+euclideanSize(b)}.
14607`Approximate numbers with arbitrarily large exponents
14658`Approximate numbers for which the user can set the precision for subsequent calculations.
14710`The class of all integral domains such that we can choose a canonical representative for each class of associate elements,{} that is \spad{associates?(a,{}b)} returns \spad{true} if and only if \spad{unitCanonical(a) = unitCanonical(b)}.
14763`The class of all integral domains such that \spad{unitCanonical(a)*unitCanonical(b) = unitCanonical(a*b)}.
14813`The class of all aggregates with a finite number of elements.
14862`The class of all euclidean domains such that \spad{euclideanSize(a*b)=euclideanSize(a)*euclideanSize(b)}.
14919`The class of all semirings such that \spad{x * y \~~= 0} implies both \spad{x} and \spad{y} are non-zero
14967`The class of all parially ordered sets,{} that is domains with \spadop{<} which is transitive,{} but \spad{not(a < b or a = b)} does not necessarily imply \spad{b<a}.
15020`The class of all domains which have have immediate components that are updateable (mutable).
15070`The class of all monoids (multiplicative semigroups with a 1) such that the operation \spadfun{recip} can only return "failed" if its argument is not a unit.
15114`AlgebraGivenByStructuralConstants implements finite rank algebras over a commutative ring,{} given by the structural constants \spad{gamma} with respect to a fixed basis \spad{[a1,{}..,{}an]},{} where \spad{gamma} is an \spad{n}-vector of \spad{n} by \spad{n} matrices \spad{[(gammaijk) for k in 1..rank()]} defined by \spad{\spad{ai} * aj
15114`-- = gammaij1 * a1 + ... + gammaijn * an}. The symbols for the fixed basis have to be given as a list of symbols.
15281`Function field defined by \spad{f}(\spad{x},{} \spad{y}) = 0.
15483`Algebraic closure of the rational numbers,{} with mathematical =
15544`This domain implements anonymous functions
15595`The domain of antisymmetric polynomials.
15678`\spadtype{Any} implements a type that packages up objects and their types in objects of \spadtype{Any}. Roughly speaking that means that if \spad{s : S} then when converted to \spadtype{Any},{} the new object will include both the original object and its type. This is a way of converting arbitrary objects into a single type without losing any of the original information. Any object can be converted to one of \spadtype{Any}.
15724`A stack represented as a flexible array.
15800`\indented{1}{AssociatedJordanAlgebra takes an algebra \spad{A} and uses \spadfun{*\$A}} \indented{1}{to define the new multiplications \spad{a*b := (a *\$A b + b *\$A a)/2}} \indented{1}{(anticommutator).} \indented{1}{The usual notation \spad{{a,{}b}_+} cannot be used due to} \indented{1}{restrictions in the current language.} \indented{1}{This domain only gives a Jordan 
15800`--algebra if the} \indented{1}{Jordan-identity \spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} holds} \indented{1}{for all \spad{a},{}\spad{b},{}\spad{c} in \spad{A}.} \indented{1}{This relation can be checked by} \indented{1}{\spadfun{jordanAdmissible?()\$A}.} \blankline If the underlying algebra is of type \spadtype{FramedNonAssociativeAlgebra(R)} (\spadignore{i.e.} a non associative algebra over \spad{R} which is a free \spad{R}-module of finite rank,{} together with a fixed \spad{R}-module basis),{} 
15800`--then the same is \spad{true} for the associated Jordan algebra. Moreover,{} if the underlying algebra is of type \spadtype{FiniteRankNonAssociativeAlgebra(R)} (\spadignore{i.e.} a non associative algebra over \spad{R} which is a free \spad{R}-module of finite rank),{} then the same \spad{true} for the associated Jordan algebra.
15931`AssociatedLieAlgebra takes an algebra \spad{A} and uses \spadfun{*\$A} to define the Lie bracket \spad{a*b := (a *\$A b - b *\$A a)} (commutator). Note that the notation \spad{[a,{}b]} cannot be used due to restrictions of the current compiler. This domain only gives a Lie algebra if the Jacobi-identity \spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} holds for all \spad{a},{}\spad{b},{}\
15931`--spad{c} in \spad{A}. This relation can be checked by \spad{lieAdmissible?()\$A}. \blankline If the underlying algebra is of type \spadtype{FramedNonAssociativeAlgebra(R)} (\spadignore{i.e.} a non associative algebra over \spad{R} which is a free \spad{R}-module of finite rank,{} together with a fixed \spad{R}-module basis),{} then the same is \spad{true} for the associated Lie algebra. Also,{} if the underlying algebra is of type \spadtype{FiniteRankNonAssociativeAlgebra(R)} (\spadignore{i.e.}
15931`-- a non associative algebra over \spad{R} which is a free \spad{R}-module of finite rank),{} then the same is \spad{true} for the associated Lie algebra.
16056`\spadtype{AssociationList} implements association lists. These may be viewed as lists of pairs where the first part is a key and the second is the stored value. For example,{} the key might be a string with a persons employee identification number and the value might be a record with personnel data.
16170`Automorphism \spad{R} is the multiplicative group of automorphisms of \spad{R}.
16230`\spadtype{BalancedBinaryTree(S)} is the domain of balanced binary trees (bbtree). A balanced binary tree of \spad{2^k} leaves,{} for some \spad{k > 0},{} is symmetric,{} that is,{} the left and right subtree of each interior node have identical shape. In general,{} the left and right subtree of a given node can differ by at most leaf node.
16318`Stream-based implementation of \spad{Zp:} \spad{p}-adic numbers are represented as sum(\spad{i} = 0..,{} a[\spad{i}] * p^i),{} where the a[\spad{i}] lie in -(\spad{p} - 1)\spad{/2},{}...,{}(\spad{p} - 1)\spad{/2}.
16396`Stream-based implementation of \spad{Qp:} numbers are represented as sum(\spad{i} = \spad{k}..,{} a[\spad{i}] * p^i),{} where the a[\spad{i}] lie in -(\spad{p} - 1)\spad{/2},{}...,{}(\spad{p} - 1)\spad{/2}.
16510`A basic operator is an object that can be applied to a list of arguments from a set,{} the result being a kernel over that set.
16565`This domain allows rational numbers to be presented as repeating binary expansions.
16645`This domain provides an implementation of binary files. Data is accessed one byte at a time as a small integer.
16727`BinarySearchTree(\spad{S}) is the domain of a binary trees where elements are ordered across the tree. A binary search tree is either empty or has a value which is an \spad{S},{} and a right and left which are both BinaryTree(\spad{S}) Elements are ordered across the tree.
16812`\spadtype{BinaryTournament(S)} is the domain of binary trees where elements are ordered down the tree. A binary search tree is either empty or is a node containing a \spadfun{value} of type \spad{S},{} and a \spadfun{right} and a \spadfun{left} which are both \spadtype{BinaryTree(S)}
16897`\spadtype{BinaryTree(S)} is the domain of all binary trees. A binary tree over \spad{S} is either empty or has a \spadfun{value} which is an \spad{S} and a \spadfun{right} and \spadfun{left} which are both binary trees.
16976`\spadtype{Bits} provides logical functions for Indexed Bits.
17025`\indented{1}{\spadtype{Boolean} is the elementary logic with 2 values:} \spad{true} and \spad{false}
17078`Members of the domain CardinalNumber are values indicating the cardinality of sets,{} both finite and infinite. Arithmetic operations are defined on cardinal numbers as follows. \blankline If \spad{x = \#X} and \spad{y = \#Y} then \indented{2}{\spad{x+y\space{2}= \#(X+Y)}\space{3}\tab{30}disjoint union} \indented{2}{\spad{x-y\space{2}= \#(X-Y)}\space{3}\tab{30}relative complement} \indented{2}{\spad{x*y\space{2}= \#(X*Y)}\space{3}\tab{30}cartesi
17078`--an product} \indented{2}{\spad{x^y = \#(X^Y)}\space{2}\tab{30}\spad{X^Y = \{g| g:Y->X\}}} \blankline The non-negative integers have a natural construction as cardinals \indented{2}{\spad{0 = \#\{\}},{} \spad{1 = \{0\}},{} \spad{2 = \{0,{} 1\}},{} ...,{} \spad{n = \{i| 0 <= i < n\}}.} \blankline That \spad{0} acts as a zero for the multiplication of cardinals is equivalent to the axiom of choice. \blankline The generalized continuum hypothesis asserts \center{\spad{2^Aleph i = Aleph(i+1)}} and 
17078`--is independent of the axioms of set theory [Goedel 1940]. \blankline Three commonly encountered cardinal numbers are \indented{3}{\spad{a = \#Z}\space{7}\tab{30}countable infinity} \indented{3}{\spad{c = \#R}\space{7}\tab{30}the continuum} \indented{3}{\spad{f = \#\{g| g:[0,{}1]->R\}}} \blankline In this domain,{} these values are obtained using \indented{3}{\spad{a := Aleph 0},{} \spad{c := 2^a},{} \spad{f := 2^c}.} \blankline
17135`CartesianTensor(minix,{}dim,{}\spad{R}) provides Cartesian tensors with components belonging to a commutative ring \spad{R}. These tensors can have any number of indices. Each index takes values from \spad{minix} to \spad{minix + dim - 1}.
17281`This domain allows classes of characters to be defined and manipulated efficiently.
17341`This domain provides the basic character data type.
17396`CliffordAlgebra(\spad{n},{} \spad{K},{} bLin) defines a module of dimension \spad{2^n} over \spad{K},{} given a bilinear form bLin on \spad{K^n}. \blankline Examples of Clifford Algebras are: gaussians,{} quaternions,{} exterior algebras and spin algebras.
17501`Color() specifies a domain of 27 colors provided in the \Language{} system (the colors mix additively).
17556`A type for basic commutators
17610`This is a low-level domain which implements matrices (two dimensional arrays) of complex double precision floating point numbers. Indexing is 0 based,{} there is no bound checking (unless provided by lower level).
17755`This is a low-level domain which implements vectors (one dimensional arrays) of complex double precision floating point numbers. Indexing is 0 based,{} there is no bound checking (unless provided by lower level).
17850`\spadtype {Complex(R)} creates the domain of elements of the form \spad{a + b * i} where \spad{a} and \spad{b} come from the ring \spad{R},{} and \spad{i} is a new element such that \spad{i^2 = -1}.
17929`\spadtype{ContinuedFraction} implements general \indented{1}{continued fractions.\space{2}This version is not restricted to simple,{}} \indented{1}{finite fractions and uses the \spadtype{Stream} as a} \indented{1}{representation.\space{2}The arithmetic functions assume that the} \indented{1}{approximants alternate below/above the convergence point.} \indented{1}{This is enforced by ensuring the partial numerators and pa
17929`--rtial} \indented{1}{denominators are greater than 0 in the Euclidean domain view of \spad{R}} \indented{1}{(\spadignore{i.e.} \spad{sizeLess?(0,{} x)}).}
18011`This domain provides some nice functions on lists
18082`\indented{1}{This domain implements a simple view of a database whose fields are} indexed by symbols
18158`The deRham complex of Euclidean space,{} that is,{} the class of differential forms of arbitary degree over a coefficient ring. See Flanders,{} Harley,{} Differential Forms,{} With Applications to the Physical Sciences,{} New York,{} Academic Press,{} 1963.
18287`This domain allows rational numbers to be presented as repeating decimal expansions.
18369`\indented{1}{A Denavit-Hartenberg Matrix is a 4x4 Matrix of the form:} \indented{1}{\spad{nx ox ax px}} \indented{1}{\spad{ny oy ay py}} \indented{1}{\spad{nz oz az pz}} \indented{2}{\spad{0\space{2}0\space{2}0\space{2}1}} (\spad{n},{} \spad{o},{} and a are the direction cosines)
18484`Linked list implementation of a Dequeue
18560`\spadtype{DifferentialSparseMultivariatePolynomial} implements an ordinary differential polynomial ring by combining a domain belonging to the category \spadtype{DifferentialVariableCategory} with the domain \spadtype{SparseMultivariatePolynomial}. \blankline
18748`This constructor provides a direct product type with a left matrix-module view.
18947`This constructor provides a direct product of \spad{R}-modules with an \spad{R}-module view.
19073`\indented{2}{This type represents the finite direct or cartesian product of an} underlying component type. This contrasts with simple vectors in that the members can be viewed as having constant length. Thus many categorical properties can by lifted from the underlying component type. Component extraction operations are provided but no updating operations. Thus new direct product elements can either be c
19073`--reated by converting vector elements using the \spadfun{directProduct} function or by taking appropriate linear combinations of basis vectors provided by the \spad{unitVector} operation.
19172`DirichletRing is the ring of arithmetical functions with Dirichlet convolution as multiplication
19235`\indented{2}{This type supports distributed multivariate polynomials} whose variables are from a user specified list of symbols. The coefficient ring may be non commutative,{} but the variables are assumed to commute. The term ordering is lexicographic specified by the variable list parameter with the most significant variable fi
19235`--rst in the list.
19410`This is a low-level domain which implements matrices (two dimensional arrays) of double precision floating point numbers. Indexing is 0 based,{} there is no bound checking (unless provided by lower level).
19524`This is a low-level domain which implements vectors (one dimensional arrays) of double precision floating point numbers. Indexing is 0 based,{} there is no bound checking (unless provided by lower level).
19602`\indented{1}{\spadtype{DoubleFloat} is intended to make accessible} hardware floating point arithmetic in \Language{},{} either native double precision,{} or IEEE. On most machines,{} there will be hardware support for the arithmetic operations: \spadfunFrom{+}{DoubleFloat},{} \spadfunFrom{*}{DoubleFloat},{} \spadfunFrom{/}{DoubleFloat} and possibly also the \spadfunFrom{sqrt}{DoubleFloat} operation. The operations \spadfunFrom{exp}{Doub
19602`--leFloat},{} \spadfunFrom{log}{DoubleFloat},{} \spadfunFrom{sin}{DoubleFloat},{} \spadfunFrom{cos}{DoubleFloat},{} \spadfunFrom{atan}{DoubleFloat} are normally coded in software based on minimax polynomial/rational approximations. Some general comments about the accuracy of the operations: the operations \spadfunFrom{+}{DoubleFloat},{} \spadfunFrom{*}{DoubleFloat},{} \spadfunFrom{/}{DoubleFloat} and \spadfunFrom{sqrt}{DoubleFloat} are expected to be fully accurate. The operations \spadfunFrom{e
19602`--xp}{DoubleFloat},{} \spadfunFrom{log}{DoubleFloat},{} \spadfunFrom{sin}{DoubleFloat},{} \spadfunFrom{cos}{DoubleFloat} and \spadfunFrom{atan}{DoubleFloat} are not expected to be fully accurate. In particular,{} \spadfunFrom{sin}{DoubleFloat} and \spadfunFrom{cos}{DoubleFloat} will lose all precision for large arguments. \blankline The \spadtype{Float} domain provides an alternative to the \spad{DoubleFloat} domain. It provides an arbitrary precision model of floating point arithmetic. This mea
19602`--ns that accuracy problems like those above are eliminated by increasing the working precision where necessary. \spadtype{Float} provides some special functions such as \spadfunFrom{erf}{DoubleFloat},{} the error function in addition to the elementary functions. The disadvantage of \spadtype{Float} is that it is much more expensive than small floats when the latter can be used.
19667`DrawOption allows the user to specify defaults for the creation and rendering of plots.
19722`\indented{1}{This package provides elementary functions on any Laurent series} domain over a field which was constructed from a Taylor series domain. These functions are implemented by calling the corresponding functions on the Taylor series domain. We
19722`-- also provide 'partial functions' which compute transcendental functions of Laurent series when possible and return "failed" when this is not possible.
19976`\indented{1}{This package provides elementary functions on any Laurent series} domain over a field which was constructed from a Taylor series domain. These functions are implemented by calling the correspo
19976`--nding functions on the Taylor series domain. We also provide 'partial functions' which compute transcendental functions of Laurent series when possible and return "failed" when this is not possible.
20277`\spad{Enumeration(a,b,...,c)} is a primitive type in FriCAS used to represent the object composed of the symbols \spad{a},\spad{b},..., and \spad{c}.
20325`This domain provides tables where the keys are compared using \spadfun{eq?}. Thus keys are considered equal only if they are the same instance of a structure.
20431`Equations as mathematical objects. All properties of the basis domain,{} \spadignore{e.g.} being an abelian group are carried over the equation domain,{} by performing the structural operations on the left and on the right hand side.
20481`These domains are used for the factorization and gcds of univariate polynomials over the integers in order to work modulo different primes. See \spadtype{ModularRing},{} \spadtype{ModularField}
20722`A function which does not return directly to its caller should have Exit as its return type. \blankline Note: It is convenient to have a formal \spad{coerce} into each type from type Exit. This allows,{} for example,{} errors to be raised in one half of a type-balanced \spad{if}.
20760`UnivariatePuiseuxSeriesWithExponentialSingularity is a domain used to represent essential singularities of functions. Objects in this domain are quotients of sums,{} where each term in the sum is a univariate Puiseux series times the exponential of a univariate Puiseux s
20760`--eries.
20995`ExponentialOfUnivariatePuiseuxSeries is a domain used to represent essential singularities of functions. An object in this domain is a function of the form \spad{exp(f(x))},{} where \spad{f(x)} is a Puiseux series with no terms of non-negative degree. Objects are ordered according to order of singularity,{} with functions which tend more rapidly to 
20995`--zero or infinity considered to be larger. Thus,{} if \spad{order(f(x)) < order(g(x))},{} \spadignore{i.e.} the first non-zero term of \spad{f(x)} has lower degree than the first non-zero term of \spad{g(x)},{} then \spad{exp(f(x)) > exp(g(x))}. If \spad{order(f(x)) = order(g(x))},{} then the ordering is essentially random. This domain is used in computing limits involving functions with essential singularities.
21150`Expressions involving symbolic functions.
21222`A domain used in the construction of the exterior algebra on a set \spad{X} over a ring \spad{R}. This domain represents the set of all ordered subsets of the set \spad{X},{} assumed to be in correspondance with {1,{}2,{}3,{} ...}. The ordered subsets are themselves ordered lexicographically and are in bijective correspondance with an ordered basis of the exterior algebra. In this domain we are dealing strictly with the exponents of basis elements w
21222`--hich can only be 0 or 1. \blankline The multiplicative identity element of the exterior algebra corresponds to the empty subset of \spad{X}. A coerce from List Integer to an ordered basis element is provided to allow the convenient input of expressions. Another exported function forgets the ordered structure and simply returns the list corresponding to an ordered subset.
21275`\spadtype{Factored} creates a domain whose objects are kept in factored form as long as possible. Thus certain operations like multiplication and \spad{gcd} are relatively easy to do. Others,{} like addition require somewhat more work,{} and unless the argument domain provides a factor function,{} the result may not be completely factored. Each object consists of a unit and a list of factors,{} where a factor has a member of \spad{R
21275`--} (the "base"),{} and exponent and a flag indicating what is known about the base. A flag may be one of "nil",{} "sqfr",{} "irred" or "prime",{} which respectively mean that nothing is known about the base,{} it is square-free,{} it is irreducible,{} or it is prime. The current restriction to integral domains allows simplification to be performed without worrying about multiplication order.
21345`This domain provides an interface to names in the file system.
21393`This domain provides a basic model of files to save arbitrary values. The operations provide sequential access to the contents.
21468`This domains implements finite rational divisors on a curve,{} that is finite formal sums SUM(\spad{n} * \spad{P}) where the \spad{n}\spad{'s} are integers and the \spad{P}\spad{'s} are finite rational points on the curve.
21686`FiniteFieldCyclicGroupExtensionByPolynomial(\spad{GF},{}defpol) implements a finite extension field of the ground field {\em GF}. Its elements are represented by powers of a primitive element,{} \spadignore{i.e.} a generator of the multiplicative (cyclic) group. As primitive element we choose the root of the extension polynomial {
21686`--\em defpol},{} which MUST be primitive (user responsibility). Zech logarithms are stored in a table of size half of the field size,{} and use \spadtype{SingleInteger} for representing field elements,{} hence,{} there are restrictions on the size of the field.
21860`FiniteFieldCyclicGroupExtension(\spad{GF},{}\spad{n}) implements a extension of degree \spad{n} over the ground field {\em GF}. Its elements are represented by powers of a primitive element,{} \spadignore{i.e.} a generator of the multiplicative (cyclic) group. As primitive element we choose the root of the extension polynomial,{} which is created by {\em creat
21860`--ePrimitivePoly} from \spadtype{FiniteFieldPolynomialPackage}. Zech logarithms are stored in a table of size half of the field size,{} and use \spadtype{SingleInteger} for representing field elements,{} hence,{} there are restrictions on the size of the field.
22004`FiniteFieldCyclicGroup(\spad{p},{}\spad{n}) implements a finite field extension of degee \spad{n} over the prime field with \spad{p} elements. Its elements are represented by powers of a primitive element,{} \spadignore{i.e.} a generator of the multiplicative (cyclic) group. As primitive element we choose the root of the extension polynomial,{} which is created by
22004`-- {\em createPrimitivePoly} from \spadtype{FiniteFieldPolynomialPackage}. The Zech logarithms are stored in a table of size half of the field size,{} and use \spadtype{SingleInteger} for representing field elements,{} hence,{} there are restrictions on the size of the field.
22144`FiniteFieldExtensionByPolynomial(\spad{GF},{} defpol) implements the extension of the finite field {\em GF} generated by the extension polynomial {\em defpol} which MUST be irreducible. Note: the user has the responsibility to ensure that {\em defpol} is irreducible.
22295`FiniteFieldExtensionByPolynomial(\spad{GF},{} \spad{n}) implements an extension of the finite field {\em GF} of degree \spad{n} generated by the extension polynomial constructed by \spadfunFrom{createIrreduciblePoly}{FiniteFieldPolynomialPackage} from \spadtype{FiniteFieldPolynomialPackage}.
22411`FiniteFieldNormalBasisExtensionByPolynomial(\spad{GF},{}uni) implements a finite extension of the ground field {\em GF}. The elements are represented by coordinate vectors with respect to. a normal basis,{} \spadignore{i.e.} a basis consisting of the conjugates (\spad{q}-power
22411`--s) of an element,{} in this case called normal element,{} where \spad{q} is the size of {\em GF}. The normal element is chosen as a root of the extension polynomial,{} which MUST be normal over {\em GF} (user responsibility)
22640`FiniteFieldNormalBasisExtensionByPolynomial(\spad{GF},{}\spad{n}) implements a finite extension field of degree \spad{n} over the ground field {\em GF}. The elements are represented by coordinate vectors with respect to a normal basis,{} \spadignore{i.e.} a basis consisting of the conjugates (\spad{q}-powers) of an element,{} in this case called normal element
22640`--. This is chosen as a root of the extension polynomial,{} created by {\em createNormalPoly} from \spadtype{FiniteFieldPolynomialPackage}
22784`FiniteFieldNormalBasis(\spad{p},{}\spad{n}) implements a finite extension field of degree \spad{n} over the prime field with \spad{p} elements. The elements are represented by coordinate vectors with respect to a normal basis,{} \spadignore{i.e.} a basis consisting of the conjugates (\spad{q}-powers) of an element,{} in this case called normal element. This is cho
22784`--sen as a root of the extension polynomial created by \spadfunFrom{createNormalPoly}{FiniteFieldPolynomialPackage}.
22924`FiniteField(\spad{p},{}\spad{n}) implements finite fields with \spad{p^n} elements. This packages checks that \spad{p} is prime. For a non-checking version,{} see \spadtype{InnerFiniteField}.
23036`\indented{1}{A FlexibleArray is the notion of an array intended to allow for growth} at the end only. Hence the following efficient operations \indented{2}{\spad{concat!(a,{} x)} meaning append item \spad{x} at the end of the array \spad{a}} \indented{2}{\spad{delete!(a,{} n)} meaning delete the last item from the array \spad{a}} Flexible arrays support the other operations inherited from \spadtype{ExtensibleLinearAg
23036`--gregate}. However,{} these are not efficient. Flexible arrays combine the \spad{O(1)} access time property of arrays with growing and shrinking at the end in \spad{O(1)} (average) time. This is done by using an ordinary array which may have zero or more empty slots at the end. When the array becomes full it is copied into a new larger (50\% larger) array. Conversely,{} when the array becomes less than 1/2 full,{} it is copied into a smaller array. Flexible arrays provide for an efficient imple
23036`--mentation of many data structures in particular heaps,{} stacks and sets.
23122`\spadtype{Float} implements arbitrary precision floating point arithmetic. The number of significant digits of each operation can be set to an arbitrary value (the default is 20 decimal digits). The operation \spad{float(mantissa,{}exponent,{}\spadfunFrom{base}{FloatingPointSystem})} for integer \spad{mantissa},{} \spad{exponent} specifies the number \spad{mantissa * \spadfunFrom{base}{FloatingPointSystem} ^ exponent} The underlying representat
23122`--ion for floats is binary not decimal. The implications of this are described below. \blankline The model adopted is that arithmetic operations are rounded to to nearest unit in the last place,{} that is,{} accurate to within \spad{2^(-\spadfunFrom{bits}{FloatingPointSystem})}. Also,{} the elementary functions and constants are accurate to one unit in the last place. A float is represented as a record of two integers,{} the mantissa and the exponent. The \spadfunFrom{base}{FloatingPointSystem} 
23122`--of the representation is binary,{} hence a \spad{Record(m:mantissa,{}e:exponent)} represents the number \spad{m * 2 ^ e}. Though it is not assumed that the underlying integers are represented with a binary \spadfunFrom{base}{FloatingPointSystem},{} the code will be most efficient when this is the the case (this is \spad{true} in most implementations of Lisp). The decision to choose the \spadfunFrom{base}{FloatingPointSystem} to be binary has some unfortunate consequences. First,{} decimal numb
23122`--ers like 0.3 cannot be represented exactly. Second,{} there is a further loss of accuracy during conversion to decimal for output. To compensate for this,{} if \spad{d} digits of precision are specified,{} \spad{1 + ceiling(log2 d)} bits are used. Two numbers that are displayed identically may therefore be not equal. On the other hand,{} a significant efficiency loss would be incurred if we chose to use a decimal \spadfunFrom{base}{FloatingPointSystem} when the underlying integer base is binar
23122`--y. \blankline Algorithms used: For the elementary functions,{} the general approach is to apply identities so that the taylor series can be used,{} and,{} so that it will converge within \spad{O( sqrt n )} steps. For example,{} using the identity \spad{exp(x) = exp(x/2)^2},{} we can compute \spad{exp(1/3)} to \spad{n} digits of precision as follows. We have \spad{exp(1/3) = exp(2 ^ (-sqrt s) / 3) ^ (2 ^ sqrt s)}. The taylor series will converge in less than sqrt \spad{n} steps and the exponent
23122`--iation requires sqrt \spad{n} multiplications for a total of \spad{2 sqrt n} multiplications. Assuming integer multiplication costs \spad{O( n^2 )} the overall running time is \spad{O( sqrt(n) n^2 )}. This approach is the best known approach for precisions up to about 10,{}000 digits at which point the methods of Brent which are \spad{O( log(n) n^2 )} become competitive. Note also that summing the terms of the taylor series for the elementary functions is done using integer operations. This av
23122`--oids the overhead of floating point operations and results in efficient code at low precisions. This implementation makes no attempt to reuse storage,{} relying on the underlying system to do \spadgloss{garbage collection}. \spad{I} estimate that the efficiency of this package at low precisions could be improved by a factor of 2 if in-place operations were available. \blankline Running times: in the following,{} \spad{n} is the number of bits of precision \indented{5}{\spad{*},{} \spad{/},{} \
23122`--spad{sqrt},{} \spad{\spad{pi}},{} \spad{exp1},{} \spad{log2},{} \spad{log10}: \spad{ O( n^2 )}} \indented{5}{\spad{exp},{} \spad{log},{} \spad{sin},{} \spad{atan}:\space{2}\spad{ O( sqrt(n) n^2 )}} The other elementary functions are coded in terms of the ones above.
23180`This domain builds representations of program code segments for use with the FortranProgram domain.
23233`A domain of expressions involving functions which can be translated into standard Fortran-77,{} with some extra extensions from the NAG Fortran Library.
23388`\axiomType{FortranProgram} allows the user to build and manipulate simple models of FORTRAN subprograms. These can then be transformed into actual FORTRAN notation.
23567`Creates and manipulates objects which correspond to the basic FORTRAN data types: REAL,{} INTEGER,{} COMPLEX,{} LOGICAL and CHARACTER
23639`Code to manipulate Fortran templates
23716`Creates and manipulates objects which correspond to FORTRAN data types,{} including array dimensions.
23769`\indented{1}{Author: James Davenport} Date Created: 17 April 1992 Date Last Updated: 12 June 1992 Basic Functions: Related Constructors: Also See: AMS Classifications: Keywords: References: Description:
23842`\indented{1}{Author: James Davenport} Date Created: 17 April 1992 Date Last Updated: Basic Functions: Related Constructors: Also See: AMS Classifications: Keywords: References: Description:
23954`\spad{Canonical} means that equal elements are in fact identical.
24036`Fractional ideals in a framed algebra.
24213`Module representation of fractional ideals.
24393`The free abelian group on a set \spad{S} is the monoid of finite sums of the form \spad{reduce(+,{}[\spad{ni} * \spad{si}])} where the \spad{si}\spad{'s} are in \spad{S},{} and the \spad{ni}\spad{'s} are integers. The operation is commutative.
24471`The free abelian monoid on a set \spad{S} is the monoid of finite sums of the form \spad{reduce(+,{}[\spad{ni} * \spad{si}])} where the \spad{si}\spad{'s} are in \spad{S},{} and the \spad{ni}\spad{'s} are non-negative integers. The operation is commutative.
24576`The free group on a set \spad{S} is the group of finite products of the form \spad{reduce(*,{}[\spad{si} ^ \spad{ni}])} where the \spad{si}\spad{'s} are in \spad{S},{} and the \spad{ni}\spad{'s} are integers. The multiplication is not commutative.
24639`A \spad{bi}-module is a free module over a ring with generators indexed by an ordered set. Each element can be expressed as a finite linear combination of generators. Only non-zero terms are stored. old domain \spad{FreeModule1} was merged to it in May 2009 The description of the latter: \indented{2}{This domain implements linear combinations} \indented{2}{of elements from the domain \spad{S} with coefficients} \indented{2}{in
24639`-- the domain \spad{R} where \spad{S} is an ordered set} \indented{2}{and \spad{R} is a ring (which may be non-commutative).} \indented{2}{This domain is used by domains of non-commutative algebra such as:} \indented{6}{\spadtype{XDistributedPolynomial},{}} \indented{6}{\spadtype{XRecursivePolynomial}.} \indented{2}{Author: Michel Petitot (petitot@lifl.\spad{fr})}
24715`The free monoid on a set \spad{S} is the monoid of finite products of the form \spad{reduce(*,{}[\spad{si} ^ \spad{ni}])} where the \spad{si}\spad{'s} are in \spad{S},{} and the \spad{ni}\spad{'s} are nonnegative integers. The multiplication is not commutative.
24781`Generate the Free Lie Algebra over a ring \spad{R} with identity; A \spad{P}. Hall basis is generated by a package call to HallBasis.
24920`\indented{1}{Full partial fraction expansion of rational functions} Author: Manuel Bronstein Date Created: 9 December 1992 Date Last Updated: 6 October 1993 References: \spad{M}.Bronstein & \spad{B}.Salvy,{} \indented{12}{Full Partial Fraction Decomposition of Rational Functions,{}} \indented{12}{in Proceedings of \spad{ISSAC'93},{} Kiev,{} ACM Press.}
25054`This domain implements named functions
25125`\indented{2}{This type supports distributed multivariate polynomials} whose variables are from a user specified list of symbols. The coefficient ring may be non commutative,{} but the variables are assumed to commute. The term ordering is specified by its third parameter. Suggested types which define term
25125`-- orderings include: \spadtype{DirectProduct},{} \spadtype{HomogeneousDirectProduct},{} \spadtype{SplitHomogeneousDirectProduct} and finally \spadtype{OrderedDirectProduct} which accepts an arbitrary user function to define a term ordering.
25325`This package \undocumented
25626`A domain for polynomial sets.
25823`\spadtype{GeneralQuaternion} implements general quaternions \indented{2}{over a commutative ring. The main constructor function is} \indented{2}{\spadfun{quatern} which takes 4 arguments: the real part,{}} \indented{2}{the \spad{i} imaginary part,{} the \spad{j} imaginary part and the \spad{k} imaginary part.}
25913`A sparse table has a default entry,{} which is returned if no other value has been explicitly stored for a key.
26065`A domain constructor of the category \axiomType{TriangularSetCategory}. The only requirement for a list of polynomials to be a member of such a domain is the following: no polynomial is constant and two distinct polynomials have distinct main variables. Such a triangular set may not be auto-reduced or consistent. Triangular sets 
26065`--are stored as sorted lists \spad{w}.\spad{r}.\spad{t}. the main variables of their members but they are displayed in reverse order.\newline References : \indented{1}{[1] \spad{P}. AUBRY,{} \spad{D}. LAZARD and \spad{M}. MORENO MAZA "On the Theories} \indented{5}{of Triangular Sets" Journal of Symbol. Comp. (to appear)}
26240`This is a category of univariate Puiseux series constructed from univariate Laurent series. A Puiseux series is represented by a pair \spad{[r,{}f(x)]},{} where \spad{r} is a positive rational number and \spad{f(x)} is a Laurent series. This pair represents the Puiseux series \spad{f(x\^r)}.
26384`AlgebraGenericElementPackage allows you to create generic elements of an algebra,{} \spadignore{i.e.} the scalars are extended to include symbolic coefficients
26580`TwoDimensionalGraph creates virtual two dimensional graphs (to be displayed on TwoDimensionalViewports).
26637`\spad{GuessOptionFunctions0} provides operations that extract the values of options for \spadtype{Guess}.
26703`GuessOption is a domain whose elements are various options used by \spadtype{Guess}.
26758`\spadtype{HtmlFormat} provides a coercion from \spadtype{OutputForm} to html.
26816`This domain provides access to the underlying Lisp hash tables. By varying the hashfn parameter,{} tables suited for different purposes can be obtained.
26940`Heap implemented in a flexible array to allow for insertions
27015`This domain allows rational numbers to be presented as repeating hexadecimal expansions.
27101`\indented{2}{This type represents the finite direct or cartesian product of an} underlying ordered component type. The vectors are ordered first by the sum of their components,{} and then refined using a reverse lexicographic ordering. This type is a suitable third argument for \spadtype{GeneralDistributedMultivariatePolynomial}.
27226`\indented{2}{This type supports distributed multivariate polynomials} whose variables are from a user specified list of symbols. The coefficient ring may be non commutative,{} but the variables are assumed to commute. The term ordering is total degree ordering refined by reverse lexicographic ordering with 
27226`--respect to the position that the variables appear in the list of variables parameter.
27424`This domains implements finite rational divisors on an hyperelliptic curve,{} that is finite formal sums SUM(\spad{n} * \spad{P}) where the \spad{n}\spad{'s} are integers and the \spad{P}\spad{'s} are finite rational points on the curve. The equation of the curve must be \spad{y^2
27424`--} = \spad{f}(\spad{x}) and \spad{f} must have odd degree.
27649`\indented{1}{This domain implements a container of information} about the FriCAS library
27702`\spadtype{IndexedBits} is a domain to compactly represent large quantities of Boolean data.
27770`\indented{1}{Indexed direct products of abelian groups over an abelian group \spad{A} of} generators indexed by the ordered set \spad{S}. All items have finite support: only non-zero terms are stored.
27878`\indented{1}{Indexed direct products of abelian monoids over an abelian monoid \spad{A} of} generators indexed by the ordered set \spad{S}. All items have finite support. Only non-zero terms are stored.
27989`\indented{1}{Indexed direct products of objects over a set \spad{A}} of generators indexed by an ordered set \spad{S}. All items have finite support.
28100`\indented{1}{Indexed direct products of ordered abelian monoid sups \spad{A},{}} generators indexed by the ordered set \spad{S}. All items have finite support: only non-zero terms are stored.
28243`\indented{1}{Indexed direct products of ordered abelian monoids \spad{A} of} generators indexed by the ordered set \spad{S}. The inherited order is lexicographical. All items have finite support: only non-zero terms are stored.
28376`converts entire exponents to OutputForm
28466`\indented{1}{Author: Michael Monagan \spad{July/87},{} modified \spad{SMW} \spad{June/91}} A FlexibleArray is the notion of an array intended to allow for growth at the end only. Hence the following efficient operations \indented{2}{\spad{concat!(a,{} x)} meaning append item \spad{x} at the end of the array \spad{a}} \indented{2}{\spad{delete!(a,{} n)} meaning delete the last item from the array \s
28466`--pad{a}} Flexible arrays support the other operations inherited from \spadtype{ExtensibleLinearAggregate}. However,{} these are not efficient. Flexible arrays combine the \spad{O(1)} access time property of arrays with growing and shrinking at the end in \spad{O(1)} (average) time. This is done by using an ordinary array which may have zero or more empty slots at the end. When the array becomes full it is copied into a new larger (50\% larger) array. Conversely,{} when the array becomes less th
28466`--an 1/2 full,{} it is copied into a smaller array. Flexible arrays provide for an efficient implementation of many data structures in particular heaps,{} stacks and sets.
28571`\spadtype{IndexedList} is a basic implementation of the functions in \spadtype{ListAggregate},{} often using functions in the underlying LISP system. The second parameter to the constructor (\spad{mn}) is the beginning index of the list. That is,{} if \spad{l} is a list,{} then \spad{elt(l,{}mn)} is the first value. This constructor is probably best viewed as the implementation of singly-linked lists that are addressable by index rat
28571`--her than as a mere wrapper for LISP lists.
28640`An \spad{IndexedMatrix} is a matrix where the minimal row and column indices are parameters of the type. The domains Row and Col are both IndexedVectors. The index of the 'first' row may be obtained by calling the function \spadfun{minRowIndex}. The index of the 'first' column may be obtained by calling the function \spadfun{minColIndex}. The index of the
28640`-- first element of a 'Row' is the same as the index of the first column in a matrix and vice versa.
28789`\indented{1}{Author Micheal Monagan \spad{Aug/87}} This is the basic one dimensional array data type.
28890`This domain implements low-level strings
28965`\indented{1}{An IndexedTwoDimensionalArray is a 2-dimensional array where} the minimal row and column indices are parameters of the type. Rows and columns are returned as IndexedOneDimensionalArray\spad{'s} with minimal indices matching those of the IndexedTwoDimensionalArray. The index of the 'first' ro
28965`--w may be obtained by calling the function 'minRowIndex'. The index of the 'first' column may be obtained by calling the function 'minColIndex'. The index of the first element of a 'Row' is the same as the index of the first column in an array and vice versa.
29166`\indented{2}{This type represents vector like objects with varying lengths} and a user-specified initial index.
29240`\indented{1}{This package implements 'infinite tuples' for the interpreter.} The representation is a stream.
29318`Algebraic closure of the rational numbers.
29385`InnerFiniteField(\spad{p},{}\spad{n}) implements finite fields with \spad{p^n} elements where \spad{p} is assumed prime but does not check. For a version which checks that \spad{p} is prime,{} see \spadtype{FiniteField}.
29508`Internal implementation of a free abelian monoid.
29634`\indented{1}{This is an internal type which provides an implementation of} 2-dimensional arrays as PrimitiveArray\spad{'s} of PrimitiveArray\spad{'s}.
29833`This domain implements \spad{Zp},{} the \spad{p}-adic completion of the integers. This is an internal domain.
29928`InnerPrimeField(\spad{p}) implements the field with \spad{p} elements. Note: argument \spad{p} MUST be a prime (this domain does not check). See \spadtype{PrimeField} for a domain that does check.
30012`InnerSparseUnivariatePowerSeries is an internal domain \indented{2}{used for creating sparse Taylor and Laurent series.}
30131`This domain is used to provide a conditional "add" domain for the implementation of \spadtype{Table}.
30288`Internal package for dense Taylor series. This is an internal Taylor series type in which Taylor series are represented by a \spadtype{Stream} of \spadtype{Ring} elements. For univariate series,{} the \spad{Stream} elements are the Taylor coefficients. For multivariate series,{} the \spad{n}th Stream element is a form of degree \spad{n} in the power series variables.
30355`Domain of parsed forms which can be passed to the interpreter. This is also the interface between algebra code and facilities in the interpreter.
30464`IntegerMod(\spad{n}) creates the ring of integers reduced modulo the integer \spad{n}.
30540`\spadtype{Integer} provides the domain of arbitrary precision integers.
30598`If a function \spad{f} has an elementary integral \spad{g},{} then \spad{g} can be written in the form \spad{g = h + c1 log(u1) + c2 log(u2) + ... + cn log(un)} where \spad{h},{} which is in the same field than \spad{f},{} is called the rational part of the integral,{} and \spad{c1 log(u1) + ... cn log(un)} is called the logarithmic part of the integral. This domain manipulates integrals represented in that form,{} by kee
30598`--ping both parts separately. The logs are not explicitly computed.
30679`\indented{1}{+ Author: Mike Dewar} + Date Created: November 1996 + Date Last Updated: + Basic Functions: + Related Constructors: + Also See: + AMS Classifications: + Keywords: + References: + Description: + This domain is an implementation of interval arithmetic and transcendental + functions over intervals.
30763`A kernel over a set \spad{S} is an operator applied to a given list of arguments from \spad{S}.
30828`This domain allows a random access file to be viewed both as a table and as a file object.
30949`\indented{1}{Univariate polynomials with negative and positive exponents.} Author: Manuel Bronstein Date Created: May 1988 Date Last Updated: 26 Apr 1990
31083`This domain provides a simple way to save values in files.
31148`Management of the Lie Group associated with a free nilpotent Lie algebra. Every Lie bracket with length greater than \axiom{Order} are assumed to be null. The implementation inherits from the \spadtype{XPBWPolynomial} domain constructor: Lyndon coordinates are exponential coordinates of the second kind. \newline Author: Michel Petitot (petitot@lifl.\spad{fr}).
31271`This type supports Lie polynomials in Lyndon basis see Free Lie Algebras by \spad{C}. Reutenauer (Oxford science publications). \newline Author: Michel Petitot (petitot@lifl.\spad{fr}).
31380`LieSquareMatrix(\spad{n},{}\spad{R}) implements the Lie algebra of the \spad{n} by \spad{n} matrices over the commutative ring \spad{R}. The Lie bracket (commutator) of the algebra is given by \spad{a*b := (a *\$SQMATRIX(n,{}R) b - b *\$SQMATRIX(n,{}R) a)},{} where \spadfun{*\$SQMATRIX(\spad{n},{}\spad{R})} is the usual matrix multiplication.
31529`\spad{LinearOrdinaryDifferentialOperator1} defines a ring of differential operators with coefficients in a differential ring A. Multiplication of operators corresponds to functional composition: \indented{4}{\spad{(L1 * L2).(f) = L1 L2 f}}
31652`\spad{LinearOrdinaryDifferentialOperator2} defines a ring of differential operators with coefficients in a differential ring A and acting on an A-module \spad{M}. Multiplication of operators corresponds to functional composition: \indented{4}{\spad{(L1 * L2).(f) = L1 L2 f}}
31815`\spad{LinearOrdinaryDifferentialOperator} defines a ring of differential operators with coefficients in a ring A with a given derivation. Multiplication of operators corresponds to functional composition: \indented{4}{\spad{(L1 * L2).(f) = L1 L2 f}}
31938`This internal package represents monoid (abelian or not,{} with or without inverses) as lists and provides some common operations to the various flavors of monoids.
32115`\spadtype{List} implements singly-linked lists that are addressable by indices; the index of the first element is 1. In addition to the operations provided by \spadtype{IndexedList},{} this constructor provides some LISP-like functions such as \spadfun{null} and \spadfun{cons}.
32175`LocalAlgebra produces the localization of an algebra,{} \spadignore{i.e.} fractions whose numerators come from some \spad{R} algebra.
32262`Localize(\spad{M},{}\spad{R}) produces fractions with numerators from an \spad{R} module \spad{M} and denominators being the nonzero elements of \spad{R}.
32343`Lyndon words over arbitrary (ordered) symbols: see Free Lie Algebras by \spad{C}. Reutenauer (Oxford science publications). A Lyndon word is a word which is smaller than any of its right factors \spad{w}.\spad{r}.\spad{t}. the pure lexicographical ordering. If \axiom{a} and \axiom{\spad{b}} are two Lyndon words such that \axiom{a < \spad{b}} holds \spad{w}.\spad{r}.\spad{t} lexicographical ordering then \axiom{a*b} is a Lyndon wor
32343`--d. Parenthesized Lyndon words can be generated from symbols by using the following rule: \axiom{[[a,{}\spad{b}],{}\spad{c}]} is a Lyndon word iff \axiom{a*b < \spad{c} \spad{<=} \spad{b}} holds. Lyndon words are internally represented by binary trees using the \spadtype{Magma} domain constructor. Two ordering are provided: lexicographic and length-lexicographic. \newline Author : Michel Petitot (petitot@lifl.\spad{fr}).
32416`A domain which models the complex number representation used by machines in the AXIOM-NAG link.
32492`A domain which models the floating point representation used by machines in the AXIOM-NAG link.
32559`A domain which models the integer representation used by machines in the AXIOM-NAG link.
32633`This type is the basic representation of parenthesized words (binary trees over arbitrary symbols) useful in \spadtype{LiePolynomial}. \newline Author: Michel Petitot (petitot@lifl.\spad{fr}).
32701`MakeCachableSet(\spad{S}) returns a cachable set which is equal to \spad{S} as a set.
32778`\spad{Mapping(T,S)} is a primitive type in FriCAS used to represent mappings from source type \spad{S} to target type \spad{T}. Similarly, \spad{Mapping(T,A,B)} denotes a mapping from source type \spad{(A,B)} to target type \spad{T}.
32820`\spadtype{MathMLFormat} provides a coercion from \spadtype{OutputForm} to MathML format.
32880`\spadtype{Matrix} is a matrix domain where 1-based indexing is used for both rows and columns.
32966`This package \undocumented
33090`\indented{1}{These domains are used for the factorization and gcds} of univariate polynomials over the integers in order to work modulo different primes. See \spadtype{ModularRing},{} \spadtype{EuclideanModularRing}
33283`These domains are used for the factorization and gcds of univariate polynomials over the integers in order to work modulo different primes. See \spadtype{EuclideanModularRing} ,{}\spadtype{ModularField}
33473`This package \undocumented
33638`Algebra of ADDITIVE operators on a module.
33718`\indented{1}{MoebiusTransform(\spad{F}) is the domain of fractional linear (Moebius)} transformations over \spad{F}.
33784`\spadtype{MonoidRing}(\spad{R},{}\spad{M}),{} implements the algebra of all maps from the monoid \spad{M} to the commutative ring \spad{R} with finite support. Multiplication of two maps \spad{f} and \spad{g} is defined to map an element \spad{c} of \spad{M} to the (convolution) sum over {\em f(a)g(b)} such that {\em ab = c}. Thus \spad{M} can be identified with a canonical basis and the maps can also be considered as formal li
33784`--near combinations of the elements in \spad{M}. Scalar multiples of a basis element are called monomials. A prominent example is the class of polynomials where the monoid is a direct product of the natural numbers with pointwise addition. When \spad{M} is \spadtype{FreeMonoid Symbol},{} one gets polynomials in infinitely many non-commuting variables. Another application area is representation theory of finite groups \spad{G},{} where modules over \spadtype{MonoidRing}(\spad{R},{}\spad{G}) are s
33784`--tudied.
33860`A multiset is a set with multiplicities.
33936`\indented{2}{This type is the basic representation of sparse recursive multivariate} polynomials whose variables are from a user specified list of symbols. The ordering is specified by the position of the variable in the list. The coefficient ring may be non commutative,{} but the variables are assumed to commute.
34095`A post-facto extension for \axiomType{\spad{SMP}} in order to speed up operations related to pseudo-division and \spad{gcd}. This domain is based on the \axiomType{NSUP} constructor which is itself a post-facto extension of the \axiomType{SUP} constructor.
34254`A post-facto extension for \axiomType{SUP} in order to speed up operations related to pseudo-division and \spad{gcd} for both \axiomType{SUP} and,{} consequently,{} \axiomType{NSMP}.
34355`\spadtype{NonNegativeInteger} provides functions for non \indented{2}{negative integers.}
34429`\spadtype{None} implements a type with no objects. It is mainly used in technical situations where such a thing is needed (\spadignore{e.g.} the interpreter and some of the internal \spadtype{Expression} code).
34468`Octonion implements octonions (Cayley-Dixon algebra) over a commutative ring,{} an eight-dimensional non-associative algebra,{} doubling the quaternions in the same way as doubling the complex numbers to get the quaternions the main constructor function is {\em octon} which takes 8 arguments: the real part,{} the \spad{i} imaginary part,{} the \spad{j} imaginary part,{} the \spad{k} imaginary part,{} (as with quaternions) and
34468`-- in addition the imaginary parts \spad{E},{} \spad{I},{} \spad{J},{} \spad{K}.
34546`This is the domain of 1-based one dimensional arrays
34639`Adjunction of a complex infinity to a set. Date Created: 4 Oct 1989 Date Last Updated: 1 Nov 1989
34719`\spadtype{OpenMathConnection} provides low-level functions for handling connections to and from \spadtype{OpenMathDevice}\spad{s}.
34766`\spadtype{OpenMathDevice} provides support for reading and writing openMath objects to files,{} strings etc. It also provides access to low-level operations from within the interpreter.
34808`\spadtype{OpenMathEncoding} is the set of valid OpenMath encodings.
34870`\spadtype{OpenMathErrorKind} represents different kinds of OpenMath errors: specifically parse errors,{} unknown \spad{CD} or symbol errors,{} and read errors.
34934`\spadtype{OpenMathError} is the domain of OpenMath errors.
34993`Algebra of ADDITIVE operators over a ring.
35044`This constructor creates the \spadtype{MonogenicLinearOperator} domain which is ``opposite\spad{''} in the ring sense to \spad{P}. That is,{} as sets \spad{P = \$} but \spad{a * b} in \spad{\$} is equal to \spad{b * a} in \spad{P}.
35173`\indented{1}{Author : Larry Lambe} Date created : 14 August 1988 Date Last Updated : 11 March 1991 Description : A domain used in order to take the free \spad{R}-module on the Integers \spad{I}. This is actually the forgetful functor from OrderedRings to OrderedSets applied to \spad{I}
35226`Adjunction of two real infinites quantities to a set. Date Created: 4 Oct 1989 Date Last Updated: 1 Nov 1989
35305`\indented{2}{This type represents the finite direct or cartesian product of an} underlying ordered component type. The ordering on the type is determined by its third argument which represents the less than function on vectors. This type is a suitable third argument for \spadtype{GeneralDistributedMultivariatePolynomial}.
35460`\indented{3}{The free monoid on a set \spad{S} is the monoid of finite products of} the form \spad{reduce(*,{}[\spad{si} ^ \spad{ni}])} where the \spad{si}\spad{'s} are in \spad{S},{} and the \spad{ni}\spad{'s} are non-negative integers. The multiplication is not commutative. For two elements \spad{x} and \spad{y} the relation \spad{x < y} holds if either \spad{length(x) < length(y)} holds or if these lengths are equal and
35460`-- if \spad{x} is smaller than \spad{y} \spad{w}.\spad{r}.\spad{t}. the lexicographical ordering induced by \spad{S}. This domain inherits implementation from \spadtype{FreeMonoid}.
35541`This domain implements ordered variables
35633`\spadtype{OrderlyDifferentialPolynomial} implements an ordinary differential polynomial ring in arbitrary number of differential indeterminates,{} with coefficients in a ring. The ranking on the differential indeterminate is orderly. This is analogous to the domain \spadtype{Polynomial}. \blankline
35834`\spadtype{OrderlyDifferentialVariable} adds a commonly used orderly ranking to the set of derivatives of an ordered list of differential indeterminates. An orderly ranking is a ranking \spadfun{<} of the derivatives with the property that for two derivatives \spad{u} and \spad{v},{} \spad{u} \spadfun{<} \spad{v} if the \spadfun{order} of \spad{u} is less than that of \spad{v}. This domain belongs to \spadtyp
35834`--e{DifferentialVariableCategory}. It defines \spadfun{weight} to be just \spadfun{order},{} and it defines an orderly ranking \spadfun{<} on derivatives \spad{u} via the lexicographic order on the pair (\spadfun{order}(\spad{u}),{} \spadfun{variable}(\spad{u})).
35930`This constructor produces an ordinary differential ring from a partial differential ring by specifying a variable.
36079`This domain represents truncated weighted polynomials over the "Polynomial" type. The variables must be specified,{} as must the weights. The representation is sparse in the sense that only non-zero terms are represented.
36221`This domain is used to create and manipulate mathematical expressions for output. It is intended to provide an insulating layer between the expression rendering software (\spadignore{e.g.} TeX,{} or Script) and the output coercions in the various domains.
36279`Stream-based implementation of \spad{Zp:} \spad{p}-adic numbers are represented as sum(\spad{i} = 0..,{} a[\spad{i}] * p^i),{} where the a[\spad{i}] lie in 0,{}1,{}...,{}(\spad{p} - 1).
36349`This is the category of stream-based representations of \spad{Qp}.
36486`Stream-based implementation of \spad{Qp:} numbers are represented as sum(\spad{i} = \spad{k}..,{} a[\spad{i}] * p^i) where the a[\spad{i}] lie in 0,{}1,{}...,{}(\spad{p} - 1).
36585`This domain describes four groups of color shades (palettes).
36640`ParametricPlaneCurve is used for plotting parametric plane curves in the affine plane.
36714`ParametricSpaceCurve is used for plotting parametric space curves in affine 3-space.
36788`ParametricSurface is used for plotting parametric surfaces in affine 3-space.
36858`The domain \spadtype{PartialFraction} implements partial fractions over a euclidean domain \spad{R}. This requirement on the argument domain allows us to normalize the fractions. Of particular interest are the 2 forms for these fractions. The ``compact\spad{''} form has only one fractional term per prime in the denominator,{} while the \spad{``p}-adic\spad{''} form expands each numerator \spad{p}-adically via the prime \spad{p} in t
36858`--he denominator. For computational efficiency,{} the compact form is used,{} though the \spad{p}-adic form may be gotten by calling the function \spadfunFrom{padicFraction}{PartialFraction}. For a general euclidean domain,{} it is not known how to factor the denominator. Thus the function \spadfunFrom{partialFraction}{PartialFraction} takes as its second argument an element of \spadtype{Factored(R)}.
36929`\indented{1}{Partition is an OrderedCancellationAbelianMonoid which is used} as the basis for symmetric polynomial representation of the sums of powers in SymmetricPolynomial. Thus,{} \spad{(5 2 2 1)} will represent \spad{s5 * s2^2 * s1}.
37008`A PatternMatchListResult is an object internally returned by the pattern matcher when matching on lists. It is either a failed match,{} or a pair of PatternMatchResult,{} one for atoms (elements of the list),{} and one for lists.
37129`A PatternMatchResult is an object internally returned by the pattern matcher; It is either a failed match,{} or a list of matches of the form (var,{} expr) meaning that the variable var matches the expression expr.
37224`Patterns for use by the pattern matcher.
37293`\indented{1}{A PendantTree(\spad{S})is either a leaf? and is an \spad{S} or has} a left and a right both PendantTree(\spad{S})\spad{'s}
37383`PermutationGroup implements permutation groups acting on a set \spad{S},{} \spadignore{i.e.} all subgroups of the symmetric group of \spad{S},{} represented as a list of permutations (generators). Note that therefore the objects are not members of the \Language category \spadtype{Group}. Using the idea of base and strong generators by Sims,{} basic routines and algorithms are implemented so that the word problem for permutati
37383`--on groups can be solved.
37461`Permutation(\spad{S}) implements the group of all bijections \indented{2}{on a set \spad{S},{} which move only a finite number of points.} \indented{2}{A permutation is considered as a map from \spad{S} into \spad{S}. In particular} \indented{2}{multiplication is defined as composition of maps:} \indented{2}{{\em pi1 * pi2 = pi1 o pi2}.} \indented{2}{The internal representation of permuatations are two lists} \indented{2}{
37461`--of equal length representing preimages and images.}
37542`\indented{1}{Symbolic fractions in \%\spad{pi} with integer coefficients;} \indented{1}{The point for using \spad{Pi} as the default domain for those fractions} \indented{1}{is that \spad{Pi} is coercible to the float types,{} and not Expression.} Date Created: 21 Feb 1990 Date Last Updated: 12 Mai 1992
37585`\indented{1}{Plot a NON-SINGULAR plane algebraic curve \spad{p}(\spad{x},{}\spad{y}) = 0.} Author: Clifton \spad{J}. Williamson Date Created: Fall 1988 Date Last Updated: 27 April 1990 Keywords: algebraic curve,{} non-singular,{} plot Examples: References:
37671`Plot3D supports parametric plots defined over a real number system. A real number system is a model for the real numbers and as such may be an approximation. For example,{} floating point numbers and infinite continued fractions are real number systems. The facilities at this point are limited to 3-dimensional parametric plots.
37740`The Plot domain supports plotting of functions defined over a real number system. A real number system is a model for the real numbers and as such may be an approximation. For example floating point numbers and infinite continued fractions. The facilities at this point are limited to 2-dimensional plots or either a single function or a parametric function.
37805`This domain provides the internal representation of polynomials in non-commutative variables written over the Poincare-Birkhoff-Witt basis. See the \spadtype{XPBWPolynomial} domain constructor. See Free Lie Algebras by \spad{C}. Reutenauer (Oxford science publications). \newline Author: Michel Petitot (petitot@lifl.\spad{fr}).
37899`This domain implements points in coordinate space
37952`This domain represents polynomial ideals with coefficients in any field and supports the basic ideal operations,{} including intersection sum and quotient. An ideal is represented by a list of polynomials (the generators of the ideal) and a boolean that is \spad{true} if the generators are a Groebner basis. The algorithms used
37952`-- are based on Groebner basis computations. The ordering is determined by the datatype of the input polynomials. Users may use refinements of total degree orderings.
38131`This domain represents generalized polynomials with coefficients (from a not necessarily commutative ring),{} and terms indexed by their exponents (from an arbitrary ordered abelian monoid). This type is used,{} for example,{} by the \spadtype{DistributedMultivariatePolynomial} domain where the exponent domain is a direct product of non negative integers.
38237`\indented{2}{This type is the basic representation of sparse recursive multivariate} polynomials whose variables are arbitrary symbols. The ordering is alphabetic determined by the Symbol type. The coefficient ring may be non commutative,{} but the variables are assumed to commute.
38341`\spadtype{PositiveInteger} provides functions for \indented{2}{positive integers.}
38411`PrimeField(\spad{p}) implements the field with \spad{p} elements if \spad{p} is a prime number. Error: if \spad{p} is not prime. Note: this domain does not check that argument is a prime.
38490`\indented{1}{This provides a fast array type with no bound checking on elt\spad{'s}.} Minimum index is 0 in this type,{} cannot be changed
38569`This domain implements cartesian product
38654`This domain provides modest support for quadratic forms.
38742`\spadtype{QuasiAlgebraicSet} constructs a domain representing quasi-algebraic sets,{} which is the intersection of a Zariski closed set,{} defined as the common zeros of a given list of polynomials (the defining polynomials for equations),{} and a principal Zariski open set,{} defined as the complement of the common zeros of a 
38742`--polynomial \spad{f} (the defining polynomial for the inequation). This domain provides simplification of a user-given representation using groebner basis computations. There are two simplification routines: the first function \spadfun{idealSimplify} uses groebner basis of ideals alone,{} while the second,{} \spadfun{simplify} uses both groebner basis and factorization. The resulting defining equations \spad{L} always form a groebner basis,{} and the resulting defining inequation \spad{f} is al
38742`--ways reduced. The function \spadfun{simplify} may be applied several times if desired. A third simplification routine \spadfun{radicalSimplify} is provided in \spadtype{QuasiAlgebraicSet2} for comparison study only,{} as it is inefficient compared to the other two,{} as well as is restricted to only certain coefficient domains. For detail analysis and a comparison of the three methods,{} please consult the reference cited. \blankline A polynomial function \spad{q} defined on the quasi-algebrai
38742`--c set is equivalent to its reduced form with respect to \spad{L}. While this may be obtained using the usual normal form algorithm,{} there is no canonical form for \spad{q}. \blankline The ordering in groebner basis computation is determined by the data type of the input polynomials. If it is possible we suggest to use refinements of total degree orderings.
38920`\spadtype{Quaternion} implements Hamilton quaternions \indented{2}{over a commutative ring.}
38993`This domain implements simple database queries
39065`Linked List implementation of a Queue
39136`Function field defined by \spad{y^n} = \spad{f}(\spad{x}).
39376`This domain allows rational numbers to be presented as repeating decimal expansions or more generally as repeating expansions in any base.
39466`This domain implements the real closure of an ordered field.
39559`\spad{Record(a:A,b:B,...,c:C)} is a primitive type in FriCAS used to represent composite objects made up of objects of type \spad{A}, \spad{B},..., \spad{C} which are indexed by "keys" (identifiers) \spad{a},\spad{b},...,\spad{c}.
39611`\spadtype{RectangularMatrix} is a matrix domain where the number of rows and the number of columns are parameters of the domain.
39788`\indented{1}{\spadtype{Reference} is for making a changeable instance} of something.
39841`A domain for regular chains (\spadignore{i.e.} regular triangular sets) over a \spad{Gcd}-Domain and with a fix list of variables. This is just a front-end for the \spadtype{RegularTriangularSet} domain constructor.
40077`This domain provides an implementation of regular chains. Moreover,{} the operation \axiomOpFrom{zeroSetSplit}{RegularTriangularSetCategory} is an implementation of a new algorithm for solving polynomial systems by means of regular chains.\newline References : \indented{1}{[1] \spad{M}. MORENO MAZA "A new algorithm f
40077`--or computing triangular} \indented{5}{decomposition of algebraic varieties" NAG Tech. Rep. 4/98.}
40266`ResidueRing is the quotient of a polynomial ring by an ideal. The ideal is given as a list of generators. The elements of the domain are equivalence classes expressed in terms of reduced elements
40462`A domain used to return the results from a call to the NAG Library. It prints as a list of names and types,{} though the user may choose to display values automatically if he or she wishes.
40530`\indented{1}{Rules for the pattern matcher} Author: Manuel Bronstein Date Created: 24 Oct 1988 Date Last Updated: 26 October 1993 Keywords: pattern,{} matching,{} rule.
40658`\axiomType{RightOpenIntervalRootCharacterization} provides work with interval root coding.
40881`\spadtype{RomanNumeral} provides functions for converting \indented{1}{integers to roman numerals.}
40947`This domain implements named rules
41015`A ruleset is a set of pattern matching rules grouped together.
41142`\indented{1}{Domain for representing points,{} vectors,{} subspaces and properties} of components in a subspace as complex numbers
41199`an implementation of PointCategory to represent points and vectors in three dimensional coordinate space
41265`\indented{1}{number of base dimensions (excluding point at infinity and point} at origin)
41342`This domain allows the manipulation of Lisp values over arbitrary atomic types.
41519`This domain allows the manipulation of the usual Lisp values.
41618`\indented{1}{Domains for transform from SPoint to SPoint} STransform transform domain can by used to represent linear transforms for instance when working with SCartesian where its Representation is a matrix of DoubleFloat Other transform domains may not nessarily be linear and will have different representatons
41683`Generates various surfaces (2 dimentional subsets of 3D space). When 1 Dimentional subspace (a line) needs to be represented in a 3D editor then we convert the line into a tube. Many of the functions here are from NumericTubePlot and related packages. However \spad{I} needed them to be defined over SPointCategory so \spad{I} adapted them here.
41737`\indented{1}{Author: Martin Baker} Date Created: September 2010 Date Last Updated: September 2010 Basic Operations: spnt,{}sipnt,{}svec,{}sivec,{}screenCoords Related Constructors: SCarteasian,{} SConformal,{} SArgand,{} STransform Also See: AMS Classifications: Keywords: References: http://www.euclideanspace.com/maths/standards/program/mycode/graph/
41788`\spadtype{ScriptFormulaFormat} provides a coercion from \spadtype{OutputForm} to IBM SCRIPT/VS Mathematical Formula Format. The basic SCRIPT formula format object consists of three parts: a prologue,{} a formula part and an epilogue. The functions \spadfun{prologue},{} \spadfun{formula} and \spadfun{epilogue} extract these parts,{} respectively. The central parts of the expression go into the formula part. The other parts can be set (\s
41788`--padfun{setPrologue!},{} \spadfun{setEpilogue!}) so that contain the appropriate tags for printing. For example,{} the prologue and epilogue might simply contain ":df." and ":edf." so that the formula section will be printed in display math mode.
41855`This domain is used to provide the function argument syntax \spad{v=a..b}. This is used,{} for example,{} by the top-level \spadfun{draw} functions.
41917`This type is used to specify a range of values from type \spad{S}.
41982`\spadtype{SequentialDifferentialPolynomial} implements an ordinary differential polynomial ring in arbitrary number of differential indeterminates,{} with coefficients in a ring. The ranking on the differential indeterminate is sequential. \blankline
42192`\spadtype{OrderlyDifferentialVariable} adds a commonly used sequential ranking to the set of derivatives of an ordered list of differential indeterminates. A sequential ranking is a ranking \spadfun{<} of the derivatives with the property that for any derivative \spad{v},{} there are only a finite number of derivatives \spad{u} with \spad{u} \spadfun{<} \spad{v}. This domain belongs to \spadtype{Different
42192`--ialVariableCategory}. It defines \spadfun{weight} to be just \spadfun{order},{} and it defines a sequential ranking \spadfun{<} on derivatives \spad{u} by the lexicographic order on the pair (\spadfun{variable}(\spad{u}),{} \spadfun{order}(\spad{u})).
42291`\spadtype{SetOfMIntegersInOneToN} implements the subsets of \spad{M} integers in the interval \spad{[1..n]}
42390`A set over a domain \spad{S} models the usual mathematical notion of a finite set of elements from \spad{S}. Sets are unordered collections of distinct elements (that is,{} order and duplication does not matter). The notation \spad{set [a,{}b,{}c]} can be used to create a set and the usual operations such as union and intersection are available to form new sets. If \spad{S} has OrderdSet,{} \Language{} maintains the entries in sorted order. S
42390`--pecifically,{} the parts function returns the entries as a list in ascending order and the extract operation returns the maximum entry. Given two sets \spad{s} and \spad{t} where \spad{\#s = m} and \spad{\#t = n},{} the complexity of \indented{2}{\spad{s = t} is \spad{O(min(n,{}m))}} \indented{2}{\spad{s < t} is \spad{O(max(n,{}m))}} \indented{2}{\spad{union(s,{}t)},{} \spad{intersect(s,{}t)},{} \spad{minus(s,{}t)},{} \spad{symmetricDifference(s,{}t)} is \spad{O(max(n,{}m))}} \indented{2}{\spa
42390`--d{member(x,{}t)} is \spad{O(n log n)}} \indented{2}{\spad{insert(x,{}t)} and \spad{remove(x,{}t)} is \spad{O(n)}}
42451`Domain which represents simple algebraic extensions of arbitrary rings. The first argument to the domain,{} \spad{R},{} is the underlying ring,{} the second argument is a domain of univariate polynomials over \spad{K},{} while the last argument specifies the defining minimal polynomial. The elements of the domain are canonically represented as polynomials of degree less th
42451`--an that of the minimal polynomial with coefficients in \spad{R}. The second argument is both the type of the third argument and the underlying representation used by \spadtype{SAE} itself.
42583`\axiomType{SimpleFortranProgram(\spad{f},{}type)} provides a simple model of some FORTRAN subprograms,{} making it possible to coerce objects of various domains into a FORTRAN subprogram called \axiom{\spad{f}}. These can then be translated into legal FORTRAN code.
42698`SingleInteger is intended to support machine integer arithmetic.
42764`This trivial domain lets us build Univariate Polynomials in an anonymous variable
42829`\indented{1}{Domain for storing information about structure of polynomials} as vectors of exponents
43068`\indented{2}{This type is the basic representation of sparse recursive multivariate} polynomials. It is parameterized by the coefficient ring and the variable set which may be infinite. The variable ordering is determined by the variable set parameter. The coefficient ring may be non-commutative,{} but the variables are assumed to commute.
43204`This domain provides multivariate Taylor series with variables from an arbitrary ordered set. A Taylor series is represented by a stream of polynomials from the polynomial domain \spad{SMP}. The \spad{n}th element of the stream is a form of degree \spad{n}. SMTS is an internal domain.
43403`A sparse table has a default entry,{} which is returned if no other value has been explicitly stored for a key.
43506`Sparse Laurent series in one variable \indented{2}{\spadtype{SparseUnivariateLaurentSeries} is a domain representing Laurent} \indented{2}{series in one variable with coefficients in an arbitrary ring.\space{2}The} \indented{2}{parameters of the type specify the coefficient ring,{} the power series} \indented
43506`--{2}{variable,{} and the center of the power series expansion.\space{2}For example,{}} \indented{2}{\spad{SparseUnivariateLaurentSeries(Integer,{}x,{}3)} represents Laurent} \indented{2}{series in \spad{(x - 3)} with integer coefficients.}
43703`This domain represents univariate polynomials over arbitrary (not necessarily commutative) coefficient rings. The variable is unspecified so that the variable displays as \spad{?} on output. If it is necessary to specify the variable name,{} use type \spadtype{UnivariatePolynomial}. The representation is sparse in the sense that only non-zero terms are represented.
43800`Sparse Puiseux series in one variable \indented{2}{\spadtype{SparseUnivariatePuiseuxSeries} is a domain representing Puiseux} \indented{2}{series in one variable with coefficients in an arbitrary ring.\space{2}The} \indented{2}{parameters of the type specify the coefficient ring,{} the power series} \indent
43800`--ed{2}{variable,{} and the center of the power series expansion.\space{2}For example,{}} \indented{2}{\spad{SparseUnivariatePuiseuxSeries(Integer,{}x,{}3)} represents Puiseux} \indented{2}{series in \spad{(x - 3)} with \spadtype{Integer} coefficients.}
43999`This is the domain of sparse univariate skew polynomials over an Ore coefficient field. The multiplication is given by \spad{x a = \sigma(a) x + \delta a}.
44144`Sparse Taylor series in one variable \indented{2}{\spadtype{SparseUnivariateTaylorSeries} is a domain representing Taylor} \indented{2}{series in one variable with coefficients in an arbitrary ring.\space{2}The} \indented{2}{parameters of the type specify the coefficient ring,{} the power series} \indented{2}{variable,{} and the center of the power series expansion.\s
44144`--pace{2}For example,{}} \indented{2}{\spadtype{SparseUnivariateTaylorSeries}(Integer,{}\spad{x},{}3) represents Taylor} \indented{2}{series in \spad{(x - 3)} with \spadtype{Integer} coefficients.}
44281`\indented{2}{This type represents the finite direct or cartesian product of an} underlying ordered component type. The vectors are ordered as if they were split into two blocks. The \spad{dim1} parameter specifies the length of the first block. The ordering is lexicographic between the blocks but acts like \spadtype{HomogeneousDirectProduct} w
44281`--ithin each block. This type is a suitable third argument for \spadtype{GeneralDistributedMultivariatePolynomial}.
44443`This domain exports a modest implementation for the vertices of splitting trees. These vertices are called here splitting nodes. Every of these nodes store 3 informations. The first one is its value,{} that is the current expression to evaluate. The second one is its condition,{} that is the hypothesis under which the value has to be evaluated. The last one is its status,{} that is a boolean f
44443`--lag which is \spad{true} iff the value is the result of its evaluation under its condition. Two splitting vertices are equal iff they have the sane values and the same conditions (so their status do not matter).
44554`This domain exports a modest implementation of splitting trees. Spliiting trees are needed when the evaluation of some quantity under some hypothesis requires to split the hypothesis into sub-cases. For instance by adding some new hypothesis on one hand and its negation on another hand. The computations are terminated is a splitting tree \axiom{a} when \axiom{status(
44554`--value(a))} is \axiom{\spad{true}}. Thus,{} if for the splitting tree \axiom{a} the flag \axiom{status(value(a))} is \axiom{\spad{true}},{} then \axiom{status(value(\spad{d}))} is \axiom{\spad{true}} for any subtree \axiom{\spad{d}} of \axiom{a}. This property of splitting trees is called the termination condition. If no vertex in a splitting tree \axiom{a} is equal to another,{} \axiom{a} is said to satisfy the no-duplicates condition. The splitting tree \axiom{a} will satisfy this condition i
44554`--f nodes are added to \axiom{a} by mean of \axiom{splitNodeOf!} and if \axiom{construct} is only used to create the root of \axiom{a} with no children.
44692`This domain provides an implementation of square-free regular chains. Moreover,{} the operation \axiomOpFrom{zeroSetSplit}{SquareFreeRegularTriangularSetCategory} is an implementation of a new algorithm for solving polynomial systems by means of regular chains.\newline References : \indented{1}{[
44692`--1] \spad{M}. MORENO MAZA "A new algorithm for computing triangular} \indented{5}{decomposition of algebraic varieties" NAG Tech. Rep. 4/98.} \indented{2}{Version: 2}
44902`\spadtype{SquareMatrix} is a matrix domain of square matrices,{} where the number of rows (= number of columns) is a parameter of the type.
45057`Linked List implementation of a Stack
45128`A stream is an implementation of an infinite sequence using a list of terms that have been computed and a function closure to compute additional terms when needed.
45199`This domain provides tables where the keys are strings. A specialized hash function for strings is used.
45333`This domain implements some global properties of subspaces.
45407`This domain \undocumented
45491`This domain implements "such that" forms
45577`This domain builds representations of boolean expressions for use with the \axiomType{FortranCode} domain.
45642`Create and manipulate a symbol table for generated FORTRAN code
45712`Basic and scripted symbols.
45764`This domain implements symmetric polynomial
45863`This is the general purpose table type. The keys are hashed to look up the entries. This creates a \spadtype{HashTable} if equal for the Key domain is consistent with Lisp EQUAL otherwise an \spadtype{AssociationList}
45968`\indented{1}{The tableau domain is for printing Young tableaux,{} and} coercions to and from List List \spad{S} where \spad{S} is a set.
46019`\spadtype{TaylorSeries} is a general multivariate Taylor series domain over the ring Coef and with variables of type Symbol.
46118`Tensor powers of a free module over a commutative ring. It is represented as a free module over the cartesian power of the basis.
46276`Tensor product of free modules over a commutative ring. It is represented as a free module over the direct product of the respective bases. The factor domains must provide operations \spad{listOfTerms},{} whose result is assumed to be stored in reverse order.
46471`\spadtype{TexFormat} provides a coercion from \spadtype{OutputForm} to \TeX{} format. The particular dialect of \TeX{} used is \LaTeX{}. The basic object consists of three parts: a prologue,{} a tex part and an epilogue. The functions \spadfun{prologue},{} \spadfun{tex} and \spadfun{epilogue} extract these parts,{} respectively. The main guts of the expression go into the tex part. The other parts can be set (\spadfun{setPrologue!},{} \spadfun{setEpi
46471`--logue!}) so that contain the appropriate tags for printing. For example,{} the prologue and epilogue might simply contain \spad{``}\verb+\spad{\[}+\spad{''} and \spad{``}\verb+\spad{\]}+\spad{''},{} respectively,{} so that the TeX section will be printed in LaTeX display math mode.
46524`This domain provides an implementation of text files. Text is stored in these files using the native character set of the computer.
46599`Creates and manipulates one global symbol table for FORTRAN code generation,{} containing details of types,{} dimensions,{} and argument lists.
46670`This domain represents three dimensional matrices over a general object type
46828`The domain ThreeSpace is used for creating three dimensional objects using functions for defining points,{} curves,{} polygons,{} constructs and the subspaces containing them.
46892`\spadtype{Tree(S)} is a basic domains of tree structures. Each tree is either empty or else is a {\it node} consisting of a value and a list of (sub)trees.
46965`\indented{2}{Package for constructing tubes around 3-dimensional parametric curves.} Domain of tubes around 3-dimensional parametric curves.
47034`\indented{1}{This domain is used to interface with the interpreter\spad{'s} notion} of comma-delimited sequences of values.
47111`\indented{1}{A TwoDimensionalArray is a two dimensional array with} 1-based indexing for both rows and columns.
47239`TwoDimensionalViewport creates viewports to display graphs.
47308`This is a low-level domain which implements vectors (one dimensional arrays) of unsigned 32-bit numbers. Indexing is 0 based,{} there is no bound checking (unless provided by lower level).
47380`\spad{Union(A,B,...,C)} is a primitive type in FriCAS used to represent objects of type \spad{A} or of type \spad{B} or...or of type \spad{C}.
47526`This package enables one to construct a univariate Laurent series domain from a univariate Taylor series domain. Univariate Laurent series are represented by a pair \spad{[n,{}f(x)]},{} where \spad{n} is an arbitrary integer and \spad{f(x)} is a Taylor series. This pair represents the Laurent series \spad{x^n * f(x)}.
47699`Dense Laurent series in one variable \indented{2}{\spadtype{UnivariateLaurentSeries} is a domain representing Laurent} \indented{2}{series in one variable with coefficients in an arbitrary ring.\space{2}The} \indented{2}{parameters of the type specify the coefficient ring,{} the power series} \indented{2}{variable,{} and 
47699`--the center of the power series expansion.\space{2}For example,{}} \indented{2}{\spad{UnivariateLaurentSeries(Integer,{}x,{}3)} represents Laurent series in} \indented{2}{\spad{(x - 3)} with integer coefficients.}
47883`This domain represents univariate polynomials in some symbol over arbitrary (not necessarily commutative) coefficient rings. The representation is sparse in the sense that only non-zero terms are represented.
47984`This package enables one to construct a univariate Puiseux series domain from a univariate Laurent series domain. Univariate Puiseux series are represented by a pair \spad{[r,{}f(x)]},{} where \spad{r} is a positive rational number and \spad{f(x)} is a Laurent series. This pair represents the Puiseux series \spad{f(x^r)}.
48159`UnivariatePuiseuxSeriesWithExponentialSingularity is a domain used to represent functions with essential singularities. Objects in this domain are sums,{} where each term in the sum is a univariate Puiseux series times 
48159`--the exponential of a univariate Puiseux series. Thus,{} the elements of this domain are sums of expressions of the form \spad{g(x) * exp(f(x))},{} where \spad{g}(\spad{x}) is a univariate Puiseux series and \spad{f}(\spad{x}) is a univariate Puiseux series with no terms of non-negative degree.
48447`Dense Puiseux series in one variable \indented{2}{\spadtype{UnivariatePuiseuxSeries} is a domain representing Puiseux} \indented{2}{series in one variable with coefficients in an arbitrary ring.\space{2}The} \indented{2}{parameters of the type specify the coefficient ring,{} the power series} \indented{2}{variable,{} an
48447`--d the center of the power series expansion.\space{2}For example,{}} \indented{2}{\spad{UnivariatePuiseuxSeries(Integer,{}x,{}3)} represents Puiseux series in} \indented{2}{\spad{(x - 3)} with \spadtype{Integer} coefficients.}
48633`This is the domain of univariate skew polynomials over an Ore coefficient field in a named variable. The multiplication is given by \spad{x a = \sigma(a) x + \delta a}.
48780`Dense Taylor series in one variable \spadtype{UnivariateTaylorSeries} is a domain representing Taylor series in one variable with coefficients in an arbitrary ring. The parameters of the type specify the coefficient ring,{} the power series variable,{} and the center of the power series expansion. For example,{} \spadtype{UnivariateTaylorSeries}(Integer,{}\spad{x},{}3) repre
48780`--sents Taylor series in \spad{(x - 3)} with \spadtype{Integer} coefficients.
48910`This domain provides segments which may be half open. That is,{} ranges of the form \spad{a..} or \spad{a..b}.
48987`This domain implements variables
49055`This domain supports modular methods based on \indented{1}{evaluation and rational reconstruction.\space{2}Each evaluation} \indented{1}{is done modulo machine sized prime \spad{p}.\space{2}Both Chinese} \indented{1}{remaindering and (linear) Hensel lift are supported.} \indented{1}{Once enough evaluations are known rational reconstruction} \indented{1}{produces vector of rational numbers or integers.}
49111`This domain supports modular methods based on \indented{1}{evaluation and rational reconstruction.\space{2}All computation} \indented{1}{are done on polynomials modulo machine sized prime \spad{p} \spad{
49111`--} \spad{p} must} \indented{1}{be choosen small enough to avoid overflow in intermediate} \indented{1}{calculations. Each evaluation is supposed to produce vector of} \indented{1}{values. Once enough evaluations are known rational reconstruction} 
49111`--\indented{1}{produces vector of rational functions or multivariate polynomials.}
49167`The domain \spad{VectorSpaceBasis} implements subspaces of vector spaces. A subspace is represented by a basis which is a list of vectors. Elements of basis are linearly independent so the list list describes the space with only the null vector. The scalar ring must be a field in order to avoid lattice (diofantine) questions.
49221`This type represents vector like objects with varying lengths and indexed by a finite segment of integers starting at 1.
49287`This type is used when no value is needed,{} \spadignore{e.g.} in the \spad{then} part of a one armed \spad{if}. All values can be coerced to type Void. Once a value has been coerced to Void,{} it cannot be recovered.
49318`This domain represents truncated weighted polynomials over a general (not necessarily commutative) polynomial type. The variables must be specified,{} as must the weights. The representation is sparse in the sense that only non-zero terms are represented.
49541`A domain constructor of the category \axiomType{GeneralTriangularSet}. The only requirement for a list of polynomials to be a member of such a domain is the following: no polynomial is constant and two distinct polynomials have distinct main variables. Such a triangular set may not be auto-reduced or consistent. The 
49541`--\axiomOpFrom{construct}{WuWenTsunTriangularSet} operation does not check the previous requirement. Triangular sets are stored as sorted lists \spad{w}.\spad{r}.\spad{t}. the main variables of their members. Furthermore,{} this domain exports operations dealing with the characteristic set method of Wu Wen Tsun and some optimizations mainly proposed by Dong Ming Wang.\newline References : \indented{1}{[1] \spad{W}. \spad{T}. WU "A Zero Structure Theorem for polynomial equations solving"} \indent
49541`--ed{6}{\spad{MM} Research Preprints,{} 1987.} \indented{1}{[2] \spad{D}. \spad{M}. WANG "An implementation of the characteristic set method in Maple"} \indented{6}{Proc. \spad{DISCO'92}. Bath,{} England.}
49730`\indented{2}{This type supports distributed multivariate polynomials} whose variables do not commute. The coefficient ring may be non-commutative too. However,{} coefficients and variables commute.
49854`This domain constructor implements polynomials in non-commutative variables written in the Poincare-Birkhoff-Witt basis from the Lyndon basis. These polynomials can be used to compute Baker-Campbell-Hausdorff relations. \newline Author: Michel Petitot (petitot@lifl.\spad{fr}).
49969`This domain represents generalized polynomials with coefficients (from a not necessarily commutative ring),{} and words belonging to an arbitrary \spadtype{OrderedMonoid}. This type is used,{} for instance,{} by the \spadtype{XDistributedPolynomial} domain constructor where the Monoid is free.
50046`\indented{2}{This type supports multivariate polynomials} whose set of variables is \spadtype{Symbol}. The representation is recursive. The coefficient ring may be non-commutative and the variables do not commute. However,{} coefficients and variables commute.
50124`\indented{2}{This type supports multivariate polynomials} whose variables do not commute. The representation is recursive. The coefficient ring may be non-commutative. Coefficients and variables commute.
50232`This package provides support for \spad{XML} files
50272`This package provides support for \spad{XML} files
50701`\spad{\# p} returns the number of terms in \spad{p}.
50763`\spad{\#((a1,{}...,{}an))} returns \spad{n}.
50835`\spad{\# u} returns the number of items in \spad{u}.
51039`\spad{c*x} returns the product of \spad{c} and the series \spad{x}.
51171`scalar multiplication
52857`\spad{p*x} \undocumented
54052`\axiom{\spad{r} * \spad{v}} computes the product of \axiom{\spad{r}} and \axiom{\spad{v}}
55524`\spad{s * x} returns the product of \spad{x} by \spad{s} on the left.
55564`\spad{s * x} returns the product of \spad{x} by \spad{s} on the left.
55605`\spad{s * x} returns the product of \spad{x} by \spad{s} on the left.
55653`\spad{smp*ts} multiplies a TaylorSeries by a monomial \spad{SMP}.
55953`\spad{x*c} returns the product of \spad{c} and the series \spad{x}.
56007`\spad{x*i} returns the product of integer \spad{i} and the series \spad{x}.
56064`\spad{p*r} returns the product of \spad{p} by \spad{r}.
56112`\spad{x * s} returns the product of \spad{x} by \spad{s} on the right.
56152`\spad{x * s} returns the product of \spad{x} by \spad{s} on the right.
56193`\spad{x * s} returns the product of \spad{x} by \spad{s} on the right.
56241`\spad{a * b} returns the input form corresponding to \spad{a * b}.
56279`\axiom{x*y} returns the tree \axiom{[\spad{x},{}\spad{y}]}.
56321`\spad{f * g} creates the equivalent infix form.
56360`\spad{a * b} returns the pattern \spad{a * b}.
56399`\spad{b1*b2} computes a basis of the intersection of two of subspaces.
56447`\spad{f*g} is the function \spad{h} \indented{1}{such that \spad{h x= f(g x)}.}
56548`\spad{e * s} returns \spad{e} times \spad{s}.
56605`\spad{n*x} is the product of \spad{x} by the integer \spad{n}.
56651`\spad{n * x} is an integer multiple.
56710`\spad{n * y} multiplies each component of the vector \spad{y} by the integer \spad{n}.
56780`\spad{n * x} is left-multiplication by a non negative integer
56838`\spad{n*x} computes the left-multiplication of \spad{x} by the positive integer \spad{n}. This is equivalent to adding \spad{x} to itself \spad{n} times.
56938`\spad{r*b} returns the product of \spad{r} by \spad{b}.
56988`\spad{r * y} multiplies the element \spad{r} times each component of the vector \spad{y}.
57057`\spad{r*g} is left module multiplication.
57102`\spad{r*x} returns the left multiplication of the module element \spad{x} by the ring element \spad{r}.
57143`\spad{r*x} is the left scalar multiple of the scalar \spad{r} and the matrix \spad{x}.
57196`\spad{r * y} multiplies the element \spad{r} times each component of the vector \spad{y}.
57254`\spad{r * x} is the product of the row vector \spad{r} and the matrix \spad{x}. Error: if the dimensions are incompatible.
57310`\spad{r * x} is the product of the row vector \spad{r} and the matrix \spad{x}. Error: if the dimensions are incompatible.
57377`\spad{s*r} returns the product \spad{r*s} used by \spadtype{XRecursivePolynomial}
57427`\spad{x*eqn} produces a new equation by multiplying both sides of equation eqn by \spad{x}.
57482`\spad{x * c} is the product of the matrix \spad{x} and the column vector \spad{c}. Error: if the dimensions are incompatible.
57538`\spad{x * c} is the product of the matrix \spad{x} and the column vector \spad{c}. Error: if the dimensions are incompatible.
57605`\spad{t*p} applies the dhmatrix \spad{t} to point \spad{p}
57672`\spad{y * r} multiplies each component of the vector \spad{y} by the element \spad{r}.
57741`\spad{g*r} is right module multiplication.
57786`\spad{x * r} is the right scalar multiple of the scalar \spad{r} and the matrix \spad{x}.
57839`\spad{x*r} returns the right multiplication of the module element \spad{x} by the ring element \spad{r}.
57881`\spad{y * r} multiplies each component of the vector \spad{y} by the element \spad{r}.
57939`\spad{x * r} returns the product of \spad{x} by \spad{r}. Usefull if \spad{R} is a non-commutative Ring.
57985`\spad{eqn*x} produces a new equation by multiplying both sides of equation eqn by \spad{x}.
58040`\spad{x * y} is the product of the matrices \spad{x} and \spad{y}. Error: if the dimensions are incompatible.
58094`\spad{a*b} is the product of \spad{a} and \spad{b} in a set with a binary operation.
58128`\spad{x*y} returns the product of \spad{x} and \spad{y}.
58166`\spad{s*t} is the inner product of the tensors \spad{s} and \spad{t} which contracts the last index of \spad{s} with the first index of \spad{t},{} \spadignore{i.e.} \indented{4}{\spad{t*s = contract(t,{}rank t,{} s,{} 1)}} \indented{4}{\spad{t*s = sum(k=1..N,{} t[i1,{}..,{}iN,{}k]*s[k,{}j1,{}..,{}jM])}} This is compatible with the use of \spad{M*v} to denote the matrix-vector inner product.
58284`\spad{v * x} returns the product of a variable \spad{x} by \spad{x}.
58712`addition
58755`\spad{a + b} returns the input form corresponding to \spad{a + b}.
58793`\spad{f + g} creates the equivalent infix form.
58832`\spad{a + b} returns the pattern \spad{a + b}.
58871`\spad{B1+B2} computes a basis of the sum of two subspaces.
58919`\spad{s + x} returns the sum of \spad{s} and \spad{x}.
58977`\spad{x+eqn} produces a new equation by adding \spad{x} to both sides of equation eqn.
59039`\spad{p + x} returns the sum of \spad{p} and \spad{x}
59102`\spad{eqn+x} produces a new equation by adding \spad{x} to both sides of equation eqn.
59164`\spad{x+y} computes the sum of \spad{x} and \spad{y}.
59209`\spad{g+h} is the sum of \spad{g} and \spad{h} in the module of elements of the same degree as \spad{g} and \spad{h}. Error: if \spad{g} and \spad{h} have different degrees.
59255`\spad{x + y} is the sum of the matrices \spad{x} and \spad{y}. Error: if the dimensions are incompatible.
59309`\spad{x + y} returns the component-wise sum of the vectors \spad{x} and \spad{y}. Error: if \spad{x} and \spad{y} are not of the same length.
59378`\spad{c1 + c2} additively mixes the two colors \spad{c1} and \spad{c2}.
59412`\spad{db1+db2} returns the merge of databases \spad{db1} and \spad{db2}
59920`\spad{- f} creates the equivalent prefix form.
59956`\spad{-x} is the additive inverse of \spad{x}.
59994`\spad{-g} is the additive inverse of \spad{g} in the module of elements of the same grade as \spad{g}.
60037`\spad{-x} returns the negative of the matrix \spad{x}.
60088`\spad{-x} negates all components of the vector \spad{x}.
60668`subtraction
60711`\spad{f - g} creates the equivalent infix form.
60750`\spad{x-eqn} produces a new equation by subtracting both sides of equation eqn from \spad{x}.
60808`\spad{eqn-x} produces a new equation by subtracting \spad{x} from both sides of equation eqn.
60866`\spad{x - y} returns an element \spad{z} such that \spad{z+y=x} or "failed" if no such element exists.
60925`\spad{x-y} is the difference of \spad{x} and \spad{y} \spadignore{i.e.} \spad{x + (-y)}.
60966`\spad{g-h} is the difference of \spad{g} and \spad{h} in the module of elements of the same degree as \spad{g} and \spad{h}. Error: if \spad{g} and \spad{h} have different degrees.
61012`\spad{x - y} is the difference of the matrices \spad{x} and \spad{y}. Error: if the dimensions are incompatible.
61066`\spad{x - y} returns the component-wise difference of the vectors \spad{x} and \spad{y}. Error: if \spad{x} and \spad{y} are not of the same length.
61131`\spad{db1-db2} returns the difference of databases \spad{db1} and \spad{db2} \spadignore{i.e.} consisting of elements in \spad{db1} but not in \spad{db2}
61209`\spadignore { /\ }returns the logical `meet',{} \spadignore{e.g.} `and'.
61244`Implement exterior grassmann product operator need to check precidence when used as an infix operator
61299`\spad{n} \spad{/\} \spad{m} returns the bit-by-bit logical {\em and} of the single integers \spad{n} and \spad{m}.
61342`\spad{a / d} divides the element \spad{a} by \spad{d}.
61424`\spad{m / d} divides the element \spad{m} by \spad{d}.
61934`\spad{x / d} divides the element \spad{x} by \spad{d}.
61979`\spad{x / d} divides the element \spad{x} by \spad{d}.
62020`\spad{p/r} returns \spad{p*(1/r)}.
62080`\spad{x/y} division (only works if minimum weight of divisor is zero,{} and if \spad{R} is a Field)
62181`\spad{x/y} division (only works if minimum weight of divisor is zero,{} and if \spad{R} is a Field)
62285`\spad{a / b} returns the input form corresponding to \spad{a / b}.
62323`\spad{f / g} creates the equivalent infix form.
62362`\spad{a / b} returns the pattern \spad{a / b}.
62401`\spad{d1 / d2} returns the fraction \spad{d1} divided by \spad{d2}.
62593`\spad{x / i} computes the division from \spad{x} by an integer \spad{i}.
62672`\spad{p/c} divides \spad{p} by the coefficient \spad{c}.
62734`\axiom{\spad{x/r}} returns the division of \axiom{\spad{x}} by \axiom{\spad{r}}.
62787`\spad{m/r} divides the elements of \spad{m} by \spad{r}. Error: if \spad{r = 0}.
62852`\spad{m/r} divides the elements of \spad{m} by \spad{r}. Error: if \spad{r = 0}.
62932`\spad{x/y} divides the vector \spad{x} by the scalar \spad{y}.
62974`\spad{x/y} divides the element \spad{x} by the element \spad{y}. Error: if \spad{y} is 0.
63008`\spad{x/y} is the same as \spad{x} times the inverse of \spad{y}.
63042`\spad{e1/e2} produces a new equation by dividing the left and right hand sides of equations \spad{e1} and \spad{e2}.
63586`\spad{f <= g} creates the equivalent infix form.
63626`\spad{x <= y} is a less than or equal test.
64093`\spad{f < g} creates the equivalent infix form.
64132`\spad{x < y} is a strict total ordering on the elements of the set.
64176`\spad{p < q} is an order relation on permutations. Note: this order is only total if and only if \spad{S} is totally ordered or \spad{S} is finite.
64232`\spad{s < t} returns \spad{true} if all elements of set aggregate \spad{s} are also elements of set aggregate \spad{t}.
65159`\spad{x=y} tests for equality
65210`\spad{a=b} tests if \spad{a} and \spad{b} are equal.
65256`\spad{B1=B2} tests if the two subspaces are equals.
65309`\spad{f = g} creates the equivalent infix form.
65348`\spad{a=b} creates an equation.
65386`\spad{x=y} tests if \spad{x} and \spad{y} are equal.
65471`\spad{f >= g} creates the equivalent infix form.
65511`\spad{x >= y} is a greater than or equal test.
65597`\spad{f > g} creates the equivalent infix form.
65636`\spad{x > y} is a greater than test.
65884`\spad{Aleph(n)} provides the named (infinite) cardinal number.
65944`\spad{An(x)} \undocumented
65993`\spad{And(n,{}m)} returns the bit-by-bit logical {\em And} of \spad{n} and \spad{m}.
66039`\spad{And(n,{}m)} returns the bit-by-bit logical {\em and} of the single integers \spad{n} and \spad{m}.
66377`\spad{s by n} creates a new segment in which only every \spad{n}\spad{-}th element is used.
66499`\spad{Beta(x,{}y)} returns the beta function applied to \spad{x} and \spad{y}
66837`\spad{Beta(x,{}y)} is \spad{Gamma(x) * Gamma(y)/Gamma(x+y)}.
67089`\spad{\spad{Ci}(f)} denotes the cosine integral
67138`\spad{\spad{Ci}(x)} returns the cosine integral of \spad{x},{} \spadignore{i.e.} the integral of \spad{cos(x) / x dx}.
67258`\spad{D()} provides the operator corresponding to a derivation in the ring \spad{A}.
67412`\spad{D(x)} returns the derivative of \spad{x}. This function is a simple differential operator where no variable needs to be specified.
67454`\spad{D(f)} returns the derivative of \spad{f}.
67908`\spad{D(x,{} deriv)} differentiates \spad{x} extending the derivation deriv on \spad{R}.
67965`\spad{D(x,{}[s1,{}...sn])} computes successive partial derivatives,{} \spadignore{i.e.} \spad{D(...D(x,{} s1)...,{} sn)}.
68025`\spad{D(x,{} n)} returns the \spad{n}-th derivative of \spad{x}.
68086`\spad{D(f,{} n)} returns the \spad{n}-th derivative of \spad{f}.
68165`\spad{D(x,{}v)} computes the partial derivative of \spad{x} with respect to \spad{v}.
68597`\spad{D(x,{} deriv,{} n)} differentiate \spad{x} \spad{n} times using a derivation which extends \spad{deriv} on \spad{R}.
68673`\spad{D(x,{} [s1,{}...,{}sn],{} [n1,{}...,{}nn])} computes multiple partial derivatives,{} \spadignore{i.e.} \spad{D(...D(x,{} s1,{} n1)...,{} sn,{} nn)}.
68758`\spad{D(x,{} s,{} n)} computes multiple partial derivatives,{} \spadignore{i.e.} \spad{n}-th derivative of \spad{x} with respect to \spad{s}.
68831`\spad{DiffAction(k,{} l,{} g)} gives the coefficient of \spad{x^k} in \spad{z^l} \spad{g}(\spad{x}),{} where \spad{z*}(a+b*x+c*x^2+d*x^3+...) = (a*x+b*x^2+c*x^3+...),{} \spadignore{i.e.} multiplication with \spad{x}.
68931`\spad{DiffC} gives the coefficients \spad{c_}{\spad{k},{}\spad{k}} in the expansion <x^k> \spad{z} \spad{g}(\spad{x}) = sum_{\spad{i=0}}\spad{^k} \spad{c_}{\spad{k},{}\spad{i}} <x^i> \spad{g}(\spad{x}),{} where \spad{z} acts on \spad{g}(\spad{x}) by shifting. In fact,{} the result is [0,{}0,{}0,{}...]
69372`\spad{\spad{Ei}(f)} denotes the exponential integral
69421`\spad{\spad{Ei}(x)} returns the exponential integral of \spad{x},{} \spadignore{i.e.} the integral of \spad{exp(x)/x dx}.
69633`\spad{F2FG(a + sqrt(-1) b)} returns \spad{a + i b}.
69700`\spad{FG2F(a + i b)} returns \spad{a + sqrt(-1) b}.
69977`\spad{Frobenius(x)} \undocumented
70100`\spad{Frobenius(a)} returns \spad{a ^ q} where \spad{q} is the \spad{size()\$F}.
70230`\spad{Frobenius(a,{}s)} returns \spad{a^(q^s)} where \spad{q} is the size()\$\spad{F}.
70791`\spad{Gamma(f)} returns the formal Gamma function applied to \spad{f}
70937`\spad{Gamma(x)} is the Euler Gamma function.
71082`\spad{Gamma(x)} is the Euler Gamma function.
71139`\spad{Gamma(x)} is the Euler Gamma function.
71192`\spad{Gamma(a,{}x)} returns the incomplete Gamma function applied to a and \spad{x}
71252`\spad{Gamma(a,{}x)} is the incomplete Gamma function.
71308`\spad{GospersMethod(b,{} n,{} new)} returns a rational function \spad{rf(n)} such that \spad{a(n) * rf(n)} is the indefinite sum of \spad{a(n)} with respect to upward difference on \spad{n},{} \spadignore{i.e.} \spad{a(n+1) * rf(n+1) - a(n) * rf(n) = a(n)},{} where \spad{b(n) = a(n)/a(n-1)} is a rational function. Returns "failed" if no such rational function \spad{rf(n)} exists. Note: \spad{new} is a nullary fu
71308`--nction returning a new \spad{V} every time. The condition on \spad{a(n)} is that \spad{a(n)/a(n-1)} is a rational function of \spad{n}.
73658`\spad{HermiteIntegrate(f,{} ')} returns \spad{[g,{}h]} such that \spad{f = g' + h} and \spad{h} has a only simple finite normal poles.
73771`\spad{Is([e1,{}...,{}en],{} pat)} matches the pattern pat on the list of expressions \spad{[e1,{}...,{}en]} and returns the result.
73891`\spad{Is(expr,{} pat)} matches the pattern pat on the expression \spad{expr} and returns a list of matches \spad{[v1 = e1,{}...,{}vn = en]}; returns an empty list if either \spad{expr} is exactly equal to pat or if pat does not match \spad{expr}.
74052`\spad{Is(expr,{} pat)} matches the pattern pat on the expression \spad{expr} and returns a list of matches \spad{[v1 = e1,{}...,{}vn = en]}; returns an empty list if either \spad{expr} is exactly equal to pat or if pat does not match \spad{expr}.
74173`\spad{Is(expr,{} pat)} matches the pattern pat on the expression \spad{expr} and returns a match of the form \spad{[v1 = e1,{}...,{}vn = en]}; returns an empty match if \spad{expr} is exactly equal to pat. returns a \spadfun{failed} match if pat does not match \spad{expr}.
75137`\spad{LODO2FUN(op)} returns the function to pass to the series ODE solver in order to solve \spad{op y = 0}.
75409`\spad{LagrangeInterpolation(l1,{}l2)} \undocumented
75590`\axiom{\spad{LazardQuotient2}(\spad{p},{}a,{}\spad{b},{}\spad{n})} returns \axiom{(a^(\spad{n}-1) * \spad{p}) exquo \spad{b^}(\spad{n}-1)} assuming that this quotient does not fail.
75710`\axiom{LazardQuotient(a,{}\spad{b},{}\spad{n})} returns \axiom{a^n exquo \spad{b^}(\spad{n}-1)} assuming that this quotient does not fail.
76163`\spad{LowTriBddDenomInv(B,{}d)} returns \spad{M},{} where \spad{B} is a non-singular lower triangular matrix and \spad{d} is an element of \spad{R} such that \spad{M = d * inv(B)} has entries in \spad{R}.
76754`\spad{NOT(x)} returns the \axiomType{Switch} expression representing \spad{\~~x}.
76788`\spad{Not(n)} returns the bit-by-bit logical {\em Not} of \spad{n}.
76831`\spad{Not(n)} returns the bit-by-bit logical {\em not} of the single integer \spad{n}.
76922`\spad{OMParseError?(u)} tests whether \spad{u} is an OpenMath parsing error.
76982`\spad{OMReadError?(u)} tests whether \spad{u} is an OpenMath read error.
77041`\spad{OMUnknownCD?(u)} tests whether \spad{u} is an OpenMath unknown \spad{CD} error.
77100`\spad{OMUnknownSymbol?(u)} tests whether \spad{u} is an OpenMath unknown symbol error.
77163`\spad{OMbindTCP}
77234`\spad{OMcloseConn}
77290`\spad{OMclose(dev)} closes \axiom{\spad{dev}},{} flushing output if necessary.
77338`\spad{OMconnInDevice:}
77407`\spad{OMconnOutDevice:}
77477`\spad{OMconnectTCP}
77558`\spad{OMencodingBinary()} is the constant for the OpenMath binary encoding.
77613`\spad{OMencodingSGML()} is the constant for the deprecated OpenMath SGML encoding.
77666`\spad{OMencodingUnknown()} is the constant for unknown encoding types. If this is used on an input device,{} the encoding will be autodetected. It is invalid to use it on an output device.
77722`\spad{OMencodingXML()} is the constant for the OpenMath \spad{XML} encoding.
77774`\spad{OMgetApp(dev)} reads a begin application token from \axiom{\spad{dev}}.
77823`\spad{OMgetAtp(dev)} reads a begin attribute pair token from \axiom{\spad{dev}}.
77872`\spad{OMgetAttr(dev)} reads a begin attribute token from \axiom{\spad{dev}}.
77922`\spad{OMgetBVar(dev)} reads a begin bound variable list token from \axiom{\spad{dev}}.
77972`\spad{OMgetBind(dev)} reads a begin binder token from \axiom{\spad{dev}}.
78022`\spad{OMgetEndApp(dev)} reads an end application token from \axiom{\spad{dev}}.
78074`\spad{OMgetEndAtp(dev)} reads an end attribute pair token from \axiom{\spad{dev}}.
78126`\spad{OMgetEndAttr(dev)} reads an end attribute token from \axiom{\spad{dev}}.
78179`\spad{OMgetEndBVar(dev)} reads an end bound variable list token from \axiom{\spad{dev}}.
78232`\spad{OMgetEndBind(dev)} reads an end binder token from \axiom{\spad{dev}}.
78285`\spad{OMgetEndError(dev)} reads an end error token from \axiom{\spad{dev}}.
78339`\spad{OMgetEndObject(dev)} reads an end object token from \axiom{\spad{dev}}.
78394`\spad{OMgetError(dev)} reads a begin error token from \axiom{\spad{dev}}.
78445`\spad{OMgetFloat(dev)} reads a float from \axiom{\spad{dev}}.
78503`\spad{OMgetInteger(dev)} reads an integer from \axiom{\spad{dev}}.
78559`\spad{OMgetObject(dev)} reads a begin object token from \axiom{\spad{dev}}.
78611`\spad{OMgetString(dev)} reads a string from \axiom{\spad{dev}}.
78665`\spad{OMgetSymbol(dev)} reads a symbol from \axiom{\spad{dev}}.
78742`\spad{OMgetType(dev)} returns the type of the next object on \axiom{\spad{dev}}.
78794`\spad{OMgetVariable(dev)} reads a variable from \axiom{\spad{dev}}.
78850`\spad{OMlistCDs()} lists all the \spad{CDs} supported by AXIOM.
78907`\spad{OMlistSymbols(cd)} lists all the symbols in \axiom{\spad{cd}}.
78974`\spad{OMmakeConn}
79038`\spad{OMopenFile(f,{}mode,{}enc)} opens file \axiom{\spad{f}} for reading or writing OpenMath objects (depending on \axiom{\spad{mode}} which can be \spad{"r"},{} \spad{"w"} or "a" for read,{} write and append respectively),{} in the encoding \axiom{\spad{enc}}.
79115`\spad{OMopenString(s,{}mode)} opens the string \axiom{\spad{s}} for reading or writing OpenMath objects in encoding \axiom{enc}.
79187`\spad{OMputApp(dev)} writes a begin application token to \axiom{\spad{dev}}.
79236`\spad{OMputAtp(dev)} writes a begin attribute pair token to \axiom{\spad{dev}}.
79285`\spad{OMputAttr(dev)} writes a begin attribute token to \axiom{\spad{dev}}.
79335`\spad{OMputBVar(dev)} writes a begin bound variable list token to \axiom{\spad{dev}}.
79385`\spad{OMputBind(dev)} writes a begin binder token to \axiom{\spad{dev}}.
79435`\spad{OMputEndApp(dev)} writes an end application token to \axiom{\spad{dev}}.
79487`\spad{OMputEndAtp(dev)} writes an end attribute pair token to \axiom{\spad{dev}}.
79539`\spad{OMputEndAttr(dev)} writes an end attribute token to \axiom{\spad{dev}}.
79592`\spad{OMputEndBVar(dev)} writes an end bound variable list token to \axiom{\spad{dev}}.
79645`\spad{OMputEndBind(dev)} writes an end binder token to \axiom{\spad{dev}}.
79698`\spad{OMputEndError(dev)} writes an end error token to \axiom{\spad{dev}}.
79752`\spad{OMputEndObject(dev)} writes an end object token to \axiom{\spad{dev}}.
79807`\spad{OMputError(dev)} writes a begin error token to \axiom{\spad{dev}}.
79858`\spad{OMputFloat(dev,{}i)} writes the float \axiom{\spad{i}} to \axiom{\spad{dev}}.
79921`\spad{OMputInteger(dev,{}i)} writes the integer \axiom{\spad{i}} to \axiom{\spad{dev}}.
79982`\spad{OMputObject(dev)} writes a begin object token to \axiom{\spad{dev}}.
80034`\spad{OMputString(dev,{}i)} writes the string \axiom{\spad{i}} to \axiom{\spad{dev}}.
80093`\spad{OMputSymbol(dev,{}cd,{}s)} writes the symbol \axiom{\spad{s}} from \spad{CD} \axiom{\spad{cd}} to \axiom{\spad{dev}}.
80159`\spad{OMputVariable(dev,{}i)} writes the variable \axiom{\spad{i}} to \axiom{\spad{dev}}.
80220`\spad{OMreadFile(f)} reads an OpenMath object from \axiom{\spad{f}} and passes it to AXIOM.
80275`\spad{OMreadStr(f)} reads an OpenMath object from \axiom{\spad{f}} and passes it to AXIOM.
80329`\spad{OMread(dev)} reads an OpenMath object from \axiom{\spad{dev}} and passes it to AXIOM.
80388`\spad{OMreceive(c)} reads an OpenMath object from connection \axiom{\spad{c}} and returns the appropriate AXIOM object.
80460`\spad{OMsend(c,{}u)} attempts to output \axiom{\spad{u}} on \aciom{\spad{c}} in OpenMath.
80534`\spad{OMserve(portnum,{}timeout)} puts AXIOM into server mode on port number \axiom{\spad{portnum}}. The parameter \axiom{\spad{timeout}} specifies the \spad{timeout} period for the connection.
80614`\spad{OMsetEncoding(dev,{}enc)} sets the encoding used for reading or writing OpenMath objects to or from \axiom{\spad{dev}} to \axiom{\spad{enc}}.
80685`\spad{OMsupportsCD?(cd)} returns \spad{true} if AXIOM supports \axiom{\spad{cd}},{} \spad{false} otherwise.
80747`\spad{OMsupportsSymbol?(s,{}cd)} returns \spad{true} if AXIOM supports symbol \axiom{\spad{s}} from \spad{CD} \axiom{\spad{cd}},{} \spad{false} otherwise.
80820`\spad{OMunhandledSymbol(s,{}cd)} raises an error if AXIOM reads a symbol which it is unable to handle. Note that this is different from an unexpected symbol.
80955`\spad{OMwrite(u)} returns the OpenMath \spad{XML} encoding of \axiom{\spad{u}} as a complete OpenMath object.
81150`\spad{OMwrite(dev,{} u)} writes the OpenMath form of \axiom{\spad{u}} to the OpenMath device \axiom{\spad{dev}} as a complete OpenMath object.
81207`\spad{OMwrite(u,{} true)} returns the OpenMath \spad{XML} encoding of \axiom{\spad{u}} as a complete OpenMath object; OMwrite(\spad{u},{} \spad{false}) returns the OpenMath \spad{XML} encoding of \axiom{\spad{u}} as an OpenMath fragment.
81345`\spad{OMwrite(dev,{} u,{} true)} writes the OpenMath form of \axiom{\spad{u}} to the OpenMath device \axiom{\spad{dev}} as a complete OpenMath object; OMwrite(\spad{dev},{} \spad{u},{} \spad{false}) writes the object as an OpenMath fragment.
81613`\spad{Or(n,{}m)} returns the bit-by-bit logical {\em Or} of \spad{n} and \spad{m}.
81658`\spad{Or(n,{}m)} returns the bit-by-bit logical {\em or} of the single integers \spad{n} and \spad{m}.
82117`\spad{Pfaffian(m)} returns the Pfaffian of the matrix \spad{m}. Error: if the matrix is not antisymmetric.
82196`\spad{Pfaffian(m)} returns the Pfaffian of the matrix \spad{m}. Error: if the matrix is not antisymmetric.
82286`Is this a valid point? if any of its elements are not valid then return \spad{true} Examples of invalid floating point numbers are when have we have divided a given element by zero or taken sqrt of negative number at some stage? Note: we can still represent a point at infinity and we can represent points using complex numbers,{} but we do this using projective or conformal space or by using the appropriate instance of SPointCategory,{} not by using inval
82286`--id floating point numbers.
82335`\spad{PollardSmallFactor(n)} returns a factor of \spad{n} or "failed" if no one is found
82422`\spad{RF2UTS(f)} converts \spad{f} to a Taylor series.
82508`\spad{ReduceOrder(op,{} s)} returns \spad{op1} such that for any solution \spad{z} of \spad{op1 z = 0},{} \spad{y = s \int z} is a solution of \spad{op y = 0}. \spad{s} must satisfy \spad{op s = 0}.
82565`\spad{ReduceOrder(op,{} [f1,{}...,{}fk])} returns \spad{[op1,{}[g1,{}...,{}gk]]} such that for any solution \spad{z} of \spad{op1 z = 0},{} \spad{y = gk \int(g_{k-1} \int(... \int(g1 \int z)...)} is a solution of \spad{op y = 0}. Each \spad{\spad{fi}} must satisfy \spad{op \spad{fi} = 0}.
82902`\axiom{RittWuCompare(a,{}\spad{b})} returns \axiom{"failed"} if \axiom{a} and \axiom{\spad{b}} have same rank \spad{w}.\spad{r}.\spad{t}. Ritt and Wu Wen Tsun ordering using the refinement of Lazard,{} otherwise returns \axiom{infRittWu?(a,{}\spad{b})}.
82998`\spad{SEGMENT(x)} creates the prefix form: \spad{x..}.
83040`\spad{l..} produces a half open segment,{} that is,{} one with no upper bound.
83090`\spad{SEGMENT(x,{}y)} creates the infix form: \spad{x..y}.
83135`\spad{l..h} creates a segment with \spad{l} and \spad{h} as the endpoints.
83539`\spad{ShiftAction(k,{} l,{} g)} gives the coefficient of \spad{x^k} in \spad{z^l} \spad{g}(\spad{x}),{} where \spad{z*(a+b*x+c*x^2+d*x^3+...) = (b*x+2*c*x^2+3*d*x^3+...)}. In terms of sequences,{} z*u(\spad{n})=n*u(\spad{n}).
83640`\spad{ShiftC} gives the coefficients \spad{c_}{\spad{k},{}\spad{k}} in the expansion <x^k> \spad{z} \spad{g}(\spad{x}) = sum_{\spad{i=0}}\spad{^k} \spad{c_}{\spad{k},{}\spad{i}} <x^i> \spad{g}(\spad{x}),{} where \spad{z} acts on \spad{g}(\spad{x}) by shifting. In fact,{} the result is [0,{}1,{}2,{}...]
83721`\spad{\spad{Si}(f)} denotes the sine integral
83770`\spad{\spad{Si}(x)} returns the sine integral of \spad{x},{} \spadignore{i.e.} the integral of \spad{sin(x) / x dx}.
83913`\spad{Somos(d)} specifies whether we want that the total degree of the differential operators is constant,{} and equal to \spad{d},{} or maxDerivative if \spad{true}. If \spad{true},{} maxDerivative must be set,{} too.
84490`\spad{UP2UTS(p)} converts \spad{p} to a Taylor series.
84742`\spad{UTS2UP(s,{} n)} converts the first \spad{n} terms of \spad{s} to a univariate polynomial.
84816`\spad{UnVectorise(v)} \undocumented
84874`\spad{UpTriBddDenomInv(B,{}d)} returns \spad{M},{} where \spad{B} is a non-singular upper triangular matrix and \spad{d} is an element of \spad{R} such that \spad{M = d * inv(B)} has entries in \spad{R}.
85368`\spad{Vectorise(x)} \undocumented
85685`\spadignore{ \/ } returns the logical `join',{} \spadignore{e.g.} `or'.
85720`Implement regressive inner,{}meet product operator need to check precidence when used as an infix operator
85775`\spad{n} \spad{\/} \spad{m} returns the bit-by-bit logical {\em or} of the single integers \spad{n} and \spad{m}.
85818`\spad{a ^ b} is the formal exponential a^b.
86594`\spad{s ^ n} returns the product of \spad{s} by itself \spad{n} times.
87179`\spad{s ^ n} returns the product of \spad{s} by itself \spad{n} times.
88494`\spad{a ^ b} returns the input form corresponding to \spad{a ^ b}.
88577`\spad{a ^ b} returns the input form corresponding to \spad{a ^ b}.
88631`\spad{a ^ n} returns the pattern \spad{a ^ n}.
88686`\spad{f ^ g} creates the equivalent infix form.
88725`\spad{a ^ b} returns the pattern \spad{a ^ b}.
88830`\spad{f(x) ^ a} computes a power of a power series. When the coefficient ring is a field,{} we may raise a series to an exponent from the coefficient ring provided that the constant coefficient of the series is 1.
88912`\spad{x ^ y} is the rational exponentiation of \spad{x} by the power \spad{y}.
88971`\spad{x^n} returns \spad{x} raised to the integer power \spad{n}.
89017`\spad{x^n} returns \spad{x} raised to the integer power \spad{n}.
89056`\spad{m^n} computes an integral power of the matrix \spad{m}. Error: if matrix is not square or if the matrix is square but not invertible.
89127`\spad{m^n} computes an integral power of the matrix \spad{m}. Error: if the matrix is not invertible.
89209`\spad{op^n} \undocumented
89262`\spad{x ^ n} computes a non-negative integral power of the matrix \spad{x}. Error: if the matrix is not square.
89332`\spad{a^n} returns the \spad{n}\spad{-}th power of \spad{a},{} defined by repeated squaring.
89390`\spad{x^n} returns the repeated product of \spad{x} \spad{n} times,{} \spadignore{i.e.} exponentiation.
89518`\spad{a^n} returns the \spad{n}\spad{-}th power of \spad{a},{} defined by repeated squaring.
89565`\spad{x^n} returns the repeated product of \spad{x} \spad{n} times,{} \spadignore{i.e.} exponentiation.
89616`\spad{x^y} returns \spad{x} to the power \spad{y}.
89671`\spad{x^y} returns \spad{\#(X^Y)} where \spad{X^Y} is defined \indented{1}{as \spad{\{g| g:Y->X\}}.}
89714`\spad{aCubic(f,{}g,{}h,{}j)} \undocumented
89780`\spad{aLinear(f,{}g)} \undocumented
89843`\spad{aQuadratic(f,{}g,{}h)} \undocumented
89911`\spad{aQuartic(f,{}g,{}h,{}i,{}k)} \undocumented
90079`\spad{abs(f)} returns the absolute value operator applied to \spad{f}
90301`\spad{abs(x)} represents the Fortran intrinsic function ABS
90381`\spad{abs(o)} computes the absolute value of an octonion,{} equal to the square root of the \spadfunFrom{norm}{Octonion}.
90450`\spad{abs(q)} computes the absolute value of quaternion \spad{q} (sqrt of norm).
90521`\spad{abs(x)} returns the absolute value of \spad{x} = sqrt(norm(\spad{x})).
90590`\spad{abs(x)} returns the absolute value of \spad{x}.
90629`\spad{abs x} returns the absolute value of \spad{x}.
90673`\spad{abs(x)} returns the absolute value of \spad{x}.
90724`\spad{absolutelyIrreducible?()} tests if the curve absolutely irreducible?
90809`\spad{absolutelyIrreducible?()} tests if the curve absolutely irreducible?
90891`\spad{acosIfCan(z)} returns acos(\spad{z}) if possible,{} and "failed" otherwise.
90972`\spad{acos(x)} applies the inverse cosine operator to \spad{x}
91354`\spad{acos(z)} returns the arc-cosine of Laurent series \spad{z}.
91441`\spad{acos(z)} returns the arc-cosine of a Puiseux series \spad{z}.
91537`\spad{acos(x)} represents the Fortran intrinsic function ACOS
91618`\spad{acos(x)} returns the arc-cosine of \spad{x}. When evaluated into some subset of the complex numbers,{} one branch cut for acos lies along the negative real axis to the left of \spad{-1} (inclusive),{} continuous with the upper half plane,{} the other along the positive real axis to the right of 1 (inclusive),{} continuous with the lower half plane.
91679`\spad{acoshIfCan(z)} returns acosh(\spad{z}) if possible,{} and "failed" otherwise.
91761`\spad{acosh(x)} applies the inverse hyperbolic cosine operator to \spad{x}
92149`\spad{acosh(z)} returns the inverse hyperbolic cosine of Laurent series \spad{z}.
92237`\spad{acosh(z)} returns the inverse hyperbolic cosine of a Puiseux series \spad{z}.
92334`\spad{acosh(x)} returns the hyperbolic arc-cosine of \spad{x}.
92393`\spad{acotIfCan(z)} returns acot(\spad{z}) if possible,{} and "failed" otherwise.
92474`\spad{acot(x)} applies the inverse cotangent operator to \spad{x}
92813`\spad{acot(z)} returns the arc-cotangent of Laurent series \spad{z}.
92900`\spad{acot(z)} returns the arc-cotangent of a Puiseux series \spad{z}.
92996`\spad{acot(x)} returns the arc-cotangent of \spad{x}.
93057`\spad{acothIfCan(z)} returns acoth(\spad{z}) if possible,{} and "failed" otherwise.
93139`\spad{acoth(x)} applies the inverse hyperbolic cotangent operator to \spad{x}
93483`\spad{acoth(z)} returns the inverse hyperbolic cotangent of Laurent series \spad{z}.
93571`\spad{acoth(z)} returns the inverse hyperbolic cotangent of a Puiseux series \spad{z}.
93668`\spad{acoth(x)} returns the hyperbolic arc-cotangent of \spad{x}.
93727`\spad{acscIfCan(z)} returns acsc(\spad{z}) if possible,{} and "failed" otherwise.
93808`\spad{acsc(x)} applies the inverse cosecant operator to \spad{x}
94149`\spad{acsc(z)} returns the arc-cosecant of Laurent series \spad{z}.
94236`\spad{acsc(z)} returns the arc-cosecant of a Puiseux series \spad{z}.
94332`\spad{acsc(x)} returns the arc-cosecant of \spad{x}.
94393`\spad{acschIfCan(z)} returns acsch(\spad{z}) if possible,{} and "failed" otherwise.
94475`\spad{acsch(x)} applies the inverse hyperbolic cosecant operator to \spad{x}
94819`\spad{acsch(z)} returns the inverse hyperbolic cosecant of Laurent series \spad{z}.
94907`\spad{acsch(z)} returns the inverse hyperbolic cosecant of a Puiseux series \spad{z}.
95004`\spad{acsch(x)} returns the hyperbolic arc-cosecant of \spad{x}.
95240`\spad{adaptive(b)} turns adaptive 2D plotting on if \spad{b} is \spad{true},{} or off if \spad{b} is \spad{false}. This option is expressed in the form \spad{adaptive == b}.
95288`\spad{adaptive(l,{}b)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{adaptive}. If the option does not exist the value,{} \spad{b} is returned.
95482`\spad{addBadValue(p,{} v)} adds \spad{v} to the list of "bad values" for \spad{p}; \spad{p} is not allowed to match any of its "bad values".
95558`\spad{addBadValue(p,{} v)} adds \spad{v} to the list of "bad values" for \spad{p}. Note: \spad{p} is not allowed to match any of its "bad values".
95608`add the specified child node \spad{'c'} below node \spad{'n'}
95656`\spad{addMatchRestricted(var,{} expr,{} r,{} val)} adds the match (\spad{var},{} \spad{expr}) in \spad{r},{} provided that \spad{expr} satisfies the predicates attached to \spad{var},{} that \spad{var} is not matched to another expression already,{} and that either \spad{var} is an optional pattern variable or that \spad{expr} is not equal to val (usually an identity).
95737`\spad{addMatch(var,{} expr,{} r)} adds the match (\spad{var},{} \spad{expr}) in \spad{r},{} provided that \spad{expr} satisfies the predicates attached to \spad{var},{} and that \spad{var} is not matched to another expression already.
97571`a convenience function which combines createSceneGroup with addChild!
97618`a convenience function which combines createSceneIFS with addChild!
97756`a convenience function which combines createSceneLine with addChild!
98482`\spad{addWarning s} is an internal function that adds \spad{s},{} together with an indication of the current testcase and the current testsuite to the list of warnings.
98540`extend reconstructor adding zeros at specified positions.
98764`\spad{addmod(a,{}b,{}p)},{} \spad{0<=a,{}b<p>1},{} means \spad{a+b mod p}.
98953`\spad{adjoint(m)} returns the ajoint matrix of \spad{m} (\spadignore{i.e.} the matrix \spad{n} such that \spad{m*n} = determinant(\spad{m})*id) and the detrminant of \spad{m}.
99069`\spad{adjoint(a)} returns the adjoint operator of a.
99146`\spad{adjoint(op)} returns the adjoint of the operator \spad{op}.
99282`\spad{adjoint(op1,{} op2)} sets the adjoint of \spad{op1} to be \spad{op2}. \spad{op1} must be a basic operator
99358`\spad{airyAiPrime(x)} returns the derivative of Airy \spad{Ai} function applied to \spad{x}
99422`\spad{airyAiPrime(x)} is the derivative of the Airy function \spad{\spad{Ai}(x)}.
99481`\spad{airyAi(x)} returns the Airy \spad{Ai} function applied to \spad{x}
99700`\spad{airyAi(x)} is the Airy function \spad{\spad{Ai}(x)}.
99754`\spad{airyBiPrime(x)} returns the derivative of Airy \spad{Bi} function applied to \spad{x}
99818`airyBiPrime is the derivative of the Airy function \spad{\spad{Bi}(x)}.
99877`\spad{airyBi(x)} returns the Airy \spad{Bi} function applied to \spad{x}
100096`\spad{airyBi(x)} is the Airy function \spad{\spad{Bi}(x)}.
100643`\spad{algebraic?(a)} tests whether an element \spad{a} is algebraic with respect to the ground field \spad{F}.
100769`\axiom{algebraic?(\spad{v},{}\spad{ts})} returns \spad{true} iff \axiom{\spad{v}} is the main variable of some polynomial in \axiom{\spad{ts}}.
100929`\spad{algebraicCoefficients?(p,{}ts)} returns \spad{true} iff every variable of \spad{p} which is not the main one of \spad{p} is algebraic \spad{w}.\spad{r}.\spad{t}. \spad{ts}.
101694`\axiom{algebraicVariables(\spad{ts})} returns the decreasingly sorted list of the main variables of the polynomials of \axiom{\spad{ts}}.
102086`\spad{allDegrees(d)} specifies whether all possibilities of the degree vector - taking into account maxDegree - should be tried. This is mainly interesting for rational interpolation. This option is expressed in the form \spad{allDegrees == d}.
103085`\axiom{allRootsOf(pol)} creates all the roots of \axiom{pol} in the Real Closure,{} assumed in order.
103181`\spad{alphabetic?(c)} tests if \spad{c} is a letter,{} \spadignore{i.e.} one of a..\spad{z} or A..\spad{Z}.
103231`\spad{alphabetic()} returns the class of all characters for which \spadfunFrom{alphabetic?}{Character} is \spad{true}.
103278`\spad{alphanumeric?(c)} tests if \spad{c} is either a letter or number,{} \spadignore{i.e.} one of 0..9,{} a..\spad{z} or A..\spad{Z}.
103330`\spad{alphanumeric()} returns the class of all characters for which \spadfunFrom{alphanumeric?}{Character} is \spad{true}.
103656`\spad{alternative?()} tests if \spad{2*associator(a,{}a,{}b) = 0 = 2*associator(a,{}b,{}b)} for all \spad{a},{} \spad{b} in the algebra. Note: we only can test this; in general we don\spad{'t} know whether \spad{2*a=0} implies \spad{a=0}.
103733`\spad{alternative?()} tests if \spad{2*associator(a,{}a,{}b) = 0 = 2*associator(a,{}b,{}b)} for all \spad{a},{} \spad{b} in the algebra. Note: we only can test this; in general we don\spad{'t} know whether \spad{2*a=0} implies \spad{a=0}.
103807`\spad{f and g} creates the equivalent infix form.
103848`\spad{a and b} returns the logical {\em and} of bit aggregates \axiom{a} and \axiom{\spad{b}}.
103891`\spad{a and b} returns the logical {\em and} of Boolean \spad{a} and \spad{b}.
104108`\spad{angerJ(v,{} z)} is the Anger \spad{J} function
104169`\spad{angerJ(v,{} z)} is the Anger \spad{J} function.
104226`\spad{antiAssociative?()} tests if multiplication in algebra is anti-associative,{} \spadignore{i.e.} \spad{(a*b)*c + a*(b*c) = 0} for all \spad{a},{}\spad{b},{}\spad{c} in the algebra.
104307`\spad{antiAssociative?()} tests if multiplication in algebra is anti-associative,{} \spadignore{i.e.} \spad{(a*b)*c + a*(b*c) = 0} for all \spad{a},{}\spad{b},{}\spad{c} in the algebra.
104385`\spad{antiCommutative?()} tests if \spad{a*a = 0} for all \spad{a} in the algebra. Note: this implies \spad{a*b + b*a = 0} for all \spad{a} and \spad{b}.
104466`\spad{antiCommutative?()} tests if \spad{a*a = 0} for all \spad{a} in the algebra. Note: this implies \spad{a*b + b*a = 0} for all \spad{a} and \spad{b}.
104599`\spad{antiCommutator(a,{}b)} returns \spad{a*b+b*a}.
104839`\spad{antipode(x)} computes the antipode of an element \spad{x}.
105037`\spad{antisymmetric?(m)} returns \spad{true} if the matrix \spad{m} is square and antisymmetric (\spadignore{i.e.} \spad{m[i,{}j] = -m[j,{}i]} for all \spad{i} and \spad{j}) and \spad{false} otherwise.
105106`\spad{antisymmetric?(m)} returns \spad{true} if the matrix \spad{m} is square and antisymmetric (\spadignore{i.e.} \spad{m[i,{}j] = -m[j,{}i]} for all \spad{i} and \spad{j}) and \spad{false} otherwise.
105894`\spad{any?(p,{}u)} tests if \axiom{\spad{p}(\spad{x})} is \spad{true} for any element \spad{x} of \spad{u}. Note: for collections,{} \axiom{any?(\spad{p},{}\spad{u}) = reduce(or,{}map(\spad{f},{}\spad{u}),{}\spad{false},{}\spad{true})}. However,{} \axiom{any?(\spad{p},{}\spad{u})} returns as soon as it finds an element for which \axiom{\spad{p}} gives \spad{true}.
105987`\spad{any(type,{}object)} is a technical function for creating an \spad{object} of \spadtype{Any}. Arugment \spad{type} is a \spadgloss{LISP} form for the \spad{type} of \spad{object}.
106094`\spad{append(u1,{}u2)} appends the elements of list \spad{u1} onto the front of list \spad{u2}. This new list and \spad{u2} will share some structure.
106946`\spad{apply(m,{}a)} defines a left operation of \spad{n} by \spad{n} matrices where \spad{n} is the rank of the algebra in terms of matrix-vector multiplication,{} this is a substitute for a left module structure. Error: if shape of matrix doesn\spad{'t} fit.
107016`\spad{apply(p,{} f,{} m)} returns \spad{p(m)} where the action is given by \spad{x m = f(m)}. \spad{f} must be an \spad{R}-pseudo linear map on \spad{M}.
107089`\spad{apply(p,{} c,{} m)} returns \spad{p(m)} where the action is given by \spad{x m = c sigma(m) + delta(m)}.
107312`\spad{approxSqrt(n)} returns an approximation \spad{x} to \spad{sqrt(n)} such that \spad{-1 < x - sqrt(n) < 1}. Compute an approximation \spad{s} to \spad{sqrt(n)} such that \indented{10}{\spad{-1 < s - sqrt(n) < 1}} A variable precision Newton iteration is used. The running time is \spad{O( log(n)^2 )}.
107664`\spad{approximate(f)} returns a truncated power series with the series variable viewed as an element of the coefficient domain.
107965`\axiom{approximate(term,{}root,{}prec)} gives an approximation of \axiom{term} over \axiom{root} with precision \axiom{prec}
108419`\spad{arg1(a,{}c)} selects its first argument.
108481`\spad{arg2(a,{}c)} selects its second argument.
108595`\spad{argumentList!(l)} declares that the argument list for the current subprogram in the global symbol table is \spad{l}.
108659`\spad{argumentList!(f,{}l)} declares that the argument list for subprogram \spad{f} in the global symbol table is \spad{l}.
108730`\spad{argumentList!(f,{}l,{}tab)} declares that the argument list for subprogram \spad{f} in symbol table \spad{tab} is \spad{l}.
108804`\spad{argumentListOf(f,{}tab)} returns the argument list of \spad{f}
108921`\spad{argument(x)} returns the argument of a given sin/cos expressions
108972`\spad{argument(op(a1,{}...,{}an))} returns \spad{[a1,{}...,{}an]}.
109019`\spad{argument(x)} returns the angle made by (0,{}1) and (0,{}\spad{x}).
109106`\spad{arity(op)} returns \spad{n} if \spad{op} is \spad{n}-ary,{} and "failed" if \spad{op} has arbitrary arity.
109350`\spad{arrayStack([x,{}y,{}...,{}z])} creates an array stack with first (top) element \spad{x},{} second element \spad{y},{}...,{}and last element \spad{z}.
109403`\spad{asecIfCan(z)} returns asec(\spad{z}) if possible,{} and "failed" otherwise.
109484`\spad{asec(x)} applies the inverse secant operator to \spad{x}
109825`\spad{asec(z)} returns the arc-secant of Laurent series \spad{z}.
109912`\spad{asec(z)} returns the arc-secant of a Puiseux series \spad{z}.
110008`\spad{asec(x)} returns the arc-secant of \spad{x}.
110069`\spad{asechIfCan(z)} returns asech(\spad{z}) if possible,{} and "failed" otherwise.
110151`\spad{asech(x)} applies the inverse hyperbolic secant operator to \spad{x}
110495`\spad{asech(z)} returns the inverse hyperbolic secant of Laurent series \spad{z}.
110583`\spad{asech(z)} returns the inverse hyperbolic secant of a Puiseux series \spad{z}.
110680`\spad{asech(x)} returns the hyperbolic arc-secant of \spad{x}.
110908`\spad{asinIfCan(z)} returns asin(\spad{z}) if possible,{} and "failed" otherwise.
110989`\spad{asin(x)} applies the inverse sine operator to \spad{x}
111371`\spad{asin(z)} returns the arc-sine of Laurent series \spad{z}.
111458`\spad{asin(z)} returns the arc-sine of a Puiseux series \spad{z}.
111554`\spad{asin(x)} represents the Fortran intrinsic function ASIN
111635`\spad{asin(x)} returns the arc-sine of \spad{x}. When evaluated into some subset of the complex numbers,{} one branch cut for asin lies along the negative real axis to the left of \spad{-1} (inclusive),{} continuous with the upper half plane,{} the other along the positive real axis to the right of 1 (inclusive),{} continuous with the lower half plane.
111696`\spad{asinhIfCan(z)} returns asinh(\spad{z}) if possible,{} and "failed" otherwise.
111778`\spad{asinh(x)} applies the inverse hyperbolic sine operator to \spad{x}
112166`\spad{asinh(z)} returns the inverse hyperbolic sine of Laurent series \spad{z}.
112254`\spad{asinh(z)} returns the inverse hyperbolic sine of a Puiseux series \spad{z}.
112351`\spad{asinh(x)} returns the hyperbolic arc-sine of \spad{x}.
112410`\spad{assert(op,{} s)} attaches property \spad{s} to \spad{op}. Argument \spad{op} is modified "in place",{} \spadignore{i.e.} no copy is made.
112461`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
112534`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
112598`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
112664`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
112737`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
112808`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
112881`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
112962`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
113034`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
113108`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
113189`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
113268`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
113349`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
113418`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
113485`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
113554`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
113607`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
113688`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
113760`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
113834`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
113915`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
113994`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
114075`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
114144`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
114211`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
114280`\spad{assign(f,{}g)} creates a form for the assignment \spad{f := g}.
114870`\spad{assoc(k,{}u)} returns the element \spad{x} in association list \spad{u} stored with key \spad{k},{} or "failed" if \spad{u} has no key \spad{k}.
115384`\spad{associates?(x,{}y)} tests whether \spad{x} and \spad{y} are associates,{} \spadignore{i.e.} differ by a unit factor.
115442`\spad{associative?()} tests if multiplication in algebra is associative.
115519`\spad{associative?()} tests if multiplication in algebra is associative.
115593`\spad{associatorDependence()} looks for the associator identities,{} \spadignore{i.e.} finds a basis of the solutions of the linear combinations of the six permutations of \spad{associator(a,{}b,{}c)} which yield 0,{} for all \spad{a},{}\spad{b},{}\spad{c} in the algebra. The order of the permutations is \spad{123 231 312 132 321 213}.
115686`\spad{associatorDependence()} looks for the associator identities,{} \spadignore{i.e.} finds a basis of the solutions of the linear combinations of the six permutations of \spad{associator(a,{}b,{}c)} which yield 0,{} for all \spad{a},{}\spad{b},{}\spad{c} in the algebra. The order of the permutations is \spad{123 231 312 132 321 213}.
115850`\spad{associator(a,{}b,{}c)} returns \spad{(a*b)*c-a*(b*c)}.
115908`\spad{atType(f)} returns the input form corresponding to \spad{f@R}
115975`\spad{atanIfCan(z)} returns atan(\spad{z}) if possible,{} and "failed" otherwise.
116056`\spad{atan(x)} applies the inverse tangent operator to \spad{x}
116382`\spad{atan(z)} returns the arc-tangent of Laurent series \spad{z}.
116469`\spad{atan(z)} returns the arc-tangent of a Puiseux series \spad{z}.
116565`\spad{atan(x)} represents the Fortran intrinsic function ATAN
116646`\spad{atan(x)} returns the arc-tangent of \spad{x}. When evaluated into some subset of the complex numbers,{} one branch cut for atan lies along the positive imaginary axis above \%\spad{i} (exclusive),{} continuous with the left half plane,{} the other along the negative imaginary axis below -\%\spad{i} (exclusive) continuous with the right half plane. The domain does not contain \%\spad{i} and -\%\spad{i}.
116707`\spad{atan(x,{}y)} computes the arc tangent from \spad{x} with phase \spad{y}.
116750`\spad{atan(x,{}y)} computes the arc tangent from \spad{x} with phase \spad{y}.
116787`\spad{atanhIfCan(z)} returns atanh(\spad{z}) if possible,{} and "failed" otherwise.
116869`\spad{atanh(x)} applies the inverse hyperbolic tangent operator to \spad{x}
117257`\spad{atanh(z)} returns the inverse hyperbolic tangent of Laurent series \spad{z}.
117345`\spad{atanh(z)} returns the inverse hyperbolic tangent of a Puiseux series \spad{z}.
117442`\spad{atanh(x)} returns the hyperbolic arc-tangent of \spad{x}.
117501`\spad{atom?(s)} is \spad{true} if \spad{s} is a Lisp atom.
117577`\spad{atoms(r)} returns the list of matches that match atoms (elements of the lists).
118146`\spad{augment(lp,{}lts)} returns the same as \spad{concat([augment(lp,{}ts) for ts in lts])}
118235`\spad{augment(lp,{}ts)} returns \spad{ts} if \spad{empty? lp},{} \spad{augment(p,{}ts)} if \spad{lp = [p]},{} otherwise \spad{augment(first lp,{} augment(rest lp,{} ts))}
118318`\spad{augment(p,{}lts)} returns the same as \spad{concat([augment(p,{}ts) for ts in lts])}
118401`\spad{augment(p,{}ts)} assumes that \spad{p} is a non-constant polynomial whose main variable is greater than any variable of \spad{ts}. This operation assumes also that if \spad{p} is added to \spad{ts} the resulting set,{} say \spad{ts+p},{} is a regular triangular set. Then it returns a split of Kalkbrener of \spad{ts+p}. This may not be \spad{ts+p} itself,{} if for instance \spad{ts+p} is required to be square-free.
118568`\axiom{autoReduced?(\spad{ts},{}redOp?)} returns \spad{true} iff every element of \axiom{\spad{ts}} is reduced \spad{w}.\spad{r}.\spad{t} to every other in the sense of \axiom{redOp?}
119101`\spad{back(q)} returns the element at the back of the queue. The queue \spad{q} is unchanged by this operation. Error: if \spad{q} is empty.
119303`\spad{badValues(p)} returns the list of "bad values" for \spad{p}; \spad{p} is not allowed to match any of its "bad values".
119417`\spad{bag([x,{}y,{}...,{}z])} creates a bag with elements \spad{x},{}\spad{y},{}...,{}\spad{z}.
119465`\spad{balancedBinaryTree(n,{} s)} creates a balanced binary tree with \spad{n} nodes each with value \spad{s}.
119547`\spad{balancedFactorisation(a,{} [b1,{}...,{}bn])} returns a factorisation \spad{a = p1^e1 ... pm^em} such that each \spad{pi} is balanced with respect to \spad{[b1,{}...,{}bm]}.
119639`\spad{balancedFactorisation(a,{} b)} returns a factorisation \spad{a = p1^e1 ... pm^em} such that each \spad{\spad{pi}} is balanced with respect to \spad{b}.
120255`\spad{base()} returns the base of the \spadfunFrom{exponent}{FloatingPointSystem}.
120314`\spad{base()} returns the base for the operations of \spad{IntegerNumberSystem}.
120590`\axiom{basicSet(\spad{ps},{}redOp?)} returns \axiom{[\spad{bs},{}\spad{ts}]} where \axiom{concat(\spad{bs},{}\spad{ts})} is \axiom{\spad{ps}} and \axiom{\spad{bs}} is a basic set in Wu Wen Tsun sense of \axiom{\spad{ps}} \spad{w}.\spad{r}.\spad{t} the reduction-test \axiom{redOp?},{} if no non-zero constant polynomial lie in \axiom{\spad{ps}},{} otherwise \axiom{"failed"} is return
120590`--ed.
120846`\axiom{basicSet(\spad{ps},{}pred?,{}redOp?)} returns the same as \axiom{basicSet(\spad{qs},{}redOp?)} where \axiom{\spad{qs}} consists of the polynomials of \axiom{\spad{ps}} satisfying property \axiom{pred?}.
120982`\spad{basisOfCenter()} returns a basis of the space of all \spad{x} of \spad{A} satisfying \spad{commutator(x,{}a) = 0} and \spad{associator(x,{}a,{}b) = associator(a,{}x,{}b) = associator(a,{}b,{}x) = 0} for all \spad{a},{}\spad{b} in \spad{A}.
121042`\spad{basisOfCentroid()} returns a basis of the centroid,{} \spadignore{i.e.} the endomorphism ring of \spad{A} considered as \spad{(A,{}A)}-bimodule.
121112`\spad{basisOfCommutingElements()} returns a basis of the space of all \spad{x} of \spad{A} satisfying \spad{0 = commutator(x,{}a)} for all \spad{a} in \spad{A}.
121183`\spad{basisOfLeftAnnihilator(a)} returns a basis of the space of all \spad{x} of \spad{A} satisfying \spad{0 = x*a}.
121253`\spad{basisOfLeftNucleus()} returns a basis of the space of all \spad{x} of \spad{A} satisfying \spad{0 = associator(x,{}a,{}b)} for all \spad{a},{}\spad{b} in \spad{A}.
121318`\spad{basisOfLeftNucloid()} returns a basis of the space of endomorphisms of \spad{A} as right module. Note: left nucloid coincides with left nucleus if \spad{A} has a unit.
121391`\spad{basisOfMiddleNucleus()} returns a basis of the space of all \spad{x} of \spad{A} satisfying \spad{0 = associator(a,{}x,{}b)} for all \spad{a},{}\spad{b} in \spad{A}.
121458`\spad{basisOfNucleus()} returns a basis of the space of all \spad{x} of \spad{A} satisfying \spad{associator(x,{}a,{}b) = associator(a,{}x,{}b) = associator(a,{}b,{}x) = 0} for all \spad{a},{}\spad{b} in \spad{A}.
121519`\spad{basisOfRightAnnihilator(a)} returns a basis of the space of all \spad{x} of \spad{A} satisfying \spad{0 = a*x}.
121590`\spad{basisOfRightNucleus()} returns a basis of the space of all \spad{x} of \spad{A} satisfying \spad{0 = associator(a,{}b,{}x)} for all \spad{a},{}\spad{b} in \spad{A}.
121656`\spad{basisOfRightNucloid()} returns a basis of the space of endomorphisms of \spad{A} as left module. Note: right nucloid coincides with right nucleus if \spad{A} has a unit.
121785`\spad{basis()} returns a fixed basis of \% as \spad{F}-vector space.
121853`\spad{basis()} returns the fixed \spad{R}-module basis.
121908`\spad{basis()} returns the fixed \spad{R}-module basis.
121974`\spad{basis LV} extracts a basis of the subspace spanned by the \spad{LV} list of vectors.
122242`\spad{basis(n)} returns a fixed basis of a subfield of \% as \spad{F}-vector space.
123444`\spad{besselI(x,{}y)} returns the besseli function applied to \spad{x} and \spad{y}
123701`\spad{besselI(v,{}z)} is the modified Bessel function of the first kind.
123759`\spad{besselJ(x,{}y)} returns the besselj function applied to \spad{x} and \spad{y}
124016`\spad{besselJ(v,{}z)} is the Bessel function of the first kind.
124074`\spad{besselK(x,{}y)} returns the besselk function applied to \spad{x} and \spad{y}
124331`\spad{besselK(v,{}z)} is the modified Bessel function of the second kind.
124389`\spad{besselY(x,{}y)} returns the bessely function applied to \spad{x} and \spad{y}
124646`\spad{besselY(v,{}z)} is the Bessel function of the second kind.
124704`\spad{bezoutDiscriminant(p)} computes the discriminant of a polynomial \spad{p} by computing the determinant of a Bezout matrix.
124796`\spad{bezoutMatrix(p,{}q)} returns the Bezout matrix for the two polynomials \spad{p} and \spad{q}.
124863`\spad{bezoutResultant(p,{}q)} computes the resultant of the two polynomials \spad{p} and \spad{q} by computing the determinant of a Bezout matrix.
124955`\spad{biRank(x)} determines the number of linearly independent elements in \spad{x},{} \spad{x*bi},{} \spad{bi*x},{} \spad{bi*x*bj},{} \spad{i,{}j=1,{}...,{}n},{} where \spad{b=[b1,{}...,{}bn]} is a basis. Note: if \spad{A} has a unit,{} then \spadfunFrom{doubleRank}{AlgebraPackage},{} \spadfunFrom{weakBiRank}{AlgebraPackage} and \spadfunFrom{biRank}{AlgebraPackage} coincide.
125103`\spad{binarySearchTree(l)} \undocumented
125168`\spad{binaryTournament(ls)} creates a binary tournament with the elements of \spad{ls} as values at the nodes.
125233`\spad{binaryTree(v)} is an non-empty binary tree with value \spad{v},{} and left and right empty.
125280`\spad{binaryTree(l,{}v,{}r)} creates a binary tree with value \spad{v} with left subtree \spad{l} and right subtree \spad{r}.
125333`\spad{binary(r)} converts a rational number to a binary expansion.
125394`\spad{binary(op,{} [a1,{}...,{}an])} returns the input form corresponding to \spad{a1 op a2 op ... op an}.
125525`\spad{binomThmExpt(p,{}q,{}n)} returns \spad{(x+y)^n} by means of the binomial theorem trick.
125634`\spad{binomial(n,{} r)} returns the number of subsets of \spad{r} objects taken among \spad{n} objects,{} \spadignore{i.e.} \spad{n!/}(\spad{r!} * (\spad{n}-\spad{r})!).
125693`\spad{binomial(n,{}f)} \undocumented
125832`\spad{binomial(n,{}m)} creates a form for the binomial coefficient of \spad{n} and \spad{m}.
125878`\spad{binomial(n,{}r)} returns the binomial coefficient \spad{C(n,{}r) = n!/(r! (n-r)!)},{} where \spad{n >= r >= 0}. This is the number of combinations of \spad{n} objects taken \spad{r} at a time.
125942`\spad{binomial(n,{}r)} returns the \spad{(n,{}r)} binomial coefficient (often denoted in the literature by \spad{C(n,{}r)}). Note: \spad{C(n,{}r) = n!/(r!(n-r)!)} where \spad{n >= r >= 0}.
126146`\spad{birth(x)} \undocumented
126242`\spad{bit?(n,{}i)} returns \spad{true} if and only if \spad{i}-th bit of \spad{n} is a 1.
126298`\spad{bitCoef(n,{}m)} returns the coefficient of 2^m in abs(\spad{n})
126359`\spad{bitLength(n)} returns the number of bits to represent abs(\spad{n})
126414`\spad{bitTruth(n,{}m)} returns \spad{true} if coefficient of 2^m in abs(\spad{n}) is 1
126476`\spad{bits()} returns ceiling\spad{'s} precision in bits.
126535`\spad{bits(n)} set the \spadfunFrom{precision}{FloatingPointSystem} to \spad{n} bits.
126635`\spad{bits(n,{}b)} creates bits with \spad{n} values of \spad{b}
127087`\spad{blankSeparate(l)} creates the form separating the elements of \spad{l} by blanks.
127141`\spad{block(l)} creates a representation of the statements in \spad{l} as a block.
127188`\spad{blue()} returns the position of the blue hue from total hues.
127220`\spad{bombieriNorm(p)} returns quadratic Bombieri\spad{'s} norm of \spad{p}.
127385`\spad{bottom!(d)} returns the element at the bottom (back) of the dequeue.
127435`\axiom{boundOfCauchy(\spad{p})} bounds the roots of \spad{p}
127643`\spad{box(f)} encloses \spad{f} in a box.
127681`\spad{box([f1,{}...,{}fn])} returns \spad{(f1,{}...,{}fn)} with a 'box' around them that prevents the \spad{fi} from being evaluated when operators are applied to them,{} and makes them applicable to a unary operator. For example,{} \spad{atan(box [x,{} 2])} returns the formal kernel \spad{atan(x,{} 2)}.
127730`\spad{box(f)} returns \spad{f} with a 'box' around it that prevents \spad{f} from being evaluated when operators are applied to it. For example,{} \spad{log(1)} returns 0,{} but \spad{log(box 1)} returns the formal kernel log(1).
127819`\spad{brace()}\$\spad{D} (otherwise written {}\$\spad{D}) creates an empty set aggregate of type \spad{D}. This form is considered obsolete. Use \axiomFun{set} instead.
127915`\spad{brace(lf)} creates the form separating the elements of \spad{lf} by commas and encloses the result in curly brackets.
127961`\spad{brace(f)} creates the form enclosing \spad{f} in braces (curly brackets).
128001`\spad{brace([x,{}y,{}...,{}z])} creates a set aggregate containing items \spad{x},{}\spad{y},{}...,{}\spad{z}. This form is considered obsolete. Use \axiomFun{set} instead.
128051`\spad{bracket(lf)} creates the form separating the elements of \spad{lf} by commas and encloses the result in square brackets.
128099`\spad{bracket(f)} creates the form enclosing \spad{f} in square brackets.
128499`\spad{branchPoint?(a)} tests whether \spad{x = a} is a branch point.
128572`\spad{branchPoint?(p)} tests whether \spad{p(x) = 0} is a branch point.
128646`\spad{branchPointAtInfinity?()} tests if there is a branch point at infinity.
128878`\spad{brillhartIrreducible?(p)} returns \spad{true} if \spad{p} can be shown to be irreducible by a remark of Brillhart,{} \spad{false} is inconclusive.
128947`\spad{brillhartIrreducible?(p,{}noLinears)} returns \spad{true} if \spad{p} can be shown to be irreducible by a remark of Brillhart,{} \spad{false} else. If \spad{noLinears} is \spad{true},{} we are being told \spad{p} has no linear factors \spad{false} does not mean that \spad{p} is reducible.
129841`\spad{bubbleSort!(a)} \undocumented
129908`\spad{bubbleSort!(a,{}f)} \undocumented
129973`\spad{build(r,{}i,{}e)} \undocumented
130359`\spad{cAcos(f)} computes the arccosine of the power series \spad{f}. For use when the coefficient ring is commutative.
130463`\spad{cAcosh(f)} computes the inverse hyperbolic cosine of the power series \spad{f}. For use when the coefficient ring is commutative.
130568`\spad{cAcot(f)} computes the arccotangent of the power series \spad{f}. For use when the coefficient ring is commutative.
130672`\spad{cAcoth(f)} computes the inverse hyperbolic cotangent of the power series \spad{f}. For use when the coefficient ring is commutative.
130777`\spad{cAcsc(f)} computes the arccosecant of the power series \spad{f}. For use when the coefficient ring is commutative.
130881`\spad{cAcsch(f)} computes the inverse hyperbolic cosecant of the power series \spad{f}. For use when the coefficient ring is commutative.
130986`\spad{cAsec(f)} computes the arcsecant of the power series \spad{f}. For use when the coefficient ring is commutative.
131090`\spad{cAsech(f)} computes the inverse hyperbolic secant of the power series \spad{f}. For use when the coefficient ring is commutative.
131195`\spad{cAsin(f)} computes the arcsine of the power series \spad{f}. For use when the coefficient ring is commutative.
131299`\spad{cAsinh(f)} computes the inverse hyperbolic sine of the power series \spad{f}. For use when the coefficient ring is commutative.
131404`\spad{cAtan(f)} computes the arctangent of the power series \spad{f}. For use when the coefficient ring is commutative.
131508`\spad{cAtanh(f)} computes the inverse hyperbolic tangent of the power series \spad{f}. For use when the coefficient ring is commutative.
131613`\spad{cCos(f)} computes the cosine of the power series \spad{f}. For use when the coefficient ring is commutative.
131716`\spad{cCosh(f)} computes the hyperbolic cosine of the power series \spad{f}. For use when the coefficient ring is commutative.
131820`\spad{cCot(f)} computes the cotangent of the power series \spad{f}. For use when the coefficient ring is commutative.
131923`\spad{cCoth(f)} computes the hyperbolic cotangent of the power series \spad{f}. For use when the coefficient ring is commutative.
132027`\spad{cCsc(f)} computes the cosecant of the power series \spad{f}. For use when the coefficient ring is commutative.
132130`\spad{cCsch(f)} computes the hyperbolic cosecant of the power series \spad{f}. For use when the coefficient ring is commutative.
132234`\spad{cExp(f)} computes the exponential of the power series \spad{f}. For use when the coefficient ring is commutative.
132337`\spad{cLog(f)} computes the logarithm of the power series \spad{f}. For use when the coefficient ring is commutative.
132440`\spad{cPower(f,{}r)} computes \spad{f^r},{} where \spad{f} has constant coefficient 1. For use when the coefficient ring is commutative.
132676`\spad{cSec(f)} computes the secant of the power series \spad{f}. For use when the coefficient ring is commutative.
132779`\spad{cSech(f)} computes the hyperbolic secant of the power series \spad{f}. For use when the coefficient ring is commutative.
132883`\spad{cSin(f)} computes the sine of the power series \spad{f}. For use when the coefficient ring is commutative.
132986`\spad{cSinh(f)} computes the hyperbolic sine of the power series \spad{f}. For use when the coefficient ring is commutative.
133090`\spad{cTan(f)} computes the tangent of the power series \spad{f}. For use when the coefficient ring is commutative.
133193`\spad{cTanh(f)} computes the hyperbolic tangent of the power series \spad{f}. For use when the coefficient ring is commutative.
133388`\spad{call(s)} creates a representation of a FORTRAN CALL statement
133432`\spad{canonicalBasis n} build the canononical basis of the vector space with \spad{n} coordinates.
133634`\spad{car((a1,{}...,{}an))} returns \spad{a1}.
133773`\spad{cardinality(u)} returns the number of elements of \spad{u}. Note: \axiom{cardinality(\spad{u}) = \#u}.
133981`\spad{cdr((a1,{}...,{}an))} returns \spad{(a2,{}...,{}an)}.
134095`\spad{ceiling(x)} returns the smallest integral element above \spad{x}.
134176`\spad{ceiling x} returns the small integer \spad{>= x}.
134224`\spad{center(f)} centers form \spad{f} in total space.
134265`\spad{center(f)} returns the point about which the series \spad{f} is expanded.
134339`\spad{center(f,{}n)} centers form \spad{f} within space of width \spad{n}.
134622`\axiom{chainSubResultants(\spad{P},{} \spad{Q})} computes the list of non zero subresultants of \axiom{\spad{P}} and \axiom{\spad{Q}}.
134782`\spad{changeBase(M,{} A,{} sig,{} der)}: computes the new matrix of a pseudo-linear transform given by the matrix \spad{M} under the change of base A
134957`\spad{changeVar(+/[\spad{ai} D^i],{} a)} returns the operator \spad{+/[\spad{ai} (D+a)\spad{^i}]}.
135020`\spad{changeWeightLevel(n)} This changes the weight level to the new value given: \spad{NB:} previously calculated terms are not affected
135124`\spad{changeWeightLevel(n)} changes the weight level to the new value given: \spad{NB:} previously calculated terms are not affected
135231`\spad{charClass(l)} creates a character class which contains exactly the characters given in the list \spad{l}.
135292`\spad{charClass(s)} creates a character class which contains exactly the characters given in the string \spad{s}.
135344`\spad{char(i)} provides a character corresponding to the integer code \spad{i}. It is always \spad{true} that \spad{ord char i = i}.
135387`\spad{char(s)} provides a character from a string \spad{s} of length one.
135429`\spad{character?(t)} tests whether \spad{t} is equivalent to the FORTRAN type CHARACTER.
135486`\spad{characteristicPolynomial(e)} returns the characteristic polynomial of \spad{e} using resultants
136178`\spad{characteristicPolynomial(a)} returns the characteristic polynomial of the regular representation of \spad{a} with respect to any basis.
137747`\spad{characteristic()} returns the characteristic of the ring.
137818`\spad{characteristic()} returns the characteristic of the ring this is the smallest positive integer \spad{n} such that \spad{n*x=0} for all \spad{x} in the ring,{} or zero if no such \spad{n} exists.
138385`\spad{charthRoot(x)} returns the \spad{p}th root of \spad{x} where \spad{p} is the characteristic of the ring.
138457`\spad{charthRoot(r)} returns the \spad{p}\spad{-}th root of \spad{r},{} or "failed" if none exists in the domain.
138568`\spad{charthRoot(a)} takes the characteristic'th root of {\em a}. Note: such a root is alway defined in finite fields.
139055`\spad{checkExtraValues(d)} specifies whether we want to check the solution beyond the order given by the degree bounds. This option is expressed in the form \spad{checkExtraValues == d}
139491`\spad{checkRur(ts,{}lus)} returns \spad{true} if \spad{lus} is a rational univariate representation of \spad{ts}.
139850`\spad{check(d)} specifies how we want to check the solution. If the value is "skip",{} we return the solutions found by the interpolation routine without checking. If the value is "MonteCarlo",{} we use a probabilistic check. This option is expressed in the form \spad{check == d}
139931`\spad{check(s)} returns lllpt,{} list of lists of lists of point information about the \spadtype{ThreeSpace} \spad{s}.
140439`\spad{child?(u,{}v)} tests if node \spad{u} is a child of node \spad{v}.
140754`\spad{children(x)} \undocumented
140806`\spad{children(u)} returns a list of the children of aggregate \spad{u}.
140866`\spad{chineseRemainder(llv,{}lm)} returns a list of values,{} each of which corresponds to the Chinese remainder of the associated element of \axiom{\spad{llv}} and axiom{\spad{lm}}. This is more efficient than applying chineseRemainder several times.
140944`\spad{chineseRemainder(lv,{}lm)} returns a value \axiom{\spad{v}} such that,{} if \spad{x} is \axiom{\spad{lv}.\spad{i}} modulo \axiom{\spad{lm}.\spad{i}} for all \axiom{\spad{i}},{} then \spad{x} is \axiom{\spad{v}} modulo \axiom{\spad{lm}(1)\spad{*lm}(2)*...\spad{*lm}(\spad{n})}.
141325`chinese__update(\spad{v},{} \spad{p},{} \spad{r}) informs \spad{r} about evaluation at \spad{p}
141490`chkLibraryError is an internal function that steps the number of tests.
141855`\spad{clearCache()} empties the cache.
141904`\spad{clearDenominator([q1,{}...,{}qn])} returns \spad{[p1,{}...,{}pn]} such that \spad{\spad{qi} = pi/d} where \spad{d} is a common denominator for the \spad{qi}\spad{'s}.
141974`\spad{clearDenominator([q1,{}...,{}qn])} returns \spad{[p1,{}...,{}pn]} such that \spad{\spad{qi} = pi/d} where \spad{d} is a common denominator for the \spad{qi}\spad{'s}.
142037`\spad{clearDenominator(q)} returns \spad{p} such that \spad{q = p/d} where \spad{d} is a common denominator for the elements of \spad{q}.
142120`\spad{clearDenominator(q)} returns \spad{p} such that \spad{q = p/d} where \spad{d} is a common denominator for the coefficients of \spad{q}.
142206`\spad{clearFortranOutputStack()} clears the Fortran output stack
142288`\axiom{clearTable!()} clears the hash-table and assumes that it will no longer be used.
142362`\spad{clearTheSymbolTable()} clears the current symbol table.
142420`\spad{clearTheSymbolTable(x)} removes the symbol \spad{x} from the table
142484`\spad{clipBoolean(l,{}b)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{clipBoolean}. If the option does not exist the value,{} \spad{b} is returned.
143848`\spad{clip(b)} turns 2D clipping on if \spad{b} is \spad{true},{} or off if \spad{b} is \spad{false}. This option is expressed in the form \spad{clip == b}.
144132`\spad{close!(f)} returns the file \spad{f} closed to input and output.
144183`\spad{close!(f)} returns the library \spad{f} closed to input and output.
144301`\spad{close(v)} closes the viewport window of the given two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport},{} and terminates the corresponding process ID.
144355`\spad{close(v)} closes the viewport window of the given three-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{ThreeDimensionalViewport},{} and terminates the corresponding process ID.
145119`\axiom{coHeight(\spad{ts})} returns \axiom{size()\spad{\$}\spad{V}} minus \axiom{\spad{\#}\spad{ts}}.
146014`\spad{coef(p,{}e)} extracts the coefficient of the monomial \spad{e}. Returns zero if \spad{e} is not present.
146205`\spad{coef(x,{}y)} returns scalar product of \spad{x} by \spad{y},{} the set of words being regarded as an orthogonal basis.
146659`\spad{coefficient(p,{}u)} returns the coefficient of the term in \spad{p} containing the basis term \spad{u} if such a term exists,{} and 0 otherwise. Error: if the second argument \spad{u} is not a basis element.
146713`\spad{coefficient(s,{} e1 a1 + ... + en an)} returns \spad{ei} such that \spad{ai} = \spad{s},{} or 0 if \spad{s} is not one of the \spad{ai}\spad{'s}.
146780`\spad{coefficient(p,{}e)} extracts the coefficient of the monomial with exponent \spad{e} from polynomial \spad{p},{} or returns zero if exponent is not present.
146916`\spad{coefficient(f,{}m)} extracts the coefficient of \spad{m} in \spad{f} with respect to the canonical basis \spad{M}.
147133`\spad{coefficient(l,{}k)} is \spad{a(k)} if \indented{2}{\spad{l = sum(monomial(a(i),{}i),{} i = 0..n)}.}
147222`\spad{coefficient(x,{}s)} returns the coefficient of the basis element \spad{s}
147784`\spad{coefficient(p,{} lv,{} ln)} views the polynomial \spad{p} as a polynomial in the variables of \spad{lv} and returns the coefficient of the term \spad{lv^ln},{} \spadignore{i.e.} \spad{prod(lv_i ^ ln_i)}.
147984`\spad{coefficient(p,{}v,{}n)} views the polynomial \spad{p} as a univariate polynomial in \spad{v} and returns the coefficient of the \spad{v^n} term.
148217`\spad{coefficients(x)} returns a stream of ring elements. \indented{1}{When \spad{x} is a univariate series,{} this is a stream of Taylor} \indented{1}{coefficients. When \spad{x} is a multivariate series,{} the} \indented{1}{\spad{n}th element of the stream is a form of} \indented{1}{degree \spad{n} in the power series variables.}
148287`\spad{coefficients(p)} gives the list of non-zero coefficients of polynomial \spad{p}.
148357`\spad{coefficients(x)} returns the list of coefficients of \spad{x}.
148422`\spad{coefficients(f)} lists all non-zero coefficients.
148487`\spad{coefficients(l)} returns the list of all the nonzero coefficients of \spad{l}.
148564`\spad{coefficients(a0 + a1 x + a2 x^2 + ...)} returns a stream of coefficients: \spad{[a0,{}a1,{}a2,{}...]}. The entries of the stream may be zero.
148925`\spad{coerceListOfPairs(lls)} coerces a list of pairs {\em lls} to a permutation. Error: if not consistent,{} \spadignore{i.e.} the set of the first elements coincides with the set of second elements. coerce(\spad{p}) generates output of the permutation \spad{p} with domain OutputForm.
148992`\spad{coerceP(v)} coerces a vector \spad{v} with entries in \spadtype{Matrix R} as vector over \spadtype{Matrix Polynomial R}
149094`\spad{coercePreimagesImages(lls)} coerces the representation {\em lls} of a permutation as a list of preimages and images to a permutation. We assume that both preimage and image do not contain repetitions.
149265`\spad{coerceToType(f)} returns the input form corresponding to f::R
150313`\spad{coerce(u)} transforms \spad{u} into a MachineComplex
150370`\spad{coerce(u)} transforms \spad{u} into a MachineComplex
150429`\spad{coerce(u)} transforms \spad{u} into a MachineComplex
150493`\spad{coerce(u)} transforms \spad{u} into a MachineComplex
150559`\spad{coerce(e)} returns \spad{1*e}
151434`\spad{coerce(x)} returns \spad{x} with coefficients in the domain
151895`\spad{coerce(f)} produces the equivalence class of \spad{f} in the residue ring
151966`\spad{coerce(e)} takes an object from \spadtype{FortranCode} and \indented{1}{uses it as the body of an ASP.}
152029`\spad{coerce(e)} takes an object from \spadtype{FortranCode} and \indented{1}{uses it as the body of an ASP.}
152090`\spad{coerce(e)} takes an object from \spadtype{FortranCode} and \indented{1}{uses it as the body of an ASP.}
152159`\spad{coerce(e)} takes an object from \spadtype{FortranCode} and \indented{1}{uses it as the body of an ASP.}
152220`\spad{coerce(e)} takes an object from \spadtype{FortranCode} and \indented{1}{uses it as the body of an ASP.}
152289`\spad{coerce(fc)} \undocumented{}
154918`\spad{coerce(e)} takes an object from \spadtype{List FortranCode} and \indented{1}{uses it as the body of an ASP.}
154987`\spad{coerce(e)} takes an object from \spadtype{List FortranCode} and \indented{1}{uses it as the body of an ASP.}
155054`\spad{coerce(e)} takes an object from \spadtype{List FortranCode} and \indented{1}{uses it as the body of an ASP.}
155129`\spad{coerce(e)} takes an object from \spadtype{List FortranCode} and \indented{1}{uses it as the body of an ASP.}
155196`\spad{coerce(e)} takes an object from \spadtype{List FortranCode} and \indented{1}{uses it as the body of an ASP.}
155271`\spad{coerce(lfc)} \undocumented{}
155478`\spad{coerce(u)} transforms a MachineInteger into a MachineFloat
155533`\spad{coerce(v)} produces an ASP which returns the value of \spad{v}.
155651`\spad{coerce(p)} coerces \spad{p} into Weighted form,{} applying weights and ignoring terms
155848`\spad{coerce(p)} coerces a Polynomial(\spad{R}) into Weighted form,{} applying weights and ignoring terms
156032`\spad{coerce(a)} makes a tuple from primitive array a
156785`\spad{coerce(e)} takes the component of \spad{e} from \spadtype{List FortranCode} and uses it as the body of the ASP,{} making the declarations in the \spadtype{SymbolTable} component.
156892`\spad{coerce(e)} takes the component of \spad{e} from \spadtype{List FortranCode} and uses it as the body of the ASP,{} making the declarations in the \spadtype{SymbolTable} component.
156997`\spad{coerce(e)} takes the component of \spad{e} from \spadtype{List FortranCode} and uses it as the body of the ASP,{} making the declarations in the \spadtype{SymbolTable} component.
157110`\spad{coerce(e)} takes the component of \spad{e} from \spadtype{List FortranCode} and uses it as the body of the ASP,{} making the declarations in the \spadtype{SymbolTable} component.
157215`\spad{coerce(e)} takes the component of \spad{e} from \spadtype{List FortranCode} and uses it as the body of the ASP,{} making the declarations in the \spadtype{SymbolTable} component.
157456`\spad{coerce(x)} \undocumented
159432`\spad{coerce(s)} returns \spad{s} viewed as an element of \%.
159480`\spad{coerce(poly)} regroups the terms by total degree and forms a series.
159864`\spad{coerce(s)} transforms the string \spad{s} into an element of FortranScalarType provided \spad{s} is one of "real",{} "double precision",{} "complex",{} "logical",{} "integer",{} "character",{} "REAL",{} "COMPLEX",{} "LOGICAL",{} "INTEGER",{} "CHARACTER",{} "DOUBLE PRECISION"
160020`\spad{coerce(s)} transforms the symbol \spad{s} into an element of FortranScalarType provided \spad{s} is one of real,{} complex,{}double precision,{} logical,{} integer,{} character,{} REAL,{} COMPLEX,{} LOGICAL,{} INTEGER,{} CHARACTER,{} DOUBLE PRECISION
160072`\spad{coerce(s)} \undocumented{}
160605`\spad{coerce(var)} converts a variable to a Taylor series
160681`\spad{coerce(v)} return \spad{v}
160824`\spad{coerce(var)} converts the series variable \spad{var} into a Laurent series.
160909`\spad{coerce(var)} converts the series variable \spad{var} into a Puiseux series.
160994`\spad{coerce(var)} converts the series variable \spad{var} into a \indented{1}{Taylor series.}
161078`\spad{coerce(var)} converts the series variable \spad{var} into a Laurent series.
161157`\spad{coerce(var)} converts the series variable \spad{var} into a Puiseux series.
161236`\spad{coerce(var)} converts the series variable \spad{var} into a \indented{1}{Taylor series.}
161314`\spad{coerce(x)} returns \spad{x} as a skew-polynomial.
161492`\spad{coerce(v)} produces an ASP which returns the value of \spad{v}.
161562`\spad{coerce(v)} coerces a vector \spad{v} with entries in \spadtype{Matrix R} as vector over \spadtype{Matrix Fraction Polynomial R}
161790`\spad{coerce(u)} transforms \spad{u} into a COmplex Float
161977`\spad{coerce(p)} coerces a partition into a list of integers
162074`\spad{coerce b} retracts a basis to this list of vectors.
162137`\axiom{coerce(\spad{x})} returns the element of \axiomType{Magma}(VarSet) corresponding to \axiom{\spad{x}}.
162197`\spad{coerce(m)} converts a matrix of type \spadtype{RectangularMatrix} to a matrix of type \spad{Matrix}.
162259`\spad{coerce(m)} converts a matrix of type \spadtype{SquareMatrix} to a matrix of type \spadtype{Matrix}.
162534`output
162632`\spad{coerce(x)} provides a printable form for \spad{x}
162726`output
162779`output
162883`\spad{coerce b} prepares the display of a basis.
162941`convert back into a \spad{"P"},{} ignoring weights
163018`\spad{coerce(p)} converts back into a Polynomial(\spad{R}),{} ignoring weights
163202`\spad{coerce(x)} \undocumented
163286`\spad{coerce(x)} \undocumented
163360`\spad{coerce(x)} \undocumented
163507`\spad{coerce(x)} returns the \spad{s}-expression associated with \spad{x}
163661`\spad{coerce(x)} returns the symbol associated with \spad{x}
163713`\spad{coerce(x)} returns the symbol
163761`\spad{coerce(x)} returns a table view of \spad{x}
164150`\spad{coerce A} changes its argument into a \indented{1}{nullary function.}
164201`\spad{coerce(a)} coerces the element \spad{a} of the algebra \spad{A} to an element of the Jordan algebra \spadtype{AssociatedJordanAlgebra}(\spad{R},{}A).
164259`\spad{coerce(a)} coerces the element \spad{a} of the algebra \spad{A} to an element of the Lie algebra \spadtype{AssociatedLieAlgebra}(\spad{R},{}A).
164314`\spad{coerce(c)} returns \spad{c} as a string \spad{s} with the character \spad{c}.
164367`\spad{coerce(col)} converts the column \spad{col} to a column matrix.
164532`\spad{coerce(e)} is never really evaluated. This coercion is used for formal type correctness when a function will not return directly to its caller.
164591`\spad{coerce(x)} is the homomorphic image of \spad{x} from {\em F1} in {\em F2}. Thus {\em coerce} is a field homomorphism between the fields extensions {\em F1} and {\em F2} both over ground field {\em GF} (the second argument to the package). Error: if the extension degree of {\em F1} doesn\spad{'t} divide the extension degree of {\em F2}. Note that the other coercion function in the \spadtype{FiniteFieldHomomorphisms} is a left inverse
164591`--.
164656`\spad{coerce(x)} is the homomorphic image of \spad{x} from {\em F2} in {\em F1},{} where {\em coerce} is a field homomorphism between the fields extensions {\em F2} and {\em F1} both over ground field {\em GF} (the second argument to the package). Error: if the extension degree of {\em F2} doesn\spad{'t} divide the extension degree of {\em F1}. Note that the other coercion function in the \spadtype{FiniteFieldHomomorphisms} is a left inve
164656`--rse.
164721`\spad{coerce(c)} converts sin/cos terms into Fourier Series
164850`\spad{coerce(rf)} converts \spad{rf},{} a fraction of polynomial \spad{p} with algebraic number coefficients to \spadtype{Expression Integer}.
165122`\spad{coerce(n)} coerces the integer \spad{n} to an element of the ring.
165176`\spad{coerce(i)} converts the integer \spad{i} to a member of the given domain.
165216`\spad{coerce(i)} returns the element corresponding to \spad{i}
165334`\spad{coerce(lls)} coerces a list of cycles {\em lls} to a permutation,{} each cycle being a list with no repetitions,{} is coerced to the permutation,{} which maps {\em ls.i} to {\em ls.i+1},{} indices modulo the length of the list,{} then these permutations are mutiplied. Error: if repetitions occur in one cycle.
165453`\spad{coerce([r_1,{}...,{}r_dim])} allows tensors to be constructed using lists.
165592`\spad{coerce(l)} creates a datalist from \spad{l}
165639`\spad{coerce(l)} makes a database out of a list
165686`\spad{coerce(ls)} coerces a cycle {\em ls},{} \spadignore{i.e.} a list with not repetitions to a permutation,{} which maps {\em ls.i} to {\em ls.i+1},{} indices modulo the length of the list. Error: if repetitions occur.
165776`\spad{coerce([t_1,{}...,{}t_dim])} allows tensors to be constructed using lists.
165987`\spad{coerce(f)} returns \spad{f} as an Expression(\spad{R}).
166043`\spad{coerce(p)} converts the polynomial \spad{p} with algebraic number coefficients to \spadtype{Expression Integer}.
166346`\spad{coerce(r)} maps the ring element \spad{r} to a member of the algebra.
166386`\spad{coerce(r)} returns \spad{r} * 1 where 1 is the identity of the left algebra.
166430`\spad{coerce(r)} equals \spad{r*1}.
166471`\spad{coerce(r)} converts coefficients into Fourier Series
166519`\spad{coerce(r)} views \spad{r} as a value in the ordinary differential ring.
166588`\spad{coerce(s)} creates an object of \spadtype{Any} from the object \spad{s} of type \spad{S}.
166635`\spad{coerce(x)} changes \spad{x} into an object of type \spadtype{None}.
166684`\spad{coerce(s)} provides a direct coercion from an expression \spad{s} of domain \spad{S} to SCRIPT formula format. This allows the user to skip the step of first manually coercing the object to standard output format before it is coerced to SCRIPT formula format.
166754`\spad{coerce(s)} provides a direct coercion from a domain \spad{S} to TeX format. This allows the user to skip the step of first manually coercing the object to standard output format before it is coerced to TeX format.
166804`\spad{coerce(s)} throws all information about \spad{s} away. This coercion allows values of any type to appear in contexts where they will not be used. For example,{} it allows the resolution of different types in the \spad{then} and \spad{else} branches when an \spad{if} is in a context where the resulting value is not used.
166863`\spad{coerce(s)} returns \spad{s},{} viewed as the zero-th order derivative of \spad{s}.
166924`\spad{coerce(a)} transforms a into an element of \%.
166970`\spad{coerce(x)} allows \spadtype{Segment} values to be used as \%.
167028`\spad{coerce(x = a..b)} returns \spad{x = c..d} where \spad{c} and \spad{d} are the numerical values of \spad{a} and \spad{b}.
167374`\spad{coerce(s)} converts a string to a file name according to operating system-dependent conventions.
167425`\spad{coerce(s)} converts \axiom{\spad{s}} into an \axiom{IndexCard}. Warning: if \axiom{\spad{s}} is not of the right format then an error will occur when using it.
167469`\spad{coerce(s)} converts the string \spad{s} to a symbol.
167510`\spad{coerce(u)} creates an OpenMath error object of an appropriate type if \axiom{\spad{u}} is one of \axiom{OMParseError},{} \axiom{OMReadError},{} \axiom{OMUnknownCD} or \axiom{OMUnknownSymbol},{} otherwise it raises a runtime error.
167562`\spad{coerce(s)} converts a variable to a Taylor series
167615`\spad{coerce(f(x))} converts the Laurent series \spad{f(x)} to a Puiseux series.
167699`\spad{coerce(f(x))} converts the Taylor series \spad{f(x)} to a Laurent series.
167886`\axiom{coerce(\spad{x})} returns \axiom{\spad{x}} as a Lie polynomial.
167945`\spad{coerce(var)} converts the series variable \spad{var} into a Puiseux series.
168029`\spad{coerce(x)} converts the variable \spad{x} to a univariate polynomial.
168094`\spad{coerce(v)} converts a vector to a member of the algebra by forming a linear combination with the basis element. Note: the vector is assumed to have length equal to the dimension of the algebra.
168179`\spad{coerce(b)} converts a binary expansion to a rational number.
168240`\spad{coerce(d)} converts a decimal expansion to a rational number.
168302`\spad{coerce(h)} converts a hexadecimal expansion to a rational number.
168368`\spad{coerce(rx)} converts a radix expansion to a rational number.
168432`\spad{coerce(p)} sums up the components of the partial fraction and returns a single fraction.
168553`\spad{coerce(x)} returns the list of elements in \spad{x}
168699`\spad{coerce(v)} coerces void object to outputForm.
168742`\spad{coerce(p)} views \spad{p} as a valie in the partial differential ring.
168811`\spad{coerce(d)} converts a decimal expansion to a radix expansion with base 10.
168874`\spad{coerce(h)} converts a hexadecimal expansion to a radix expansion with base 16.
168941`\spad{coerce(b)} converts a binary expansion to a radix expansion with base 2.
169002`\spad{coerce(a)} transforms a into an element of \spad{S}.
169046`\spad{coerce(fn)} produces a string for a file name according to operating system-dependent conventions.
169097`\spad{coerce(x)} \undocumented
169305`\spad{coerce(v)} returns \spad{v}.
169468`Measure of how parallel two vectors are independant of the length of the vectors: 1.0 is complelely parallel 0.0 is completely perpendicular returned value will be between these
169598`\axiom{collectQuasiMonic(\spad{ts})} returns the subset of \axiom{\spad{ts}} consisting of the polynomials with initial in \axiom{\spad{R}}.
169810`\axiom{collectUnder(\spad{ps},{}\spad{v})} returns the set consisting of the polynomials of \axiom{\spad{ps}} with main variable less than \axiom{\spad{v}}.
170029`\axiom{collectUpper(\spad{ps},{}\spad{v})} returns the set consisting of the polynomials of \axiom{\spad{ps}} with main variable greater than \axiom{\spad{v}}.
170178`\axiom{collect(\spad{ps},{}\spad{v})} returns the set consisting of the polynomials of \axiom{\spad{ps}} with \axiom{\spad{v}} as main variable.
170600`\spad{color(i)} returns a color of the indicated hue \spad{i}.
170640`\spad{columnSpace M} builds a basis of the kernel of the linear transformation defined by the matrix \spad{M}. This function calls the \spad{columnSpace} in the \spad{Matrix} domain. Only the signature changes.
170767`\spad{columnSpace(m)} returns a sublist of columns of the matrix \spad{m} forming a basis of its column space
170857`\spad{columnSpace(m)} returns a sublist of columns of the matrix \spad{m} forming a basis of its column space.
171037`\spad{column(m,{}j)} returns the \spad{j}th column of the matrix \spad{m}. Error: if the index outside the proper range.
171117`\spad{column(m,{}j)} returns the \spad{j}th column of \spad{m} error check to determine if index is in proper ranges
171195`\spad{commaSeparate(l)} creates the form separating the elements of \spad{l} by commas.
171249`\spad{comment(s)} creates a representation of the Strings \spad{s} as a multi-line FORTRAN comment.
171302`\spad{comment(s)} creates a representation of the String \spad{s} as a single FORTRAN comment.
171349`\spad{commonDenominator([q1,{}...,{}qn])} returns a common denominator \spad{d} for \spad{q1},{}...,{}\spad{qn}.
171420`\spad{commonDenominator([q1,{}...,{}qn])} returns a common denominator \spad{d} for \spad{q1},{}...,{}\spad{qn}.
171484`\spad{commonDenominator(q)} returns a common denominator \spad{d} for the elements of \spad{q}.
171560`\spad{commonDenominator(q)} returns a common denominator \spad{d} for the coefficients of \spad{q}.
171646`\spad{common(name,{}contents)} creates a representation a named common block.
171705`\spad{commutative?()} tests if multiplication in the algebra is commutative.
171782`\spad{commutative?()} tests if multiplication in the algebra is commutative.
171856`\spad{commutativeEquality(x,{}y)} returns \spad{true} if \spad{x} and \spad{y} are equal assuming commutativity
172019`\spad{commutator(p,{}q)} computes \spad{inv(p) * inv(q) * p * q}.
172062`\spad{commutator(a,{}b)} returns \spad{a*b-b*a}.
172188`\spad{comp(f,{}g,{}x)} is \spad{f(g x)}.
172264`\spad{compactFraction(p)} normalizes the partial fraction \spad{p} to the compact representation. In this form,{} the partial fraction has only one fractional term per prime in the denominator.
172434`\spad{comparison(op,{} foo?)} attaches foo? as the "\%less?" property to \spad{op}. If \spad{op1} and \spad{op2} have the same name,{} and one of them has a "\%less?" property \spad{f},{} then \spad{f(op1,{} op2)} is called to decide whether \spad{op1 < op2}.
172629`\spad{compile(f,{} [t1,{}...,{}tn])} forces the interpreter to compile the function \spad{f} with signature \spad{(t1,{}...,{}tn) -> ?}. returns the symbol \spad{f} if successful. Error: if \spad{f} was not defined beforehand in the interpreter,{} or if the \spad{ti}\spad{'s} are not valid types,{} or if the compiler fails.
172859`\spad{complementSpace Lv} build a complement space of the subspace spaned by the list of vectors \spad{LV}. This function fails with the empty basis.
172931`\spad{complementSpace B} build a complement space of the basis \spad{B} with some vectors of the canonical basis. This function fails with the empty basis.
172990`\spad{complementSpace(Lv,{}n)} build a complement space of the subspace spaned by the list of vectors \spad{LV}. The integer \spad{n} gives the length of the vectors if the basis is empty.
173081`\spad{complementSpace(B,{}n)} build a complement space of the basis \spad{B} with some vectors of the canonical basis \spad{B}. The integer \spad{n} gives the length of the vectors if the basis is empty.
173211`\spad{complement(u)} returns the complement of the set \spad{u},{} \spadignore{i.e.} the set of all values not in \spad{u}.
173367`\spad{complementaryBasis(b1,{}...,{}bn)} returns the complementary basis \spad{(b1',{}...,{}bn')} of \spad{(b1,{}...,{}bn)}.
174089`\spad{complete(st)} causes all entries of 'st' to be computed. this function should only be called on streams which are known to be finite.
174144`\spad{complete(x)} forces the computation of all digits.
174200`\spad{complete(f)} causes all terms of \spad{f} to be computed. Note: this results in an infinite loop if \spad{f} has infinitely many terms.
174268`\spad{complete(x)} causes all entries in \spadvar{\spad{x}} to be computed. Normally entries are only computed as needed. If \spadvar{\spad{x}} is an infinite continued fraction,{} a user-initiated interrupt is necessary to stop the computation.
174321`\spad{complex?(t)} tests whether \spad{t} is equivalent to the FORTRAN type COMPLEX.
174685`\spad{complexElementary(f)} rewrites \spad{f} in terms of the 2 fundamental complex transcendental elementary functions: \spad{log,{} exp}.
174763`\spad{complexElementary(f)} rewrites \spad{f} in terms of the 2 fundamental complex transcendental elementary functions: \spad{log,{} exp}.
175273`\spad{complexForm(f)} returns \spad{[real f,{} imag f]}.
176014`\spad{complexNormalize(f)} rewrites \spad{f} using the least possible number of complex independent kernels.
176091`\spad{complexNormalize(f)} rewrites \spad{f} using the least possible number of complex independent kernels.
176312`\spad{complexNumericIfCan(x)} returns a complex approximation of \spad{x},{} or "failed" if \axiom{\spad{x}} is not a constant.
176458`\spad{complexNumericIfCan(x)} returns a complex approximation of \spad{x},{} or "failed" if \axiom{\spad{x}} is not a constant.
176595`\spad{complexNumericIfCan(x)} returns a complex approximation of \spad{x},{} or "failed" if \axiom{\spad{x}} is not a constant.
176728`\spad{complexNumericIfCan(x)} returns a complex approximation of \spad{x},{} or "failed" if \axiom{\spad{x}} is not a constant.
176852`\spad{complexNumericIfCan(x)} returns a complex approximation of \spad{x},{} or "failed" if \axiom{\spad{x}} is not constant.
176976`\spad{complexNumericIfCan(x)} returns a complex approximation of \spad{x},{} or "failed" if \axiom{\spad{x}} is not a constant.
177080`\spad{complexNumericIfCan(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places,{} or "failed" if \axiom{\spad{x}} is not a constant.
177242`\spad{complexNumericIfCan(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places,{} or "failed" if \axiom{\spad{x}} is not a constant.
177395`\spad{complexNumericIfCan(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places,{} or "failed" if \axiom{\spad{x}} is not a constant.
177544`\spad{complexNumericIfCan(x,{} n)} returns a complex approximation of \spad{x},{} or "failed" if \axiom{\spad{x}} is not a constant.
177684`\spad{complexNumericIfCan(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places,{} or "failed" if \axiom{\spad{x}} is not a constant.
177824`\spad{complexNumericIfCan(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places,{} or "failed" if \axiom{\spad{x}} is not a constant.
177944`\spad{complexNumeric(x)} returns a complex approximation of \spad{x}.
178035`\spad{complexNumeric(x)} returns a complex approximation of \spad{x}.
178160`\spad{complexNumeric(x)} returns a complex approximation of \spad{x}.
178276`\spad{complexNumeric(x)} returns a complex approximation of \spad{x}.
178388`\spad{complexNumeric(x)} returns a complex approximation of \spad{x}.
178491`\spad{complexNumeric(x)} returns a complex approximation of \spad{x}.
178594`\spad{complexNumeric(x)} returns a complex approximation of \spad{x}.
178677`\spad{complexNumeric(x)} returns a complex approximation of \spad{x}.
178737`\spad{complexNumeric(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places.
178844`\spad{complexNumeric(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places.
178985`\spad{complexNumeric(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places.
179117`\spad{complexNumeric(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places.
179245`\spad{complexNumeric(x,{} n)} returns a complex approximation of \spad{x}
179364`\spad{complexNumeric(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places.
179483`\spad{complexNumeric(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places.
179582`\spad{complexNumeric(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places.
180739`\spad{complex(x,{}y)} constructs \spad{x} + \%i*y.
181179`\spad{compose(p,{}q)} \undocumented
181475`\spad{composite(f,{} q)} returns \spad{h} if \spad{f} = \spad{h}(\spad{q}),{} and "failed" is no such \spad{h} exists.
181600`\spad{composite(p,{} q)} returns \spad{h} if \spad{p = h(q)},{} and "failed" no such \spad{h} exists.
181762`combine two transforms
181976`\spad{computeCycleEntry(x,{}cycElt)},{} where \spad{cycElt} is a pointer to a node in the cyclic part of the cyclic stream \spad{x},{} returns a pointer to the first node in the cycle
182044`\spad{computeCycleLength(s)} returns the length of the cycle of a cyclic stream \spad{t},{} where \spad{s} is a pointer to a node in the cyclic part of \spad{t}.
182278`\spad{computePowers()} \undocumented
182557`\spad{concat!(u,{}x)} destructively adds element \spad{x} to the end of \spad{u}.
182619`\spad{concat!(u,{}x)} destructively adds element \spad{x} to the end of \spad{u}. Note: \axiom{concat!(a,{}\spad{x}) = setlast!(a,{}[\spad{x}])}.
182703`\spad{concat!(u,{}v)} destructively concatenates doubly-linked aggregate \spad{v} to the end of doubly-linked aggregate \spad{u}.
182786`\spad{concat!(u,{}v)} destructively appends \spad{v} to the end of \spad{u}. \spad{v} is unchanged
182849`\spad{concat!(u,{}v)} destructively concatenates \spad{v} to the end of \spad{u}. Note: \axiom{concat!(\spad{u},{}\spad{v}) = setlast_!(\spad{u},{}\spad{v})}.
183161`\spad{concat(u)},{} where \spad{u} is a lists of aggregates \axiom{[a,{}\spad{b},{}...,{}\spad{c}]},{} returns a single aggregate consisting of the elements of \axiom{a} followed by those of \spad{b} followed ... by the elements of \spad{c}. Note: \axiom{concat(a,{}\spad{b},{}...,{}\spad{c}) = concat(a,{}concat(\spad{b},{}...,{}\spad{c}))}.
184026`\spad{concat(x,{}u)} returns aggregate \spad{u} with additional element at the front. Note: for lists: \axiom{concat(\spad{x},{}\spad{u}) \spad{==} concat([\spad{x}],{}\spad{u})}.
184077`\spad{concat(x,{}u)} returns aggregate consisting of \spad{x} followed by the elements of \spad{u}. Note: if \axiom{\spad{v} = concat(\spad{x},{}\spad{u})} then \axiom{\spad{x} = first \spad{v}} and \axiom{\spad{u} = rest \spad{v}}.
184136`\spad{concat(u,{}x)} returns aggregate \spad{u} with additional element \spad{x} at the end. Note: for lists,{} \axiom{concat(\spad{u},{}\spad{x}) \spad{==} concat(\spad{u},{}[\spad{x}])}
184187`\spad{concat(u,{}v)} returns an aggregate consisting of the elements of \spad{u} followed by the elements of \spad{v}. Note: if \axiom{\spad{w} = concat(\spad{u},{}\spad{v})} then \axiom{\spad{w}.\spad{i} = \spad{u}.\spad{i} for \spad{i} in indices \spad{u}} and \axiom{\spad{w}.(\spad{j} + maxIndex \spad{u}) = \spad{v}.\spad{j} for \spad{j} in indices \spad{v}}.
184239`\spad{concat(u,{}v)} returns an aggregate \spad{w} consisting of the elements of \spad{u} followed by the elements of \spad{v}. Note: \axiom{\spad{v} = rest(\spad{w},{}\#a)}.
184299`\spad{cond(s,{}e)} creates a representation of the FORTRAN expression IF (\spad{s}) THEN \spad{e}.
184346`\spad{cond(s,{}e,{}f)} creates a representation of the FORTRAN expression IF (\spad{s}) THEN \spad{e} ELSE \spad{f}.
184572`\spad{conditionP(mat)},{} given a matrix representing a homogeneous system of equations,{} returns a vector whose characteristic'th powers is a non-trivial solution,{} or "failed" if no such vector exists.
184658`\spad{conditionP(m)} returns a vector of elements,{} not all zero,{} whose \spad{p}\spad{-}th powers (\spad{p} is the characteristic of the domain) are a solution of the homogenous linear system represented by \spad{m},{} or "failed" is there is no such vector.
184785`\axiom{condition(\spad{n})} returns the condition of the node \spad{n}.
184836`\spad{conditionsForIdempotents()} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the fixed \spad{R}-module basis
184963`\spad{conditionsForIdempotents()} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the fixed \spad{R}-module basis.
185060`\spad{conditionsForIdempotents()} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the fixed \spad{R}-module basis.
185255`\spad{conditionsForIdempotents([v1,{}...,{}vn])} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the \spad{R}-module basis \spad{v1},{}...,{}\spad{vn}
185392`\spad{conditionsForIdempotents([v1,{}...,{}vn])} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the \spad{R}-module basis \spad{v1},{}...,{}\spad{vn}.
185500`\axiom{conditions(a)} returns the list of the conditions of the leaves of a
185624`implements Clifford conjugate for a multivector by involution and reverse of each term seperately using: grade: 0 1 2 3... multi: 1 \spad{-1} \spad{-1} 1...
185735`\spad{conjug(x)}should be local but conditional
185953`\spad{conjugate(p)} returns the conjugate partition of a partition \spad{p}
186072`\spad{conjugate(x + \%i y)} returns \spad{x} - \%\spad{i} \spad{y}.
186124`\spad{conjugate(o)} negates the imaginary parts \spad{i},{}\spad{j},{}\spad{k},{}\spad{E},{}\spad{I},{}\spad{J},{}\spad{K} of octonian \spad{o}.
186177`\spad{conjugate(q)} negates the imaginary parts of quaternion \spad{q}.
186270`\spad{conjugate(p,{}q)} computes \spad{inv(q) * p * q}; this is 'right action by conjugation'.
186479`\spad{cons(element,{}u)} appends \spad{element} onto the front of list \spad{u} and returns the new list. This new list and the old one will share some structure.
186517`\spad{cons(a,{}s)} returns a stream whose \spad{first} is \spad{a} and whose \spad{rest} is \spad{s}. Note: \spad{cons(a,{}s) = concat(a,{}s)}.
186850`\spad{const c} is a function which produces \spad{c} when \indented{1}{applied to its argument.}
186947`\spad{constant?(p)} tests whether the polynomial \spad{p} belongs to the coefficient ring.
187006`\spad{constant?(x)} returns \spad{true} if \spad{x} is constant.
187175`\spad{constantIfCan(k)} \undocumented
187256`\spad{constantKernel(r)} \undocumented
187322`\spad{constantLeft(f)} is the function \spad{g} \indented{1}{such that \spad{g (a,{}b)= f b}.}
187550`\spad{constantRight(f)} is the function \spad{g} \indented{1}{such that \spad{g (a,{}b)= f a}.}
187714`\spad{constant(p)} return the constant term of \spad{p}.
187766`\spad{vu(f)} is the function \spad{g} \indented{1}{such that \spad{g a= f ()}.}
187826`\spad{constant(x)} tells the pattern matcher that \spad{x} should match only the symbol \spad{'x} and no other quantity. Error: if \spad{x} is not a symbol.
187956`\spad{constant(x)} returns the constant term of \spad{x}.
188231`\spad{construct([v1,{}e1],{}...,{}[vn,{}en])} returns the match result containing the matches (\spad{v1},{}\spad{e1}),{}...,{}(\spad{vn},{}en).
188966`\axiom{construct(\spad{x},{}\spad{y},{}...,{}\spad{z})} returns the collection of elements \axiom{\spad{x},{}\spad{y},{}...,{}\spad{z}} ordered as given. Equivalently written as \axiom{[\spad{x},{}\spad{y},{}...,{}\spad{z}]\$\spad{D}},{} where \spad{D} is the domain. \spad{D} may be omitted for those of type List.
189018`\spad{construct(t)} converts an infinite tuple to a stream.
189075`\spad{construct(i,{}e)} \undocumented
189292`\spad{construct(s,{}t)} makes a form \spad{s:t}
189344`\axiom{construct(\spad{v},{}\spad{t})} returns the same as \axiom{construct(\spad{v},{}\spad{t},{}\spad{false})}
189397`\axiom{construct(\spad{v},{}\spad{lt})} returns the same as \axiom{[construct(\spad{v},{}\spad{t}) for \spad{t} in \spad{lt}]}
189533`\axiom{construct(\spad{x},{}\spad{y})} returns the Lie bracket of \axiom{\spad{x}} and \axiom{\spad{y}}.
189923`\axiom{construct(\spad{v},{}\spad{t},{}la)} creates a splitting tree with value (\spadignore{i.e.} root vertex) given by \axiom{[\spad{v},{}\spad{t}]\$\spad{S}} and with \axiom{la} as children list.
189985`\axiom{construct(\spad{v1},{}\spad{t},{}\spad{v2},{}\spad{lt})} creates a splitting tree with value (\spadignore{i.e.} root vertex) given by \axiom{[\spad{v},{}\spad{t}]\$\spad{S}} and with children list given by \axiom{[[[\spad{v},{}\spad{t}]\$\spad{S}]\$\% for \spad{s} in \spad{ls}]}.
190048`\spad{contains?(i,{}f)} returns \spad{true} if \axiom{\spad{f}} is contained within the interval \axiom{\spad{i}},{} \spad{false} otherwise.
190398`\spad{content(p)} gives the \spad{gcd} of the coefficients of polynomial \spad{p}.
190473`\spad{content(l)} returns the \spad{gcd} of all the coefficients of \spad{l}.
190701`\spad{content(p,{}v)} is the \spad{gcd} of the coefficients of the polynomial \spad{p} when \spad{p} is viewed as a univariate polynomial with respect to the variable \spad{v}. Thus,{} for polynomial 7*x^2*y + 14*x*y^2,{} the \spad{gcd} of the coefficients with respect to \spad{x} is 7*y.
190786`\spad{continue(l)} creates a representation of a FORTRAN CONTINUE labelled with \spad{l}
191280`\spad{continuedFraction(b0,{}a,{}b)} constructs a continued fraction in the following way: if \spad{a = [a1,{}a2,{}...]} and \spad{b = [b1,{}b2,{}...]} then the result is the continued fraction \spad{b0 + a1/(b1 + a2/(b2 + ...))}.
192039`\spad{contract(t,{}i,{}j)} is the contraction of tensor \spad{t} which sums along the \spad{i}\spad{-}th and \spad{j}\spad{-}th indices. For example,{} if \spad{r = contract(t,{}1,{}3)} for a rank 4 tensor \spad{t},{} then \spad{r} is the rank 2 \spad{(= 4 - 2)} tensor given by \indented{4}{\spad{r(i,{}j) = sum(h=1..dim,{}t(h,{}i,{}h,{}j))}.}
192116`\spad{contract(t,{}i,{}s,{}j)} is the inner product of tenors \spad{s} and \spad{t} which sums along the \spad{k1}\spad{-}th index of \spad{t} and the \spad{k2}\spad{-}th index of \spad{s}. For example,{} if \spad{r = contract(s,{}2,{}t,{}1)} for rank 3 tensors rank 3 tensors \spad{s} and \spad{t},{} then \spad{r} is the rank 4 \spad{(= 3 + 3 - 2)} tensor given by \indented{4}{\spad{r(i,{}j,{}k,{}l) = sum(h=1..dim,{}s(i,{}h
192116`--,{}j)*t(h,{}k,{}l))}.}
192702`\spad{convert(cs)} converts the complex expression \spad{cs} to a pattern
192884`\spad{convert([a1,{}...,{}an])} returns the pattern \spad{[a1,{}...,{}an]}.
195231`\spad{convert(s)} makes \spad{s} into an input form.
195732`\spad{convert(x)} returns the Lisp atom \spad{x}.
195807`\spad{convert(f1\^e1 ... fm\^em)} returns \spad{(f1)\^e1 ... (fm)\^em} as an element of \%,{} using formal kernels created using a \spadfunFrom{paren}{ExpressionSpace}.
195886`\spad{convert(x)} returns the Lisp atom \spad{x}.
195960`\spad{convert(x)} returns the Lisp atom \spad{x}.
196034`\spad{convert(l)} takes a list of elements,{} \spad{l},{} from the domain Ring and returns the form of point category.
196087`\spad{convert([a1,{}...,{}an])} returns the \spad{S}-expression \spad{(a1,{}...,{}an)}.
196498`\axiom{convert(\spad{p})} returns the same as \axiom{retract(\spad{p})}.
196658`\axiom{convert(\spad{p})} returns the same as \axiom{retract(\spad{p})}.
196808`\axiom{convert(\spad{p})} returns the same as \axiom{retract(\spad{p})}.
197096`\axiom{convert(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if all its variables belong to \axiom{\spad{V}},{} otherwise an error is produced.
197409`\spad{convert(i)} creates the segment \spad{i..i}.
197885`\spad{convert(x)} returns the Lisp atom \spad{x}.
197959`\spad{convert(x)} returns the Lisp atom \spad{x}.
198033`\spad{convert(n)} creates a roman numeral for symbol \spad{n}.
198081`\spad{convert(up)} converts the univariate polynomial \spad{up} to an algebra element,{} reducing by the \spad{definingPolynomial()} if necessary.
198135`\spad{convert([a1,{}..,{}an])} returns \spad{a1*v1 + ... + an*vn},{} where \spad{v1},{} ...,{} \spad{vn} are the elements of the fixed basis.
198193`\spad{convert([a1,{}...,{}an])} returns \spad{a1*v1 + ... + an*vn},{} where \spad{v1},{} ...,{} \spad{vn} are the elements of the fixed \spad{R}-module basis.
198262`\spad{convert(a)} transforms a into an element of \spad{S}.
198309`\spad{convert(a)} returns the coordinates of \spad{a} with respect to the fixed \spad{R}-module basis.
198367`\spad{convert(a)} returns the coordinates of \spad{a} with respect to the fixed \spad{R}-module basis.
199291`\spad{coordinatesIfCan(v,{}B)} either computes the coordinates of the vector \spad{v} in the basis \spad{B} either returns \spad{"failed"} if vector \spad{v} isn\spad{'t} in the subspace described by \spad{B}.
200013`\spad{coordinates([v1,{}...,{}vm])} returns the coordinates of the \spad{vi}\spad{'s} with to the fixed basis. The coordinates of \spad{vi} are contained in the \spad{i}th row of the matrix returned by this function.
200096`\spad{coordinates([v1,{}...,{}vm])} returns the coordinates of the \spad{vi}\spad{'s} with to the fixed basis. The coordinates of \spad{vi} are contained in the \spad{i}th row of the matrix returned by this function.
200166`\spad{coordinates([a1,{}...,{}am])} returns a matrix whose \spad{i}-th row is formed by the coordinates of \spad{\spad{ai}} with respect to the fixed \spad{R}-module basis.
200247`\spad{coordinates(a)} returns the coordinates of \spad{a} with respect to the fixed \spad{F}-vector space basis.
200322`\spad{coordinates(a)} returns the coordinates of \spad{a} with respect to the fixed \spad{R}-module basis.
200384`\spad{coordinates(a)} returns the coordinates of \spad{a} with respect to the fixed \spad{R}-module basis.
200831`\spad{coordinates(v,{}B)} computes the coordinates of the vector \spad{v} in the basis \spad{B} either returns \spad{"failed"},{} and produces an error if vector \spad{v} isn\spad{'t} in the subspace described
201407`\spad{coordinates([v1,{}...,{}vm],{} basis)} returns the coordinates of the \spad{vi}\spad{'s} with to the basis \spad{basis}. The coordinates of \spad{vi} are contained in the \spad{i}th row of the matrix returned by this function.
201492`\spad{coordinates([a1,{}...,{}am],{}[v1,{}...,{}vn])} returns a matrix whose \spad{i}-th row is formed by the coordinates of \spad{\spad{ai}} with respect to the \spad{R}-module basis \spad{v1},{}...,{}\spad{vn}.
201588`\spad{coordinates(a,{}basis)} returns the coordinates of \spad{a} with respect to the \spad{basis} \spad{basis}.
201665`\spad{coordinates(a,{}[v1,{}...,{}vn])} returns the coordinates of \spad{a} with respect to the \spad{R}-module basis \spad{v1},{}...,{}\spad{vn}.
201809`\spad{coproduct(x)} computes the coproduct of an element \spad{x}
202071`\spad{copyInto!(u,{}v,{}i)} returns aggregate \spad{u} containing a copy of \spad{v} inserted at element \spad{i}.
202643`\spad{copy(p)} returns a recursive copy of \spad{p}.
202682`\axiom{copy(\spad{n})} returns a copy of \spad{n}.
202729`\spad{copy(x)} \undocumented
202783`\spad{copy(u)} returns a top-level (non-recursive) copy of \spad{u}. Note: for collections,{} \axiom{copy(\spad{u}) \spad{==} [\spad{x} for \spad{x} in \spad{u}]}.
202821`\spad{copy(n)} gives a copy of \spad{n}.
202869`\spad{copy(s)} returns a new \spadtype{ThreeSpace} that is an exact copy of \spad{s}.
202919`\spad{copy(op)} returns a copy of \spad{op}.
203218`\spad{cos2sec(f)} converts every \spad{cos(u)} appearing in \spad{f} into \spad{1/sec(u)}.
203280`\spad{cosIfCan(z)} returns cos(\spad{z}) if possible,{} and "failed" otherwise.
203429`\spad{cos(x)} makes a cos kernel for use in Fourier series
203475`\spad{cos(x)} applies the cosine operator to \spad{x}
203796`\spad{cos(z)} returns the cosine of Laurent series \spad{z}.
203882`\spad{cos(z)} returns the cosine of a Puiseux series \spad{z}.
203977`\spad{cos(x)} represents the Fortran intrinsic function COS
204057`\spad{cos(x)} returns the cosine of \spad{x}.
204114`\spad{cosh2sech(f)} converts every \spad{cosh(u)} appearing in \spad{f} into \spad{1/sech(u)}.
204178`\spad{coshIfCan(z)} returns cosh(\spad{z}) if possible,{} and "failed" otherwise.
204259`\spad{cosh(x)} applies the hyperbolic cosine operator to \spad{x}
204637`\spad{cosh(z)} returns the hyperbolic cosine of Laurent series \spad{z}.
204724`\spad{cosh(z)} returns the hyperbolic cosine of a Puiseux series \spad{z}.
204820`\spad{cosh(x)} represents the Fortran intrinsic function COSH
204901`\spad{cosh(x)} returns the hyperbolic cosine of \spad{x}.
204956`\spad{cot2tan(f)} converts every \spad{cot(u)} appearing in \spad{f} into \spad{1/tan(u)}.
205018`\spad{cot2trig(f)} converts every \spad{cot(u)} appearing in \spad{f} into \spad{cos(u)/sin(u)}.
205081`\spad{cotIfCan(z)} returns cot(\spad{z}) if possible,{} and "failed" otherwise.
205161`\spad{cot(x)} applies the cotangent operator to \spad{x}
205494`\spad{cot(z)} returns the cotangent of Laurent series \spad{z}.
205580`\spad{cot(z)} returns the cotangent of a Puiseux series \spad{z}.
205675`\spad{cot(x)} returns the cotangent of \spad{x}.
205732`\spad{coth2tanh(f)} converts every \spad{coth(u)} appearing in \spad{f} into \spad{1/tanh(u)}.
205796`\spad{coth2trigh(f)} converts every \spad{coth(u)} appearing in \spad{f} into \spad{cosh(u)/sinh(u)}.
205861`\spad{cothIfCan(z)} returns coth(\spad{z}) if possible,{} and "failed" otherwise.
205942`\spad{coth(x)} applies the hyperbolic cotangent operator to \spad{x}
206277`\spad{coth(z)} returns the hyperbolic cotangent of Laurent series \spad{z}.
206364`\spad{coth(z)} returns the hyperbolic cotangent of a Puiseux series \spad{z}.
206460`\spad{coth(x)} returns the hyperbolic cotangent of \spad{x}.
206515`\spad{counit(x)} evaluates the counit at an element \spad{x}
207749`\spad{count(p,{}u)} returns the number of elements \spad{x} in \spad{u} such that \axiom{\spad{p}(\spad{x})} is \spad{true}. For collections,{} \axiom{count(\spad{p},{}\spad{u}) = reduce(+,{}[1 for \spad{x} in \spad{u} | \spad{p}(\spad{x})],{}0)}.
207854`\spad{count(x,{}u)} returns the number of occurrences of \spad{x} in \spad{u}. For collections,{} \axiom{count(\spad{x},{}\spad{u}) = reduce(+,{}[x=y for \spad{y} in \spad{u}],{}0)}.
207972`\spad{countable?(\spad{a})} determines whether \spad{a} is a countable cardinal,{} \spadignore{i.e.} an integer or \spad{Aleph 0}.
208026`\spad{create3Space()} creates a \spadtype{ThreeSpace} object capable of holding point,{} curve,{} mesh components and any combination.
209133`\spad{createNormalElement()} computes a normal element over the ground field \spad{F},{} that is,{} \spad{a^(q^i),{} 0 <= i < extensionDegree()} is an \spad{F}-basis,{} where \spad{q = size()\$F}. Reference: Such an element exists Lidl/Niederreiter: Theorem 2.35.
210699`\spad{createPrimitiveElement()} computes a generator of the (cyclic) multiplicative group of the field.
211442`Constructs a group node,{} this node does not do anything itself but contains other nodes
211490`Constructs an indexed face set node,{} this defines a surface represented by a set of polygons in \spad{n}-dimensional space.
211628`Constructs a line node,{} this contains a line (represented by a list of points) in \spad{n}-dimensional space. The space is implicit in the \spad{PT} and \spad{TR} parameters supplied when this Scene was created.
212045`Constructs the root node,{} all other nodes are contained in a tree structure under this node
212539`\spad{createThreeSpace()} creates a \spadtype{ThreeSpace(DoubleFloat)} object capable of holding point,{} curve,{} mesh components and any combination.
212776`\spad{credPol }\undocumented
212869`\spad{critB }\undocumented
213406`\spad{critM }\undocumented
214203`\spad{cross(p,{}q)} computes the cross product of the two points \spad{p} and \spad{q}. Error if the \spad{p} and \spad{q} are not 3 dimensional
214252`\spad{cross(u,{}v)} constructs the cross product of \spad{u} and \spad{v}. Error: if \spad{u} and \spad{v} are not of length 3.
214391`\spad{csc2sin(f)} converts every \spad{csc(u)} appearing in \spad{f} into \spad{1/sin(u)}.
214453`\spad{cscIfCan(z)} returns \spad{csc}(\spad{z}) if possible,{} and "failed" otherwise.
214533`\spad{csc(x)} applies the cosecant operator to \spad{x}
214866`\spad{csc(z)} returns the cosecant of Laurent series \spad{z}.
214952`\spad{csc(z)} returns the cosecant of a Puiseux series \spad{z}.
215047`\spad{csc(x)} returns the cosecant of \spad{x}.
215104`\spad{csch2sinh(f)} converts every \spad{csch(u)} appearing in \spad{f} into \spad{1/sinh(u)}.
215168`\spad{cschIfCan(z)} returns csch(\spad{z}) if possible,{} and "failed" otherwise.
215249`\spad{csch(x)} applies the hyperbolic cosecant operator to \spad{x}
215584`\spad{csch(z)} returns the hyperbolic cosecant of Laurent series \spad{z}.
215671`\spad{csch(z)} returns the hyperbolic cosecant of a Puiseux series \spad{z}.
215767`\spad{csch(x)} returns the hyperbolic cosecant of \spad{x}.
216209`\spad{currentSubProgram()} returns the name of the current subprogram being processed
216267`\spad{curryLeft(f,{}a)} is the function \spad{g} \indented{1}{such that \spad{g b = f(a,{}b)}.}
216335`\spad{curryRight(f,{}b)} is the function \spad{g} such that \indented{1}{\spad{g a = f(a,{}b)}.}
216404`\spad{cu(f,{}a)} is the function \spad{g} \indented{1}{such that \spad{g ()= f a}.}
216599`\spad{curveColor(v)} specifies a color,{} \spad{v},{} for 2D graph curves. This option is expressed in the form \spad{curveColor == v}.
216888`\spad{curve(c1,{}c2)} creates a plane curve from 2 component functions \spad{c1} and \spad{c2}.
217111`\spad{curve(c1,{}c2,{}c3)} creates a space curve from 3 component functions \spad{c1},{} \spad{c2},{} and \spad{c3}.
217231`\spad{cycleElt(s)} returns a pointer to a node in the cycle if the stream \spad{s} is cyclic and returns "failed" if \spad{s} is not cyclic
217413`\spad{cycleEntry(u)} returns the head of a top-level cycle contained in aggregate \spad{u},{} or \axiom{empty()} if none exists.
217620`\spad{cycleLength(u)} returns the length of a top-level cycle contained in aggregate \spad{u},{} or 0 is \spad{u} has no such cycle.
217698`\spad{cyclePartition(p)} returns the cycle structure of a permutation \spad{p} including cycles of length 1 only if \spad{S} is finite.
217758`\spad{cycleRagits(rx)} returns the cyclic part of the ragits of the fractional part of a radix expansion. For example,{} if \spad{x = 3/28 = 0.10 714285 714285 ...},{} then \spad{cycleRagits(x) = [7,{}1,{}4,{}2,{}8,{}5]}.
217881`\spad{cycleSplit!(u)} splits the aggregate by dropping off the cycle. The value returned is the cycle entry,{} or nil if none exists. For example,{} if \axiom{\spad{w} = concat(\spad{u},{}\spad{v})} is the cyclic list where \spad{v} is the head of the cycle,{} \axiom{cycleSplit!(\spad{w})} will drop \spad{v} off \spad{w} thus destructively changing \spad{w} to \spad{u},{} and returning \spad{v}.
218075`\spad{cycleTail(u)} returns the last node in the cycle,{} or empty if none exists.
218135`\spad{cycle(ls)} coerces a cycle {\em ls},{} \spadignore{i.e.} a list with not repetitions to a permutation,{} which maps {\em ls.i} to {\em ls.i+1},{} indices modulo the length of the list. Error: if repetitions occur.
218192`\spad{cycles(lls)} coerces a list list of cycles {\em lls} to a permutation,{} each cycle being a list with not repetitions,{} is coerced to the permutation,{} which maps {\em ls.i} to {\em ls.i+1},{} indices modulo the length of the list,{} then these permutations are mutiplied. Error: if repetitions occur in one cycle.
218433`\spad{cyclic?(u)} tests if \spad{u} has a cycle.
218491`\spad{cyclic?(t)} tests if \spad{t} is a cyclic tree.
218535`\spad{cyclicCopy(l)} makes a copy of a (possibly) cyclic tree \spad{l}.
218577`\spad{cyclicEntries(t)} returns a list of top-level cycles in tree \spad{t}.
218628`\spad{cyclicEqual?(t1,{} t2)} tests of two cyclic trees have the same structure.
218860`\spad{cyclicParents(t)} returns a list of cycles that are parents of \spad{t}.
219831`\spad{datalist(l)} creates a datalist from \spad{l}
220130`\spad{debug(d)} specifies whether we want additional output on the progress. This option is expressed in the form \spad{debug == d}.
220176`decFatal is an internal function that declares that the preceding test did not fail with a fatal error.
220222`like decFatal,{} but for expected failures.
220270`\spad{decXfPass()} is an internal function that declares that the preceding test did not pass unexpectedly. This function should only be used for testing library errors
220317`\spad{dec(x)} returns \spad{x - 1}.
220364`\spad{decimal(r)} converts a rational number to a decimal expansion.
220427`\spad{declare!(u,{}t)} declares the parameter \spad{u} to have type \spad{t} in the current level of the symbol table.
220572`\spad{declare!(u,{}t,{}asp)} declares the parameter \spad{u} to have type \spad{t} in \spad{asp}.
220651`\spad{declare!(u,{}t,{}tab)} creates a new entry in \spad{tab},{} declaring \spad{u} to be of type \spad{t}
220723`\spad{declare!(u,{}t,{}asp,{}tab)} declares the parameters \spad{u} of subprogram \spad{asp} to have type \spad{t} in symbol table \spad{tab}.
220811`\spad{declare!(u,{}t,{}asp,{}tab)} declares the parameter \spad{u} of subprogram \spad{asp} to have type \spad{t} in symbol table \spad{tab}.
220893`\spad{declare(t)} returns a name \spad{f} such that \spad{f} has been declared to the interpreter to be of type \spad{t},{} but has not been assigned a value yet. Note: \spad{t} should be created as \spad{devaluate(T)\$Lisp} where \spad{T} is the actual type of \spad{f} (this hack is required for the case where \spad{T} is a mapping type).
221206`\spad{decompose(up)} \undocumented
222162`\spad{decreasePrecision(n)} decreases the current \spadfunFrom{precision}{FloatingPointSystem} precision by \spad{n} decimal digits.
222267`\spad{deepCopy(x)} \undocumented
222447`\axiom{deepestInitial(\spad{p})} returns an error if \axiom{\spad{p}} belongs to \axiom{\spad{R}},{} otherwise returns the last term of \axiom{iteratedInitials(\spad{p})}.
222586`\axiom{deepestTail(\spad{p})} returns \axiom{0} if \axiom{\spad{p}} belongs to \axiom{\spad{R}},{} otherwise returns tail(\spad{p}),{} if \axiom{tail(\spad{p})} belongs to \axiom{\spad{R}} or \axiom{mvar(tail(\spad{p})) < mvar(\spad{p})},{} otherwise returns \axiom{deepestTail(tail(\spad{p}))}.
222754`\spad{definingEquations(s)} returns a list of defining polynomials for equations,{} that is,{} for the Zariski closed part of \spad{s}.
222841`\spad{definingInequation(s)} returns a single defining polynomial for the inequation,{} that is,{} the Zariski open part of \spad{s}.
223007`\spad{definingPolynomial()} returns the polynomial used to define the field extension.
223107`\spad{definingPolynomial()} returns the minimal polynomial which \spad{generator()} satisfies.
223170`\spad{definingPolynomial(f)} returns the defining polynomial of \spad{f} as an element of \spad{F}. Error: if \spad{f} is not a kernel.
223262`\axiom{definingPolynomial(aRoot)} gives a polynomial such that \axiom{definingPolynomial(aRoot).aRoot = 0}
223360`\spad{definingPolynomial(x)} returns an expression \spad{p} such that \spad{p(x) = 0}.
224655`\spad{degree(p)} returns the maximum of the exponents of the terms of \spad{p}.
224707`\spad{degree(g)} names the degree of \spad{g}. The set of all elements of a given degree form an \spad{R}-module.
224754`\spad{degree(f)} returns the exponent of the lowest order term of \spad{f}.
224999`\axiom{degree(\spad{x})} returns the greatest length of a word in the support of \axiom{\spad{x}}.
225138`\axiom{degree(\spad{ts})} returns the product of main degrees of the members of \axiom{\spad{ts}}.
225215`\spad{degree(l)} is \spad{n} if \indented{2}{\spad{l = sum(monomial(a(i),{}i),{} i = 0..n)}.}
225297`\spad{degree(p)} returns the degree of \spad{p}. \indented{1}{Note that the degree of a word is its length.}
225365`\spad{degree(p)} retuns the number of points moved by the permutation \spad{p}.
225487`\spad{degree(a)} returns the degree of minimal polynomial of an element \spad{a} if \spad{a} is algebraic with respect to the ground field \spad{F},{} and \spad{infinity} otherwise.
225568`\spad{degree(a)} returns the degree of the minimal polynomial of an element \spad{a} over the ground field \spad{F}.
226229`\spad{degree(p,{}lv)} gives the list of degrees of polynomial \spad{p} with respect to each of the variables in the list \spad{lv}.
226325`\spad{degree(p,{} s)} returns the maximum degree of the differential polynomial \spad{p} viewed as a differential polynomial in the differential indeterminate \spad{s} alone.
226413`\spad{degree(p,{}v)} gives the degree of polynomial \spad{p} with respect to the variable \spad{v}.
226497`\spad{delay(f)} creates a stream with a lazy evaluation defined by function \spad{f}. Caution: This function can only be called in compiled code.
226662`\spad{delete!(u,{}i)} destructively deletes the \axiom{\spad{i}}th element of \spad{u}.
226730`\spad{delete!(u,{}i..j)} destructively deletes elements \spad{u}.\spad{i} through \spad{u}.\spad{j}.
226816`\spad{deleteProperty!(op,{} s)} unattaches property \spad{s} from \spad{op}. Argument \spad{op} is modified "in place",{} \spadignore{i.e.} no copy is made.
227302`\spad{delete(u,{}i)} returns a copy of \spad{u} with the \axiom{\spad{i}}th element deleted. Note: for lists,{} \axiom{delete(a,{}\spad{i}) \spad{==} concat(a(0..\spad{i} - 1),{}a(\spad{i} + 1,{}..))}.
227359`\spad{delete(u,{}i..j)} returns a copy of \spad{u} with the \axiom{\spad{i}}th through \axiom{\spad{j}}th element deleted. Note: \axiom{delete(a,{}\spad{i}..\spad{j}) = concat(a(0..\spad{i}-1),{}a(\spad{j+1}..))}.
227688`\spad{denomRicDE(op)} returns a polynomial \spad{d} such that any rational solution of the associated Riccati equation of \spad{op y = 0} is of the form \spad{p/d + q'/q + r} for some polynomials \spad{p} and \spad{q} and a reduced \spad{r}. Also,{} \spad{deg(p) < deg(d)} and {\spad{gcd}(\spad{d},{}\spad{q}) = 1}.
227750`\spad{denom(1/d * (f1,{}...,{}fn))} returns \spad{d}.
227804`\spad{denom x} returns the denominator of \spad{x}.
227850`\spad{denom x} returns the denominator of \spad{x}.
227983`\spad{denom(x)} returns the denominator of the fraction \spad{x}.
228328`\spad{denominator(f)} returns the denominator of \spad{f} converted to \%.
228401`\spad{denominator(x)} is the denominator of the fraction \spad{x} converted to \%.
228461`\spad{denominators(x)} returns the stream of denominators of the approximants of the continued fraction \spadvar{\spad{x}}. If the continued fraction is finite,{} then the stream will be finite.
228525`\spad{depth(p)} returns the nesting level of \spad{p}.
228581`\spad{depth(s)} returns the number of elements of stack \spad{s}. Note: \axiom{depth(\spad{s}) = \spad{#s}}.
228644`\spad{dequeue! s} destructively extracts the first (top) element from queue \spad{q}. The element previously second in the queue becomes the first element. Error: if \spad{q} is empty.
228693`\spad{dequeue()}\$\spad{D} creates an empty dequeue of type \spad{D}.
228742`\spad{dequeue([x,{}y,{}...,{}z])} creates a dequeue with first (top or front) element \spad{x},{} second element \spad{y},{}...,{}and last (bottom or back) element \spad{z}.
228798`\spad{dequeue([x,{}y,{}...,{}z])} creates a dequeue with first (top or front) element \spad{x},{} second element \spad{y},{}...,{}and last (bottom or back) element \spad{z}.
228845`\spad{deref(n)} is equivalent to \spad{elt(n)}.
229043`\spad{derivationCoordinates(b,{} ')} returns \spad{M} such that \spad{b' = M b}.
229426`\spad{destruct(r)} returns the list of matches (var,{} expr) in \spad{r}. Error: if \spad{r} is a failed match.
229512`\spad{destruct((a1,{}...,{}an))} returns the list [\spad{a1},{}...,{}an].
229592`\spad{determinant(m)} returns the determinant of the matrix \spad{m}. an error message is returned if the matrix is not square.
229672`\spad{determinant(m)} returns the determinant of the matrix \spad{m}. an error message is returned if the matrix is not square.
229747`\spad{determinant(m)} returns the determinant of the matrix \spad{m}. Error: if the matrix is not square.
229829`\spad{determinant(m)} returns the determinant of the matrix \spad{m}.
229922`\spad{diag(f)} is the function \spad{g} \indented{1}{such that \spad{g a = f(a,{}a)}.}
230118`\spad{diagonal?(m)} returns \spad{true} if the matrix \spad{m} is square and diagonal (\spadignore{i.e.} all entries of \spad{m} not on the diagonal are zero) and \spad{false} otherwise.
230182`\spad{diagonal?(m)} returns \spad{true} if the matrix \spad{m} is square and diagonal (\spadignore{i.e.} all entries of \spad{m} not on the diagonal are zero) and \spad{false} otherwise.
230393`\spad{diagonalMatrix(l)} returns a diagonal matrix with the elements of \spad{l} on the diagonal.
230462`\spad{diagonalMatrix(l)} returns a diagonal matrix with the elements of \spad{l} on the diagonal.
230542`\spad{diagonalMatrix([m1,{}...,{}mk])} creates a block diagonal matrix \spad{M} with block matrices {\em m1},{}...,{}{\em mk} down the diagonal,{} with 0 block matrices elsewhere. More precisly: if \spad{\spad{ri} := nrows \spad{mi}},{} \spad{\spad{ci} := ncols \spad{mi}},{} then \spad{m} is an (\spad{r1+}..\spad{+rk}) by (\spad{c1+}..\spad{+ck}) - matrix with entries \spad{m.i.j = ml.(i-r1-..-r(l-1)).(j-n1-..-n(l-1))},{} if \spad{(r
230542`--1+..+r(l-1)) < i <= r1+..+rl} and \spad{(c1+..+c(l-1)) < i <= c1+..+cl},{} \spad{m.i.j} = 0 otherwise.
230612`\spad{diagonalMatrix(v)} returns a diagonal matrix where the elements of \spad{v} appear on the diagonal.
230806`\spad{diagonalProduct(m)} returns the product of the elements on the diagonal of the matrix \spad{m}.
230948`\spad{diagonal(m)} returns a row consisting of the elements on the diagonal of the matrix \spad{m}.
231176`\spad{dictionary()}\$\spad{D} creates an empty dictionary of type \spad{D}.
231342`\spad{dictionary([x,{}y,{}...,{}z])} creates a dictionary consisting of entries \axiom{\spad{x},{}\spad{y},{}...,{}\spad{z}}.
244398`\spad{difference(u,{}x)} returns the set aggregate \spad{u} with element \spad{x} removed. If \spad{u} does not contain \spad{x},{} a copy of \spad{u} is returned. Note: \axiom{difference(\spad{s},{} \spad{x}) = difference(\spad{s},{} {\spad{x}})}.
244450`\spad{difference(u,{}v)} returns the set aggregate \spad{w} consisting of elements in set aggregate \spad{u} but not in set aggregate \spad{v}. If \spad{u} and \spad{v} have no elements in common,{} \axiom{difference(\spad{u},{}\spad{v})} returns a copy of \spad{u}. Note: equivalent to the notation (not currently supported) \axiom{{\spad{x} for \spad{x} in \spad{u} | not member?(\spad{x},{}\spad{v})}}.
244590`\spad{differentialVariables(p)} returns a list of differential indeterminates occurring in a differential polynomial \spad{p}.
245519`\spad{differentiate(x)} returns the derivative of \spad{x}. This function is a simple differential operator where no variable needs to be specified.
245573`\spad{differentiate(v)} returns the derivative of \spad{v}.
245642`\spad{differentiate(f)} returns the derivative of \spad{f}.
249701`\spad{differentiate(ir,{}D)} differentiates \spad{ir} with respect to the derivation \spad{D}.
249765`\spad{differentiate(f,{}n)} creates a form for the \spad{n}th derivative of \spad{f},{} \spadignore{e.g.} \spad{f'},{} \spad{f''},{} \spad{f'''},{} \spad{"f} super \spad{iv}".
249832`\spad{differentiate(ir,{}x)} differentiates \spad{ir} with respect to \spad{x}
250013`\spad{differentiate(f(x),{}x)} returns the derivative of \spad{f(x)} with respect to \spad{x}.
250108`\spad{differentiate(f(x),{}x)} returns the derivative of \spad{f(x)} with respect to \spad{x}.
250203`\spad{differentiate(f(x),{}x)} computes the derivative of \spad{f(x)} with respect to \spad{x}.
250297`\spad{differentiate(f(x),{}x)} returns the derivative of \spad{f(x)} with respect to \spad{x}.
250386`\spad{differentiate(f(x),{}x)} returns the derivative of \spad{f(x)} with respect to \spad{x}.
250475`\spad{differentiate(f(x),{}x)} computes the derivative of \spad{f(x)} with respect to \spad{x}.
250563`\spad{differentiate(x,{} deriv)} differentiates \spad{x} extending the derivation deriv on \spad{R}.
250632`\spad{differentiate(x,{} d)} extends the derivation \spad{d} from UP to \$ and applies it to \spad{x}.
250711`\spad{differentiate(x,{}[s1,{}...sn])} computes successive partial derivatives,{} \spadignore{i.e.} \spad{differentiate(...differentiate(x,{} s1)...,{} sn)}.
250783`\spad{differentiate(x,{} n)} returns the \spad{n}-th derivative of \spad{x}.
250856`\spad{differentiate(v,{} n)} returns the \spad{n}-th derivative of \spad{v}.
251030`\spad{differentiate(x,{}v)} computes the partial derivative of \spad{x} with respect to \spad{v}.
251096`\spad{differentiate(f(x),{}x)} returns the derivative of \spad{f(x)} with respect to \spad{x}.
254896`\spad{differentiate(x,{} deriv,{} n)} differentiate \spad{x} \spad{n} times using a derivation which extends \spad{deriv} on \spad{R}.
254984`\spad{differentiate(p,{} d,{} x')} extends the \spad{R}-derivation \spad{d} to an extension \spad{D} in \spad{R[x]} where \spad{Dx} is given by \spad{x'},{} and returns \spad{Dp}.
255063`\spad{differentiate(x,{} [s1,{}...,{}sn],{} [n1,{}...,{}nn])} computes multiple partial derivatives,{} \spadignore{i.e.}
255160`\spad{differentiate(x,{} s,{} n)} computes multiple partial derivatives,{} \spadignore{i.e.} \spad{n}-th derivative of \spad{x} with respect to \spad{s}.
255245`\spad{digamma(x)} returns the digamma function applied to \spad{x}
255467`\spad{digamma(x)} is the logarithmic derivative of \spad{Gamma(x)} (often written \spad{psi(x)} in the literature).
255522`\spad{digit?(c)} tests if \spad{c} is a digit character,{} \spadignore{i.e.} one of 0..9.
255567`\spad{digit()} returns the class of all characters for which \spadfunFrom{digit?}{Character} is \spad{true}.
255609`\spad{digits()} returns ceiling\spad{'s} precision in decimal digits.
255674`\spad{digits()} returns ceiling\spad{'s} precision in decimal digits.
255735`\spad{digits(d)} set the \spadfunFrom{precision}{FloatingPointSystem} to \spad{d} digits.
255815`\spad{digits(d)} set the \spadfunFrom{precision}{FloatingPointSystem} to \spad{d} digits.
256247`\spad{dilog(f)} denotes the dilogarithm
256299`\spad{dilog(x)} returns the dilogarithm of \spad{x},{} \spadignore{i.e.} the integral of \spad{log(x) / (1 - x) dx}.
256635`\spad{dimension()} returns the dimensionality of the vector space.
256763`\spad{dimension(p)} returns the number of dimensions that make up the point category \spad{p}.
256895`\spad{dimension(s)} returns the dimension of the point category \spad{s}.
257042`\spad{dimensionsOf(t)} returns the dimensions of \spad{t}
257683`\spad{directProduct(v)} converts the vector \spad{v} to become a direct product. Error: if the length of \spad{v} is different from dim.
257756`\spad{directSum(a,{}b)} computes an operator \spad{c} of minimal order such that the nullspace of \spad{c} is generated by all the sums of a solution of \spad{a} by a solution of \spad{b}.
257850`\spad{directSum(a,{}b,{}D)} computes an operator \spad{c} of minimal order such that the nullspace of \spad{c} is generated by all the sums of a solution of \spad{a} by a solution of \spad{b}. \spad{D} is the derivation to use.
257934`\spad{direction(s)} \undocumented
257992`\spad{directory(f)} returns the directory part of the file name.
258115`\spad{discreteLog(a)} computes the discrete logarithm of \spad{a} with respect to \spad{primitiveElement()} of the field.
258273`\spad{discreteLog(b,{}a)} computes \spad{s} with \spad{b^s = a} if such an \spad{s} exists.
258370`\axiom{discriminantEuclidean(\spad{P})} carries out the equality \axiom{\spad{coef1} * \spad{P} + \spad{coef2} * \spad{D}(\spad{P}) = discriminant(\spad{P})}.
258491`\spad{discriminant()} returns the discriminant of the integral closure of \spad{Z} in the quotient field of the framed algebra \spad{F}.
258611`\spad{discriminant()} = determinant(traceMatrix()).
258667`\spad{discriminant()} = determinant(traceMatrix()).
258903`\axiom{discriminant(\spad{P},{} \spad{Q})} returns the discriminant of \axiom{\spad{P}} and \axiom{\spad{Q}}.
258972`\spad{discriminant([v1,{}..,{}vn])} returns \spad{determinant(traceMatrix([v1,{}..,{}vn]))}.
259039`\spad{discriminant(p)} returns the discriminant of the polynomial \spad{p}.
259198`\spad{discriminant(p,{}v)} returns the disriminant of the polynomial \spad{p} with respect to the variable \spad{v}.
259366`\spad{displayAsGF(d)} specifies whether the result is a generating function or a recurrence. This option should not be set by the user,{} but rather by the \spad{HP}-specification.
259418`list this instance to the command line
259639`\spad{display(db)} prints a summary line for each entry in \axiom{\spad{db}}.
259684`\spad{display(ic)} prints a summary of the information contained in \axiom{\spad{ic}}.
259727`\spad{display(t)} outputs the formatted code \spad{t} so that each line has length less than or equal to the value set by the system command \spadsyscom{set output length}.
259780`\spad{display(t)} outputs the TeX formatted code \spad{t} so that each line has length less than or equal to the value set by the system command \spadsyscom{set output length}.
260330`returns the distance between the points \spad{p1} and \spad{p2} there are 2 metrics associated with any SPointCategory the underlying space and the space that it is embedded in this is the distance in the underlying space.
260455`returns the distance between the points \spad{p1} and \spad{p2} there are 2 metrics associated with any SPointCategory the underlying space and the space that it is embedded in this is the distance in the underlying space.
260514`\spad{distance(u,{}v)} returns the path length (an integer) from node \spad{u} to \spad{v}.
260742`\spad{distribute(f)} expands all the kernels in \spad{f} that are formally enclosed by a \spadfunFrom{box}{ExpressionSpace} or \spadfunFrom{paren}{ExpressionSpace} expression.
260841`\spad{distribute(f,{} g)} expands all the kernels in \spad{f} that contain \spad{g} in their arguments and that are formally enclosed by a \spadfunFrom{box}{ExpressionSpace} or a \spadfunFrom{paren}{ExpressionSpace} expression.
260894`\spad{f div g} creates the equivalent infix form.
260935`\spad{divergence(vf,{}xlist)} computes the divergence of the vector field \spad{vf},{} \spad{vf} a vector function of the variables listed in \spad{xlist}.
261114`\spad{divideExponents(p,{}n)} returns a new polynomial resulting from dividing all exponents of the polynomial \spad{p} by the non negative integer \spad{n},{} or "failed" if some exponent is not exactly divisible by \spad{n}.
261220`\spad{divideIfCan!(matrix,{}matrixOut,{}prime,{}n)} attempts to divide the entries of \spad{matrix} by \spad{prime} and store the result in \spad{matrixOut}. If it is successful,{} 1 is returned and if not,{} \spad{prime} is returned. Here both \spad{matrix} and \spad{matrixOut} are \spad{n}-by-\spad{n} upper triangular matrices.
261659`\spad{divide(x,{} y)} returns the left and right exact quotients of \spad{x} by \spad{y},{} \spadignore{i.e.} \spad{[l,{} r]} such that \spad{x = l * y * r},{} "failed" if \spad{x} is not of the form \spad{l * y * r}.
261739`\spad{divide(x,{} y)} returns the left and right exact quotients of \spad{x} by \spad{y},{} that is \spad{[l,{} r]} such that \spad{x = l * y * r}. "failed" is returned iff \spad{x} is not of the form \spad{l * y * r}.
261826`\axiom{divide(\spad{F},{}\spad{G})} computes quotient and rest of the exact euclidean division of \axiom{\spad{F}} by \axiom{\spad{G}}.
261929`\spad{divide(x,{}y)} divides \spad{x} by \spad{y} producing a record containing a \spad{quotient} and \spad{remainder},{} where the remainder is smaller (see \spadfunFrom{sizeLess?}{EuclideanDomain}) than the divisor \spad{y}.
262008`\spad{divide(a,{}b)} returns a record containing both remainder and quotient.
262404`\spad{divisor(g)} returns the divisor of the function \spad{g}.
262469`\spad{divisor(a,{} b)} makes the divisor \spad{P:} \spad{(x = a,{} y = b)}. Error: if \spad{P} is singular.
262536`\spad{divisor(a,{} b,{} n)} makes the divisor \spad{nP} where \spad{P:} \spad{(x = a,{} y = b)}. \spad{P} is allowed to be singular if \spad{n} is a multiple of the rank.
262611`\spad{divisor(h,{} d,{} d',{} g,{} r)} returns the sum of all the finite points where \spad{h/d} has residue \spad{r}. \spad{h} must be integral. \spad{d} must be squarefree. \spad{d'} is some derivative of \spad{d} (not necessarily dd/dx). \spad{g = gcd(d,{}discriminant)} contains the ramified zeros of \spad{d}
264524`\spad{dom(a)} returns a \spadgloss{LISP} form of the type of the original object that was converted to \spadtype{Any}.
264564`\spad{domainOf(a)} returns a printable form of the type of the original object that was converted to \spadtype{Any}.
264898`\spad{dot(f)} creates the form with a one dot overhead.
265059`\spad{dot(f,{}n)} creates the form \spad{f} with \spad{n} dots overhead.
265116`\spad{dot(x,{}y)} computes the inner product of the vectors \spad{x} and \spad{y}.
265185`\spad{dot(x,{}y)} computes the inner product of the two vectors \spad{x} and \spad{y}. Error: if \spad{x} and \spad{y} are not of the same length.
265243`\spad{double?(t)} tests whether \spad{t} is equivalent to the FORTRAN type DOUBLE PRECISION
265297`\spad{doubleComplex?(t)} tests whether \spad{t} is equivalent to the (non-standard) FORTRAN type DOUBLE COMPLEX.
265428`change the output format for doublefloats using lisp format strings
265489`\spad{doubleRank(x)} determines the number of linearly independent elements in \spad{b1*x},{}...,{}\spad{x*bn},{} where \spad{b=[b1,{}...,{}bn]} is a basis.
265558`\spad{doubleResultant(f,{} ')} returns \spad{p}(\spad{x}) whose roots are rational multiples of the residues of \spad{f} at all its finite poles. Argument ' is the derivation to use.
265641`\spad{double(i,{} r)} multiplies \spad{r} by \spad{i} using repeated doubling.
265705`\spad{doublyTransitive?(p)} is \spad{true} if \spad{p} is irreducible over over the field \spad{K} generated by its coefficients,{} and if \spad{p(X) / (X - a)} is irreducible over \spad{K(a)} where \spad{p(a) = 0}.
270754`\spad{droot(l)} should be a non-exported function.
270819`\spad{duplicates?(d)} tests if dictionary \spad{d} has duplicate entries.
270882`\spad{duplicates(d)} returns a list of values which have duplicates in \spad{d}
271054`unit pseudoscalar
271906`\spad{eisensteinIrreducible?(p)} returns \spad{true} if \spad{p} can be shown to be irreducible by Eisenstein\spad{'s} criterion,{} \spad{false} is inconclusive.
272598`\spad{ellipticE(m)} is the complete elliptic integral of the second kind: \spad{ellipticE(m) = integrate(sqrt(1-m*t^2)/sqrt(1-t^2),{} t = 0..1)}
272660`\spad{ellipticE(m)} is the complete elliptic integral of the second kind: \spad{ellipticE(m) = integrate(sqrt(1-m*t^2)/sqrt(1-t^2),{} t = 0..1)}.
272717`\spad{ellipticE(z,{} m)} is the incomplete elliptic integral of the second kind: \spad{ellipticE(z,{} m) = integrate(sqrt(1-m*t^2)/sqrt(1-t^2),{} t = 0..z)}
272781`\spad{ellipticE(z,{} m)} expands the elliptic integral \spad{E} as a Taylor series.
272907`\spad{ellipticE(z,{} m)} is the incomplete elliptic integral of the second kind: \spad{ellipticE(z,{} m) = integrate(sqrt(1-m*t^2)/sqrt(1-t^2),{} t = 0..z)}.
272967`\spad{ellipticF(z,{} m)} is the incomplete elliptic integral of the first kind : \spad{ellipticF(z,{} m) = integrate(1/sqrt((1-t^2)*(1-m*t^2)),{} t = 0..z)}
273031`\spad{ellipticF(z,{} m)} expands the elliptic integral \spad{F} as a Taylor series.
273157`\spad{ellipticF(z,{} m)} is the incomplete elliptic integral of the first kind : \spad{ellipticF(z,{} m) = integrate(1/sqrt((1-t^2)*(1-m*t^2)),{} t = 0..z)}.
273217`\spad{ellipticK(m)} is the complete elliptic integral of the first kind: \spad{ellipticK(m) = integrate(1/sqrt((1-t^2)*(1-m*t^2)),{} t = 0..1)}
273279`\spad{ellipticK(m)} is the complete elliptic integral of the first kind: \spad{ellipticK(m) = integrate(1/sqrt((1-t^2)*(1-m*t^2)),{} t = 0..1)}.
273336`\spad{ellipticPi(z,{} n,{} m)} is the incomplete elliptic integral of the third kind: \spad{ellipticPi(z,{} n,{} m) = integrate(1/((1-n*t^2)*sqrt((1-t^2)*(1-m*t^2))),{} t = 0..z)}
273403`\spad{ellipticPi(z,{} n,{} m)} expands the elliptic integral \spad{Pi} as a Taylor series.
273535`\spad{ellipticPi(z,{} n,{} m)} is the incomplete elliptic integral of the third kind: \spad{ellipticPi(z,{} n,{} m) = integrate(1/((1-n*t^2)*sqrt((1-t^2)*(1-m*t^2))),{} t = 0..z)}.
273599`\spad{elliptic()} returns \spad{p(x)} if the curve is the elliptic defined by \spad{y^2 = p(x)},{} "failed" otherwise.
273681`\spad{elliptic()} returns \spad{p(x)} if the curve is the elliptic defined by \spad{y^2 = p(x)},{} "failed" otherwise.
274044`\spad{elt(n)} returns the object \spad{n}.
274083`\spad{elt(t)} gives the component of a rank 0 tensor.
276353`\spad{elt(op,{}l)} creates a form for application of \spad{op} to list of arguments \spad{l}.
276400`\spad{elt(x,{}r)} or \spad{x}.\spad{r} \undocumented
276593`\spad{elt(a,{}b)} evaluates the fraction of univariate polynomials \spad{a} with the distinguished variable replaced by \spad{b}.
276706`\spad{elt(a,{}r)} evaluates the fraction of univariate polynomials \spad{a} with the distinguished variable replaced by the constant \spad{r}.
276788`\axiom{\spad{l}."count"} returns the number of elements in \axiom{\spad{l}}.
276851`\spad{elt(u,{}"first")} (also written: \axiom{\spad{u} . first}) is equivalent to first \spad{u}.
276912`\spad{elt(u,{}"last")} (also written: \axiom{\spad{u} . last}) is equivalent to last \spad{u}.
276972`\spad{elt(u,{}"left")} (also written: \axiom{a . left}) is equivalent to \axiom{left(a)}.
277034`\spad{elt(\%,{}"rest")} (also written: \axiom{\spad{u}.rest}) is equivalent to \axiom{rest \spad{u}}.
277095`\spad{elt(a,{}"right")} (also written: \axiom{a . right}) is equivalent to \axiom{right(a)}.
277158`\axiom{\spad{l}.sort} returns \axiom{\spad{l}} with elements sorted. Note: \axiom{\spad{l}.sort = sort(\spad{l})}
277204`\axiom{\spad{l}.unique} returns \axiom{\spad{l}} with duplicates removed. Note: \axiom{\spad{l}.unique = removeDuplicates(\spad{l})}.
277252`\spad{elt(u,{}"value")} (also written: \axiom{a. value}) is equivalent to \axiom{value(a)}.
277367`\spad{elt(f(x),{}r)} returns the coefficient of the term of degree \spad{r} in \spad{f(x)}. This is the same as the function \spadfun{coefficient}.
277444`\spad{elt(a,{}i)} returns the \spad{i}-th coefficient of \spad{a} with respect to the fixed \spad{R}-module basis.
277567`\spad{elt((a1,{}...,{}an),{} i)} returns \spad{\spad{ai}}.
277708`\spad{elt((a1,{}...,{}an),{} [i1,{}...,{}im])} returns \spad{(a_i1,{}...,{}a_im)}.
277910`\spad{elt(db,{}q)} returns all elements of \axiom{\spad{db}} which satisfy \axiom{\spad{q}}.
277963`\spad{elt(u,{}i)} (also written: \spad{u} . \spad{i}) returns the element of \spad{u} indexed by \spad{i}. Error: if \spad{i} is not an index of \spad{u}.
278012`\spad{elt(p,{} el)} returns the image of {\em el} under the permutation \spad{p}.
278063`\spad{elt(lib,{}k)} or \spad{lib}.\spad{k} extracts the value corresponding to the key \spad{k} from the library \spad{lib}.
278106`\spad{elt(db,{}s)} returns the \axiom{\spad{s}} field of each element of \axiom{\spad{db}}.
278166`\spad{elt(ic,{}s)} selects a particular field from \axiom{\spad{ic}}. Valid fields are \axiom{name,{} nargs,{} exposed,{} type,{} abbreviation,{} kind,{} origin,{} params,{} condition,{} doc}.
278214`\spad{elt(u,{}i..j)} (also written: \axiom{a(\spad{i}..\spad{j})}) returns the aggregate of elements \axiom{\spad{u}} for \spad{k} from \spad{i} to \spad{j} in that order. Note: in general,{} \axiom{a.\spad{s} = [a.\spad{k} for \spad{i} in \spad{s}]}.
278286`\spad{elt(s,{}t)} returns the concatenation of \spad{s} and \spad{t}. It is provided to allow juxtaposition of strings to work as concatenation. For example,{} \axiom{"smoo" "shed"} returns \axiom{"smooshed"}.
279150`\spad{elt(u,{} x,{} y)} applies \spad{u} to \spad{x} if \spad{x} is in the domain of \spad{u},{} and returns \spad{y} otherwise. For example,{} if \spad{u} is a polynomial in \axiom{\spad{x}} over the rationals,{} \axiom{elt(\spad{u},{}\spad{n},{}0)} may define the coefficient of \axiom{\spad{x}} to the power \spad{n},{} returning 0 when \spad{n} is out of range.
279209`\spad{elt(f,{}a,{}b)} or \spad{f}(a,{} \spad{b}) returns the value of \spad{f} at the point \spad{(x = a,{} y = b)} if it is not singular.
279272`\spad{elt(r,{}f,{}n)} or \spad{r}(\spad{f},{} \spad{n}) applies the rule \spad{r} to \spad{f} at most \spad{n} times.
279338`\spad{elt(r,{}f,{}n)} or \spad{r}(\spad{f},{} \spad{n}) applies all the rules of \spad{r} to \spad{f} at most \spad{n} times.
279400`\spad{elt(m,{}i,{}j)} returns the element in the \spad{i}th row and \spad{j}th column of the matrix \spad{m}. Error: if indices are outside the proper ranges.
279483`\spad{elt(m,{}i,{}j)} returns the element in the \spad{i}th row and \spad{j}th column of the array \spad{m} error check to determine if indices are in proper ranges
279630`\spad{elt(x,{}rowList,{}colList)} returns an \spad{m}-by-\spad{n} matrix consisting of elements of \spad{x},{} where \spad{m = \# rowList} and \spad{n = \# colList}. If \spad{rowList = [i<1>,{}i<2>,{}...,{}i<m>]} and \spad{colList = [j<1>,{}j<2>,{}...,{}j<n>]},{} then the \spad{(k,{}l)}th entry of \spad{elt(x,{}rowList,{}colList)} is \spad{x(i<k>,{}j<l>)}.
280210`\spad{elt(m,{}i,{}j,{}r)} returns the element in the \spad{i}th row and \spad{j}th column of the matrix \spad{m},{} if \spad{m} has an \spad{i}th row and a \spad{j}th column,{} and returns \spad{r} otherwise.
280295`\spad{elt(m,{}i,{}j,{}r)} returns the element in the \spad{i}th row and \spad{j}th column of the array \spad{m},{} if \spad{m} has an \spad{i}th row and a \spad{j}th column,{} and returns \spad{r} otherwise
281435`\spad{eltable? r} returns whether the operator determines (most likely) the sequence completely.
281538`\spad{empty?(s)} returns \spad{true} if the quasialgebraic set \spad{s} has no points,{} and \spad{false} otherwise.
281700`\spad{empty?(u)} tests if \spad{u} has 0 elements.
281745`\spad{empty()} creates an empty form.
281783`\spad{empty()} returns the empty quasi-algebraic set
281847`\axiom{empty()} returns the same as \axiom{[empty()\$\spad{V},{}empty()\$\spad{C},{}\spad{false}]\$\%}
281893`\spad{empty()} returns a new,{} empty symbol table
281932`\spad{empty()} creates a new,{} empty symbol table.
281974`\spad{empty()}\$\spad{D} creates an aggregate of type \spad{D} with 0 elements. Note: The {\em \$D} can be dropped if understood by context,{} \spadignore{e.g.} \axiom{u: \spad{D} \spad{:=} empty()}.
282011`\spad{empty(n)} produces recinstructor with \spad{n} slots
282072`\spad{empty(n,{} p)} initializes reconstructor with \spad{n} slots working modulo \spad{p}
282141`\spad{endOfFile?(f)} tests whether the file \spad{f} is positioned after the end of all text. If the file is open for output,{} then this test is always \spad{true}.
282189`\spad{endSubProgram()} asserts that we are no longer processing the current subprogram.
282243`\spad{enqueue!(x,{}q)} inserts \spad{x} into the queue \spad{q} at the back end.
282294`\spad{enterInCache(x,{} f)} enters \spad{x} in the cache,{} calling \spad{f(y)} to determine whether \spad{x} is equal to \spad{y}. It returns \spad{x} with an integer associated with it.
282356`\spad{enterInCache(x,{} f)} enters \spad{x} in the cache,{} calling \spad{f(x,{} y)} to determine whether \spad{x < y (f(x,{}y) < 0),{} x = y (f(x,{}y) = 0)},{} or \spad{x > y (f(x,{}y) > 0)}. It returns \spad{x} with an integer associated with it.
282692`\spad{entries(u)} returns a list of all the entries of aggregate \spad{u} in no assumed order.
282830`\spad{entry?(x,{}u)} tests if \spad{x} equals \axiom{\spad{u} . \spad{i}} for some index \spad{i}.
282994`\spad{enumerate()} returns a vector of all the sets of \spad{M} integers in \spad{1..n}.
283061`\spad{enumerate()} returns list of elements of the set.
283251`\spad{eq?(u,{}v)} tests if \spad{u} and \spad{v} are same objects.
283296`\spad{eq(s,{} t)} is \spad{true} if EQ(\spad{s},{}\spad{t}) is \spad{true} in Lisp.
283372`\spad{equality(op,{} foo?)} attaches foo? as the "\%equal?" property to \spad{op}. If \spad{op1} and \spad{op2} have the same name,{} and one of them has an "\%equal?" property \spad{f},{} then \spad{f(op1,{} op2)} is called to decide whether \spad{op1} and \spad{op2} should be considered equal.
283435`\spad{equation(a,{}b)} creates an equation.
283480`\spad{equation(v,{}a..b)} creates a segment binding value with variable \spad{v} and segment \spad{a..b}. Note that the interpreter parses \spad{v=a..b} to this form.
283545`\spad{equation(s,{}"a")} creates a new equation.
283602`\spad{erf(f)} denotes the error function
283652`\spad{erf(x)} returns the error function of \spad{x},{} \spadignore{i.e.} \spad{2 / sqrt(\%\spad{pi})} times the integral of \spad{exp(-x^2) dx}.
283707`\spad{errorInfo(u)} returns information about the error \spad{u}.
283764`\spad{errorKind(u)} returns the type of error which \spad{u} represents.
283826`\spad{error(lmsg)} displays error message \spad{lmsg} and terminates.
283882`\spad{error(msg)} displays error message \spad{msg} and terminates.
283932`\spad{error(nam,{}lmsg)} displays error messages \spad{lmsg} preceded by a message containing the name \spad{nam} of the function in which the error is contained.
283995`\spad{error(nam,{}msg)} displays error message \spad{msg} preceded by a message containing the name \spad{nam} of the function in which the error is contained.
284052`\spad{escape()} provides the escape character,{} \spad{_},{} which is used to allow quotes and other characters {\em within} strings.
284090`\spad{euclideanGroebner(lp)} computes a groebner basis for a polynomial ideal over a euclidean domain generated by the list of polynomials \spad{lp}.
284200`\spad{euclideanGroebner(lp,{} infoflag)} computes a groebner basis for a polynomial ideal over a euclidean domain generated by the list of polynomials \spad{lp}. During computation,{} additional information is printed out if infoflag is given as either "info" (for summary information) or "redcrit" (for reduced critical pairs)
284317`\spad{euclideanGroebner(lp,{} "info",{} "redcrit")} computes a groebner basis for a polynomial ideal generated by the list of polynomials \spad{lp}. If the second argument is \spad{"info"},{} a summary is given of the critical pairs. If the third argument is "redcrit",{} critical pairs are printed.
284441`\spad{euclideanNormalForm(poly,{}gb)} reduces the polynomial \spad{poly} modulo the precomputed groebner basis \spad{gb} giving a canonical representative of the residue class.
284831`\spad{euclideanSize(x)} returns the euclidean size of the element \spad{x}. Error: if \spad{x} is zero.
286090`\spad{eval(f)} unquotes all the quoted operators in \spad{f}.
287449`\spad{eval(m,{}x)} returns \spad{(a*x + b)/(c*x + d)} where \spad{m = moebius(a,{}b,{}c,{}d)} (see \spadfunFrom{moebius}{MoebiusTransform}).
287826`\spad{eval(x,{}f)} evaluates the bivariate function \spad{f} linearly on the tensor product.
287898`\spad{eval(f,{}a)} evaluates a power series at a value in the ground ring by returning a stream of partial sums.
288022`\spad{eval(f,{}x = v)} replaces \spad{x} by \spad{v} in \spad{f}.
288074`\spad{eval(f,{} [x1 = v1,{}...,{}xn = vn])} replaces \spad{xi} by \spad{vi} in \spad{f}.
288216`\spad{eval(eqn,{} [x1=v1,{} ... xn=vn])} replaces \spad{xi} by \spad{vi} in equation \spad{eqn}.
288307`\spad{eval(p,{} el)} returns the image of {\em el} under the permutation \spad{p}.
288437`\spad{eval(eqn,{} x=f)} replaces \spad{x} by \spad{f} in equation \spad{eqn}.
292537`\spad{eval(f,{} x,{} v)} replaces \spad{x} by \spad{v} in \spad{f}.
292718`\spad{eval(f,{} [x1,{}...,{}xn],{} [v1,{}...,{}vn])} replaces \spad{xi} by \spad{vi} in \spad{f}.
293075`\axiom{eval(\spad{p},{} [\spad{x1},{}...,{}\spad{xn}],{} [\spad{v1},{}...,{}\spad{vn}])} replaces \axiom{\spad{xi}} by \axiom{\spad{vi}} in \axiom{\spad{p}}.
293266`\axiom{eval(\spad{p},{} \spad{x},{} \spad{v})} replaces \axiom{\spad{x}} by \axiom{\spad{v}} in \axiom{\spad{p}}.
294460`\spad{evaluateInverse(x,{}f)} \undocumented
294923`\spad{evaluate(f,{} u +-> g u)} attaches the map \spad{g} to \spad{f}. \spad{f} must be a basic operator \spad{g} MUST be additive,{} \spadignore{i.e.} \spad{g(a + b) = g(a) + g(b)} for any \spad{a},{} \spad{b} in \spad{M}. This implies that \spad{g(n a) = n g(a)} for any \spad{a} in \spad{M} and integer \spad{n > 0}.
295082`\spad{even? x} is \spad{true} if \spad{x} is an even integer.
295162`\spad{even?(n)} returns \spad{true} if and only if \spad{n} is even.
295216`\spad{even?(p)} returns \spad{true} if and only if \spad{p} is an even permutation,{} \spadignore{i.e.} {\em sign(p)} is 1.
295351`\spad{evenInfiniteProduct(f(x))} computes \spad{product(n=2,{}4,{}6...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
295440`\spad{evenInfiniteProduct(f(x))} computes \spad{product(n=2,{}4,{}6...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
295527`\spad{evenInfiniteProduct(f(x))} computes \spad{product(n=2,{}4,{}6...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
295748`\spad{evenlambert(f(x))} returns \spad{f(x^2) + f(x^4) + f(x^6) + ...}. \indented{1}{\spad{f(x)} should have a zero constant coefficient.} \indented{1}{This function is used for computing infinite products.} \indented{1}{If \spad{f(x)} is a Taylor series with constant term 1,{} then} \indented{1}{\spad{product(n=1..infinity,{}f(x^(2*n))) = exp(log(evenlambert(f(x))))}.}
296290`\spad{every?(f,{}u)} tests if \spad{p}(\spad{x}) is \spad{true} for all elements \spad{x} of \spad{u}. Note: for collections,{} \axiom{every?(\spad{p},{}\spad{u}) = reduce(and,{}map(\spad{f},{}\spad{u}),{}\spad{true},{}\spad{false})}. However,{} \axiom{every?(\spad{p},{}\spad{u})} returns as soon as it finds an element for which \axiom{\spad{p}} gives \spad{false}.
296385`\spad{exQuo(x,{}y)} \undocumented
296488`\spad{exQuo(x,{}y)} \undocumented
296581`\spad{exQuo(x,{}y)} \undocumented
296815`\axiom{exactQuotient!(\spad{p},{}\spad{r})} replaces \axiom{\spad{p}} by \axiom{exactQuotient(\spad{p},{}\spad{r})}.
296911`\axiom{exactQuotient!(a,{}\spad{b})} replaces \axiom{a} by \axiom{exactQuotient(a,{}\spad{b})}
297148`\axiom{exactQuotient(\spad{p},{}\spad{r})} computes the exact quotient of \axiom{\spad{p}} by \axiom{\spad{r}},{} which is assumed to be a divisor of \axiom{\spad{p}}. No error is returned if this exact quotient fails!
297243`\axiom{exactQuotient(a,{}\spad{b})} computes the exact quotient of \axiom{a} by \axiom{\spad{b}},{} which is assumed to be a divisor of \axiom{a}. No error is returned if this exact quotient fails!
297339`\spad{exists?(f)} tests if the file exists in the file system.
297392`\spad{exp1()} returns the natural log base \spad{2.718281828...}.
297430`\spad{exp1()} returns exp 1: \spad{2.7182818284...}.
297462`\spad{expIfCan(z)} returns exp(\spad{z}) if possible,{} and "failed" otherwise.
297646`\spad{exp(x)} applies the exponential operator to \spad{x}
298092`\spad{exp(z)} returns the exponential of Laurent series \spad{z}.
298178`\spad{exp(z)} returns the exponential of a Puiseux series \spad{z}.
298273`\spad{exp(x)} represents the Fortran intrinsic function EXP
298353`\spad{exp(x)} returns \%\spad{e} to the power \spad{x}.
298588`\spad{expandLog(f)} converts every \spad{log(a/b)} appearing in \spad{f} into \spad{log(a) - log(b)},{} and every \spad{log(a*b)} into \spad{log(a) + log(b)}..
298652`\spad{expandPower(f)} converts every power \spad{(a/b)^c} appearing in \spad{f} into \spad{a^c * b^(-c)}.
298718`\spad{expandTrigProducts(e)} replaces \axiom{sin(\spad{x})*sin(\spad{y})} by \spad{(cos(x-y)-cos(x+y))/2},{} \axiom{cos(\spad{x})*cos(\spad{y})} by \spad{(cos(x-y)+cos(x+y))/2},{} and \axiom{sin(\spad{x})*cos(\spad{y})} by \spad{(sin(x-y)+sin(x+y))/2}. Note that this operation uses the pattern matcher and so 
298718`--is relatively expensive. To avoid getting into an infinite loop the transformations are applied at most ten times.
299071`\spad{expand(f)} performs the following expansions on \spad{f:}\begin{items} \item 1. logs of products are expanded into sums of logs,{} \item 2. trigonometric and hyperbolic trigonometric functions of sums are expanded into sums of products of trigonometric and hyperbolic trigonometric functions. \item 3. formal powers of the form \spad{(a/b)^c} are expanded into \spad{a^c * b^(-c)}. \end{items}
299328`\spad{expand(l)} creates a new value of type \spad{L} in which each segment \spad{l..h by k} is replaced with \spad{l,{} l+k,{} ... lN},{} where \spad{lN <= h < lN+k}. For example,{} \spad{expand [1..4,{} 7..9] = [1,{}2,{}3,{}4,{}7,{}8,{}9]}.
299393`\spad{expand(l..h by k)} creates value of type \spad{L} with elements \spad{l,{} l+k,{} ... lN} where \spad{lN <= h < lN+k}. For example,{} \spad{expand(1..5 by 2) = [1,{}3,{}5]}.
299452`\spad{expand(f)} multiplies the unit and factors together,{} yielding an "unfactored" object. Note: this is purposely not called \spadfun{coerce} which would cause the interpreter to do this automatically.
300232`\spad{explicitEntries?(s)} returns \spad{true} if the stream \spad{s} has explicitly computed entries,{} and \spad{false} otherwise.
300300`\spad{explicitlyEmpty?(s)} returns \spad{true} if the stream is an (explicitly) empty stream. Note: this is a null test which will not cause lazy evaluation.
300496`\spad{explicitlyFinite?(s)} tests if the stream has a finite number of elements,{} and \spad{false} otherwise. Note: for many datatypes,{} \axiom{explicitlyFinite?(\spad{s}) = not possiblyInfinite?(\spad{s})}.
300919`\spad{exponent(x)} \undocumented
301128`\spad{exponent(x)} returns the \spadfunFrom{exponent}{FloatingPointSystem} part of \spad{x}.
301185`\spad{exponent(u)} returns the exponent of the first factor of \spadvar{\spad{u}},{} or 0 if the factored form consists solely of a unit.
301234`\spad{exponential1()} \undocumented
301296`\spad{exponentialOrder(exp(c * x ^(-n) + ...))} returns \spad{-n}. exponentialOrder(0) returns \spad{0}.
301400`\spad{exponential(f)} \undocumented
302285`\spad{expr(s)} returns \spad{s} as an element of Expr; Error: if \spad{s} is not an atom that also belongs to Expr.
302442`\spad{expressIdealMember([f1,{}...,{}fn],{}h)} returns a representation of \spad{h} as a linear combination of the \spad{fi} or "failed" if \spad{h} is not in the ideal generated by the \spad{fi}.
302789`\spad{expt(r,{} i)} computes r^i by repeated squaring
303473`\axiom{\spad{v} exquo \spad{r}} computes the exact quotient of \axiom{\spad{v}} by \axiom{\spad{r}}
303556`\spad{exquo(f,{}g)} creates the equivalent infix form.
303599`\spad{exquo(x,{} r)} returns the exact quotient of \spad{x} by \spad{r},{} or "failed" if \spad{r} does not divide \spad{x} exactly.
303686`\spad{exquo(p,{}r)} returns the exact quotient of polynomial \spad{p} by \spad{r},{} or "failed" if none exists.
303783`\spad{exquo(m,{}r)} computes the exact quotient of the elements of \spad{m} by \spad{r},{} returning \axiom{"failed"} if this is not possible.
303877`\spad{exquo(m,{}r)} computes the exact quotient of the elements of \spad{m} by \spad{r},{} returning \axiom{"failed"} if this is not possible.
303986`\spad{exquo(l,{} a)} returns the exact quotient of \spad{l} by a,{} returning \axiom{"failed"} if this is not possible.
304090`\spad{exquo(a,{}b)} either returns an element \spad{c} such that \spad{c*b=a} or "failed" if no such element can be found.
304153`\spad{exquo(a,{}b)} returns the quotient of \spad{a} and \spad{b},{} or "failed" if \spad{b} is zero or \spad{a} rem \spad{b} is zero.
304220`\axiom{extendIfCan(\spad{ts},{}\spad{p})} returns a triangular set which encodes the simple extension by \axiom{\spad{p}} of the extension of the base field defined by \axiom{\spad{ts}},{} according to the properties of triangular sets of the current domain. If the required properties do not hold then "failed" is returned. This operation encodes in some sense the properties of the triangular sets of the current category
304220`--. Is is used to implement the \axiom{construct} operation to guarantee that every triangular set build from a list of polynomials has the required properties.
304798`\spad{extend(lp,{}lts)} returns the same as \spad{concat([extend(lp,{}ts) for ts in lts])|}
304886`\spad{extend(lp,{}ts)} returns \spad{ts} if \spad{empty? lp} \spad{extend(p,{}ts)} if \spad{lp = [p]} else \spad{extend(first lp,{} extend(rest lp,{} ts))}
304968`\spad{extend(p,{}lts)} returns the same as \spad{concat([extend(p,{}ts) for ts in lts])|}
305050`\spad{extend(p,{}ts)} assumes that \spad{p} is a non-constant polynomial whose main variable is greater than any variable of \spad{ts}. Then it returns a split of Kalkbrener of \spad{ts+p}. This may not be \spad{ts+p} itself,{} if for instance \spad{ts+p} is not a regular triangular set.
305126`\spad{extend(f,{}n)} causes all terms of \spad{f} of degree \spad{<=} \spad{n} to be computed.
305204`\spad{extend(st,{}n)} causes entries to be computed,{} if necessary,{} so that 'st' will have at least \spad{'n'} explicit entries or so that all entries of 'st' will be computed if 'st' is finite with length \spad{<=} \spad{n}.
305322`\spad{extend(x,{}n)} causes the first \spadvar{\spad{n}} entries in the continued fraction \spadvar{\spad{x}} to be computed. Normally entries are only computed as needed.
305381`\spad{extend(x,{}l,{}r)} \undocumented
305523`\axiom{extend(\spad{ts},{}\spad{p})} returns a triangular set which encodes the simple extension by \axiom{\spad{p}} of the extension of the base field defined by \axiom{\spad{ts}},{} according to the properties of triangular sets of the current category If the required properties do not hold an error is returned.
305586`returns extended coordinates as a list
305744`\spad{extendedEuclidean(x,{}y)} returns a record rec where \spad{rec.coef1*x+rec.coef2*y = rec.generator} and rec.generator is a \spad{gcd} of \spad{x} and \spad{y}. The \spad{gcd} is unique only up to associates if \spadatt{canonicalUnitNormal} is not asserted. \spadfun{principalIdeal} provides a version of this operation which accepts an arbitrary length list of arguments.
305936`\spad{extendedEuclidean(x,{}y,{}z)} either returns a record rec where \spad{rec.coef1*x+rec.coef2*y=z} or returns "failed" if \spad{z} cannot be expressed as a linear combination of \spad{x} and \spad{y}.
306240`\axiom{extendedResultant(a,{}\spad{b})} returns \axiom{[\spad{r},{}ca,{}\spad{cb}]} such that \axiom{\spad{r}} is the resultant of \axiom{a} and \axiom{\spad{b}} and \axiom{\spad{r} = ca * a + \spad{cb} * \spad{b}}
306373`\axiom{extendedSubResultantGcd(a,{}\spad{b})} returns \axiom{[\spad{g},{}ca,{} \spad{cb}]} such that \axiom{\spad{g}} is a \spad{gcd} of \axiom{a} and \axiom{\spad{b}} in \axiom{\spad{R^}(\spad{-1}) \spad{P}} and \axiom{\spad{g} = ca * a + \spad{cb} * \spad{b}}
306507`\axiom{extendedSubResultantGcd(a,{}\spad{b})} returns \axiom{[ca,{}\spad{cb},{}\spad{r}]} such that \axiom{\spad{r}} is \axiom{subResultantGcd(a,{}\spad{b})} and we have \axiom{ca * a + \spad{cb} * \spad{cb} = \spad{r}} .
306884`\spad{extensionDegree()} returns the degree of field extension.
306970`\spad{extensionDegree()} returns the degree of the field extension if the extension is algebraic,{} and \spad{infinity} if it is not.
307058`\spad{extensionDegree()} returns the degree of field extension.
307141`\spad{extension(f)} returns the type part of the file name.
307195`\spad{exteriorDifferential(df)} returns the exterior derivative (gradient,{} curl,{} divergence,{} ...) of the differential form \spad{df}.
307274`\spad{external?(u)} returns \spad{true} if \spad{u} is declared to be EXTERNAL
307457`\spad{extract!(u)} destructively removes a (random) item from bag \spad{u}.
307504`\spad{extractBottom!(d)} destructively extracts the bottom (back) element from the dequeue \spad{d}. Error: if \spad{d} is empty.
307612`\axiom{extractIfCan(\spad{x})} searches the item whose key is \axiom{\spad{x}}.
307890`\axiom{extractSplittingLeaf(a)} returns the left most leaf (as a tree) whose status is \spad{false} if any,{} else "failed" is returned.
307969`\spad{extractTop!(d)} destructively extracts the top (front) element from the dequeue \spad{d}. Error: if \spad{d} is empty.
308294`\spad{factorAndSplit(eq)} make the right hand side 0 and factors the new left hand side. Each factor is equated to 0 and put into the resulting list without repetitions.
311853`\spad{factorSquareFree(p)} factors an extended squareFree polynomial \spad{p} over the rational numbers.
311925`\spad{factorSquareFree(p)} returns the factorization of \spad{p} which is supposed not having any repeated factor (this is not checked).
312070`\spad{factorSquareFree(p)} produces the complete factorization of the square free polynomial \spad{p}.
312904`\spad{factor(p)} factors the multivariate polynomial \spad{p} over its coefficient domain
312987`\spad{factor(p)} factors the multivariate polynomial \spad{p} with coefficients which are fractions of elements of \spad{R}.
313054`\spad{factor(p)} factors a polynomial with polynomial coefficients.
313131`\spad{factor(p)} factors an extended polynomial \spad{p} over the rational numbers.
313797`\spad{factor(p)} returns the factorization of \spad{p} over the integers.
314014`\spad{factor(p)} produces the complete factorization of the polynomial \spad{p}.
314084`\spad{factor(p)} factors the multivariate polynomial \spad{p} over its coefficient domain
314154`\spad{factor(p)} produces the complete factorization of the multivariate polynomial \spad{p} over a finite field.
314225`\spad{factor(p)} factorizes the polynomial \spad{p} with complex coefficients.
314293`\spad{factor(prf)} factors a polynomial with rational function coefficients.
314674`\spad{factor(p)} returns a prime factorisation of \spad{p}.
314741`\spad{factor(p)} returns a prime factorisation of \spad{p}.
314824`\spad{factor(p)} returns a prime factorisation of \spad{p}.
314912`\spad{factor(x)} returns the factorization of \spad{x} into irreducibles.
316454`\spad{factorial(n)} returns the factorial of \spad{n},{} \spadignore{i.e.} \spad{n!}.
316562`\spad{factorial(n)} returns \spad{n!}. this is the product of all integers between 1 and \spad{n} (inclusive). Note: \spad{0!} is defined to be 1.
316625`\spad{factorial(n)} computes the factorial of \spad{n} (denoted in the literature by \spad{n!}) Note: \spad{n! = n (n-1)! when n > 0}; also,{} \spad{0! = 1}.
316688`\spad{factorials(f)} rewrites the permutations and binomials in \spad{f} in terms of factorials.
316747`\spad{factorials(f)} rewrites the permutations and binomials in \spad{f} in terms of factorials.
316867`\spad{factorials(f,{} x)} rewrites the permutations and binomials in \spad{f} involving \spad{x} in terms of factorials.
316933`\spad{factorsOfCyclicGroupSize()} returns the factorization of size()\spad{-1}
317145`\spad{factors(a1\^e1,{}...,{}an\^en)} returns \spad{[[a1,{} e1],{}...,{}[an,{} en]]}.
317218`\spad{factors(a1\^e1,{}...,{}an\^en)} returns \spad{[[a1,{} e1],{}...,{}[an,{} en]]}.
317390`\spad{factors(u)} returns a list of the factors in a form suitable for iteration. That is,{} it returns a list where each element is a record containing a base and exponent. The original object is the product of all the factors and the unit (which can be extracted by \axiom{unit(\spad{u})}).
317546`\spad{failed?(r)} tests if \spad{r} is a failed match.
317612`\spad{failed?(r)} tests if \spad{r} is a failed match.
317672`\spad{failed()} returns a failed match.
317730`\spad{failed()} returns a failed match.
317782`\spad{false} is a logical constant.
318444`\spad{filename(d,{}n,{}e)} creates a file name with \spad{d} as its directory,{} \spad{n} as its name and \spad{e} as its extension. This is a portable way to create file names. When \spad{d} or \spad{t} is the empty string,{} a default is used.
318684`\spad{fill!(u,{}x)} replaces each entry in aggregate \spad{u} by \spad{x}. The modified \spad{u} is returned as value.
318773`\spad{fill!(m,{}r)} fills \spad{m} with \spad{r}\spad{'s}
318843`\spad{fillPascalTriangle()} fills the stored table.
318909`\spad{filterUntil(p,{}t)} returns \spad{[x for x in t while not p(x)]}.
318974`\spad{filterUntil(p,{}s)} returns \spad{[x0,{}x1,{}...,{}x(n)]} where \spad{s = [x0,{}x1,{}x2,{}..]} and \spad{n} is the smallest index such that \spad{p(xn) = true}.
319032`\spad{filterWhile(p,{}t)} returns \spad{[x for x in t while p(x)]}.
319097`\spad{filterWhile(p,{}s)} returns \spad{[x0,{}x1,{}...,{}x(n-1)]} where \spad{s = [x0,{}x1,{}x2,{}..]} and \spad{n} is the smallest index such that \spad{p(xn) = false}.
319637`\spad{find(p,{}u)} returns the first \spad{x} in \spad{u} such that \axiom{\spad{p}(\spad{x})} is \spad{true},{} and "failed" otherwise.
319707`\spad{finite?(\spad{a})} determines whether \spad{a} is a finite cardinal,{} \spadignore{i.e.} an integer.
319863`\spad{finiteBasis(d)} returns a basis for \spad{d} as a module over {\em K[x]}.
319932`\spad{fintegrate(f,{}v,{}c)} is the integral of \spad{f()} with respect \indented{1}{to \spad{v} and having \spad{c} as the constant of integration.} \indented{1}{The evaluation of \spad{f()} is delayed.}
320060`\spad{fintegrate(f,{}v,{}c)} is the integral of \spad{f()} with respect \indented{1}{to \spad{v} and having \spad{c} as the constant of integration.} \indented{1}{The evaluation of \spad{f()} is delayed.}
320222`\spad{firstNumer(p)} extracts the numerator of the first fractional term. This returns 0 if there is no fractional part (use \spadfunFrom{wholePart}{PartialFraction} to get the whole part).
320768`\spad{first(x)} returns the first letter of \spad{x}.
320817`\axiom{first(\spad{x})} returns the first entry of the tree \axiom{\spad{x}}.
320864`\spad{first(u)} returns the first element \spad{x} of \spad{u}. Note: for collections,{} \axiom{first([\spad{x},{}\spad{y},{}...,{}\spad{z}]) = \spad{x}}. Error: if \spad{u} is empty.
320947`\spad{first(u)} returns the first element of \spad{u} (equivalently,{} the value at the current node).
321002`\axiom{first(\spad{ts})} returns the polynomial of \axiom{\spad{ts}} with greatest main variable if \axiom{\spad{ts}} is not empty,{} otherwise returns \axiom{"failed"}.
321207`\spad{first(u,{}n)} returns a copy of the first \spad{n} (\axiom{\spad{n} \spad{>=} 0}) elements of \spad{u}.
321282`\spad{fixPredicate(f)} returns \spad{g} defined by \spad{g}(a) = \spad{f}(a::B).
321467`\spad{fixedPointExquo(f,{}g)} computes the exact quotient of \spad{f} and \spad{g} using a fixed point computation.
321554`\spad{fixedPoint f} is the fixed point of function \spad{f}. \indented{1}{\spadignore{i.e.} such that \spad{fixedPoint f = f(fixedPoint f)}.} \indented{1}{Unused and buggy.}
321610`\spad{fixedPoint(f,{}n)} is the fixed point of function \indented{1}{\spad{f} which is assumed to transform a list of length} \indented{1}{\spad{n}.} \indented{1}{Unused and buggy.}
321692`\spad{fixedPoints(p)} returns the points fixed by the permutation \spad{p}.
321843`\spad{flatten(s)} returns an input form corresponding to \spad{s} with all the nested operations flattened to triples using new local variables. If \spad{s} is a piece of code,{} this speeds up the compilation tremendously later on.
321884`\spad{flexible?()} tests if \spad{2*associator(a,{}b,{}a) = 0} for all \spad{a},{} \spad{b} in the algebra. Note: we only can test this; in general we don\spad{'t} know whether \spad{2*a=0} implies \spad{a=0}.
321958`\spad{flexible?()} tests if \spad{2*associator(a,{}b,{}a) = 0} for all \spad{a},{} \spad{b} in the algebra. Note: we only can test this; in general we don\spad{'t} know whether \spad{2*a=0} implies \spad{a=0}.
322029`\spad{flexibleArray(l)} creates a flexible array from the list of elements \spad{l}
322152`\spad{float?(s)} is \spad{true} if \spad{s} is an atom and belong to \spad{Flt}.
322229`\spad{float(s)} returns \spad{s} as an element of \spad{Flt}; Error: if \spad{s} is not an atom that also belongs to \spad{Flt}.
322361`\spad{float(a,{}e)} returns \spad{a * base() ^ e}.
322499`\spad{float(a,{}e,{}b)} returns \spad{a * b ^ e}.
322620`\spad{floor(x)} returns the largest integral element below \spad{x}.
322699`\spad{floor x} returns the largest integer \spad{<= x}.
322745`\spad{flush(f)} make sure that buffered data in written out to the operating system.
322797`\spad{fmecg(p1,{}e,{}r,{}p2)} finds \spad{X} : \spad{p1} - \spad{r} * X^e * \spad{p2}
322993`\spad{fmecg(p1,{}e,{}r,{}p2)} finds \spad{X} : \spad{p1} - \spad{r} * X^e * \spad{p2}
323205`\spad{fmecg(p1,{}e,{}r,{}p2)} finds \spad{X} : \spad{p1} - \spad{r} * X^e * \spad{p2}
323508`\spad{fortranCarriageReturn()} produces a carriage return on the current Fortran output stream
323569`\spad{fortranCharacter()} returns CHARACTER,{} an element of FortranType
323619`\spad{fortranComplex()} returns COMPLEX,{} an element of FortranType
323667`\spad{fortranDoubleComplex()} returns DOUBLE COMPLEX,{} an element of FortranType
323721`\spad{fortranDouble()} returns DOUBLE PRECISION,{} an element of FortranType
323768`\spad{fortranInteger()} returns INTEGER,{} an element of FortranType
323816`\spad{fortranLiteralLine(s)} writes \spad{s} to the current Fortran output stream,{} followed by a carriage return
323880`\spad{fortranLiteral(s)} writes \spad{s} to the current Fortran output stream
323940`\spad{fortranLogical()} returns LOGICAL,{} an element of FortranType
323988`\spad{fortranReal()} returns REAL,{} an element of FortranType
324033`\spad{fortranTypeOf(u,{}tab)} returns the type of \spad{u} in \spad{tab}
324481`\spad{fractRadix(pre,{}cyc)} creates a fractional radix expansion from a list of prefix ragits and a list of cyclic ragits. For example,{} \spad{fractRadix([1],{}[6])} will return \spad{0.16666666...}.
324559`\spad{fractRagits(rx)} returns the ragits of the fractional part of a radix expansion.
324626`\spad{fractionFreeGauss(m)} performs the fraction free gaussian elimination on the matrix \spad{m}.
324836`\spad{fractionPart(b)} returns the fractional part of a binary expansion.
324903`\spad{fractionPart(d)} returns the fractional part of a decimal expansion.
324971`\spad{fractionPart(h)} returns the fractional part of a hexadecimal expansion.
325043`\spad{fractionPart(rx)} returns the fractional part of a radix expansion.
325113`\spad{fractionPart(x)} returns the fractional part of \spad{x}. \spad{x} = wholePart(\spad{x}) + fractionPart(\spad{x})
325196`\spad{fractionPart x} returns the fractional part of \spad{x}.
325412`\spad{freeOf?(x,{} y)} tests if \spad{x} does not contain any occurrence of \spad{y},{} where \spad{y} is a single kernel.
325467`\spad{fresnelC(f)} denotes the Fresnel integral \spad{C}
325522`fresnelC is the Fresnel integral \spad{C},{} defined by \spad{C(x) = integrate(cos(t^2),{}t=0..x)}
325582`\spad{fresnelS(f)} denotes the Fresnel integral \spad{S}
325637`fresnelS is the Fresnel integral \spad{S},{} defined by \spad{S(x) = integrate(sin(t^2),{}t=0..x)}
325697`\spad{frobenius(x)} \undocumented
325772`\spad{front(q)} returns the element at the front of the queue. The queue \spad{q} is unchanged by this operation. Error: if \spad{q} is empty.
325954`\spad{frst(s)} returns the first element of stream \spad{s}. Caution: this function should only be called after a \spad{empty?} test has been made since there no error check.
326004`\spad{fullDisplay(db)} prints full details of each entry in \axiom{\spad{db}}.
326053`\spad{fullDisplay(ic)} prints all of the information contained in \axiom{\spad{ic}}.
326100`\spad{fullDisplay(db,{}start,{}end )} prints full details of entries in the range \axiom{\spad{start}..end} in \axiom{\spad{db}}.
326336`\spad{functionName(d)} specifies the name of the function given by the algebraic differential equation or recurrence. This option is expressed in the form \spad{functionName == d}.
326441`\spad{function(code,{} [x1,{}...,{}xn],{} f)} returns the input form corresponding to \spad{f(x1,{}...,{}xn) == code}.
326696`\spad{gbasis }\undocumented
326805`\spad{gcdBasis(v)} returns basis part \spad{o}
328232`\spad{gcdPrimitive lp} computes the \spad{gcd} of the list of primitive polynomials \spad{lp}.
328303`\spad{gcdPrimitive(p,{}q)} computes the \spad{gcd} of the primitive polynomials \spad{p} and \spad{q}.
328516`\spad{gcd([f1,{}..,{}fk])} = \spad{gcd} of the polynomials \spad{fi}.
328560`\spad{gcd(lp)} computes the \spad{gcd} of the list of polynomials \spad{lp}.
328921`\spad{gcd(l)} returns the common \spad{gcd} of the elements in the list \spad{l}.
328964`\spad{gcd(p,{}q)} computes the \spad{gcd} of the two polynomials \spad{p} and \spad{q}.
329389`\axiom{\spad{gcd}(\spad{P},{} \spad{Q})} returns the \spad{gcd} of \axiom{\spad{P}} and \axiom{\spad{Q}}.
329473`\axiom{\spad{gcd}(\spad{r},{}\spad{p})} returns the \spad{gcd} of \axiom{\spad{r}} and the content of \axiom{\spad{p}}.
329552`\spad{gcd(x,{}y)} returns the greatest common divisor of \spad{x} and \spad{y}.
329592`\spad{gcd(a,{}b)} computes the greatest common divisor of two non negative integers \spad{a} and \spad{b}.
329641`\spad{gcd(a,{}b)} computes the greatest common divisor of two positive integers \spad{a} and \spad{b}.
329942`\spad{gcdcofact([f1,{}..fk])} = \spad{gcd} and cofactors of \spad{k} univariate polynomials.
329998`\spad{gcdcofactprim([f1,{}..fk])} = \spad{gcd} and cofactors of \spad{k} primitive polynomials.
330058`\spad{gcdprim([f1,{}..,{}fk])} = \spad{gcd} of \spad{k} PRIMITIVE univariate polynomials
330190`like genVectorStream,{} but skips every second vector.
330344`\spad{genVectorStream(sumEta,{} maxEta,{} k)} generates stream of all possible non-increasing lists \spad{eta} with maximal entry \spad{maxEta} and sum of entries at most \spad{sumEta}.
332507`\spad{generalizedContinuumHypothesisAssumed?()} tests if the hypothesis is currently assumed.
332587`\spad{generalizedContinuumHypothesisAssumed(bool)} is used to dictate whether the hypothesis is to be assumed.
333379`\spad{generalizedInverse(m)} returns the generalized (Moore
333379`--Penrose) inverse of the matrix \spad{m},{} \spadignore{i.e.} the matrix \spad{h} such that m*h*m=h,{} h*m*h=m,{} \spad{m*h} and \spad{h*m} are both symmetric matrices.
333572`\spad{generate(f)} creates an infinite stream all of whose elements are equal to \spad{f()}. Note: \spad{generate(f) = [f(),{}f(),{}f(),{}...]}.
333617`\spad{generate(f,{}s)} returns \spad{[s,{}f(s),{}f(f(s)),{}...]}.
333672`\spad{generate(f,{}x)} creates an infinite stream whose first element is \spad{x} and whose \spad{n}th element (\spad{n > 1}) is \spad{f} applied to the previous element. Note: \spad{generate(f,{}x) = [x,{}f(x),{}f(f(x)),{}...]}.
333862`\spad{generator()} returns a root of the defining polynomial. This element generates the field as an algebra over the ground field.
333939`\spad{generator()} returns the generator for this domain.
334205`\spad{generator(d)} returns \spad{f} if \spad{(f) = d},{} "failed" if \spad{d} is not principal.
335989`\spad{generic()} returns a generic element,{} \spadignore{i.e.} the linear combination of the fixed basis with the symbolic coefficients \spad{\%x1,{}\%x2,{}..}
336061`\spad{generic(s)} returns a generic element,{} \spadignore{i.e.} the linear combination of the fixed basis with the symbolic coefficients \spad{s1,{}s2,{}..}
336139`\spad{generic(vs)} returns a generic element,{} \spadignore{i.e.} the linear combination of the fixed basis with the symbolic coefficients \spad{vs}; error,{} if the vector of symbols is too short
336225`\spad{generic(ve)} returns a generic element,{} \spadignore{i.e.} the linear combination of \spad{ve} basis with the symbolic coefficients \spad{\%x1,{}\%x2,{}..}
336307`\spad{generic(s,{}v)} returns a generic element,{} \spadignore{i.e.} the linear combination of \spad{v} with the symbolic coefficients \spad{s1,{}s2,{}..}
336396`\spad{generic(vs,{}ve)} returns a generic element,{} \spadignore{i.e.} the linear combination of \spad{ve} with the symbolic coefficients \spad{vs} error,{} if the vector of symbols is shorter than the vector of elements
336493`\spad{genus()} returns the genus of one absolutely irreducible component
336572`\spad{genus()} returns the genus of one absolutely irreducible component
336648`\spad{geometric(f)} \undocumented
336729`\spad{getBadValues(p)} returns the list of "bad values" for \spad{p}. Note: \spad{p} is not allowed to match any of its "bad values".
336830`\spad{getCurve(t)} returns the \spadtype{PlottableSpaceCurveCategory} representing the parametric curve of the given tube plot \spad{t}.
336881`\spad{getDatabase("char")} returns a list of appropriate entries in the browser database. The legal values for \spad{"char"} are "o" (operations),{} \spad{"k"} (constructors),{} \spad{"d"} (domains),{} \spad{"c"} (categories) or \spad{"p"} (packages).
336953`\spad{getEq f} returns the defining equation,{} if \spad{f} represents the coefficient of an ADE or a recurrence.
337161`\spad{getMatch(var,{} r)} returns the expression that \spad{var} matches in the result \spad{r},{} and "failed" if \spad{var} is not matched in \spad{r}.
337243`\spad{getMultiplicationMatrix()} returns the multiplication table in form of a matrix.
337439`\spad{getMultiplicationMatrix()} returns the multiplication table in form of a matrix.
337936`\spad{getOrder()} returns \spad{[[b1,{}...,{}bm],{} [a1,{}...,{}an]]} such that the partial ordering on \spad{S} was given by \spad{setOrder([b1,{}...,{}bm],{}[a1,{}...,{}an])}.
338301`\spad{getType()} returns the input form corresponding to \spad{R}
338360`\spad{getVariableOrder()} returns \spad{[[b1,{}...,{}bm],{} [a1,{}...,{}an]]} such that the ordering on the variables was given by \spad{setVariableOrder([b1,{}...,{}bm],{} [a1,{}...,{}an])}.
338466`\spad{getZechTable()} returns the zech logarithm table of the field. This table is used to perform additions in the field quickly.
338681`\spad{getZechTable()} returns the zech logarithm table of the field. This table is used to perform additions in the field quickly.
338883`\spad{goodPoint(p,{} q)} returns an integer a such that a is neither a pole of \spad{p(x,{}y)} nor a branch point of \spad{q(x,{}y) = 0}.
338950`\spad{goto(l)} creates a representation of a FORTRAN GOTO statement
339001`\spad{x} = ((\spad{-1})^grade(\spad{x}))\spad{*x}
339132`\spad{gradient(v,{}xlist)} computes the gradient,{} the vector of first partial derivatives,{} of the scalar field \spad{v},{} \spad{v} a function of the variables listed in \spad{xlist}.
339221`\spad{graeffe p} determines \spad{q} such that \spad{q(-z^2) = p(z)*p(-z)}. Note that the roots of \spad{q} are the squares of the roots of \spad{p}.
339708`\spad{graphImage()} returns an empty graph with 0 point lists of the domain \spadtype{GraphImage}. A graph image contains the graph data component of a two dimensional viewport.
340367`\spad{green()} returns the position of the green hue from total hues.
341636`\spad{groebner(lp)} computes a groebner basis for a polynomial ideal generated by the list of polynomials \spad{lp}.
341788`\spad{groebner(lp,{} infoflag)} computes a groebner basis for a polynomial ideal generated by the list of polynomials \spad{lp}. Argument infoflag is used to get information on the computation. If infoflag is "info",{} then summary information is displayed for each \spad{s}-polynomial generated. If infoflag is "redcrit",{} the reduced critical pairs are displayed. If infoflag is any other string,{} no informat
341788`--ion is printed during computation.
341882`\spad{groebner(lp,{} "info",{} "redcrit")} computes a groebner basis for a polynomial ideal generated by the list of polynomials \spad{lp},{} displaying both a summary of the critical pairs considered (\spad{"info"}) and the result of reducing each critical pair ("redcrit"). If the second or third arguments have any other string value,{} the indicated information is suppressed.
342095`\spad{ground?(p)} tests if polynomial \spad{p} is a member of the coefficient ring.
342160`\spad{ground?(f)} tests if \spad{f} is an element of \spad{R}.
342311`\spad{ground(p)} retracts polynomial \spad{p} to the coefficient ring.
342369`\spad{ground(f)} returns \spad{f} as an element of \spad{R}. An error occurs if \spad{f} is not an element of \spad{R}.
365008`\spad{hMonic }\undocumented
365255`\axiom{\spad{halfExtendedResultant1}(a,{}\spad{b})} returns \axiom{[\spad{r},{}ca]} such that \axiom{extendedResultant(a,{}\spad{b})} returns \axiom{[\spad{r},{}ca,{} \spad{cb}]}
365384`\axiom{\spad{halfExtendedResultant2}(a,{}\spad{b})} returns \axiom{[\spad{r},{}ca]} such that \axiom{extendedResultant(a,{}\spad{b})} returns \axiom{[\spad{r},{}ca,{} \spad{cb}]}
365513`\axiom{\spad{halfExtendedSubResultantGcd1}(a,{}\spad{b})} returns \axiom{[\spad{g},{}ca]} such that \axiom{extendedSubResultantGcd(a,{}\spad{b})} returns \axiom{[\spad{g},{}ca,{} \spad{cb}]}
365643`\axiom{\spad{halfExtendedSubResultantGcd1}(a,{}\spad{b})} returns \axiom{[\spad{g},{}ca]} if \axiom{extendedSubResultantGcd(a,{}\spad{b})} returns \axiom{[\spad{g},{}ca,{}\spad{cb}]} otherwise produces an error.
365775`\axiom{\spad{halfExtendedSubResultantGcd2}(a,{}\spad{b})} returns \axiom{[\spad{g},{}\spad{cb}]} such that \axiom{extendedSubResultantGcd(a,{}\spad{b})} returns \axiom{[\spad{g},{}ca,{} \spad{cb}]}
365905`\axiom{\spad{halfExtendedSubResultantGcd2}(a,{}\spad{b})} returns \axiom{[\spad{g},{}\spad{cb}]} if \axiom{extendedSubResultantGcd(a,{}\spad{b})} returns \axiom{[\spad{g},{}ca,{}\spad{cb}]} otherwise produces an error.
366037`\spad{hankelH1(v,{} z)} is first Hankel function (Bessel function of the third kind)
366100`\spad{hankelH1(v,{} z)} is first Hankel function (Bessel function of the third kind).
366159`\spad{hankelH2(v,{} z)} is the second Hankel function (Bessel function of the third kind)
366222`\spad{hankelH2(v,{} z)} is the second Hankel function (Bessel function of the third kind).
366357`\spad{has?(op,{} s)} tests if property \spad{s} is attached to \spad{op}.
366411`\spad{hasHi(s)} tests whether the segment \spad{s} has an upper bound.
366513`\spad{hasSolution?(A,{}B)} tests if the linear system \spad{AX = B} has a solution.
366825`\spad{hash(x)} returns the hash key for \spad{x}
366870`\spad{hash(s)} calculates a hash code for \spad{s}.
366921`\spad{hash(n)} returns the hash code of \spad{n}.
367094`\spad{hclf(x,{} y)} returns the highest common left factor of \spad{x} and \spad{y},{} \spadignore{i.e.} the largest \spad{d} such that \spad{x = d a} and \spad{y = d b}.
367139`\spad{hclf(x,{} y)} returns the highest common left factor of \spad{x} and \spad{y},{} that is the largest \spad{d} such that \spad{x = d a} and \spad{y = d b}.
367191`\spad{hconcat(u)} horizontally concatenates all forms in list \spad{u}.
367239`\spad{hconcat(f,{}g)} horizontally concatenate forms \spad{f} and \spad{g}.
367284`\spad{hcrf(x,{} y)} returns the highest common right factor of \spad{x} and \spad{y},{} \spadignore{i.e.} the largest \spad{d} such that \spad{x = a d} and \spad{y = b d}.
367329`\spad{hcrf(x,{} y)} returns the highest common right factor of \spad{x} and \spad{y},{} that is the largest \spad{d} such that \spad{x = a d} and \spad{y = b d}.
367739`\axiom{headReduce(\spad{p},{}\spad{ts})} returns a polynomial \axiom{\spad{r}} such that \axiom{headReduce?(\spad{r},{}\spad{ts})} holds and there exists some product \axiom{\spad{h}} of \axiom{initials(\spad{ts})} such that \axiom{\spad{h*p} - \spad{r}} lies in the ideal generated by \axiom{\spad{ts}}.
367805`\axiom{headReduce(a,{}\spad{b})} returns a polynomial \axiom{\spad{r}} such that \axiom{headReduced?(\spad{r},{}\spad{b})} holds and there exists an integer \axiom{\spad{e}} such that \axiom{init(\spad{b})^e a - \spad{r}} is zero modulo \axiom{\spad{b}}.
367946`\spad{headReduced?(ts)} returns \spad{true} iff the head of every element of \axiom{\spad{ts}} is reduced \spad{w}.\spad{r}.\spad{t} to any other element of \axiom{\spad{ts}}.
368245`\axiom{headReduced?(\spad{p},{}\spad{ts})} returns \spad{true} iff the head of \axiom{\spad{p}} is reduced \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{ts}}.
368319`\axiom{headReduced?(\spad{q},{}\spad{lp})} returns \spad{true} iff \axiom{headReduced?(\spad{q},{}\spad{p})} holds for every \axiom{\spad{p}} in \axiom{\spad{lp}}.
368404`\axiom{headReduced?(a,{}\spad{b})} returns \spad{true} iff \axiom{degree(head(a),{}mvar(\spad{b})) < mdeg(\spad{b})}.
368572`\axiom{headRemainder(a,{}\spad{ps})} returns \axiom{[\spad{b},{}\spad{r}]} such that the leading monomial of \axiom{\spad{b}} is reduced in the sense of Groebner bases \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{ps}} and \axiom{r*a - \spad{b}} lies in the ideal generated by \axiom{\spad{ps}}.
368744`\spad{head(l)} returns the first element of a doubly-linked aggregate \spad{l}. Error: if \spad{l} is empty.
368797`\axiom{head(\spad{p})} returns \axiom{\spad{p}} if \axiom{\spad{p}} belongs to \axiom{\spad{R}},{} otherwise returns its leading term (monomial in the FriCAS sense),{} where \axiom{\spad{p}} is viewed as a univariate polynomial in its main variable.
368931`\spad{heap(ls)} creates a heap of elements consisting of the elements of \spad{ls}.
368972`\spad{height()} returns the height of the display area (an integer).
369076`\spad{height(p)} returns the maximal absolute value of the coefficients of the polynomial \spad{p}.
369147`\spad{height(f)} returns the height of form \spad{f} (an integer).
369244`\spad{height(d)} returns the number of elements in dequeue \spad{d}. Note: \axiom{height(\spad{d}) = \# \spad{d}}.
369611`one step of Hensel lifting
369868`\spad{hermite(m)} returns the Hermite normal form of the matrix \spad{m}.
369926`\spad{hessian(v,{}xlist)} computes the hessian,{} the matrix of second partial derivatives,{} of the scalar field \spad{v},{} \spad{v} a function of the variables listed in \spad{xlist}.
370014`\spad{hexDigit?(c)} tests if \spad{c} is a hexadecimal numeral,{} \spadignore{i.e.} one of 0..9,{} a..\spad{f} or A..\spad{F}.
370062`\spad{hexDigit()} returns the class of all characters for which \spadfunFrom{hexDigit?}{Character} is \spad{true}.
370107`\spad{hex(r)} converts a rational number to a hexadecimal expansion.
370170`\spad{\spad{hi}(s)} returns the second endpoint of \spad{s}. Note: \spad{\spad{hi}(l..h) = h}.
370214`\spad{highCommonTerms(e1 a1 + ... + en an,{} f1 b1 + ... + fm bm)} returns \indented{2}{\spad{reduce(+,{}[max(\spad{ei},{} \spad{fi}) \spad{ci}])}} where \spad{ci} ranges in the intersection of \spad{{a1,{}...,{}an}} and \spad{{b1,{}...,{}bm}}.
370314`\spad{high(s)} returns the second endpoint of \spad{s}. Note: \spad{high(l..h) = h}.
370423`\spad{homogeneous?(p)} tests if all of the terms of \spad{p} have the same degree.
370481`\spad{homogeneous?(df)} tests if all of the terms of differential form \spad{df} have the same degree.
370652`\spad{homogeneous(d)} specifies whether we allow only homogeneous algebraic differential equations. This option is expressed in the form \spad{homogeneous == d}. If \spad{true},{} then maxPower must be set,{} too,{} and ADEs with constant total degree are allowed. If a PositiveInteger is given,{} only ADE\spad{'s} with this total degree are allowed.
370786`\spad{horizConcat(x,{}y)} horizontally concatenates two matrices with an equal number of rows. The entries of \spad{y} appear to the right of the entries of \spad{x}. Error: if the matrices do not have the same number of rows.
370850`\spad{hspace(n)} creates white space of width \spad{n}.
370896`\spad{htrigs(f)} converts all the exponentials in \spad{f} into hyperbolic sines and cosines.
371066`\spad{hyperelliptic()} returns \spad{p(x)} if the curve is the hyperelliptic defined by \spad{y^2 = p(x)},{} "failed" otherwise.
371153`\spad{hyperelliptic()} returns \spad{p(x)} if the curve is the hyperelliptic defined by \spad{y^2 = p(x)},{} "failed" otherwise.
371452`\spad{hypergeometricF(la,{} lb,{} z)} is the generalized hypergeometric function
371565`\spad{hypergeometricF(la,{} lb,{} z)} is the generalized hypergeometric function.
371676`\spad{iAiryAiPrime(x)} should be local but conditional.
371741`\spad{iAiryAi(x)} should be local but conditional.
371801`\spad{iAiryBiPrime(x)} should be local but conditional.
371866`\spad{iAiryBi(x)} should be local but conditional.
371926`\spad{iCompose(f,{}g)} returns \spad{f(g(x))}. This is an internal function which should only be called for Taylor series \spad{f(x)} and \spad{g(x)} such that the constant coefficient of \spad{g(x)} is zero.
372091`\spad{id x} is \spad{x}.
372134`\spad{idealSimplify(s)} returns a different and presumably simpler representation of \spad{s} with the defining polynomials for the equations forming a groebner basis,{} and the defining polynomial for the inequation reduced with respect to the basis,{} using Buchberger\spad{'s} algorithm.
372938`returns the identity element which is do nothing transform
372983`\spad{identity()} create the identity dhmatrix
373040`\spad{iexactQuo(x,{}y)} should be local but conditional
373112`\spad{iexactQuo(x,{}y)} should be local but conditional
373183`\spad{iexactQuo(x,{}y)} should be local but conditional
373273`\spad{iexactQuo(x,{}y)} should be local but conditional
373427`\spad{iiAiryAiPrime(x)} should be local but conditional.
373493`\spad{iiAiryAi(x)} should be local but conditional.
373554`\spad{iiAiryBiPrime(x)} should be local but conditional.
373620`\spad{iiAiryBi(x)} should be local but conditional.
373681`\spad{iiBesselI(x)} should be local but conditional.
373749`\spad{iiBesselJ(x)} should be local but conditional.
373817`\spad{iiBesselK(x)} should be local but conditional.
373885`\spad{iiBesselY(x)} should be local but conditional.
373953`\spad{iiBeta(x)} should be local but conditional.
374018`\spad{iiGamma(x)} should be local but conditional.
374078`\spad{iiPolylog(x)} should be local but condition
374142`\spad{iiabs(x)} should be local but conditional.
374200`\spad{iiacos(x)} should be local but conditional
374252`\spad{iiacosh(x)} should be local but conditional
374305`\spad{iiacot(x)} should be local but conditional
374357`\spad{iiacoth(x)} should be local but conditional
374410`\spad{iiacsc(x)} should be local but conditional
374462`\spad{iiacsch(x)} should be local but conditional
374515`\spad{iiasec(x)} should be local but conditional
374567`\spad{iiasech(x)} should be local but conditional
374620`\spad{iiasin(x)} should be local but conditional
374672`\spad{iiasinh(x)} should be local but conditional
374725`\spad{iiatan(x)} should be local but conditional
374777`\spad{iiatanh(x)} should be local but conditional
374830`\spad{iibinom(l)} should be local but conditional.
374892`\spad{iicos(x)} should be local but conditional
374943`\spad{iicosh(x)} should be local but conditional
374995`\spad{iicot(x)} should be local but conditional
375046`\spad{iicoth(x)} should be local but conditional
375098`\spad{iicsc(x)} should be local but conditional
375149`\spad{iicsch(x)} should be local but conditional
375201`\spad{iidigamma(x)} should be local but conditional.
375263`\spad{iidprod(l)} should be local but conditional.
375325`\spad{iidsum(l)} should be local but conditional.
375386`\spad{iiexp(x)} should be local but conditional
375437`\spad{iifact(x)} should be local but conditional.
375492`\spad{iilog(x)} should be local but conditional
375610`\spad{iiperm(l)} should be local but conditional.
375671`\spad{iipolygamma(x)} should be local but conditional.
375741`\spad{iipow(l)} should be local but conditional.
375896`\spad{iisec(x)} should be local but conditional
375947`\spad{iisech(x)} should be local but conditional
375999`\spad{iisin(x)} should be local but conditional
376050`\spad{iisinh(x)} should be local but conditional
376102`\spad{iisqrt2()} should be local but conditional
376154`\spad{iisqrt3()} should be local but conditional
376206`\spad{iitan(x)} should be local but conditional
376257`\spad{iitanh(x)} should be local but conditional
376309`\spad{imagE(o)} extracts the imaginary \spad{E} part of octonion \spad{o}.
376357`\spad{imagI(o)} extracts the imaginary \spad{I} part of octonion \spad{o}.
376405`\spad{imagI(q)} extracts the imaginary \spad{i} part of quaternion \spad{q}.
376455`\spad{imagJ(o)} extracts the imaginary \spad{J} part of octonion \spad{o}.
376503`\spad{imagJ(q)} extracts the imaginary \spad{j} part of quaternion \spad{q}.
376553`\spad{imagK(o)} extracts the imaginary \spad{K} part of octonion \spad{o}.
376601`\spad{imagK(q)} extracts the imaginary \spad{k} part of quaternion \spad{q}.
376723`\spad{imag(f)} returns the imaginary part of \spad{f} where \spad{f} is a complex function.
376781`\spad{imag(x)} returns imaginary part of \spad{x}.
376827`\spad{imagi(o)} extracts the \spad{i} part of octonion \spad{o}.
376922`\spad{imaginary()} = sqrt(\spad{-1}) = \%\spad{i}.
376972`\spad{imagj(o)} extracts the \spad{j} part of octonion \spad{o}.
377020`\spad{imagk(o)} extracts the \spad{k} part of octonion \spad{o}.
377068`\spad{implies(a,{}b)} returns the logical implication of Boolean \spad{a} and \spad{b}.
377178`return \spad{true} if param is inside bounding box where bounding box is specified by min and max
377237`\spad{inGroundField?(a)} tests whether an element \spad{a} is already in the ground field \spad{F}.
377490`\spad{incFail(s,{} o)} is an internal function that steps the number of failed tests and records in- and output.
377558`\spad{incFatal s} is an internal function that steps the number of fatal tests and records input.
377610`incLibraryError is an internal function that steps the number of failed tests,{} but not the number of tests.
377686`incPass is an internal function that steps the number of passed tests.
377731`\spad{incXfFail()} is an internal function that steps the number expected failures.
377778`like incFatal,{} but for expected failures.
377832`like incLibraryError,{} but using expected failures.
377910`\spad{incXfPass(s,{} o)} is an internal function that steps the number of tests that passed but were expected to fail and records in- and output.
377980`\spad{inc(x)} returns \spad{x + 1}.
378208`\spad{incr(s)} returns \spad{n},{} where \spad{s} is a segment in which every \spad{n}\spad{-}th element is used. Note: \spad{incr(l..h by n) = n}.
378260`\spad{increasePrecision(n)} increases the current \spadfunFrom{precision}{FloatingPointSystem} by \spad{n} decimal digits.
378365`\spad{incrementBy(n)} produces a function which adds \spad{n} to whatever argument it is given. For example,{} if {\spad{f} \spad{:=} increment(\spad{n})} then \spad{f x} is \spad{x+n}.
378521`\spad{increment()} produces a function which adds \spad{1} to whatever argument it is given. For example,{} if {\spad{f} \spad{:=} increment()} then \spad{f x} is \spad{x+1}.
378760`\spad{index?(i,{}u)} tests if \spad{i} is an index of aggregate \spad{u}.
378900`\spad{indexName(d)} specifies the index variable used for the formulas. This option is expressed in the form \spad{indexName == d}.
379158`\spad{index(x)} \undocumented
379211`\spad{index(i)} takes a positive integer \spad{i} less than or equal to \spad{size()} and returns the \spad{i}\spad{-}th element of the set. This operation establishs a bijection between the elements of the finite set and \spad{1..size()}.
379850`\spad{indices(u)} returns a list of indices of aggregate \spad{u} in no particular order.
380145`\spad{indicialEquation(op,{} a)} returns the indicial equation of \spad{op} at \spad{a}.
380212`\spad{indicialEquation(op,{} a)} returns the indicial equation of \spad{op} at \spad{a}.
381061`\axiom{infRittWu?(a,{}\spad{b})} returns \spad{true} if \axiom{a} is less than \axiom{\spad{b}} \spad{w}.\spad{r}.\spad{t}. the Ritt and Wu Wen Tsun ordering using the refinement of Lazard.
381138`\axiom{infRittWu?(\spad{ts1},{}\spad{ts2})} returns \spad{true} iff \axiom{\spad{ts2}} has higher rank than \axiom{\spad{ts1}} in Wu Wen Tsun sense.
381211`\spad{inf(u)} returns the infinum of \axiom{\spad{u}}.
381670`\spad{infiniteProduct(f(x))} computes \spad{product(n=1,{}2,{}3...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
381755`\spad{infiniteProduct(f(x))} computes \spad{product(n=1,{}2,{}3...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
381838`\spad{infiniteProduct(f(x))} computes \spad{product(n=1,{}2,{}3...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
381915`\spad{infinityNorm(f)} returns the maximal absolute value of the coefficients of the polynomial \spad{f}.
381992`\spad{infinity()} returns infinity.
382056`\spad{infinity()} returns infinity.
382108`\spad{infix?(op)} returns \spad{true} if \spad{op} is an infix operator,{} and \spad{false} otherwise.
382154`\spad{infix(f,{}l)} creates a form depicting the \spad{n}-ary application of infix operation \spad{f} to a tuple of arguments \spad{l}.
382203`\spad{infix(op,{} a,{} b)} creates a form which prints as: a \spad{op} \spad{b}.
382249`\axiom{initTable!()} initializes the hash-table.
382513`\spad{init()} chooses an initial object for stepping.
382610`\axiom{init(\spad{p})} returns an error if \axiom{\spad{p}} belongs to \axiom{\spad{R}},{} otherwise returns its leading coefficient,{} where \axiom{\spad{p}} is viewed as a univariate polynomial in its main variable.
382740`\spad{initial(p)} returns the leading coefficient when the differential polynomial \spad{p} is written as a univariate polynomial in its leader.
382811`\spad{initializeGroupForWordProblem(gp)} initializes the group {\em gp} for the word problem. Notes: it calls the other function of this name with parameters 0 and 1: {\em initializeGroupForWordProblem(gp,{}0,{}1)}. Notes: (1) be careful: invoking this routine will destroy the possibly information about your group (but will recompute it again) (2) users need not call this function normally for the soultion of the word problem.
383112`\axiom{initiallyReduce(\spad{p},{}\spad{ts})} returns a polynomial \axiom{\spad{r}} such that \axiom{initiallyReduced?(\spad{r},{}\spad{ts})} holds and there exists some product \axiom{\spad{h}} of \axiom{initials(\spad{ts})} such that \axiom{\spad{h*p} - \spad{r}} lies in the ideal generated by \axiom{\spad{ts}}.
383183`\axiom{initiallyReduce(a,{}\spad{b})} returns a polynomial \axiom{\spad{r}} such that \axiom{initiallyReduced?(\spad{r},{}\spad{b})} holds and there exists an integer \axiom{\spad{e}} such that \axiom{init(\spad{b})^e a - \spad{r}} is zero modulo \axiom{\spad{b}}.
383334`\spad{initiallyReduced?(ts)} returns \spad{true} iff for every element \axiom{\spad{p}} of \axiom{\spad{ts}} \axiom{\spad{p}} and all its iterated initials are reduced \spad{w}.\spad{r}.\spad{t}. to the other elements of \axiom{\spad{ts}} with the same main variable.
383653`\axiom{initiallyReduced?(\spad{p},{}\spad{ts})} returns \spad{true} iff \axiom{\spad{p}} and all its iterated initials are reduced \spad{w}.\spad{r}.\spad{t}. to the elements of \axiom{\spad{ts}} with the same main variable.
383732`\axiom{initiallyReduced?(\spad{q},{}\spad{lp})} returns \spad{true} iff \axiom{initiallyReduced?(\spad{q},{}\spad{p})} holds for every \axiom{\spad{p}} in \axiom{\spad{lp}}.
383822`\axiom{initiallyReduced?(a,{}\spad{b})} returns \spad{false} iff there exists an iterated initial of \axiom{a} which is not reduced \spad{w}.\spad{r}.\spad{t} \axiom{\spad{b}}.
383971`\axiom{initials(\spad{ts})} returns the list of the non-constant initials of the members of \axiom{\spad{ts}}.
385024`\axiom{insert!(\spad{x},{}\spad{y})} stores the item whose key is \axiom{\spad{x}} and whose entry is \axiom{\spad{y}}.
385184`\spad{insert!(x,{}u)} inserts item \spad{x} into bag \spad{u}.
385233`\spad{insert!(x,{}b)} inserts element \spad{x} as leaves into binary search tree \spad{b}.
385286`\spad{insert!(x,{}b)} inserts element \spad{x} as leaves into binary tournament \spad{b}.
385447`\spad{insert!(x,{}u,{}i)} destructively inserts \spad{x} into \spad{u} at position \spad{i}.
385517`\spad{insert!(x,{}d,{}n)} destructively inserts \spad{n} copies of \spad{x} into dictionary \spad{d}.
385588`\spad{insert!(v,{}u,{}i)} destructively inserts aggregate \spad{v} into \spad{u} at position \spad{i}.
385659`\spad{insertBottom!(x,{}d)} destructively inserts \spad{x} into the dequeue \spad{d} at the bottom (back) of the dequeue.
385717`\spad{insertMatch(var,{} expr,{} r)} adds the match (\spad{var},{} \spad{expr}) in \spad{r},{} without checking predicates or previous matches for \spad{var}.
385789`\spad{insertRoot!(x,{}b)} inserts element \spad{x} as a root of binary search tree \spad{b}.
385846`\spad{insertTop!(x,{}d)} destructively inserts \spad{x} into the dequeue \spad{d},{} that is,{} at the top (front) of the dequeue. The element previously at the top of the dequeue becomes the second in the dequeue,{} and so on.
386395`\spad{insert(x,{}u,{}i)} returns a copy of \spad{u} having \spad{x} as its \axiom{\spad{i}}th element. Note: \axiom{insert(\spad{x},{}a,{}\spad{k}) = concat(concat(a(0..\spad{k}-1),{}\spad{x}),{}a(\spad{k}..))}.
386454`\spad{insert(v,{}u,{}k)} returns a copy of \spad{u} having \spad{v} inserted beginning at the \axiom{\spad{i}}th element. Note: \axiom{insert(\spad{v},{}\spad{u},{}\spad{k}) = concat( \spad{u}(0..\spad{k}-1),{} \spad{v},{} \spad{u}(\spad{k}..) )}.
386514`\spad{insertionSort! }\undocumented
386584`\spad{insertionSort!(a,{}f)} \undocumented
386731`\spad{inspect(u)} returns an (random) element from a bag.
386777`\spad{intBasis LLv} computes a basis of the intersection of list of subspaces.
386848`\spad{intBasis Lb} computes a basis of the intersection of list of subspaces.
386906`\spad{intBasis(Lv1,{}Lv2)} computes a basis of the intersection of the two subspaces spanned by the vectors of \spad{Lv1} and \spad{Lv2}
386987`\spad{intBasis(B1,{}B2)} computes a basis of the intersection of the two subspaces.
387710`\spad{int(expr)} creates the form prefixing \spad{expr} with an integral sign.
387795`\spad{int(expr,{}lowerlimit)} creates the form prefixing \spad{expr} by an integral sign with a \spad{lowerlimit}.
387836`\spad{int(expr,{}lowerlimit,{}upperlimit)} creates the form prefixing \spad{expr} by an integral sign with both a \spad{lowerlimit} and \spad{upperlimit}.
388125`\spad{integer?(t)} tests whether \spad{t} is equivalent to the FORTRAN type INTEGER.
388180`\spad{integer?(x)} is \spad{true} if \spad{x} is an integer,{} \spad{false} otherwise.
388238`\spad{integer?(s)} is \spad{true} if \spad{s} is an atom and belong to Int.
388377`\spad{integerIfCan(x)} returns \spad{x} as an integer,{} "failed" if \spad{x} is not an integer.
388455`\spad{integer(x)} returns \spad{x} as an integer; error if \spad{x} is not an integer.
388512`\spad{integer(s)} returns \spad{s} as an element of Int. Error: if \spad{s} is not an atom that also belongs to Int.
388731`\spad{integral?()} tests if \spad{f} is integral over \spad{k[x]}.
388943`\spad{integral?(f,{} a)} tests whether \spad{f} is locally integral at \spad{x = a}.
389016`\spad{integral?(f,{} p)} tests whether \spad{f} is locally integral at \spad{p(x) = 0}.
389168`\spad{integralAtInfinity?()} tests if \spad{f} is locally integral at infinity.
389249`\spad{integralBasisAtInfinity()} returns the local integral basis at infinity.
389830`\spad{integralBasis()} returns the integral basis for the curve.
389906`\spad{integralCoordinates(f)} returns \spad{[[A1,{}...,{}An],{} D]} such that \spad{f = (A1 w1 +...+ An wn) / D} where \spad{(w1,{}...,{}wn)} is the integral basis returned by \spad{integralBasis()}.
390009`\spad{integralDerivationMatrix(d)} extends the derivation \spad{d} from UP to \$ and returns (\spad{M},{} \spad{Q}) such that the i^th row of \spad{M} divided by \spad{Q} form the coordinates of \spad{d(\spad{wi})} with respect to \spad{(w1,{}...,{}wn)} where \spad{(w1,{}...,{}wn)} is the integral basis returned by integralBasis().
390413`\spad{integralRepresents([A1,{}...,{}An],{} D)} returns \spad{(A1 w1+...+An wn)/D} where \spad{(w1,{}...,{}wn)} is the integral basis of \spad{integralBasis()}.
390499`\spad{integral(f,{}x)} returns the formal integral of \spad{f} with respect to \spad{x}
390553`\spad{integral(f,{}x = a..b)} denotes the definite integral of \spad{f} with respect to \spad{x} from \spad{a} to \spad{b}.
390683`\spad{integral(f,{}x)} returns the formal integral of \spad{f} with respect to \spad{x}
390770`\spad{integral(f,{} x = a..b)} returns the formal definite integral of \spad{f} \spad{dx} for \spad{x} between \spad{a} and \spad{b}.
390847`\spad{integral(f,{} x)} returns the formal integral of \spad{f} \spad{dx}.
391066`\spad{integrate(f(x))} returns an anti-derivative of the power series \spad{f(x)} with constant coefficient 0. Warning: function does not check for a term of degree \spad{-1}.
391174`\spad{integrate(f(x))} returns an anti-derivative of the power series \spad{f(x)} with constant coefficient 1. We may integrate a series when we can divide coefficients by integers.
391281`\spad{integrate(p)} integrates the univariate polynomial \spad{p} with respect to its distinguished variable.
391379`\spad{integrate(f(x))} returns an anti-derivative of the power series \spad{f(x)} with constant coefficient 1. We may integrate a series when we can divide coefficients by rational numbers.
391486`\spad{integrate(f(x))} returns an anti-derivative of the power series \spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.
391812`\spad{integrate(f(x))} returns an anti-derivative of the power series \spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.
391939`\spad{integrate(f(x))} returns an anti-derivative of the power series \spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.
392066`\spad{integrate(f(x),{}x)} returns an anti-derivative of the power series \spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.
392192`\spad{integrate(f(x))} returns an anti-derivative of the power series \spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.
392313`\spad{integrate(f(x))} returns an anti-derivative of the power series \spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.
392434`\spad{integrate(f(x),{}x)} returns an anti-derivative of the power series \spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.
393502`\spad{integrate(f(x),{}y)} returns an anti-derivative of the power series \spad{f(x)} with respect to the variable \spad{y}.
393750`\spad{integrate(f(x),{}y)} returns an anti-derivative of the power series \spad{f(x)} with respect to the variable \spad{y}.
393968`\spad{integrate(f(x),{}y)} returns an anti-derivative of the power series \spad{f(x)} with respect to the variable \spad{y}.
394216`\spad{integrate(f(x),{}y)} returns an anti-derivative of the power series \spad{f(x)} with respect to the variable \spad{y}.
394434`\spad{integrate(f(x),{}y)} returns an anti-derivative of the power series \spad{f(x)} with respect to the variable \spad{y}.
394681`\spad{integrate(f(x),{}y)} returns an anti-derivative of the power series \spad{f(x)} with respect to the variable \spad{y}.
394898`\spad{integrate(p,{}x)} computes the integral of \spad{p*dx},{} \spadignore{i.e.} integrates the polynomial \spad{p} with respect to the variable \spad{x}.
394985`\spad{integrate(f,{}x)} returns the anti-derivative of the power series \spad{f(x)} with respect to the variable \spad{x} with constant coefficient 1. We may integrate a series when we can divide coefficients by integers.
395101`\spad{integrate(f(x))} returns an anti-derivative of the power series \spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.
395227`\spad{integrate(s,{}v,{}c)} is the integral of \spad{s} with respect \indented{1}{to \spad{v} and having \spad{c} as the constant of integration.}
396089`\spad{integrate(s,{}v,{}c)} is the integral of \spad{s} with respect \indented{1}{to \spad{v} and having \spad{c} as the constant of integration.}
396376`\spad{internalAugment(lp,{}ts)} returns \spad{ts} if \spad{lp} is empty otherwise returns \spad{internalAugment(rest lp,{} internalAugment(first lp,{} ts))}
396461`\spad{internalAugment(p,{}ts)} assumes that \spad{augment(p,{}ts)} returns a singleton and returns it.
398728`\axiom{internalSubQuasiComponent?(\spad{ts},{}us)} returns a boolean \spad{b} value if the fact that the regular zero set of \axiom{us} contains that of \axiom{\spad{ts}} can be decided (and in that case \axiom{\spad{b}} gives this inclusion) otherwise returns \axiom{"failed"}.
398836`\axiom{internalSubQuasiComponent?(\spad{ts},{}us)} returns a boolean \spad{b} value if the fact the regular zero set of \axiom{us} contains that of \axiom{\spad{ts}} can be decided (and in that case \axiom{\spad{b}} gives this inclusion) otherwise returns \axiom{"failed"}.
399804`\spad{interpretString(s)} treats a string as a piece of FriCAS input,{} by parsing and interpreting it.
399866`\spad{interpret(f)} passes \spad{f} to the interpreter,{} and transforms the result into an object of type \spad{R}.
399928`\spad{interpret(f)} passes \spad{f} to the interpreter.
400421`\spad{intersect(lp,{}lts)} returns the same as \spad{concat([intersect(lp,{}ts) for ts in lts])|}
400512`\spad{intersect(lp,{}ts)} returns \spad{lts} a split of Lazard of the intersection of the affine variety associated with \spad{lp} and the regular zero set of \spad{ts}.
400597`\spad{intersect(p,{}lts)} returns the same as \spad{intersect([p],{}lts)}
400682`\spad{intersect(p,{}ts)} returns the same as \spad{intersect([p],{}ts)}
400761`\spad{intersect(u,{}v)} returns the set aggregate \spad{w} consisting of elements common to both set aggregates \spad{u} and \spad{v}. Note: equivalent to the notation (not currently supported) {\spad{x} for \spad{x} in \spad{u} | member?(\spad{x},{}\spad{v})}.
400882`\spad{interval(f)} creates a new interval around \spad{f}.
400949`\spad{interval(f)} creates a new interval around \spad{f}.
401000`\spad{interval(inf,{}sup)} creates a new interval,{} either \axiom{[\spad{inf},{}\spad{sup}]} if \axiom{\spad{inf} \spad{<=} \spad{sup}} or \axiom{[\spad{sup},{}in]} otherwise.
401422`\spad{inv(x)} \undocumented
401504`\spad{inv(x)} \undocumented
401575`\spad{inv x} returns the multiplicative inverse of \spad{x}. Error: if \spad{x} is 0.
401615`\spad{inv(x)} returns the inverse of \spad{x}.
401648`\spad{inv(o)} returns the inverse of \spad{o} if it exists.
401707`\spad{inv(x)} returns the multiplicative inverse of \spad{x}.
402212`\spad{inverse(m)} returns the inverse of the matrix \spad{m}. If the matrix is not invertible,{} "failed" is returned. Error: if the matrix is not square.
402304`\spad{inverse(m)} returns the inverse of the matrix \spad{m}. If the matrix is not invertible,{} "failed" is returned. Error: if the matrix is not square. Note: the result will have entries in the quotient field.
402413`\spad{inverse(m)} returns the inverse of the matrix. If the matrix is not invertible,{} "failed" is returned. Error: if the matrix is not square.
402512`\spad{inverse(m)} returns the inverse of the matrix \spad{m}. If the matrix is not invertible,{} "failed" is returned. Error: if the matrix is not square.
402597`\spad{inverse(m)} returns the inverse of the matrix \spad{m},{} if that matrix is invertible and returns "failed" otherwise.
402693`\spad{inverse(m)} returns the inverse of the matrix \spad{m}. If the matrix is not invertible,{} "failed" is returned. Error: if the matrix is not square.
402762`\spad{invertIfCan(m)} returns the inverse of \spad{m} over \spad{R}
402874`\spad{invertible?(p,{}ts)} returns \spad{true} iff \spad{p} is invertible in the tower associated with \spad{ts}.
402954`\spad{invertible?(p,{}ts)} returns \spad{lbwt} where \spad{lbwt.i} is the result of \spad{invertibleElseSplit?(p,{}lbwt.i.tower)} and the list of the \spad{(lqrwt.i).tower} is a split of Kalkbrener of \spad{ts}.
403061`\spad{invertibleElseSplit?(p,{}ts)} returns \spad{true} (resp. \spad{false}) if \spad{p} is invertible in the tower associated with \spad{ts} or returns a split of Kalkbrener of \spad{ts}.
403166`\spad{invertibleSet(p,{}ts)} returns a split of Kalkbrener of the quotient ideal of the ideal \axiom{\spad{I}} by \spad{p} where \spad{I} is the radical of saturated of \spad{ts}.
403298`\spad{invmod(a,{}b)},{} \spad{0<=a<b>1},{} \spad{(a,{}b)=1} means \spad{1/a mod b}.
403659`\spad{ipow(l)} should be local but conditional.
403718`\axiom{iprint(\spad{s})} prints \axiom{\spad{s}} at the current position of the cursor.
403985`\spad{irreducible?(p)} tests whether the polynomial \spad{p} is irreducible.
404056`\spad{irreducibleFactor(base,{}exponent)} creates a factored object with a single factor whose \spad{base} is asserted to be irreducible (flag = "irred").
404743`\spad{is?(op(a1,{}...,{}an),{} s)} tests if the name of op is \spad{s}.
404792`\spad{is?([e1,{}...,{}en],{} pat)} tests if the list of expressions \spad{[e1,{}...,{}en]} matches the pattern pat.
404870`\spad{is?(expr,{} pat)} tests if the expression \spad{expr} matches the pattern pat.
405049`\spad{is?(op,{} s)} tests if the name of \spad{op} is \spad{s}.
405308`\spad{isBasis? Lv} tests if a list of vector is a basis,{} without linear relations between the vectors.
405378`\spad{isExpt(p)} returns \spad{[x,{} n]} if \spad{p = x^n} and \spad{n <> 0},{} "failed" otherwise.
405915`\spad{isExpt(p)} returns \spad{[x,{} n]} if polynomial \spad{p} has the form \spad{x^n} and \spad{n > 0}.
406482`\spad{isList(p)} returns \spad{[a1,{}...,{}an]} if \spad{p = [a1,{}...,{}an]},{} "failed" otherwise.
406906`\spad{isPlus(p)} returns [\spad{m1},{}...,{}\spad{mn}] if \spad{p = m1 + ... + mn} and \spad{n > 1},{} "failed" otherwise.
407147`\spad{isPlus(p)} returns \spad{[a1,{}...,{}an]} if \spad{n > 1} \indented{1}{and \spad{p = a1 + ... + an},{}} and "failed" otherwise.
407210`\spad{isPlus(p)} returns \spad{[m1,{}...,{}mn]} if \spad{p = m1 +...+ mn} and \spad{n > 1}.
407302`\spad{isPlus(p)} returns \spad{[m1,{}...,{}mn]} if polynomial \spad{p = m1 + ... + mn} and \spad{n >= 2} and each \spad{mi} is a nonzero monomial.
407385`\spad{isPower(p)} returns \spad{[x,{} n]} if \spad{p = x^n} and \spad{n <> 0},{} "failed" otherwise.
407595`\spad{isPower(p)} returns \spad{[a,{} b]} if \spad{p = a ^ b},{} and "failed" otherwise.
407851`\spad{isTimes(p)} returns \spad{[a1,{}...,{}an]} if \spad{p = a1 ... an} and \spad{n > 1},{} "failed" otherwise.
408095`\spad{isTimes(p)} returns \spad{[a1,{}...,{}an]} if \spad{n > 1} and \spad{p = a1 * ... * an},{} and "failed" otherwise.
408159`\spad{isTimes(p)} returns \spad{[a1,{}...,{}an]} if \spad{p = a1*...*an} and \spad{n > 1}.
408245`\spad{isTimes(p)} returns \spad{[a1,{}...,{}an]} if polynomial \spad{p = a1 ... an} and \spad{n >= 2},{} and,{} for each \spad{i},{} \spad{ai} is either a nontrivial constant in \spad{R} or else of the form \spad{x^e},{} where \spad{e > 0} is an integer and \spad{x} in a member of VarSet.
408404`\spad{isobaric?(p)} returns \spad{true} if every differential monomial appearing in the differential polynomial \spad{p} has same weight,{} and returns \spad{false} otherwise.
408638`\axiom{iteratedInitials(\spad{p})} returns \axiom{[]} if \axiom{\spad{p}} belongs to \axiom{\spad{R}},{} otherwise returns the list of the iterated initials of \axiom{\spad{p}}.
408719`\spad{jacobiCn(z,{} m)} is the Jacobi elliptic \spad{cn} function,{} defined by \spad{jacobiCn(z,{} m)^2 + jacobiSn(z,{} m)^2 = 1} and \spad{jacobiCn(0,{} m) = 1}
408782`\spad{jacobiCn(z,{} m)} expands the elliptic function \spad{cn} as a Taylor \indented{1}{series.}
408871`\spad{jacobiCn(z,{} m)} is the Jacobi elliptic \spad{cn} function,{} defined by \spad{jacobiCn(z,{} m)^2 + jacobiSn((z,{} m)^2 = 1} and \spad{jacobiCn(0,{} m) = 1}.
408930`\spad{jacobiDn(z,{} m)} is the Jacobi elliptic \spad{dn} function,{} defined by \spad{jacobiDn(z,{} m)^2 + m*jacobiSn(z,{} m)^2 = 1} and \spad{jacobiDn(0,{} m) = 1}
408993`\spad{jacobiDn(z,{} m)} expands the elliptic function \spad{dn} as a Taylor \indented{1}{series.}
409082`\spad{jacobiDn(z,{} m)} is the Jacobi elliptic \spad{dn} function,{} defined by \spad{jacobiDn(z,{} m)^2 + m*jacobiSn((z,{} m)^2 = 1} and \spad{jacobiDn(0,{} m) = 1}.
409141`\spad{jacobiIdentity?()} tests if \spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} for all \spad{a},{}\spad{b},{}\spad{c} in the algebra. For example,{} this holds for crossed products of 3-dimensional vectors.
409221`\spad{jacobiIdentity?()} tests if \spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} for all \spad{a},{}\spad{b},{}\spad{c} in the algebra. For example,{} this holds for crossed products of 3-dimensional vectors.
409298`\spad{jacobiSn(z,{} m)} is the Jacobi elliptic \spad{sn} function,{} defined by the formula \spad{jacobiSn(ellipticF(z,{} m),{} m) = z}
409361`\spad{jacobiSn(z,{} m)} expands the elliptic function \spad{sn} as a Taylor \indented{1}{series.}
409450`\spad{jacobiSn(z,{} m)} is the Jacobi elliptic \spad{sn} function,{} defined by the formula \spad{jacobiSn(ellipticF(z,{} m),{} m) = z}.
409509`\spad{jacobiTheta(q,{} z)} is the third Jacobi Theta function
409575`\spad{jacobiTheta(q,{} z)} is the third Jacobi Theta function.
409709`\spad{jacobian(vf,{}xlist)} computes the jacobian,{} the matrix of first partial derivatives,{} of the vector field \spad{vf},{} \spad{vf} a vector function of the variables listed in \spad{xlist}.
409956`\spad{jordanAdmissible?()} tests if 2 is invertible in the coefficient domain and the multiplication defined by \spad{(1/2)(a*b+b*a)} determines a Jordan algebra,{} \spadignore{i.e.} satisfies the Jordan identity. The property of \spadatt{CommutativeStar)} follows from by definition.
410038`\spad{jordanAdmissible?()} tests if 2 is invertible in the coefficient domain and the multiplication defined by \spad{(1/2)(a*b+b*a)} determines a Jordan algebra,{} \spadignore{i.e.} satisfies the Jordan identity. The property of \spadatt{CommutativeStar)} follows from by definition.
410117`\spad{jordanAlgebra?()} tests if the algebra is commutative,{} characteristic is not 2,{} and \spad{(a*b)*a^2 - a*(b*a^2) = 0} for all \spad{a},{}\spad{b},{}\spad{c} in the algebra (Jordan identity). Example: for every associative algebra \spad{(A,{}+,{}@)} we can construct a Jordan algebra \spad{(A,{}+,{}*)},{} where \spad{a*b := (a@b+b@a)/2}.
410196`\spad{jordanAlgebra?()} tests if the algebra is commutative,{} characteristic is not 2,{} and \spad{(a*b)*a^2 - a*(b*a^2) = 0} for all \spad{a},{}\spad{b},{}\spad{c} in the algebra (Jordan identity). Example: for every associative algebra \spad{(A,{}+,{}@)} we can construct a Jordan algebra \spad{(A,{}+,{}*)},{} where \spad{a*b := (a@b+b@a)/2}.
410387`\spad{karatsubaDivide(p,{}n)} returns the same as \spad{monicDivide(p,{}monomial(1,{}n))}
410507`\spad{karatsuba(a,{}b)} returns \spad{a*b} by applying Karatsuba\spad{'s} trick once. The other multiplications are performed by calling \spad{*} from \spad{U}.
410591`\spad{karatsuba(a,{}b,{}l,{}k)} returns \spad{a*b} by applying Karatsuba\spad{'s} trick provided that both \spad{a} and \spad{b} have at least \spad{l} terms and \spad{k > 0} holds and by calling \spad{noKaratsuba} otherwise. The other multiplications are performed by recursive calls with the same third argument and \spad{k-1} as fourth argument.
410709`\spad{kelvinBei(v,{} z)} is the Kelvin bei function defined by equality \spad{kelvinBei(v,{} z) = imag(besselJ(v,{} exp(3*\%pi*\%i/4)*z))} for \spad{z} and \spad{v} real
410773`\spad{kelvinBei(v,{} z)} is the Kelvin bei function defined by equality \spad{kelvinBei(v,{} z) = imag(besselJ(v,{} exp(3*\%pi*\%i/4)*z)} for \spad{z} and \spad{v} real.
410833`\spad{kelvinBer(v,{} z)} is the Kelvin ber function defined by equality \spad{kelvinBer(v,{} z) = real(besselJ(v,{} exp(3*\%pi*\%i/4)*z))} for \spad{z} and \spad{v} real
410897`\spad{kelvinBer(v,{} z)} is the Kelvin ber function defined by equality \spad{kelvinBer(v,{} z) = real(besselJ(v,{} exp(3*\%pi*\%i/4)*z)} for \spad{z} and \spad{v} real.
410957`\spad{kelvinKei(v,{} z)} is the Kelvin kei function defined by equality \spad{kelvinKei(v,{} z) = imag(exp(-v*\%pi*\%i/2)*besselK(v,{} exp(\%pi*\%i/4)*z))} for \spad{z} and \spad{v} real
411021`\spad{kelvinKei(v,{} z)} is the Kelvin kei function defined by equality \spad{kelvinKei(v,{} z) = imag(exp(-v*\%pi*\%i/2)*besselK(v,{} exp(\%pi*\%i/4)*z)} for \spad{z} and \spad{v} real.
411081`\spad{kelvinKer(v,{} z)} is the Kelvin kei function defined by equality \spad{kelvinKer(v,{} z) = real(exp(-v*\%pi*\%i/2)*besselK(v,{} exp(\%pi*\%i/4)*z))} for \spad{z} and \spad{v} real
411145`\spad{kelvinKer(v,{} z)} is the Kelvin kei function defined by equality \spad{kelvinKer(v,{} z) = real(exp(-v*\%pi*\%i/2)*besselK(v,{} exp(\%pi*\%i/4)*z)} for \spad{z} and \spad{v} real.
411205`\spad{kernel(x)} returns \spad{x} viewed as a kernel.
411733`\spad{key?(k,{}t)} tests if \spad{k} is a key in table \spad{t}.
411996`\spad{keys(t)} returns the list the keys in table \spad{t}.
412494`\spad{kroneckerDelta()} is the rank 2 tensor defined by \indented{3}{\spad{kroneckerDelta()(i,{}j)}} \indented{6}{\spad{= 1\space{2}if i = j}} \indented{6}{\spad{= 0 if\space{2}i \~= j}}
412641`\spad{kummerM(a,{} b,{} z)} is the Kummer \spad{M} function
412705`\spad{kummerM(mu,{} nu,{} z)} is the Kummer \spad{M} function.
412766`\spad{kummerU(a,{} b,{} z)} is the Kummer \spad{U} function
412830`\spad{kummerU(mu,{} nu,{} z)} is the Kummer \spad{U} function.
412891`\spad{lSpaceBasis(d)} returns a basis for \spad{L(d) = {f | (f) >= -d}} as a module over \spad{K[x]}.
412960`\spad{label(n,{}f)} gives form \spad{f} an equation label \spad{n}.
413137`\spad{lagrange(g(x))} produces the Taylor series for \spad{f(x)} \indented{1}{where \spad{f(x)} is implicitly defined as \spad{f(x) = x*g(f(x))}.}
413478`\spad{lambda(code,{} [x1,{}...,{}xn])} returns the input form corresponding to \spad{(x1,{}...,{}xn) +-> code} if \spad{n > 1},{} or to \spad{x1 +-> code} if \spad{n = 1}.
413531`\spad{lambertW(x)} is the Lambert \spad{W} function at \spad{x}
413592`\spad{lambertW(z)} = \spad{w} is the principial branch of the solution to the equation \spad{we^w = z}.
413780`\spad{lambert(f(x))} returns \spad{f(x) + f(x^2) + f(x^3) + ...}. \indented{1}{This function is used for computing infinite products.} \indented{1}{\spad{f(x)} should have zero constant coefficient.} \indented{1}{If \spad{f(x)} is a Taylor series with constant term 1,{} then} \indented{1}{\spad{product(n = 1..infinity,{}f(x^n)) = exp(log(lambert(f(x))))}.}
413908`\spad{laplacian(v,{}xlist)} computes the laplacian of the scalar field \spad{v},{} \spad{v} a function of the variables listed in \spad{xlist}.
413990`\spad{largest l} returns the largest element of \spad{l} where the partial ordering induced by setOrder is completed into a total one by the ordering on \spad{S}.
414072`\spad{largest(l,{} fn)} returns the largest element of \spad{l} where the partial ordering induced by setOrder is completed into a total one by \spad{fn}.
414152`\spad{lastSubResultantElseSplit(p1,{}p2,{}ts)} returns either \spad{g} a quasi-monic \spad{gcd} of \spad{p1} and \spad{p2} \spad{w}.\spad{r}.\spad{t}. the \spad{ts} or a split of Kalkbrener of \spad{ts}. This assumes that \spad{p1} and \spad{p2} have the same maim variable and that this variable is greater that any variable occurring in \spad{ts}.
414258`\axiom{lastSubResultantEuclidean(\spad{P},{} \spad{Q})} computes the last non zero subresultant \axiom{\spad{S}} and carries out the equality \axiom{coef1*P + coef2*Q = \spad{S}}.
414391`\axiom{lastSubResultant(a,{}\spad{b})} returns \axiom{resultant(a,{}\spad{b})} if \axiom{a} and \axiom{\spad{b}} has no non-trivial \spad{gcd} in \axiom{\spad{R^}(\spad{-1}) \spad{P}} otherwise the non-zero sub-resultant with smallest index.
414488`\axiom{lastSubResultant(\spad{P},{} \spad{Q})} computes the last non zero subresultant of \axiom{\spad{P}} and \axiom{\spad{Q}}
414569`\axiom{lastSubResultant(a,{}\spad{b})} returns the last non-zero subresultant of \axiom{a} and \axiom{\spad{b}} where \axiom{a} and \axiom{\spad{b}} are assumed to have the same main variable \axiom{\spad{v}} and are viewed as univariate polynomials in \axiom{\spad{v}}.
414668`\spad{lastSubResultant(p1,{}p2,{}ts)} returns \spad{lpwt} such that \spad{lpwt.i.val} is a quasi-monic \spad{gcd} of \spad{p1} and \spad{p2} \spad{w}.\spad{r}.\spad{t}. \spad{lpwt.i.tower},{} for every \spad{i},{} and such that the list of the \spad{lpwt.i.tower} is a split of Kalkbrener of \spad{ts}. Moreover,{} if \spad{p1} and \spad{p2} do not have a non-trivial \spad{gcd} \spad{w}.\spad{r}.\s
414668`--pad{t}. \spad{lpwt.i.tower} then \spad{lpwt.i.val} is the resultant of these polynomials \spad{w}.\spad{r}.\spad{t}. \spad{lpwt.i.tower}. This assumes that \spad{p1} and \spad{p2} have the same maim variable and that this variable is greater that any variable occurring in \spad{ts}.
414945`\spad{last(l)} returns the last element of a doubly-linked aggregate \spad{l}. Error: if \spad{l} is empty.
414997`\spad{last(u)} resturn the last element of \spad{u}. Note: for lists,{} \axiom{last(\spad{u}) = \spad{u} . (maxIndex \spad{u}) = \spad{u} . (\# \spad{u} - 1)}.
415051`\axiom{last(\spad{ts})} returns the polynomial of \axiom{\spad{ts}} with smallest main variable if \axiom{\spad{ts}} is not empty,{} otherwise returns \axiom{"failed"}.
415261`\spad{last(u,{}n)} returns a copy of the last \spad{n} (\axiom{\spad{n} \spad{>=} 0}) nodes of \spad{u}. Note: \axiom{last(\spad{u},{}\spad{n})} is a list of \spad{n} elements.
415378`\spad{latex(s)} returns a LaTeX-printable output representation of \spad{s}.
415423`\spad{laurentIfCan(f(x))} converts the Puiseux series \spad{f(x)} to a Laurent series if possible. If this is not possible,{} "failed" is returned.
415529`\spad{laurentRep(f(x))} returns \spad{g(x)} where the Puiseux series \spad{f(x) = g(x^r)} is represented by \spad{[r,{}g(x)]}.
415617`\spad{laurent(f)} returns a Laurent expansion of the expression \spad{f}. Note: \spad{f} should have only one variable; the series will be expanded in powers of that variable.
415760`\spad{laurent(f(x))} converts the Puiseux series \spad{f(x)} to a Laurent series if possible. Error: if this is not possible.
416398`\spad{lazy?(s)} returns \spad{true} if the first node of the stream \spad{s} is a lazy evaluation mechanism which could produce an additional entry to \spad{s}.
416455`\spad{lazyEvaluate(s)} causes one lazy evaluation of stream \spad{s}. Caution: the first node must be a lazy evaluation mechanism (satisfies \spad{lazy?(s) = true}) as there is no error check. Note: a call to this function may or may not produce an explicit first entry
416949`\axiom{lazyPquo(a,{}\spad{b})} returns the polynomial \axiom{\spad{q}} such that \axiom{lazyPseudoDivide(a,{}\spad{b})} returns \axiom{[\spad{c},{}\spad{g},{}\spad{q},{}\spad{r}]}.
417086`\axiom{lazyPquo(a,{}\spad{b},{}\spad{v})} returns the polynomial \axiom{\spad{q}} such that \axiom{lazyPseudoDivide(a,{}\spad{b},{}\spad{v})} returns \axiom{[\spad{c},{}\spad{g},{}\spad{q},{}\spad{r}]}.
417282`\axiom{lazyPremWithDefault(a,{}\spad{b})} returns \axiom{[\spad{c},{}\spad{g},{}\spad{r}]} such that \axiom{\spad{r} = lazyPrem(a,{}\spad{b})} and \axiom{(\spad{c^g})\spad{*r} = prem(a,{}\spad{b})}.
417538`\axiom{lazyPremWithDefault(a,{}\spad{b},{}\spad{v})} returns \axiom{[\spad{c},{}\spad{g},{}\spad{r}]} such that \axiom{\spad{r} = lazyPrem(a,{}\spad{b},{}\spad{v})} and \axiom{(\spad{c^g})\spad{*r} = prem(a,{}\spad{b},{}\spad{v})}.
417735`\axiom{lazyPrem(a,{}\spad{b})} returns the polynomial \axiom{\spad{r}} reduced \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{b}} and such that \axiom{\spad{b}} divides \axiom{init(\spad{b})^e a - \spad{r}} where \axiom{\spad{e}} is the number of steps of this pseudo-division.
417872`\axiom{lazyPrem(a,{}\spad{b},{}\spad{v})} returns the polynomial \axiom{\spad{r}} reduced \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{b}} viewed as univariate polynomials in the variable \axiom{\spad{v}} such that \axiom{\spad{b}} divides \axiom{init(\spad{b})^e a - \spad{r}} where \axiom{\spad{e}} is the number of steps of this pseudo-division.
418207`\axiom{lazyPseudoDivide(a,{}\spad{b})} returns \axiom{[\spad{c},{}\spad{g},{}\spad{q},{}\spad{r}]} such that \axiom{[\spad{c},{}\spad{g},{}\spad{r}] = lazyPremWithDefault(a,{}\spad{b})} and \axiom{\spad{q}} is the pseudo-quotient computed in this lazy pseudo-division.
418480`\axiom{lazyPseudoDivide(a,{}\spad{b},{}\spad{v})} returns \axiom{[\spad{c},{}\spad{g},{}\spad{q},{}\spad{r}]} such that \axiom{\spad{r} = lazyPrem(a,{}\spad{b},{}\spad{v})},{} \axiom{(\spad{c^g})\spad{*r} = prem(a,{}\spad{b},{}\spad{v})} and \axiom{\spad{q}} is the pseudo-quotient computed in this lazy pseudo-division.
418621`\axiom{lazyPseudoQuotient(a,{}\spad{b})} returns \axiom{\spad{q}} if \axiom{lazyPseudoDivide(a,{}\spad{b})} returns \axiom{[\spad{c},{}\spad{g},{}\spad{q},{}\spad{r}]}
418699`\axiom{lazyPseudoRemainder(a,{}\spad{b})} returns \axiom{\spad{r}} if \axiom{lazyResidueClass(a,{}\spad{b})} returns \axiom{[\spad{r},{}\spad{c},{}\spad{n}]}. This lazy pseudo-remainder is computed by means of the \axiomOpFrom{fmecg}{NewSparseUnivariatePolynomial} operation.
419020`\axiom{lazyResidueClass(a,{}\spad{b})} returns \axiom{[\spad{p},{}\spad{q},{}\spad{n}]} where \axiom{\spad{p} / \spad{q^n}} represents the residue class of \axiom{a} modulo \axiom{\spad{b}} and \axiom{\spad{p}} is reduced \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{b}} and \axiom{\spad{q}} is \axiom{init(\spad{b})}.
419299`left contraction inner product
419394`\spad{lcm(l)} returns the least common multiple of the elements of the list \spad{l}.
419473`\spad{lcm(x,{}y)} returns the least common multiple of \spad{x} and \spad{y}.
419685`\spad{leader(p)} returns the derivative of the highest rank appearing in the differential polynomial \spad{p} Note: an error occurs if \spad{p} is in the ground ring.
419754`\spad{leadingBasisTerm(p)} returns the leading basis term of antisymmetric polynomial \spad{p}.
419811`\spad{leadingBasisTerm(df)} returns the leading basis term of differential form \spad{df}.
420242`\spad{leadingCoefficient(p)} returns the leading coefficient of antisymmetric polynomial \spad{p}.
420300`\spad{leadingCoefficient(x)} \undocumented
420381`\spad{leadingCoefficient }\undocumented
420446`\spad{leadingCoefficient(z)} returns the coefficient of the leading (with respect to the ordering on the indexing set) monomial of \spad{z}. Error: if \spad{z} has no support.
420521`\spad{leadingCoefficient(f)} returns the coefficient of the lowest order term of \spad{f}
420601`\spad{leadingCoefficient(p)} returns the coefficient highest degree term of \spad{p}.
420665`\spad{leadingCoefficient(x)} returns the first coefficient which appears in \spad{listOfTerms(x)}.
420730`\spad{leadingCoefficient(l)} is \spad{a(n)} if \indented{2}{\spad{l = sum(monomial(a(i),{}i),{} i = 0..n)}.}
420798`\spad{leadingCoefficient(f)} gives the coefficient of \spad{f},{} whose corresponding monoid element is the greatest among all those with non-zero coefficients.
420880`\spad{leadingCoefficient(l)} is \spad{a(n)} if \indented{2}{\spad{l = sum(monomial(a(i),{}i),{} i = 0..n)}.}
421032`\axiom{leadingCoefficient(\spad{p},{}\spad{v})} returns the leading coefficient of \axiom{\spad{p}},{} where \axiom{\spad{p}} is viewed as A univariate polynomial in \axiom{\spad{v}}.
421111`\spad{leadingExponent(x)} \undocumented
421258`\spad{leadingIndex(x)} \undocumented
421506`\spad{leadingMonomial(f)} gives the monomial of \spad{f} whose corresponding monoid element is the greatest among all those with non-zero coefficients.
421585`\spad{leadingMonomial(x)} returns the first element from \spad{S} which appears in \spad{listOfTerms(x)}.
421647`\spad{leadingMonomial(p)} returns the monomial of \spad{p} with the highest degree.
421709`\spad{leadingMonomial(f)} returns the monomial of \spad{f} of lowest order.
421784`\spad{leadingSupport(z)} returns the index of leading (with respect to the ordering on the indexing set) monomial of \spad{z}. Error: if \spad{z} has no support.
422146`\spad{leaf?(u)} tests if \spad{u} is a terminal node.
422397`\axiom{leastMonomial(\spad{p})} returns an error if \axiom{\spad{p}} is \axiom{\spad{O}},{} otherwise,{} if \axiom{\spad{p}} belongs to \axiom{\spad{R}} returns \axiom{1},{} otherwise,{} the monomial of \axiom{\spad{p}} with lowest degree,{} where \axiom{\spad{p}} is viewed as a univariate polynomial in its main variable.
422636`\spad{leaves(t)} returns the list of values in obtained by visiting the nodes of tree \axiom{\spad{t}} in left-to-right order.
422693`\spad{leftAlternative?()} tests if \spad{2*associator(a,{}a,{}b) = 0} for all \spad{a},{} \spad{b} in the algebra. Note: we only can test this; in general we don\spad{'t} know whether \spad{2*a=0} implies \spad{a=0}.
422774`\spad{leftAlternative?()} tests if \spad{2*associator(a,{}a,{}b) = 0} for all \spad{a},{} \spad{b} in the algebra. Note: we only can test this; in general we don\spad{'t} know whether \spad{2*a=0} implies \spad{a=0}.
422963`\spad{leftCharacteristicPolynomial(a)} returns the characteristic polynomial of the left regular representation of \spad{a} with respect to any basis.
423077`\spad{leftDiscriminant()} returns the determinant of the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by the left trace of the product \spad{vi*vj},{} where \spad{v1},{}...,{}\spad{vn} are the elements of the fixed \spad{R}-module basis. Note: the same as \spad{determinant(leftTraceMatrix())}.
423148`\spad{leftDiscriminant()} returns the determinant of the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by the left trace of the product \spad{vi*vj},{} where \spad{v1},{}...,{}\spad{vn} are the elements of the fixed \spad{R}-module basis. Note: the same as \spad{determinant(leftTraceMatrix())}.
423370`\spad{leftDiscriminant([v1,{}...,{}vn])} returns the determinant of the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by the left trace of the product \spad{vi*vj}. Note: the same as \spad{determinant(leftTraceMatrix([v1,{}...,{}vn]))}.
423452`\spad{leftDivide(a,{}b)} returns the pair \spad{[q,{}r]} such that \spad{a = b*q + r} and the degree of \spad{r} is less than the degree of \spad{b}. This process is called ``left division\spad{''}.
423551`\spad{leftDivide(a,{}b)} returns the pair \spad{[q,{}r]} such that \spad{a = b*q + r} and the degree of \spad{r} is less than the degree of \spad{b}. This process is called ``left division\spad{''}.
423793`\spad{leftExactQuotient(a,{}b)} computes the value \spad{q},{} if it exists,{} \indented{1}{such that \spad{a = b*q}.}
423977`\spad{leftExactQuotient(a,{}b)} computes the value \spad{q},{} if it exists,{} \indented{1}{such that \spad{a = b*q}.}
424192`\spad{leftExtendedGcd(a,{}b)} returns \spad{[c,{}d]} such that \spad{g = a * c + b * d = leftGcd(a,{} b)}.
424318`\spad{leftFactorIfCan(f,{}h)} returns the left factor (\spad{g} in \spad{f} = \spad{g} \spad{o} \spad{h}) of the functional decomposition of the polynomial \spad{f} with given \spad{h} or \spad{"failed"} if \spad{g} does not exist.
424423`\spad{leftFactor(p,{}q)} \undocumented
424506`\spad{leftGcd(a,{}b)} computes the value \spad{g} of highest degree such that \indented{3}{\spad{a = aa*g}} \indented{3}{\spad{b = bb*g}} for some values \spad{aa} and \spad{bb}. The value \spad{g} is computed using left-division.
424638`\spad{leftGcd(a,{}b)} computes the value \spad{g} of highest degree such that \indented{3}{\spad{a = g*aa}} \indented{3}{\spad{b = g*bb}} for some values \spad{aa} and \spad{bb}. The value \spad{g} is computed using left-division.
424720`\spad{leftLcm(a,{}b)} computes the value \spad{m} of lowest degree such that \spad{m = a*aa = b*bb} for some values \spad{aa} and \spad{bb}. The value \spad{m} is computed using left-division.
424852`\spad{leftLcm(a,{}b)} computes the value \spad{m} of lowest degree such that \spad{m = aa*a = bb*b} for some values \spad{aa} and \spad{bb}. The value \spad{m} is computed using right-division.
425038`\spad{leftMinimalPolynomial(a)} returns the polynomial determined by the smallest non-trivial linear combination of left powers of \spad{a}. Note: the polynomial never has a constant term as in general the algebra has no unit.
425166`\spad{leftMult(s,{} a)} returns \spad{s * a} where \spad{*} is the monoid operation,{} which is assumed non-commutative.
425285`\spad{leftNorm(a)} returns the determinant of the left regular representation of \spad{a}.
425351`\spad{leftOne(eq)} divides by the left hand side.
425422`\spad{leftOne(eq)} divides by the left hand side.
425669`\spad{leftPower(a,{}n)} returns the \spad{n}\spad{-}th left power of \spad{a},{} \spadignore{i.e.} \spad{leftPower(a,{}n) := a * leftPower(a,{}n-1)} and \spad{leftPower(a,{}0) := 1}.
425735`\spad{leftPower(a,{}n)} returns the \spad{n}\spad{-}th left power of \spad{a},{} \spadignore{i.e.} \spad{leftPower(a,{}n) := a * leftPower(a,{}n-1)} and \spad{leftPower(a,{}1) := a}.
425790`\spad{leftQuotient(a,{}b)} computes the pair \spad{[q,{}r]} such that \spad{a = b*q + r} and the degree of \spad{r} is less than the degree of \spad{b}. The value \spad{q} is returned.
425932`\spad{leftQuotient(a,{}b)} computes the pair \spad{[q,{}r]} such that \spad{a = b*q + r} and the degree of \spad{r} is less than the degree of \spad{b}. The value \spad{q} is returned.
426167`\spad{leftRankPolynomial()} calculates the left minimal polynomial of the generic element in the algebra,{} defined by the same structural constants over the polynomial ring in symbolic coefficients with respect to the fixed basis.
426280`\spad{leftRankPolynomial()} calculates the left minimal polynomial of the generic element in the algebra,{} defined by the same structural constants over the polynomial ring in symbolic coefficients with respect to the fixed basis.
426402`\spad{leftRank(x)} determines the number of linearly independent elements in \spad{x*b1},{}...,{}\spad{x*bn},{} where \spad{b=[b1,{}...,{}bn]} is a basis.
426549`\spad{leftRecip(a)} returns an element,{} which is a left inverse of \spad{a},{} or \spad{"failed"} if there is no unit element,{} if such an element doesn\spad{'t} exist or cannot be determined (see unitsKnown).
426654`\spad{leftRecip(a)} returns an element,{} which is a left inverse of \spad{a},{} or \spad{"failed"} if such an element doesn\spad{'t} exist or cannot be determined (see unitsKnown).
426801`\spad{leftRegularRepresentation(a)} returns the matrix of the linear map defined by left multiplication by \spad{a} with respect to the fixed \spad{R}-module basis.
427080`\spad{leftRegularRepresentation(a,{}[v1,{}...,{}vn])} returns the matrix of the linear map defined by left multiplication by \spad{a} with respect to the \spad{R}-module basis \spad{[v1,{}...,{}vn]}.
427182`\spad{leftRemainder(a,{}b)} computes the pair \spad{[q,{}r]} such that \spad{a = b*q + r} and the degree of \spad{r} is less than the degree of \spad{b}. The value \spad{r} is returned.
427326`\spad{leftRemainder(a,{}b)} computes the pair \spad{[q,{}r]} such that \spad{a = b*q + r} and the degree of \spad{r} is less than the degree of \spad{b}. The value \spad{r} is returned.
427511`\spad{leftTraceMatrix()} is the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by left trace of the product \spad{vi*vj},{} where \spad{v1},{}...,{}\spad{vn} are the elements of the fixed \spad{R}-module basis.
427589`\spad{leftTraceMatrix()} is the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by left trace of the product \spad{vi*vj},{} where \spad{v1},{}...,{}\spad{vn} are the elements of the fixed \spad{R}-module basis.
427832`\spad{leftTraceMatrix([v1,{}...,{}vn])} is the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by the left trace of the product \spad{vi*vj}.
427985`\spad{leftTrace(a)} returns the trace of the left regular representation of \spad{a}.
428052`\spad{leftTrim(s,{}c)} returns \spad{s} with all leading characters \spad{c} deleted. For example,{} \axiom{leftTrim(" abc ",{} char " ")} returns \axiom{"abc "}.
428110`\spad{leftTrim(s,{}cc)} returns \spad{s} with all leading characters in \spad{cc} deleted. For example,{} \axiom{leftTrim("(abc)",{} charClass "()")} returns \axiom{"abc)"}.
428247`\spad{leftUnit()} returns a left unit of the algebra (not necessarily unique),{} or \spad{"failed"} if there is none.
428457`\spad{leftUnits()} returns the affine space of all left units of the algebra,{} or \spad{"failed"} if there is none
428581`\spad{leftUnits()} returns the affine space of all left units of the algebra,{} or \spad{"failed"} if there is none.
428718`\spad{leftZero(eq)} subtracts the left hand side.
428781`\axiom{left(\spad{x})} returns left subtree of \axiom{\spad{x}} or error if \axiomOpFrom{retractable?}{LyndonWord}(\axiom{\spad{x}}) is \spad{true}.
428828`\axiom{left(\spad{x})} returns left subtree of \axiom{\spad{x}} or error if \axiomOpFrom{retractable?}{Magma}(\axiom{\spad{x}}) is \spad{true}.
428870`\spad{left(f)} left-justifies form \spad{f} in total space.
428909`\axiom{left(rootChar)} is the left bound of the isolating interval
429001`\spad{left(u)} returns the left child.
429057`\spad{left(f,{}n)} left-justifies form \spad{f} within space of width \spad{n}.
429215`\spad{legendreP(nu,{} mu,{} z)} is the Legendre \spad{P} function
429281`\spad{legendreP(nu,{} mu,{} z)} is the Legendre \spad{P} function.
429344`\spad{legendreQ(nu,{} mu,{} z)} is the Legendre \spad{Q} function
429410`\spad{legendreQ(nu,{} mu,{} z)} is the Legendre \spad{Q} function.
429698`\spad{length(p)} returns the sum of the absolute values of the coefficients of the polynomial \spad{p}.
429912`\spad{length(x)} returns the number of elements in tuple \spad{x}
430024`\axiom{length(\spad{x})} returns the number of entries in \axiom{\spad{x}}.
430081`\spad{length(q)} returns the number of elements in the queue. Note: \axiom{length(\spad{q}) = \spad{#q}}.
430145`\spad{length(v)} computes the sqrt(dot(\spad{v},{}\spad{v})),{} \spadignore{i.e.} the magnitude
430231`\spad{length(a)} length of \spad{a} in digits.
430281`\spad{lepol }\undocumented
430364`\spad{lerchPhi(z,{} s,{} a)} is the Lerch Phi function
430429`\spad{lerchPhi(a,{} b,{} z)} is the Lerch Phi function.
430791`\spad{less?(a,{} b)} compares \spad{a} and \spad{b} in the partial ordering induced by setOrder.
430872`\spad{less?(u,{}n)} tests if \spad{u} has less than \spad{n} elements.
430935`\spad{less?(a,{} b,{} fn)} compares \spad{a} and \spad{b} in the partial ordering induced by setOrder,{} and returns \spad{fn(a,{} b)} if \spad{a} and \spad{b} are not comparable in that ordering.
431069`\spad{leviCivitaSymbol()} is the rank \spad{dim} tensor defined by \spad{leviCivitaSymbol()(i1,{}...idim) = +1/0/-1} if \spad{i1,{}...,{}idim} is an even/is nota /is an odd permutation of \spad{minix,{}...,{}minix+dim-1}.
431443`\axiom{lexico(\spad{x},{}\spad{y})} returns \axiom{\spad{true}} iff \axiom{\spad{x}} is smaller than \axiom{\spad{y}} \spad{w}.\spad{r}.\spad{t}. the lexicographical ordering induced by \axiom{VarSet}.
431500`\axiom{lexico(\spad{x},{}\spad{y})} returns \axiom{\spad{true}} iff \axiom{\spad{x}} is smaller than \axiom{\spad{y}} \spad{w}.\spad{r}.\spad{t}. the lexicographical ordering induced by \axiom{VarSet}. \spad{N}.\spad{B}. This operation does not take into account the tree structure of its arguments. Thus this is not a total ordering.
431552`\spad{lexico(x,{}y)} returns \spad{true} iff \spad{x} is smaller than \spad{y} \spad{w}.\spad{r}.\spad{t}. the pure lexicographical ordering induced by \spad{S}.
432204`\spad{lhs(f)} returns the left side of \spad{f}
432247`\spad{lhs(r)} returns the left hand side of the rule \spad{r}.
432295`\spad{lhs(eqn)} returns the left hand side of equation \spad{eqn}.
432333`\spad{\spad{li}(f)} denotes the logarithmic integral
432382`\spad{\spad{li}(x)} returns the logarithmic integral of \spad{x},{} \spadignore{i.e.} the integral of \spad{dx / log(x)}.
432436`\spad{library(ln)} creates a new library file.
432481`\spad{lieAdmissible?()} tests if the algebra defined by the commutators is a Lie algebra,{} \spadignore{i.e.} satisfies the Jacobi identity. The property of anticommutativity follows from definition.
432560`\spad{lieAdmissible?()} tests if the algebra defined by the commutators is a Lie algebra,{} \spadignore{i.e.} satisfies the Jacobi identity. The property of anticommutativity follows from definition.
432636`\spad{lieAlgebra?()} tests if the algebra is anticommutative and \spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} for all \spad{a},{}\spad{b},{}\spad{c} in the algebra (Jacobi identity). Example: for every associative algebra \spad{(A,{}+,{}@)} we can construct a Lie algebra \spad{(A,{}+,{}*)},{} where \spad{a*b := a@b-b@a}.
432712`\spad{lieAlgebra?()} tests if the algebra is anticommutative and \spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} for all \spad{a},{}\spad{b},{}\spad{c} in the algebra (Jacobi identity). Example: for every associative algebra \spad{(A,{}+,{}@)} we can construct a Lie algebra \spad{(A,{}+,{}*)},{} where \spad{a*b := a@b-b@a}.
432856`\spad{lift(x)} return the canonical representative of the equivalence class \spad{x}
432925`\spad{lift(x)} \undocumented
432970`\spad{lift(z)} returns a minimal degree univariate polynomial up such that \spad{z=reduce up}.
436551`\spad{linearAssociatedExp(a,{}f)} is linear over {\em F},{} \spadignore{i.e.} for elements {\em a} from {\em \$},{} {\em c,{}d} form {\em F} and {\em f,{}g} univariate polynomials over {\em F} we have \spadfun{linearAssociatedExp}(a,{}cf+dg) equals {\em c} times \spadfun{linearAssociatedExp}(a,{}\spad{f}) plus {\em d} times \spadfun{linearAssociatedExp}(a,{}\spad{g}). Therefore \spadfu
436551`--n{linearAssociatedExp} is defined completely by its action on monomials from {\em F[X]}: \spadfun{linearAssociatedExp}(a,{}monomial(1,{}\spad{k})\spad{\$}SUP(\spad{F})) is defined to be \spadfun{Frobenius}(a,{}\spad{k}) which is {\em a^(q^k)},{} where {\em q=size()\$F}.
436770`\spad{linearAssociatedLog(a)} returns a polynomial {\em g},{} such that \spadfun{linearAssociatedExp}(normalElement(),{}\spad{g}) equals {\em a}.
437004`\spad{linearAssociatedLog(b,{}a)} returns a polynomial {\em g},{} such that the \spadfun{linearAssociatedExp}(\spad{b},{}\spad{g}) equals {\em a}. If there is no such polynomial {\em g},{} then \spadfun{linearAssociatedLog} fails.
437241`\spad{linearAssociatedOrder(a)} retruns the monic polynomial {\em g} of least degree,{} such that \spadfun{linearAssociatedExp}(a,{}\spad{g}) is 0.
437462`\spad{linearDependence([v1,{}...,{}vn])} returns \spad{[c1,{}...,{}cn]} if \spad{c1*v1 + ... + cn*vn = 0} and not all the \spad{ci}\spad{'s} are 0,{} "failed" if the \spad{vi}\spad{'s} are linearly independent over \spad{S}.
437653`\spad{linearExtend:(f,{}x)} returns the linear extension of a map defined on the basis of \spad{M2} applied to a linear combination
437930`\spad{linearSearch(x,{} f)} searches \spad{x} in the cache,{} calling \spad{f(y)} to determine whether \spad{x} is equal to \spad{y}. It returns \spad{y} from cache such that \spad{f}(\spad{y}) or failed is no such \spad{y} exists.
438133`\spad{linearlyDependent?([v1,{}...,{}vn])} returns \spad{true} if the \spad{vi}\spad{'s} are linearly dependent over \spad{S},{} \spad{false} otherwise.
438209`\spad{linearlyDependentOverZ?([v1,{}...,{}vn])} returns \spad{true} if the \spad{vi}\spad{'s} are linearly dependent over the integers,{} \spad{false} otherwise.
438865`\spad{list?(s)} is \spad{true} if \spad{s} is a Lisp list,{} possibly ().
439485`\spad{listOfLists(m)} returns the rows of the matrix \spad{m} as a list of lists.
439557`\spad{listOfLists(m)} returns the rows of the matrix \spad{m} as a list of lists.
440297`\spad{list(x)} returns the list of one element \spad{x}.
440341`\spad{list(sy)} takes a scripted symbol and produces a list of the name followed by the scripts.
440450`\spad{lists(r)} returns the list of matches that match lists.
440761`\spad{lo(s)} returns the first endpoint of \spad{s}. Note: \spad{lo(l..h) = l}.
440805`\spad{localAbs(fcn)} = \spad{abs(fcn)} or \spad{sqrt(fcn^2)} depending on whether or not FE has a function \spad{abs}. This should be a local function,{} but the compiler won\spad{'t} allow it.
440885`\spad{localAbs(fcn)} = \spad{abs(fcn)} or \spad{sqrt(fcn^2)} depending on whether or not FE has a function \spad{abs}. This should be a local function,{} but the compiler won\spad{'t} allow it.
441502`\spad{localReal?(x)} should be local but conditional
441564`\spad{localUnquote(f,{}ls)} is a local function.
441632`\spad{log10()} returns \spad{ln 10}: \spad{2.3025809299...}.
441665`\spad{log10(x)} represents the Fortran intrinsic function \spad{LOG10}
441747`\spad{log10(x)} computes the logarithm with base 10 for \spad{x}.
441788`\spad{log10(x)} computes the logarithm for \spad{x} to base 10.
441823`\spad{log2()} returns \spad{ln 2},{} \spadignore{i.e.} \spad{0.6931471805...}.
441855`\spad{log2(x)} computes the logarithm with base 2 for \spad{x}.
441895`\spad{log2(x)} computes the logarithm for \spad{x} to base 2.
442156`\spad{logGamma(x)} is the natural log of \spad{Gamma(x)}.
442307`\spad{logGamma(x)} is the natural log of \spad{Gamma(x)}.
442367`\spad{logIfCan(z)} returns log(\spad{z}) if possible,{} and "failed" otherwise.
442447`\spad{log(x)} applies the logarithm operator to \spad{x}
442864`\spad{log(z)} returns the logarithm of Laurent series \spad{z}.
442950`\spad{log(z)} returns the logarithm of a Puiseux series \spad{z}.
443120`\spad{log(x)} represents the Fortran intrinsic function LOG
443200`\spad{log(x)} returns the natural logarithm of \spad{x}. When evaluated into some subset of the complex numbers,{} the branch cut lies along the negative real axis,{} continuous with quadrant II. The domain does not contain the origin.
443435`\spad{logical?(t)} tests whether \spad{t} is equivalent to the FORTRAN type LOGICAL.
443646`\spad{lommelS1(mu,{} nu,{} z)} is the Lommel \spad{s} function
443711`\spad{lommelS1(mu,{} nu,{} z)} is the Lommel \spad{s} function.
443773`\spad{lommelS2(mu,{} nu,{} z)} is the Lommel \spad{S} function
443838`\spad{lommelS2(mu,{} nu,{} z)} is the Lommel \spad{S} function.
444193`\spad{lookup(x)} returns a positive integer such that \spad{x = index lookup x}.
444398`\spad{low(s)} returns the first endpoint of \spad{s}. Note: \spad{low(l..h) = l}.
444443`\spad{lowerCase!(s)} destructively replaces the alphabetic characters in \spad{s} by lower case.
444493`\spad{lowerCase?(c)} tests if \spad{c} is an lower case letter,{} \spadignore{i.e.} one of a..\spad{z}.
444542`\spad{lowerCase()} returns the class of all characters for which \spadfunFrom{lowerCase?}{Character} is \spad{true}.
444634`\spad{lowerCase(s)} returns the string with all characters in lower case.
444683`\spad{lowerCase(c)} converts an upper case letter to the corresponding lower case letter. If \spad{c} is not an upper case letter,{} then it is returned unchanged.
444973`\spad{lquo(x,{} s)} returns the exact left quotient of \spad{x} by \spad{s}.
445040`\spad{lquo(x,{} y)} returns the exact left quotient of \spad{x} by \spad{y} \spadignore{i.e.} \spad{q} such that \spad{x = y * q},{} "failed" if \spad{x} is not of the form \spad{y * q}.
445101`\spad{lquo(x,{} y)} returns the exact left quotient of \spad{x} \indented{1}{by \spad{y} that is \spad{q} such that \spad{x = y * q},{}} "failed" if \spad{x} is not of the form \spad{y * q}.
445340`\spad{lquo(x,{}y)} returns the left simplification of \spad{x} by \spad{y}.
445390`\spad{lquo(x,{}v)} returns the left simplification of \spad{x} by the variable \spad{v}.
445715`\spad{magnitude(v)} computes the sqrt(dot(\spad{v},{}\spad{v})),{} \spadignore{i.e.} the length
446028`\axiom{mainCoefficients(\spad{p})} returns an error if \axiom{\spad{p}} is \axiom{\spad{O}},{} otherwise,{} if \axiom{\spad{p}} belongs to \axiom{\spad{R}} returns [\spad{p}],{} otherwise returns the list of the coefficients of \axiom{\spad{p}},{} where \axiom{\spad{p}} is viewed as a univariate polynomial in its main variable.
446175`\axiom{mainContent(\spad{p})} returns the content of \axiom{\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \axiom{\spad{R}}.
446707`\axiom{mainMonomial(\spad{p})} returns an error if \axiom{\spad{p}} is \axiom{\spad{O}},{} otherwise,{} if \axiom{\spad{p}} belongs to \axiom{\spad{R}} returns \axiom{1},{} otherwise,{} \axiom{mvar(\spad{p})} raised to the power \axiom{mdeg(\spad{p})}.
446852`\axiom{mainMonomials(\spad{p})} returns an error if \axiom{\spad{p}} is \axiom{\spad{O}},{} otherwise,{} if \axiom{\spad{p}} belongs to \axiom{\spad{R}} returns [1],{} otherwise returns the list of the monomials of \axiom{\spad{p}},{} where \axiom{\spad{p}} is viewed as a univariate polynomial in its main variable.
447002`\axiom{mainPrimitivePart(\spad{p})} returns the primitive part of \axiom{\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \axiom{\spad{R}}.
447167`\axiom{mainSquareFreePart(\spad{p})} returns the square free part of \axiom{\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \axiom{\spad{R}}.
447430`\axiom{mainVariable?(\spad{v},{}\spad{ps})} returns \spad{true} iff \axiom{\spad{v}} is the main variable of some polynomial in \axiom{\spad{ps}}.
447515`\spad{mainVariable(f)} returns the highest variable appearing in the numerator or the denominator of \spad{f},{} "failed" if \spad{f} has no variables.
447804`\spad{mainVariable(p)} returns the biggest variable which actually occurs in the polynomial \spad{p},{} or "failed" if no variables are present. fails precisely if polynomial satisfies ground?
447971`\axiom{mainVariables(\spad{ps})} returns the decreasingly sorted list of the variables which are main variables of some polynomial in \axiom{\spad{ps}}.
448054`\spad{makeCos(e,{}r)} makes a sin expression with given argument and coefficient
448416`\spad{evalADE(f,{} x,{} n,{} eq,{} parameters,{} values)} creates an expression that stands for the coefficient of \spad{x^n} in the Taylor expansion of \spad{f}(\spad{x}),{} where \spad{f}(\spad{x}) is given by the functional equation \spad{eq}. The argument values specifies the first few Taylor coefficients.
449027`\spad{makeGraphImage(\spad{gi})} takes the given graph,{} \spad{\spad{gi}} of the domain \spadtype{GraphImage},{} and sends it\spad{'s} data to the viewport manager where it waits to be included in a two-dimensional viewport window. \spad{\spad{gi}} cannot be an empty graph,{} and it\spad{'s} elements must have been created using the \spadfun{point} or \spadfun{component} functions,{} not by a previous \spadfun{makeGraphImage}.
451886`\spad{makeRec(u,{} n,{} eq,{} parameters,{} values)} creates an expression that stands for \spad{u}(\spad{n}),{} where \spad{u}(\spad{n}) is given by the equation \spad{eq}. The argument \spad{parameters} has to contain all variables (or kernels) that appear eventually in the stream of values. The argument values specifies the initial values of the recurrence \spad{u}(0),{} \spad{u}(1),{}... For the moment we do
451886`--n\spad{'t} allow recursions that contain \spad{u} inside of another operator.
451978`\spad{makeRecord(a,{}b)} creates a record object with type Record(part1:S,{} part2:R),{} where \spad{part1} is \spad{a} and \spad{part2} is \spad{b}.
452050`\spad{makeResult(r1,{}r2)} makes the combined result [\spad{r1},{}\spad{r2}].
452246`\spad{makeSUP(p)} converts the polynomial \spad{p} to be of type SparseUnivariatePolynomial over the same coefficients.
452472`\spad{makeSin(e,{}r)} makes a sin expression with given argument and coefficient
452661`\spad{makeTerm(s,{} e)} returns the monomial \spad{s} exponentiated by \spad{e} (\spadignore{e.g.} s^e or \spad{e} * \spad{s}).
452716`\spad{makeUnit()} returns the unit element of the monomial.
452956`\spad{makeVariable(s)} views \spad{s} as a differential indeterminate,{} in such a way that the \spad{n}-th derivative of \spad{s} may be simply referenced as \spad{z}.\spad{n} where \spad{z} :=makeVariable(\spad{s}). Note: In the interpreter,{} \spad{z} is given as an internal map,{} which may be ignored.
453053`\spad{makeVariable(p)} views \spad{p} as an element of a differential ring,{} in such a way that the \spad{n}-th derivative of \spad{p} may be simply referenced as \spad{z}.\spad{n} where \spad{z} \spad{:=} makeVariable(\spad{p}). Note: In the interpreter,{} \spad{z} is given as an internal map,{} which may be ignored.
453174`\spad{makeVariable(s,{} n)} returns the \spad{n}-th derivative of a differential indeterminate \spad{s} as an algebraic indeterminate.
453260`\spad{makeViewport2D(v)} takes the given two-dimensional viewport,{} \spad{v},{} of the domain \spadtype{TwoDimensionalViewport} and displays a viewport window on the screen which contains the contents of \spad{v}.
453402`\spad{makeViewport3D(v)} takes the given three-dimensional viewport,{} \spad{v},{} of the domain \spadtype{ThreeDimensionalViewport} and displays a viewport window on the screen which contains the contents of \spad{v}.
453886`\spad{makeprod(a,{}b)} \undocumented
453932`\axiom{makingStats?()} returns \spad{true} iff the statisitics process is running.
454055`\spad{mantissa(x)} returns the mantissa part of \spad{x}.
454591`\spad{map!(f,{}a)} assign \spad{a(i,{}j)} to \spad{f(a(i,{}j))} for all \spad{i,{} j}
454665`\spad{map!(f,{}u)} destructively replaces each element \spad{x} of \spad{u} by \axiom{\spad{f}(\spad{x})}.
454880`\spad{mapCoef(f,{} e1 a1 +...+ en an)} returns \spad{f(e1) a1 +...+ f(en) an}.
454949`\spad{mapDown!(t,{}p,{}f)} returns \spad{t} after traversing \spad{t} in "preorder" (node then left then right) fashion replacing the successive interior nodes as follows. The root value \spad{x} is replaced by \spad{q} \spad{:=} \spad{f}(\spad{p},{}\spad{x}). The mapDown!(\spad{l},{}\spad{q},{}\spad{f}) and mapDown!(\spad{r},{}\spad{q},{}\spad{f}) are evaluated for the left and right subtrees \spad{l} and \spad{r} of \spad{t}.
455014`\spad{mapDown!(t,{}p,{}f)} returns \spad{t} after traversing \spad{t} in "preorder" (node then left then right) fashion replacing the successive interior nodes as follows. Let \spad{l} and \spad{r} denote the left and right subtrees of \spad{t}. The root value \spad{x} of \spad{t} is replaced by \spad{p}. Then \spad{f}(value \spad{l},{} value \spad{r},{} \spad{p}),{} where \spad{l} and \spad{r} denote the left and right subtrees o
455014`--f \spad{t},{} is evaluated producing two values \spad{pl} and \spad{pr}. Then \spad{mapDown!(l,{}pl,{}f)} and \spad{mapDown!(l,{}pr,{}f)} are evaluated.
455087`\spad{mapExpon(f,{} a1\^e1 ... an\^en)} returns \spad{a1\^f(e1) ... an\^f(en)}.
455148`\spad{mapExpon(f,{} a1\^e1 ... an\^en)} returns \spad{a1\^f(e1) ... an\^f(en)}.
455362`\spad{mapExponents(fn,{}u)} maps function \spad{fn} onto the exponents of the non-zero monomials of polynomial \spad{u}.
455434`\spad{mapGen(f,{} a1\^e1 ... an\^en)} returns \spad{f(a1)\^e1 ... f(an)\^en}.
455484`\spad{mapGen(f,{} a1\^e1 ... an\^en)} returns \spad{f(a1)\^e1 ... f(an)\^en}.
455535`\spad{mapGen(f,{} a1\^e1 ... an\^en)} returns \spad{f(a1)\^e1 ... f(an)\^en}.
455594`\spad{mapGen(f,{} e1 a1 +...+ en an)} returns \spad{e1 f(a1) +...+ en f(an)}.
456280`\spad{mapUp!(t,{}f)} traverses balanced binary tree \spad{t} in an "endorder" (left then right then node) fashion returning \spad{t} with the value at each successive interior node of \spad{t} replaced by \spad{f}(\spad{l},{}\spad{r}) where \spad{l} and \spad{r} are the values at the immediate left and right nodes.
456340`\spad{mapUp!(t,{}t1,{}f)} traverses \spad{t} in an "endorder" (left then right then node) fashion returning \spad{t} with the value at each successive interior node of \spad{t} replaced by \spad{f}(\spad{l},{}\spad{r},{}\spad{l1},{}\spad{r1}) where \spad{l} and \spad{r} are the values at the immediate left and right nodes. Values \spad{l1} and \spad{r1} are values at the corresponding nodes of a balanced binary tree \spad{t1},{} of ide
456340`--ntical shape at \spad{t}.
456986`\spad{map(f,{}p)} changes each coefficient of \spad{p} by the application of \spad{f}.
457037`\spad{map(fn,{}x)} returns \spad{Sum(fn(r_i) w_i)} if \spad{x} writes \spad{Sum(r_i w_i)}.
457092`\axiom{map(func,{} poly)} creates a new polynomial by applying func to every non-zero coefficient of the polynomial poly.
457231`\spad{map}(\spad{f},{} a) applies the map \spad{f} to each coefficient in a. It is assumed that \spad{f} maps 0 to 0
457316`\spad{map(f,{} p)} lifts \spad{f} to the domain of \spad{p} then applies it to \spad{p}.
457724`\spad{map(f,{}x)} \undocumented
457816`\spad{map(f,{}z)} returns the new element created by applying the function \spad{f} to each component of the direct product element \spad{z}.
458302`\spad{map(f,{} [(v1,{}a1),{}...,{}(vn,{}an)])} returns the matching result [(\spad{v1},{}\spad{f}(\spad{a1})),{}...,{}(\spad{vn},{}\spad{f}(an))].
458503`\spad{map(func,{}frac)} applies the function \spad{func} to the numerator and denominator of \spad{frac}.
458803`\spad{map(f,{}x)} \undocumented
458927`\spad{map(f,{}x)} \undocumented
459051`\spad{map(f,{}x)} \undocumented
459166`\spad{map(f,{}g(x))} applies the map \spad{f} to the coefficients of \indented{1}{the Taylor series \spad{g(x)}.}
459726`\spad{map(f,{} k)} returns \spad{g = op(f(a1),{}...,{}f(an))} where \spad{k = op(a1,{}...,{}an)}.
460241`\spad{map(fn,{}u)} maps function \spad{fn} onto the coefficients of the non-zero monomials of \spad{u}.
460298`\spad{map(f,{} ex)} evaluates ex,{} applying \spad{f} to values of type \spad{R} in ex.
460353`\spad{map(fn,{}u)} maps function \spad{fn} onto the coefficients of the non-zero monomials of \spad{u}.
460411`\spad{map(f,{}a)} returns \spad{b},{} where \spad{b(i,{}j) = a(i,{}j)} for all \spad{i},{} \spad{j}.
460486`\spad{map(f,{}a)} returns \spad{b},{} where \spad{b(i,{}j) = f(a(i,{}j))} for all \spad{i,{} j}
460559`\spad{map(fn,{}x)} returns \spad{Sum(fn(r_i) w_i)} if \spad{x} writes \spad{Sum(r_i w_i)}.
460612`\spad{map(fn,{}u)} maps the function \userfun{\spad{fn}} across the factors of \spadvar{\spad{u}} and creates a new factored object. Note: this clears the information flags (sets them to "nil") because the effect of \userfun{\spad{fn}} is clearly not known in general.
460658`\spad{map(f,{} a)} applies \spad{f} to all the constants in \spad{R} appearing in \spad{a}.
460723`\spad{map(f,{}u)} maps \spad{f} onto the coordinates of \spad{u} to get an element in \spad{AS} via identification of the basis of \spad{AR} as beginning part of the basis of \spad{AS}.
460806`\spad{map(f,{}u)} maps \spad{f} onto real and imaginary parts of \spad{u}.
460879`\spad{map(f,{} e)} applies \spad{f} to all the constants appearing in \spad{e}.
460961`\spad{map(fn,{}u)} is used to apply the function \userfun{\spad{fn}} to every factor of \spadvar{\spad{u}}. The new factored object will have all its information flags set to "nil". This function is used,{} for example,{} to coerce every factor base to another type.
461037`\spad{map(f,{}u)} maps \spad{f} onto the coefficients \spad{f} the element \spad{u} of the monoid ring to create an element of a monoid ring with the same monoid \spad{b}.
461125`\spad{map(f,{}u)} maps \spad{f} onto the component parts of the octonion \spad{u}.
461396`\spad{map(f,{}p)} \undocumented
461473`\spad{map(f,{} p)} takes a function \spad{f} from \spad{R} to \spad{S},{} and applies it to each (non-zero) coefficient of a polynomial \spad{p} over \spad{R},{} getting a new polynomial over \spad{S}. Note: since the map is not applied to zero elements,{} it may map zero to zero.
461557`\spad{map(f,{} p)} applies \spad{f} to all the leaves of \spad{p} and returns the result as a pattern over \spad{S}.
461630`\spad{map(f,{} p)} produces a new polynomial as a result of applying the function \spad{f} to every coefficient of the polynomial \spad{p}.
461712`\spad{map(f,{}u)} maps \spad{f} onto the component parts of the quaternion \spad{u}.
461786`\spad{map(f,{}s)} expands the segment \spad{s},{} applying \spad{f} to each value. For example,{} if \spad{s = l..h by k},{} then the list \spad{[f(l),{} f(l+k),{}...,{} f(lN)]} is computed,{} where \spad{lN <= h < lN+k}.
461874`\spad{map(f,{}l..h)} returns a new segment \spad{f(l)..f(h)}.
461947`\spad{map(f,{}v=a..b)} returns the value given by \spad{v=f(a)..f(b)}.
462041`\spad{map(func,{} poly)} creates a new polynomial by applying \spad{func} to every non-zero coefficient of the polynomial poly.
462171`\spad{map(func,{} poly)} creates a new polynomial by applying \spad{func} to every non-zero coefficient of the polynomial poly.
462291`\spad{map(f,{}s)} expands the segment \spad{s},{} applying \spad{f} to each value.
462399`\spad{map(f,{}seg)} returns the new segment obtained by applying \spad{f} to the endpoints of \spad{seg}.
462499`\spad{map(f,{} p)} lifts \spad{f} to \spad{F1} and applies it to \spad{p}.
462600`\spad{map(f,{}d)} \undocumented{}
462749`\spad{map(f,{}i)} \undocumented{}
462888`\spad{map(f,{}m)} applies the function \spad{f} to the elements of the matrix \spad{m}.
462982`\spad{map(f,{}m)} applies the function \spad{f} to the elements of the matrix \spad{m}.
463091`\spad{map(f,{}p)} \undocumented
463164`\spad{map(f,{}m)} applies the function \spad{f} to the elements of the matrix \spad{m}.
463290`\spad{map(f,{}a)} applies function \spad{f} to each member of aggregate \spad{a} resulting in a new aggregate over a possibly different underlying domain.
463363`\spad{map(f,{}a)} applies function \spad{f} to each member of aggregate \spad{a},{} creating a new aggregate with a possibly different underlying domain.
463433`\spad{map(f,{}eq)} returns an equation where \spad{f} is applied to the sides of \spad{eq}
463509`\spad{map(f,{}l..h by k)} produces a value of type \spad{L} by applying \spad{f} to each of the succesive elements of the segment,{} that is,{} \spad{[f(l),{} f(l+k),{} ...,{} f(lN)]},{} where \spad{lN <= h < lN+k}.
463572`\spad{map(f,{}u)} returns a copy of \spad{u} with each element \spad{x} replaced by \spad{f}(\spad{x}). For collections,{} \axiom{map(\spad{f},{}\spad{u}) = [\spad{f}(\spad{x}) for \spad{x} in \spad{u}]}.
463630`\spad{map(f,{}eqn)} constructs a new equation by applying \spad{f} to both sides of \spad{eqn}.
463676`\spad{map(f,{}t)} replaces the tuple \spad{t} by \spad{[f(x) for x in t]}.
463976`\spad{map(f,{} p,{} k)} uses the property \spad{p} of the operator of \spad{k},{} in order to lift \spad{f} and apply it to \spad{k}.
464244`\spad{map(varmap,{} coefmap,{} p)} takes a \spad{varmap},{} a mapping from the variables of polynomial \spad{p} into \spad{S},{} \spad{coefmap},{} a mapping from coefficients of \spad{p} into \spad{S},{} and \spad{p},{} and produces a member of \spad{S} using the corresponding arithmetic. in \spad{S}.
464769`\spad{map(fn,{}t1,{}t2)} creates a new table \spad{t} from given tables \spad{t1} and \spad{t2} with elements \spad{fn}(\spad{x},{}\spad{y}) where \spad{x} and \spad{y} are corresponding elements from \spad{t1} and \spad{t2} respectively.
464969`\spad{map(f,{}a,{}b)} returns \spad{c},{} where \spad{c} is such that \spad{c(i,{}j) = f(a(i,{}j),{}b(i,{}j))} for all \spad{i},{} \spad{j}.
465049`\spad{map(f,{}a,{}b)} returns \spad{c},{} where \spad{c(i,{}j) = f(a(i,{}j),{}b(i,{}j))} for all \spad{i,{} j}
465127`\spad{map(f,{}u,{}v)} returns a new collection \spad{w} with elements \axiom{\spad{z} = \spad{f}(\spad{x},{}\spad{y})} for corresponding elements \spad{x} and \spad{y} from \spad{u} and \spad{v}. Note: for linear aggregates,{} \axiom{\spad{w}.\spad{i} = \spad{f}(\spad{u}.\spad{i},{}\spad{v}.\spad{i})}.
465400`\spad{map(f,{}a,{}b,{}r)} returns \spad{c},{} where \spad{c(i,{}j) = f(a(i,{}j),{}b(i,{}j))} when both \spad{a(i,{}j)} and \spad{b(i,{}j)} exist; else \spad{c(i,{}j) = f(r,{} b(i,{}j))} when \spad{a(i,{}j)} does not exist; else \spad{c(i,{}j) = f(a(i,{}j),{}r)} when \spad{b(i,{}j)} does not exist; otherwise \spad{c(i,{}j) = f(r,{}r)}.
465700`\spad{mask(n)} returns \spad{2^n-1} (an \spad{n} bit mask).
465748`\spad{match?(s,{}t,{}c)} tests if \spad{s} matches \spad{t} except perhaps for multiple and consecutive occurrences of character \spad{c}. Typically \spad{c} is the blank character.
466042`\spad{match(p,{}s,{}wc)} tests if pattern \axiom{\spad{p}} matches subject \axiom{\spad{s}} where \axiom{\spad{wc}} is a wild card character. If no match occurs,{} the index \axiom{0} is returned; otheriwse,{} the value returned is the first index of the first character in the subject matching the subject (excluding that matched by an initial wild-card). For example,{} \axiom{match("*to*",{}"yorktown",{}\spad{"*"})} returns \axio
466042`--m{5} indicating a successful match starting at index \axiom{5} of \axiom{"yorktown"}.
467036`\spad{matrixConcat3D(s,{}x,{}y)} concatenates two 3-\spad{D} matrices along a specified axis
467341`\spad{matrix(llf)} makes \spad{llf} (a list of lists of forms) into a form which displays as a matrix.
467394`\spad{matrix(l)} converts the list of lists \spad{l} to a matrix,{} where the list of lists is viewed as a list of the rows of the matrix.
467461`\spad{matrix(l)} converts the list of lists \spad{l} to a matrix,{} where the list of lists is viewed as a list of the rows of the matrix.
467602`\spad{maxColIndex(m)} returns the index of the 'last' column of the matrix \spad{m}.
467683`\spad{maxColIndex(m)} returns the index of the 'last' column of the array \spad{m}
467862`\spad{maxDegree(d)} specifies the maximum degree of the coefficient polynomials in an algebraic differential equation or a recursion with polynomial coefficients. For rational functions with an exponential term,{} \spad{maxDegree} bounds the degree of the denominator polynomial. This option is expressed in the form \spad{maxDegree == d}.
468046`\spad{maxDerivative(d)} specifies the maximum derivative in an algebraic differential equation. This option is expressed in the form \spad{maxDerivative == d}.
468302`\spad{maxIndex(u)} returns the maximum index \spad{i} of aggregate \spad{u}. Note: in general,{} \axiom{maxIndex(\spad{u}) = reduce(max,{}[\spad{i} for \spad{i} in indices \spad{u}])}; if \spad{u} is a list,{} \axiom{maxIndex(\spad{u}) = \#u}.
468487`\spad{maxLevel(d)} specifies the maximum number of recursion levels operators guessProduct and guessSum will be applied. This option is expressed in the form spad{maxLevel \spad{==} \spad{d}}.
468652`\spad{maxMixedDegree(d)} specifies the maximum \spad{q}-degree of the coefficient polynomials in a recurrence with polynomial coefficients,{} in the case of mixed shifts. Although slightly inconsistent,{} maxMixedDegree(0) specifies that no mixed shifts are allowed. This option is expressed in the form \spad{maxMixedDegree == d}.
468993`\spad{maxPower(d)} specifies the maximum degree in an algebraic differential equation. For example,{} the degree of (\spad{f}\spad{''})\spad{^3} \spad{f'} is 4. maxPower(\spad{-1}) specifies that the maximum exponent can be arbitrary. This option is expressed in the form \spad{maxPower == d}.
469069`\spad{maxRowIndex(m)} returns the index of the 'last' row of the matrix \spad{m}.
469150`\spad{maxRowIndex(m)} returns the index of the 'last' row of the array \spad{m}
469328`\spad{maxShift(d)} specifies the maximum shift in a recurrence equation. This option is expressed in the form \spad{maxShift == d}.
469503`\spad{maxSubst(d)} specifies the maximum degree of the monomial substituted into the function we are looking for. That is,{} if \spad{maxSubst == d},{} we look for polynomials such that \$\spad{p}(\spad{f}(\spad{x}),{} \spad{f}(\spad{x^2}),{} ...,{} \spad{f}(\spad{x^d}))\spad{=0}\$. equation. This option is expressed in the form \spad{maxSubst == d}.
469579`\spad{max()} returns the maximum floating point number.
469691`\spad{max()} returns the largest single integer.
469775`\spad{max(u)} returns the largest element of aggregate \spad{u}.
469840`\spad{max(q)} returns the maximum element of priority queue \spad{q}.
469929`\spad{max(x,{}y)} returns the maximum of \spad{x} and \spad{y} relative to \spad{"<"}.
469970`\spad{maxdeg(p)} returns the greatest word occurring in the polynomial \spad{p} with a non-zero coefficient. An error is produced if \spad{p} is zero.
470193`\spad{maxint()} returns the maximum integer in the model
470249`\spad{maxint(u)} sets the maximum integer in the model to \spad{u}
470814`\axiom{mdeg(\spad{p})} returns an error if \axiom{\spad{p}} is \axiom{0},{} otherwise,{} if \axiom{\spad{p}} belongs to \axiom{\spad{R}} returns \axiom{0},{} otherwise,{} returns the degree of \axiom{\spad{p}} in its main variable.
471587`\spad{meijerG(la,{} lb,{} lc,{} ld,{} z)} is the meijerG function
471708`\spad{meijerG(la,{} lb,{} lc,{} ld,{} z)} is the meijerG function.
472188`\spad{member?(v,{}B)} tests if the vector \spad{v} is in the subspace spaned by \spad{B}.
472320`\spad{member?(x,{}u)} tests if \spad{x} is a member of \spad{u}. For collections,{} \axiom{member?(\spad{x},{}\spad{u}) = reduce(or,{}[x=y for \spad{y} in \spad{u}],{}\spad{false})}. However,{} \axiom{member?(\spad{x},{}\spad{u})} returns as soon as it finds a member.
472486`\spad{members(u)} returns a list of the consecutive elements of \spad{u}. For collections,{} \axiom{parts([\spad{x},{}\spad{y},{}...,{}\spad{z}]) = (\spad{x},{}\spad{y},{}...,{}\spad{z})}.
472569`\spad{members(ms)} returns a list of the elements of \spad{ms} {\em without} their multiplicity. See also \spadfun{parts}.
472719`\spad{merge!(u,{}v)} destructively merges \spad{u} and \spad{v} in ascending order.
472798`\spad{merge!(q,{}q1)} destructively changes priority queue \spad{q} to include the values from priority queue \spad{q1}.
472989`\spad{merge!(p,{}u,{}v)} destructively merges \spad{u} and \spad{v} using predicate \spad{p}.
473283`\spad{mergeDifference(l1,{}l2)} returns a list of elements in \spad{l1} not present in \spad{l2}. Assumes lists are ordered and all \spad{x} in \spad{l2} are also in \spad{l1}.
473598`\spad{merge(ls)} a list of subspaces,{} \spad{ls},{} into one subspace.
473854`\spad{merge(s1,{}s2)} the subspaces \spad{s1} and \spad{s2} into a single subspace.
473900`\spad{merge(u,{}v)} merges \spad{u} and \spad{v} in ascending order. Note: \axiom{merge(\spad{u},{}\spad{v}) = merge(\spad{<=},{}\spad{u},{}\spad{v})}.
473974`\spad{merge(q1,{}q2)} returns combines priority queues \spad{q1} and \spad{q2} to return a single priority queue \spad{q}.
474032`\spad{merge(s1,{}s2)} will create a new \spadtype{ThreeSpace} that has the components of \spad{s1} and \spad{s2}; Groupings of components into composites are maintained.
474286`\spad{merge(p,{}a,{}b)} returns an aggregate \spad{c} which merges \axiom{a} and \spad{b}. The result is produced by examining each element \spad{x} of \axiom{a} and \spad{y} of \spad{b} successively. If \axiom{\spad{p}(\spad{x},{}\spad{y})} is \spad{true},{} then \spad{x} is inserted into the result; otherwise \spad{y} is inserted. If \spad{x} is chosen,{} the next element of \axiom{a} is examined,{} and so on. When all the elemen
474286`--ts of one aggregate are examined,{} the remaining elements of the other are appended. For example,{} \axiom{merge(<,{}[1,{}3],{}[2,{}7,{}5])} returns \axiom{[1,{}2,{}3,{}7,{}5]}.
476232`\spad{messagePrint(s)} prints \spad{s} without string quotes. Note: \spad{messagePrint(s)} is equivalent to \spad{print message(s)}.
476285`\spad{message(s)} creates an form with no string quotes from string \spad{s}.
476331`\axiom{middle(rootChar)} is the middle of the isolating interval
476774`\spad{minColIndex(m)} returns the index of the 'first' column of the matrix \spad{m}.
476855`\spad{minColIndex(m)} returns the index of the 'first' column of the array \spad{m}
476934`\spad{minGbasis }\undocumented
477149`\spad{minIndex(u)} returns the minimum index \spad{i} of aggregate \spad{u}. Note: in general,{} \axiom{minIndex(a) = reduce(min,{}[\spad{i} for \spad{i} in indices a])}; for lists,{} \axiom{minIndex(a) = 1}.
478025`\spad{minRowIndex(m)} returns the index of the 'first' row of the matrix \spad{m}.
478106`\spad{minRowIndex(m)} returns the index of the 'first' row of the array \spad{m}
478185`\spad{min()} returns the minimum floating point number.
478297`\spad{min()} returns the smallest single integer.
478381`\spad{min(u)} returns the smallest element of aggregate \spad{u}.
478446`\spad{min(u)} returns the smallest entry in the multiset aggregate \spad{u}.
478537`\spad{min(x,{}y)} returns the minimum of \spad{x} and \spad{y} relative to \spad{"<"}.
478659`\spad{mindeg(p)} returns the smallest word occurring in the polynomial \spad{p} with a non-zero coefficient. An error is produced if \spad{p} is zero.
479119`\spad{minimalPolynomial(a)} returns the minimal polynomial of an element \spad{a} over the ground field \spad{F}.
479220`\spad{minimalPolynomial(a)} returns the minimal polynomial of \spad{a}.
479411`\spad{minimalPolynomial(x,{}n)} computes the minimal polynomial of \spad{x} over the field of extension degree \spad{n} over the ground field \spad{F}.
479542`\spad{minimize(I)} returns a reduced set of generators for \spad{I}.
479682`\spad{minimumDegree(p)} gives the least exponent of a non-zero term of polynomial \spad{p}. Error: if applied to 0.
479822`\spad{minimumDegree(l)} is the smallest \spad{k} such that \spad{a(k) ~= 0} if \indented{2}{\spad{l = sum(monomial(a(i),{}i),{} i = 0..n)}.}
480131`\spad{minimumDegree(p,{} lv)} gives the list of minimum degrees of the polynomial \spad{p} with respect to each of the variables in the list \spad{lv}
480234`\spad{minimumDegree(p,{}v)} gives the minimum degree of polynomial \spad{p} with respect to \spad{v},{} \spadignore{i.e.} viewed a univariate polynomial in \spad{v}
480432`\spad{minordet(m)} computes the determinant of the matrix \spad{m} using minors. Error: if the matrix is not square.
480504`\spad{minordet(m)} computes the determinant of the matrix \spad{m} using minors. Error: if the matrix is not square.
480583`\spad{minordet(m)} computes the determinant of the matrix \spad{m} using minors.
481140`\spad{minusInfinity()} returns minusInfinity.
481208`\spad{minusInfinity()} returns -infinity.
481264`\axiom{mirror(\spad{g})} is the mirror of the internal representation of \axiom{\spad{g}}.
481326`\axiom{mirror(\spad{x})} returns the reversed word of \axiom{\spad{x}}. That is \axiom{\spad{x}} itself if \axiomOpFrom{retractable?}{Magma}(\axiom{\spad{x}}) is \spad{true} and \axiom{mirror(\spad{z}) * mirror(\spad{y})} if \axiom{\spad{x}} is \axiom{\spad{y*z}}.
481370`\spad{mirror(x)} returns the reversed word of \spad{x}.
481421`\axiom{mirror(\spad{x})} returns \axiom{Sum(r_i mirror(w_i))} if \axiom{\spad{x}} is \axiom{Sum(r_i w_i)}.
481476`\spad{mirror(x)} returns \spad{Sum(r_i mirror(w_i))} if \spad{x} writes \spad{Sum(r_i w_i)}.
482058`\spad{mkcomm(i,{}j)} \undocumented{}
482102`\spad{modTree(r,{}l)} \undocumented{}
482782`\spad{modularGcdPrimitive(listf1)} computes the \spad{gcd} of the list of primitive polynomials listf by modular methods.
482866`\spad{modularGcd(listf)} computes the \spad{gcd} of the list of polynomials \spad{listf} by modular methods.
483152`\spad{module(I)} returns \spad{I} viewed has a module over \spad{R}.
483375`\spad{modulus()} \undocumented
483474`\spad{modulus(x)} \undocumented
483561`\spad{modulus(x)} \undocumented
483638`\spad{modulus(x)} \undocumented
483781`\spad{moebius(a,{}b,{}c,{}d)} returns \spad{matrix [[a,{}b],{}[c,{}d]]}.
483904`\spad{monic?(p)} tests if \spad{p} is monic (\spadignore{i.e.} leading coefficient equal to 1).
483976`\axiom{monic?(\spad{p})} returns \spad{false} if \axiom{\spad{p}} belongs to \axiom{\spad{R}},{} otherwise returns \spad{true} iff \axiom{\spad{p}} is monic as a univariate polynomial in its main variable.
484046`\spad{monicCompleteDecompose(f)} returns a list of factors of \spad{f} for the functional decomposition ([ \spad{f1},{} ...,{} \spad{fn} ] means \spad{f} = \spad{f1} \spad{o} ... \spad{o} \spad{fn}).
484268`\spad{monicDivide(p,{}q)} divide the polynomial \spad{p} by the monic polynomial \spad{q},{} returning the pair \spad{[quotient,{} remainder]}. Error: if \spad{q} isn\spad{'t} monic.
484468`\spad{monicDivide(a,{}b,{}v)} divides the polynomial a by the polynomial \spad{b},{} with each viewed as a univariate polynomial in \spad{v} returning both the quotient and remainder. Error: if \spad{b} is not monic with respect to \spad{v}.
484574`\spad{monicLeftDivide(a,{}b)} returns the pair \spad{[q,{}r]} such that \spad{a = b*q + r} and the degree of \spad{r} is less than the degree of \spad{b}. \spad{b} must be monic. This process is called ``left division\spad{''}.
484912`\axiom{monicModulo(a,{}\spad{b})} returns \axiom{\spad{r}} such that \axiom{\spad{r}} is reduced \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{b}} and \axiom{\spad{b}} divides \axiom{a \spad{-r}} where \axiom{\spad{b}} is monic.
484983`\axiom{monicModulo(a,{}\spad{b})} computes \axiom{a mod \spad{b}},{} if \axiom{\spad{b}} is monic as univariate polynomial in its main variable.
485056`\spad{monicRightDivide(a,{}b)} returns the pair \spad{[q,{}r]} such that \spad{a = q*b + r} and the degree of \spad{r} is less than the degree of \spad{b}. \spad{b} must be monic. This process is called ``right division\spad{''}.
485924`\spad{monom(s,{}r)} returns the product of the basis element \spad{s} by the coefficient \spad{r}.
486165`\spad{monomial?(p)} tests if \spad{p} is a single monomial.
486226`\spad{monomial?(x)} returns \spad{true} if \spad{x} contains a single monomial.
486288`\spad{monomial?(f)} tests if \spad{f} is a single monomial.
486350`\spad{monomial?(x)} returns \spad{true} if \spad{x} is a monomial
486407`\spad{monomialIntPoly(p,{} ')} returns [\spad{q},{} \spad{r}] such that \spad{p = q' + r} and \spad{degree(r) < degree(t')}. Error if \spad{degree(t') < 2}.
487042`\spad{monomial(a,{}s)} constructs a direct product element with the \spad{s} component set to \spad{a}
487183`\spad{monomial(r,{}e)} makes a term from a coefficient \spad{r} and an exponent \spad{e}.
487239`\spad{monomial(r,{}m)} creates a scalar multiple of the basis element \spad{m}.
487368`\spad{monomial(c,{}k)} produces \spad{c} times the \spad{k}-th power of the generating operator,{} \spad{monomial(1,{}1)}.
488058`\spad{monomial(a,{}[x1,{}..,{}xk],{}[n1,{}..,{}nk])} computes \spad{a * x1^n1 * .. * xk^nk}.
488253`\spad{monomial(a,{}[v1..vn],{}[e1..en])} returns \spad{a*prod(vi^ei)}.
488354`\spad{monomial(a,{}x,{}n)} computes \spad{a*x^n}.
488525`\spad{monomial(a,{}x,{}n)} creates the monomial \spad{a*x^n} where \spad{a} is a polynomial,{} \spad{x} is a variable and \spad{n} is a nonnegative integer.
488680`\spad{monomials(x)} returns the list of \spad{r_i*b_i} whose sum is \spad{x}.
488743`\spad{monomials(f)} gives the list of all monomials whose sum is \spad{f}.
488806`\spad{monomials(p)} returns the list of non-zero monomials of polynomial \spad{p},{} \spadignore{i.e.} \spad{monomials(sum(a_(i) X^(i))) = [a_(1) X^(1),{}...,{}a_(n) X^(n)]}.
489176`\spad{more?(a,{} b)} compares \spad{a} and \spad{b} in the partial ordering induced by setOrder,{} and uses the ordering on \spad{S} if \spad{a} and \spad{b} are not comparable in the partial ordering.
489258`\spad{more?(u,{}n)} tests if \spad{u} has greater than \spad{n} elements.
489321`\axiom{moreAlgebraic?(\spad{ts},{}us)} returns \spad{false} iff \axiom{\spad{ts}} and \axiom{us} are both empty,{} or \axiom{\spad{ts}} has less elements than \axiom{us},{} or some variable is algebraic \spad{w}.\spad{r}.\spad{t}. \axiom{us} and is not \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{ts}}.
489401`\axiom{moreAlgebraic?(\spad{ts},{}us)} returns \spad{false} iff \axiom{\spad{ts}} and \axiom{us} are both empty,{} or \axiom{\spad{ts}} has less elements than \axiom{us},{} or some variable is algebraic \spad{w}.\spad{r}.\spad{t}. \axiom{us} and is not \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{ts}}.
489491`\spad{morphism(f)} returns the non-invertible morphism given by \spad{f}.
489543`\spad{morphism(f)} returns the morphism given by \spad{f^n(x) = f(x,{}n)}.
489603`\spad{morphism(f,{} g)} returns the invertible morphism given by \spad{f},{} where \spad{g} is the inverse of \spad{f}..
489836`\spad{movedPoints(p)} returns the set of points moved by the permutation \spad{p}.
490550`\spad{mulmod(a,{}b,{}p)},{} \spad{0<=a,{}b<p>1},{} means \spad{a*b mod p}.
490606`\spad{multMonom(r,{}e,{}x)} \undocumented
490683`\spad{multiEuclideanTree(l,{}r)} \undocumented{}
490832`\spad{multiEuclidean([f1,{}...,{}fn],{}z)} returns a list of coefficients \spad{[a1,{} ...,{} an]} such that \spad{ z / prod \spad{fi} = sum aj/fj}. If no such list of coefficients exists,{} "failed" is returned.
490917`\spad{multinomial(n,{}[m1,{}m2,{}...,{}mk])} returns the multinomial coefficient \spad{n!/(m1! m2! ... mk!)}.
491034`\spad{multiple(x)} tells the pattern matcher that \spad{x} should preferably match a multi-term quantity in a sum or product. For matching on lists,{} multiple(\spad{x}) tells the pattern matcher that \spad{x} should match a list instead of an element of a list. Error: if \spad{x} is not a symbol.
491518`\spad{multiplyExponents(f,{}r)} multiplies all exponents of the power series \spad{f} by the positive rational number \spad{r}.
491615`\spad{multiplyExponents(p,{}n)} returns a new polynomial resulting from multiplying all exponents of the polynomial \spad{p} by the non negative integer \spad{n}.
491707`\spad{multiplyExponents(f,{}n)} multiplies all exponents of the power series \spad{f} by the positive integer \spad{n}.
492055`\spad{multiset()}\$\spad{D} creates an empty multiset of domain \spad{D}.
492097`\spad{multiset(ls)} creates a multiset with elements from \spad{ls}.
492146`\spad{multiset(s)} creates a multiset with singleton \spad{s}.
492443`\spad{multivariate(sup,{}v)} converts an anonymous univariable polynomial \spad{sup} to a polynomial in the variable \spad{v}.
492544`\spad{multivariate(sup,{}v)} converts an anonymous univariable polynomial \spad{sup} to a polynomial in the variable \spad{v}.
492781`to allow entries like: 1+2*e1+3*e2+4*e1e2 = multivector[1,{}2,{}3,{}4]
493112`\axiom{mvar(\spad{p})} returns an error if \axiom{\spad{p}} belongs to \axiom{\spad{R}},{} otherwise returns its main variable \spad{w}. \spad{r}. \spad{t}. to the total ordering on the elements in \axiom{\spad{V}}.
493174`\axiom{mvar(\spad{ps})} returns the main variable of the non constant polynomial with the greatest main variable,{} if any,{} else an error is returned.
494674`\spad{name(x)} returns the symbol
494724`\spad{name(op(a1,{}...,{}an))} returns the name of op.
494766`\spad{name(x)} returns the symbol
494812`\spad{name(f)} returns the external name of the file \spad{f}.
494863`\spad{name(f)} returns the name part of the file name.
494912`\spad{name(op)} returns the name of \spad{op}.
494958`\spad{name(s)} returns \spad{s} without its scripts.
495033`\spad{nand(a,{}b)} returns the logical {\em nand} of bit aggregates \axiom{a} and \axiom{\spad{b}}.
495077`\spad{nand(a,{}b)} returns the logical negation of \spad{a} and \spad{b}.
495116`\spad{nary?(op)} tests if \spad{op} has arbitrary arity.
495247`\spad{ncols(m)} returns the number of columns in the matrix \spad{m}.
495333`\spad{ncols(m)} returns the number of columns in the array \spad{m}
495465`\spad{negative?(u)} returns \axiom{\spad{true}} if every element of \spad{u} is negative,{} \axiom{\spad{false}} otherwise.
495523`\spad{negative?(x)} tests whether \spad{x} is strictly less than 0.
495667`\axiom{negative?(pol,{}aRoot)} answers if \axiom{pol} interpreted as \axiom{aRoot} is negative
495807`\spad{newReduc()} \undocumented
495864`\spad{newSubProgram(f)} asserts that from now on type declarations are part of subprogram \spad{f}.
495974`\spad{new()} returns a new empty match result.
496029`\spad{new()} returns a new empty match result.
496078`\spad{new()} \undocumented
496117`\spad{new()} \undocumented
496168`\spad{new()} create a new,{} empty object. Use \spadfun{setPrologue!},{} \spadfun{setFormula!} and \spadfun{setEpilogue!} to set the various components of this object.
496213`\spad{new()} returns a new symbol whose name starts with \%.
496245`\spad{new()} create a new,{} empty object. Use \spadfun{setPrologue!},{} \spadfun{setTex!} and \spadfun{setEpilogue!} to set the various components of this object.
496280`\spad{new(s)} returns a new symbol whose name starts with \%\spad{s}.
496373`\spad{new(n,{}x)} returns \axiom{fill!(new \spad{n},{}\spad{x})}.
496437`\spad{new(m,{}n,{}r)} is an \spad{m}-by-\spad{n} array all of whose entries are \spad{r}
496540`\spad{new(d,{}pref,{}e)} constructs the name of a new writable file with \spad{d} as its directory,{} \spad{pref} as a prefix of its name and \spad{e} as its extension. When \spad{d} or \spad{t} is the empty string,{} a default is used. An error occurs if a new file cannot be written in the given directory.
496602`\spad{newton}(\spad{l}) returns the interpolating polynomial for the values \spad{l},{} where the \spad{x}-coordinates are assumed to be [1,{}2,{}3,{}...,{}\spad{n}] and the coefficients of the interpolating polynomial are known to be in the domain \spad{F}. \spad{I}.\spad{e}.,{} it is a very streamlined version for a special case of interpolation.
497225`\spad{nextItem(x)} returns the next item,{} or "failed" if domain is exhausted.
497925`\spad{nextPrime(n)} returns the smallest prime strictly larger than \spad{n}
498470`\spad{next(l)} returns the doubly-linked aggregate beginning with its next element. Error: if \spad{l} has no next element. Note: \axiom{next(\spad{l}) = rest(\spad{l})} and \axiom{previous(next(\spad{l})) = \spad{l}}.
498523`\axiom{\spad{nextsousResultant2}(\spad{P},{} \spad{Q},{} \spad{Z},{} \spad{s})} returns the subresultant \axiom{\spad{S_}{\spad{e}-1}} where \axiom{\spad{P} ~ \spad{S_d},{} \spad{Q} = \spad{S_}{\spad{d}-1},{} \spad{Z} = S_e,{} \spad{s} = \spad{lc}(\spad{S_d})}
498613`\axiom{\spad{nextsubResultant2}(\spad{p},{}\spad{q},{}\spad{z},{}\spad{s})} is the multivariate version of the operation \axiomOpFrom{\spad{next_sousResultant2}}{PseudoRemainderSequence} from the \axiomType{PseudoRemainderSequence} constructor.
498719`\spad{nilFactor(base,{}exponent)} creates a factored object with a single factor with no information about the kind of \spad{base} (flag = "nil").
498771`\spad{nil()} returns the empty list.
498914`\spad{noKaratsuba(a,{}b)} returns \spad{a*b} without using Karatsuba\spad{'s} trick at all.
498996`\spad{noLinearFactor?(p)} returns \spad{true} if \spad{p} can be shown to have no linear factor by a theorem of Lehmer,{} \spad{false} else. \spad{I} insist on the fact that \spad{false} does not mean that \spad{p} has a linear factor.
499236`\spad{node?(u,{}v)} tests if node \spad{u} is contained in node \spad{v} (either as a child,{} a child of a child,{} etc.).
499382`\spad{node(left,{}v,{}right)} creates a binary tree with value \spad{v},{} a binary tree \spad{left},{} and a binary tree \spad{right}.
499608`\spad{nodes(u)} returns a list of all of the nodes of aggregate \spad{u}.
499665`\spad{nonQsign(r)} \undocumented
499926`\spad{noncommutativeJordanAlgebra?()} tests if the algebra is flexible and Jordan admissible.
500019`\spad{noncommutativeJordanAlgebra?()} tests if the algebra is flexible and Jordan admissible.
500148`\spad{nor(a,{}b)} returns the logical {\em nor} of bit aggregates \axiom{a} and \axiom{\spad{b}}.
500191`\spad{nor(a,{}b)} returns the logical negation of \spad{a} or \spad{b}.
500353`\spad{normFactors(x)} \undocumented
500817`\spad{norm(p)} determines sum of absolute values of coefficients Note: this function depends on \spadfunFrom{abs}{Complex}.
500876`\spad{norm(I)} returns the norm of the ideal \spad{I}.
500929`\spad{norm(f)} returns the norm of the module \spad{f}.
500986`\spad{norm q} returns the norm of \spad{q},{} \spadignore{i.e.} the product of all the conjugates of \spad{q}.
501056`\spad{norm(a)} computes the norm of \spad{a} with respect to the field considered as an algebra with 1 over the ground field \spad{F}.
501116`\spad{norm(x)} returns \spad{x} * conjugate(\spad{x})
501162`\spad{norm(a)} returns the determinant of the regular representation of \spad{a} with respect to any basis.
501213`\spad{norm(o)} returns the norm of an octonion,{} equal to the sum of the squares of its coefficients.
501260`\spad{norm(q)} computes the norm of \spad{q} (the sum of the squares of the components).
501309`\spad{norm x} returns the same as absolute value.
501825`\spad{norm(f,{}k)} computes the norm of the algebraic number \spad{f} with respect to the extension generated by kernel \spad{k}
501885`\spad{norm(f,{}l)} computes the norm of the algebraic number \spad{f} with respect to the extension generated by kernels \spad{l}
502169`\spad{norm(f,{}k)} computes the norm of the algebraic number \spad{f} with respect to the extension generated by kernel \spad{k}
502224`\spad{norm(f,{}l)} computes the norm of the algebraic number \spad{f} with respect to the extension generated by kernels \spad{l}
502285`\spad{norm(a,{}d)} computes the norm of \spad{a} with respect to the field of extension degree \spad{d} over the ground field of size. Error: if \spad{d} does not divide the extension degree of \spad{a}. Note: norm(a,{}\spad{d}) = reduce(*,{}[a^(\spad{q^}(d*i)) for \spad{i} in 0..\spad{n/d}])
502375`\spad{normal01()} \undocumented
502665`\spad{normal?(a)} tests whether the element \spad{a} is normal over the ground field \spad{F},{} \spadignore{i.e.} \spad{a^(q^i),{} 0 <= i <= extensionDegree()-1} is an \spad{F}-basis,{} where \spad{q = size()\$F}. Implementation according to Lidl/Niederreiter: Theorem 2.39.
502946`\spad{normalElement()} returns a element,{} normal over the ground field \spad{F},{} \spadignore{i.e.} \spad{a^(q^i),{} 0 <= i < extensionDegree()} is an \spad{F}-basis,{} where \spad{q = size()\$F}. At the first call,{} the element is computed by \spadfunFrom{createNormalElement}{FiniteAlgebraicExtensionField} then cached in a global variable. On subsequent calls,{} the element is retrieved by referencing the global varia
502946`--ble.
503115`\spad{normalForm(poly,{}gb)} reduces the polynomial \spad{poly} modulo the precomputed groebner basis \spad{gb} giving a canonical representative of the residue class.
503346`\spad{normal(f,{}g)} \undocumented
503417`normalalised point has scalar value of one,{} if this is not the case then scale values
503655`\spad{normalizeAtInfinity(v)} makes \spad{v} normal at infinity.
503747`\axiom{normalizeIfCan(\spad{ts})} returns \axiom{\spad{ts}} in an normalized shape if \axiom{\spad{ts}} is zero-dimensional.
503824`\spad{normalize(f)} rewrites \spad{f} using the least possible number of real algebraically independent kernels.
503895`\spad{normalize(x)} normalizes \spad{x} at current precision.
504164`\axiom{normalized?(\spad{ts})} returns \spad{true} iff for every axiom{\spad{p}} in axiom{\spad{ts}} we have \axiom{normalized?(\spad{p},{}us)} where \axiom{us} is \axiom{collectUnder(\spad{ts},{}mvar(\spad{p}))}.
504459`\axiom{normalized?(\spad{p},{}\spad{ts})} returns \spad{true} iff \axiom{\spad{p}} and all its iterated initials have degree zero \spad{w}.\spad{r}.\spad{t}. the main variables of the polynomials of \axiom{\spad{ts}}
504532`\axiom{normalized?(\spad{q},{}\spad{lp})} returns \spad{true} iff \axiom{normalized?(\spad{q},{}\spad{p})} holds for every \axiom{\spad{p}} in \axiom{\spad{lp}}.
504616`\axiom{normalized?(a,{}\spad{b})} returns \spad{true} iff \axiom{a} and its iterated initials have degree zero \spad{w}.\spad{r}.\spad{t}. the main variable of \axiom{\spad{b}}
504694`\axiom{normalizedAssociate(\spad{p},{}\spad{ts})} returns a normalized polynomial \axiom{\spad{n}} \spad{w}.\spad{r}.\spad{t}. \spad{ts} such that \axiom{\spad{n}} and \axiom{\spad{p}} are associates \spad{w}.\spad{r}.\spad{t} \spad{ts} and assuming that \axiom{\spad{p}} is invertible \spad{w}.\spad{r}.\spad{t} \spad{ts}.
504771`\spad{normalizedDivide(n,{}d)} returns a normalized quotient and remainder such that consistently unique representatives for the residue class are chosen,{} \spadignore{e.g.} positive remainders
504872`\spad{normalizedDivide(n,{}d)} returns a normalized quotient and remainder such that consistently unique representatives for the residue class are chosen,{} \spadignore{e.g.} positive remainders
505042`\spad{not f} creates the equivalent prefix form.
505080`\spad{not(b)} returns the logical {\em not} of bit aggregate \axiom{\spad{b}}.
505120`\spad{not n} returns the negation of \spad{n}.
505155`\spad{not(n)} returns the bit-by-bit logical {\em not} of the single integer \spad{n}.
505582`\spad{nrows(m)} returns the number of rows in the matrix \spad{m}.
505668`\spad{nrows(m)} returns the number of rows in the array \spad{m}
505952`\spad{nthCoef(x,{} n)} returns the coefficient of the n^th term of \spad{x}.
506021`\spad{nthExpon(l,{} n)} returns the exponent of the n^th monomial of \spad{l}.
506082`\spad{nthExpon(x,{} n)} returns the exponent of the n^th monomial of \spad{x}.
506140`\spad{nthExpon(x,{} n)} returns the exponent of the n^th monomial of \spad{x}.
506282`\spad{nthExponent(u,{}n)} returns the exponent of the \spad{n}th factor of \spadvar{\spad{u}}. If \spadvar{\spad{n}} is not a valid index for a factor (for example,{} less than 1 or too big),{} 0 is returned.
506342`\spad{nthFactor(x,{} n)} returns the factor of the n^th monomial of \spad{x}.
506395`\spad{nthFactor(x,{} n)} returns the factor of the n^th monomial of \spad{x}.
506449`\spad{nthFactor(l,{} n)} returns the factor of the n^th monomial of \spad{l}.
506511`\spad{nthFactor(x,{} n)} returns the factor of the \spad{n-th} monomial of \spad{x}.
506572`\spad{nthFactor(u,{}n)} returns the base of the \spad{n}th factor of \spadvar{\spad{u}}. If \spadvar{\spad{n}} is not a valid index for a factor (for example,{} less than 1 or too big),{} 1 is returned. If \spadvar{\spad{u}} consists only of a unit,{} the unit is returned.
506624`\spad{nthFactor(x,{} n)} returns the factor of the n^th term of \spad{x}.
506774`\spad{nthFractionalTerm(p,{}n)} extracts the \spad{n}th fractional term from the partial fraction \spad{p}. This returns 0 if the index \spad{n} is out of range.
507176`\spad{nthRoot(x,{}n)} returns the \spad{n}th root of \spad{x}.
507356`\spad{null?(s)} is \spad{true} if \spad{s} is the \spad{S}-expression ().
507432`\spad{nullSpace(m)} returns a basis for the null space of the matrix \spad{m}.
507543`\spad{nullSpace(m)} returns a basis for the null space of the matrix \spad{m}.
507671`\spad{nullSpace M} builds a basis of the kernel of the linear transformation defined by the matrix \spad{M}. This function calls the \spad{nullSpace} in the \spad{Matrix} domain. Only the signature changes.
507731`\spad{nullSpace(m)} returns a basis for the null space of the matrix \spad{m}.
507833`\spad{nullSpace(m)} returns a basis for the null space of the matrix \spad{m}.
507920`\spad{nullSpace(m)}+ returns a basis for the null space of the matrix \spad{m}.
508022`\spad{null(u)} tests if list \spad{u} is the empty list.
508063`\spad{nullary?(op)} tests if \spad{op} is nullary.
508114`\spad{nullary A} changes its argument into a \indented{1}{nullary function.}
508166`\spad{nullity(m)} returns the mullity of the matrix \spad{m}. This is the dimension of the null space of the matrix \spad{m}.
508259`\spad{nullity(m)} returns the mullity of the matrix \spad{m}. This is the dimension of the null space of the matrix \spad{m}.
508368`\spad{nullity(m)} returns the nullity of the matrix \spad{m}. This is the dimension of the null space of the matrix \spad{m}.
508462`\spad{nullity(m)} returns the nullity of the matrix \spad{m}. This is the dimension of the null space of the matrix \spad{m}.
508651`\spad{number?(f)} tests if \spad{f} is rational
508787`\spad{numberOfComponents()} returns the number of absolutely irreducible components.
508879`\spad{numberOfComponents()} returns the number of absolutely irreducible components.
509118`\spad{numberOfComputedEntries(st)} returns the number of explicitly computed entries of stream \spad{st} which exist immediately prior to the time this function is called.
509204`\spad{numberOfCycles(p)} returns the number of non-trivial cycles of the permutation \spad{p}.
509459`\spad{numberOfFactors(u)} returns the number of factors in \spadvar{\spad{u}}.
509526`\spad{numberOfFractionalTerms(p)} computes the number of fractional terms in \spad{p}. This returns 0 if there is no fractional part.
509845`\spad{numberOfMonomials(p)} gives the number of non-zero monomials in polynomial \spad{p}.
509931`\spad{numberOfMonomials(x)} returns the number of monomials of \spad{x}.
510012`\spad{numberOfMonomials(f)} is the number of non-zero coefficients with respect to the canonical basis.
510516`\spad{numer x} returns the numerator of \spad{x}.
510562`\spad{numer x} returns the numerator of \spad{x}.
510752`\spad{numer(x)} returns the numerator of the fraction \spad{x}.
511083`\spad{numerator(f)} returns the numerator of \spad{f} converted to \%.
511144`\spad{numerator(x)} is the numerator of the fraction \spad{x} converted to \%.
511202`\spad{numerators(x)} returns the stream of numerators of the approximants of the continued fraction \spadvar{\spad{x}}. If the continued fraction is finite,{} then the stream will be finite.
511264`\spad{numericIfCan(x)} returns a real approximation of \spad{x},{} or "failed" if \axiom{\spad{x}} is not a constant.
511385`\spad{numericIfCan(x)} returns a real approximation of \spad{x},{} or "failed" if \axiom{\spad{x}} is not a constant.
511493`\spad{numericIfCan(x)} returns a real approximation of \spad{x},{} or "failed" if \axiom{\spad{x}} is not a constant.
511581`\spad{numericIfCan(x,{} n)} returns a real approximation of \spad{x} up to \spad{n} decimal places,{} or "failed" if \axiom{\spad{x}} is not a constant.
511718`\spad{numericIfCan(x,{}n)} returns a real approximation of \spad{x} up to \spad{n} decimal places,{} or "failed" if \axiom{\spad{x}} is not a constant.
511842`\spad{numericIfCan(x,{}n)} returns a real approximation of \spad{x} up to \spad{n} decimal places,{} or "failed" if \axiom{\spad{x}} is not a constant.
511946`\spad{numeric(x)} returns a real approximation of \spad{x}.
512046`\spad{numeric(x)} returns a real approximation of \spad{x}.
512133`\spad{numeric(x)} returns a real approximation of \spad{x}.
512200`\spad{numeric(x)} returns a real approximation of \spad{x}.
512244`\spad{numeric(x,{} n)} returns a real approximation of \spad{x} up to \spad{n} decimal places.
512360`\spad{numeric(x,{}n)} returns a real approximation of \spad{x} up to \spad{n} decimal places.
512463`\spad{numeric(x,{}n)} returns a real approximation of \spad{x} up to \spad{n} decimal places.
512546`\spad{numeric(x,{} n)} returns a real approximation of \spad{x} up to \spad{n} decimal places.
512606`\spad{obj(a)} essentially returns the original object that was converted to \spadtype{Any} except that the type is forced to be \spadtype{None}.
512639`\spad{objectOf(a)} returns a printable form of the original object that was converted to \spadtype{Any}.
512980`\spad{octon(re,{}\spad{ri},{}rj,{}rk,{}rE,{}rI,{}rJ,{}rK)} constructs an octonion from scalars.
513089`\spad{odd? x} is \spad{true} if \spad{x} is an odd integer.
513168`\spad{odd?(n)} returns \spad{true} if and only if \spad{n} is odd.
513221`\spad{odd?(p)} returns \spad{true} if and only if \spad{p} is an odd permutation \spadignore{i.e.} {\em sign(p)} is {\em -1}.
513354`\spad{oddInfiniteProduct(f(x))} computes \spad{product(n=1,{}3,{}5...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
513442`\spad{oddInfiniteProduct(f(x))} computes \spad{product(n=1,{}3,{}5...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
513528`\spad{oddInfiniteProduct(f(x))} computes \spad{product(n=1,{}3,{}5...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
513826`\spad{oddlambert(f(x))} returns \spad{f(x) + f(x^3) + f(x^5) + ...}. \indented{1}{\spad{f(x)} should have a zero constant coefficient.} \indented{1}{This function is used for computing infinite products.} \indented{1}{If \spad{f(x)} is a Taylor series with constant term 1,{} then} \indented{1}{\spad{product(n=1..infinity,{}f(x^(2*n-1)))=exp(log(oddlambert(f(x))))}.}
513897`\spad{ode1(f,{}c)} is the solution to \spad{y' = f(y)} such that \spad{y(a) = c}.
513981`\spad{ode2(f,{}c0,{}c1)} is the solution to \spad{y'' = f(y,{}y')} such that \spad{y(a) = c0} and \spad{y'(a) = c1}.
514164`\spad{omError(k,{}l)} creates an instance of OpenMathError.
514372`\spad{one?(a)} tests whether \spad{a} is the unit 1.
514419`\spad{one?(x)} tests if \spad{x} is equal to 1.
514525`\spad{oneDimensionalArray(l)} creates an array from a list of elements \spad{l}
514596`\spad{oneDimensionalArray(n,{}s)} creates an array from \spad{n} copies of element \spad{s}
514744`\spad{one(d)} specifies whether we are happy with one solution. This option is expressed in the form \spad{one == d}.
514788`\spad{op(p)} creates a value in \% equal to \spad{p} in \spad{P}.
514895`\spad{open(s)} returns the file \spad{s} open for input.
516009`\spad{operator(f)} makes \spad{f} into an operator with arbitrary arity.
516059`\spad{operator(f,{} n)} makes \spad{f} into an \spad{n}-ary operator.
516359`\spad{option?()} is not to be used at the top level; option? internally returns \spad{true} for drawing options which are indicated in a draw command,{} or \spad{false} for those which are not.
516419`\spad{option(l,{}s)} determines whether the indicated drawing option,{} \spad{s},{} is contained in the list of drawing options,{} \spad{l},{} which is defined by the draw command.
516509`\spad{option()} is not to be used at the top level; option determines internally which drawing options are indicated in a draw command.
516580`\spad{option(l,{} option)} returns which options are given.
516696`\spad{optional(x)} tells the pattern matcher that \spad{x} can match an identity (0 in a sum,{} 1 in a product or exponentiation). Error: if \spad{x} is not a symbol.
517088`\spad{optpair(l)} returns \spad{l} has the form \spad{[a,{} b]} and a is optional,{} and "failed" otherwise.
517158`\spad{f or g} creates the equivalent infix form.
517198`\spad{a or b} returns the logical {\em or} of bit aggregates \axiom{a} and \axiom{\spad{b}}.
517240`\spad{a or b} returns the logical inclusive {\em or} of Boolean \spad{a} and \spad{b}.
517339`\spad{orbit(p,{} el)} returns the orbit of {\em el} under the permutation \spad{p},{} \spadignore{i.e.} the set which is given by applications of the powers of \spad{p} to {\em el}.
517561`\spad{ord(c)} provides an integral code corresponding to the character \spad{c}. It is always \spad{true} that \spad{char ord c = c}.
517685`\spad{order(x)} \undocumented
518291`\spad{order(f)} is the degree of the lowest order non-zero term in \spad{f}. This will result in an infinite loop if \spad{f} has no non-zero terms.
518365`\spad{order x} is the order of magnitude of \spad{x}. Note: \spad{base ^ order x <= |x| < base ^ (1 + order x)}.
518419`\spad{order(p)} returns the order of the differential polynomial \spad{p},{} which is the maximum number of differentiations of a differential indeterminate,{} among all those appearing in \spad{p}.
518504`\spad{order(v)} returns \spad{n} if \spad{v} is the \spad{n}-th derivative of any differential indeterminate.
518645`\spad{order(p)} returns the order of a permutation \spad{p} as a group element.
518765`\spad{order(a)} computes the order of an element in the multiplicative group of the field. Error: if \spad{a} is 0.
518854`\spad{order(b)} computes the order of an element \spad{b} in the multiplicative group of the field. Error: if \spad{b} equals 0.
519159`\spad{order(f,{}n) = min(m,{}n)},{} where \spad{m} is the degree of the lowest order non-zero term in \spad{f}.
519239`\spad{order(p,{}s)} returns the order of the differential polynomial \spad{p} in differential indeterminate \spad{s}.
519413`\spad{order(p,{} q)} returns the largest \spad{n} such that \spad{q^n} divides polynomial \spad{p} \spadignore{i.e.} the order of \spad{p(x)} at \spad{q(x)=0}.
519912`\spad{outerProduct(u,{}v)} constructs the matrix whose (\spad{i},{}\spad{j})\spad{'}th element is \spad{u}(\spad{i})\spad{*v}(\spad{j}).
520133`\spad{outputAsFortran(fn)} \undocumented{}
520195`\axiom{outputAsFortran(\spad{u})} translates \axiom{\spad{u}} into a legal FORTRAN subprogram.
520259`\spad{outputAsFortran(l)} sends (for each expression in the list \spad{l}) output in FORTRAN format to the destination defined by \spadsyscom{set output fortran}.
520335`\spad{outputAsFortran(o)} sends output \spad{o} in FORTRAN format.
520405`\spad{outputAsFortran(v,{}o)} sends output \spad{v} = \spad{o} in FORTRAN format to the destination defined by \spadsyscom{set output fortran}.
520482`\spad{outputAsScript(l)} sends (for each expression in the list \spad{l}) output in Script Formula Formatter format to the destination defined. by \spadsyscom{set output forumula}.
520557`\spad{outputAsScript(o)} sends output \spad{o} in Script Formula Formatter format to the destination defined by \spadsyscom{set output formula}.
520626`\spad{outputAsTex(l)} sends (for each expression in the list \spad{l}) output in Tex format to the destination as defined by \spadsyscom{set output tex}.
520698`\spad{outputAsTex(o)} sends output \spad{o} in Tex format to the destination defined by \spadsyscom{set output tex}.
520764`\spad{outputFixed()} sets the output mode to fixed point notation; the output will contain a decimal point.
520859`\spad{outputFloating()} sets the output mode to floating (scientific) notation,{} \spadignore{i.e.} \spad{mantissa * 10 exponent} is displayed as \spad{0.mantissa E exponent}.
520960`\spad{outputForm(sf)} creates an form for small float \spad{sf}.
521014`\spad{outputForm(n)} creates an form for integer \spad{n}.
521064`\spad{outputForm(s)} creates an form for string \spad{s}.
521113`\spad{outputForm(s)} creates an form for symbol \spad{s}.
521162`\spad{outputForm(p,{}var)} converts the SparseUnivariatePolynomial \spad{p} to an output form (see \spadtype{OutputForm}) printed as a polynomial in the output form variable.
521245`\spad{outputForm(p,{} x)} returns the output form of \spad{p} using \spad{x} for the otherwise anonymous variable.
521483`\spad{outputGeneral()} sets the output mode (default mode) to general notation; numbers will be displayed in either fixed or floating (scientific) notation depending on the magnitude.
521582`\spad{outputList(l)} displays the concatenated components of the list \spad{l} on the ``algebra output\spad{''} stream,{} as defined by \spadsyscom{set output algebra}; quotes are stripped from strings.
521799`\spad{output(x)} displays the output form \spad{x} on the ``algebra output\spad{''} stream,{} as defined by \spadsyscom{set output algebra}.
521853`\spad{output(s)} displays the string \spad{s} on the ``algebra output\spad{''} stream,{} as defined by \spadsyscom{set output algebra}.
521966`\spad{output(s,{}x)} displays the string \spad{s} followed by the form \spad{x} on the ``algebra output\spad{''} stream,{} as defined by \spadsyscom{set output algebra}.
522027`\spad{over(f,{}g)} creates a form for the vertical fraction of \spad{f} over \spad{g}.
522069`\spad{overbar(f)} creates the form \spad{f} with an overbar.
522111`\spad{overlabel(x,{}f)} creates the form \spad{f} with \spad{"x} overbar" over the top.
522158`\spad{overlap(x,{} y)} returns \spad{[l,{} m,{} r]} such that \spad{x = l * m},{} \spad{y = m * r} and \spad{l} and \spad{r} have no overlap,{} \spadignore{i.e.} \spad{overlap(l,{} r) = [l,{} 1,{} r]}.
522229`\spad{overlap(x,{} y)} returns \spad{[l,{} m,{} r]} such that \spad{x = l * m} and \spad{y = m * r} hold and such that \spad{l} and \spad{r} have no overlap,{} that is \spad{overlap(l,{} r) = [l,{} 1,{} r]}.
522590`\spad{pack!(f)} reorganizes the file \spad{f} on disk to recover unused space.
522642`\spad{pack!(f)} reorganizes the file \spad{f} on disk to recover unused space.
522679`\spad{packageCall(f)} returns the input form corresponding to \spad{f}\$\spad{R}.
522748`\spad{packageCall(f,{} l)} returns the input form corresponding to \spad{f}(\spad{l}.1,{} \spad{l}.2,{}...)\$\spad{R}.
523445`\spad{padicFraction(q)} expands the fraction \spad{p}-adically in the primes \spad{p} in the denominator of \spad{q}. For example,{} \spad{padicFraction(3/(2^2)) = 1/2 + 1/(2^2)}. Use \spadfunFrom{compactFraction}{PartialFraction} to return to compact form.
523582`\spad{pair?(s)} is \spad{true} if \spad{s} has is a non-null Lisp list.
525182`\spad{palginfieldint(f,{} d)} returns an algebraic function \spad{g} such that \spad{dg = f} if such a \spad{g} exists,{} "failed" otherwise. Argument \spad{f} must be a pure algebraic function.
526498`This returns the length of the parallel component of two vectors in the case of Euclidean space this is the dot product of two vectors. We use this function to test the colinearity
526698`\spad{paren(lf)} creates the form separating the elements of \spad{lf} by commas and encloses the result in parentheses.
526786`\spad{paren(f)} creates the form enclosing \spad{f} in parentheses.
526826`\spad{paren([f1,{}...,{}fn])} returns \spad{(f1,{}...,{}fn)}. This prevents the \spad{fi} from being evaluated when operators are applied to them,{} and makes them applicable to a unary operator. For example,{} \spad{atan(paren [x,{} 2])} returns the formal kernel \spad{atan((x,{} 2))}.
526877`\spad{paren(f)} returns (\spad{f}). This prevents \spad{f} from being evaluated when operators are applied to it. For example,{} \spad{log(1)} returns 0,{} but \spad{log(paren 1)} returns the formal kernel log((1)).
526922`\spad{parent(s)} returns the subspace which is the parent of the indicated 3 dimensional subspace \spad{s}. If \spad{s} is the top level subspace an error message is returned.
526966`\spad{parse(s)} is the inverse of unparse. It parses a string to InputForm
527009`\spad{partialDenominators(x)} extracts the denominators in \spadvar{\spad{x}}. That is,{} if \spad{x = continuedFraction(b0,{} [a1,{}a2,{}a3,{}...],{} [b1,{}b2,{}b3,{}...])},{} then \spad{partialDenominators(x) = [b1,{}b2,{}b3,{}...]}.
527336`\spad{partialNumerators(x)} extracts the numerators in \spadvar{\spad{x}}. That is,{} if \spad{x = continuedFraction(b0,{} [a1,{}a2,{}a3,{}...],{} [b1,{}b2,{}b3,{}...])},{} then \spad{partialNumerators(x) = [a1,{}a2,{}a3,{}...]}.
527405`\spad{partialQuotients(x)} extracts the partial quotients in \spadvar{\spad{x}}. That is,{} if \spad{x = continuedFraction(b0,{} [a1,{}a2,{}a3,{}...],{} [b1,{}b2,{}b3,{}...])},{} then \spad{partialQuotients(x) = [b0,{}b1,{}b2,{}b3,{}...]}.
527473`\spad{particularSolution(u)} \undocumented
527886`\spad{partition(n)} returns the number of partitions of the integer \spad{n}. This is the number of distinct ways that \spad{n} can be written as a sum of positive integers.
528540`\spad{parts(m)} returns a list of the elements of \spad{m} in row major order
528613`\spad{parts(u)} returns a list of the consecutive elements of \spad{u}. For collections,{} \axiom{parts([\spad{x},{}\spad{y},{}...,{}\spad{z}]) = (\spad{x},{}\spad{y},{}...,{}\spad{z})}.
529648`\spad{patternMatch(n,{} pat,{} res)} matches the pattern \spad{pat} to the integer \spad{n}; res contains the variables of \spad{pat} which are already matched and their matches.
531085`\spad{patternMatch(expr,{} pat,{} res)} matches the pattern \spad{pat} to the expression \spad{expr}; res contains the variables of \spad{pat} which are already matched and their matches (necessary for recursion).
531213`\spad{patternMatch(expr,{} pat,{} res)} matches the pattern \spad{pat} to the expression \spad{expr}. res contains the variables of \spad{pat} which are already matched and their matches (necessary for recursion). Initially,{} res is just the result of \spadfun{new} which is an empty list of matches.
532133`\spad{perfectSqrt(n)} returns the square root of \spad{n} if \spad{n} is a perfect square and returns "failed" otherwise
532198`\spad{perfectSquare?(n)} returns \spad{true} if \spad{n} is a perfect square and \spad{false} otherwise
532927`\spad{permutation(n,{} r)} returns the number of permutations of \spad{n} objects taken \spad{r} at a time,{} \spadignore{i.e.} \spad{n!/}(\spad{n}-\spad{r})!.
533043`\spad{permutation(n)} returns \spad{!P(n,{}r) = n!/(n-r)!}. This is the number of permutations of \spad{n} objects taken \spad{r} at a time.
533110`\spad{permutation(n,{} m)} returns the number of permutations of \spad{n} objects taken \spad{m} at a time. Note: \spad{permutation(n,{}m) = n!/(n-m)!}.
533259`This returns a vector perpendicular to the two vectors in the case of 3D Euclidean space this is the cros product of the two vectors. We use this function to compute orthogonal
533554`\spad{physicalLength(x)} returns the number of elements \spad{x} can accomodate before growing
533701`\spad{\spad{pi}()} returns the \spad{pi} operator
533841`\spad{\spad{pi}(x)} represents the NAG Library function X01AAF which returns \indented{1}{an approximation to the value of \spad{pi}}
533918`\spad{\spad{pi}()} returns the constant \spad{pi}.
533973`\spad{\spad{pi}()} returns the symbolic \%\spad{pi}.
534000`\spad{pile(l)} creates the form consisting of the elements of \spad{l} which displays as a pile,{} \spadignore{i.e.} the elements begin on a new line and are indented right to the same margin.
534118`\spad{plenaryPower(a,{}n)} is recursively defined to be \spad{plenaryPower(a,{}n-1)*plenaryPower(a,{}n-1)} for \spad{n>1} and \spad{a} for \spad{n=1}.
534488`\spad{plotPolar(f,{}theta)} plots the graph of \spad{r = f(theta)} as \spad{theta} ranges from 0 to 2 \spad{pi}
534547`\spad{plotPolar(f,{}theta,{}seg)} plots the graph of \spad{r = f(theta)} as \spad{theta} ranges over an interval
535173`\spad{plot(fcn,{}x,{}seg)} plots the graph of \spad{y = f(x)} on a interval
535248`\spad{plot(f,{}g,{}t,{}seg)} plots the graph of \spad{x = f(t)},{} \spad{y = g(t)} as \spad{t} ranges over an interval.
535936`\spad{plusInfinity()} returns plusIinfinity.
536003`\spad{plusInfinity()} returns +infinity.
536058`\spad{plus(x,{} y)} returns \spad{x + y} where \spad{+} is the monoid operation,{} which is assumed commutative.
536111`\spad{plus(x,{}y)} adds two matrices,{} term by term we note that they must be the same size
536179`\spad{plus(s,{} e,{} x)} returns \spad{e * s + x} where \spad{+} is the monoid operation,{} which is assumed commutative.
536907`\spad{po(q)} creates a value in \spad{P} equal to \spad{q} in \%.
537230`\spad{pointColor(v)} specifies a color,{} \spad{v},{} for 2D graph points. This option is expressed in the form \spad{pointColor == v}.
537377`returns points
538146`\spad{point(l)} returns a point category defined by a list \spad{l} of elements from the domain \spad{R}.
538711`\spad{poisson(f)} \undocumented
539046`\spad{polarCoordinates(x)} returns (\spad{r},{} phi) such that \spad{x} = \spad{r} * exp(\%\spad{i} * phi).
539243`\spad{pole?(x)} tests if the series \spad{x} has a pole. \indented{1}{Note: this is \spad{false} when \spad{x} is a Taylor series.}
539301`\spad{pole?(f)} determines if the power series \spad{f} has a pole.
539371`\spad{polyPart(f)} returns the polynomial part of \spad{f}.
539881`\spad{polygamma(x,{}y)} returns the polygamma function applied to \spad{x} and \spad{y}
540149`\spad{polygamma(k,{}x)} is the \spad{k-th} derivative of \spad{digamma(x)},{} (often written \spad{psi(k,{}x)} in the literature).
540508`\spad{polylog(s,{} x)} is the polylogarithm of order \spad{s} at \spad{x}
540570`\spad{polylog(s,{} x)} is the polylogarithm of order \spad{s} at \spad{x}.
541074`\spad{polyred(u)} \undocumented
541205`\spad{pomopo!(p1,{}r,{}e,{}p2)} returns \spad{p1 + monomial(e,{}r) * p2} and may use \spad{p1} as workspace. The constaant \spad{r} is assumed to be nonzero.
541272`\spad{pop!(s)} returns the top element \spad{x},{} destructively removing \spad{x} from \spad{s}. Note: Use \axiom{top(\spad{s})} to obtain \spad{x} without removing it from \spad{s}. Error: if \spad{s} is empty.
541317`\spad{popFortranOutputStack()} pops the Fortran output stack
541388`\spad{position!(f,{} i)} sets the current byte-position to \spad{i}.
541457`\spad{position(x)} returns the integer \spad{n} associated to \spad{x}.
541517`\spad{position(f)} returns the current byte-position in the file \spad{f}.
541968`\spad{position(p,{}a)} returns the index \spad{i} of the first \spad{x} in \axiom{a} such that \axiom{\spad{p}(\spad{x})} is \spad{true},{} and \axiom{minIndex(a) - 1} if there is no such \spad{x}.
542043`\spad{position(x,{}a)} returns the index \spad{i} of the first occurrence of \spad{x} in a,{} and \axiom{minIndex(a) - 1} if there is no such \spad{x}.
542331`\spad{position(cc,{}t,{}i)} returns the position \axiom{\spad{j} \spad{>=} \spad{i}} in \spad{t} of the first character belonging to \spad{cc}.
542407`\spad{position(x,{}a,{}n)} returns the index \spad{i} of the first occurrence of \spad{x} in \axiom{a} where \axiom{\spad{i} \spad{>=} \spad{n}},{} and \axiom{minIndex(a) - 1} if no such \spad{x} is found.
542497`\spad{position(s,{}t,{}i)} returns the position \spad{j} of the substring \spad{s} in string \spad{t},{} where \axiom{\spad{j} \spad{>=} \spad{i}} is required.
542665`\spad{positive?(u)} returns \axiom{\spad{true}} if every element of \spad{u} is positive,{} \axiom{\spad{false}} otherwise.
542723`\spad{positive?(x)} tests whether \spad{x} is strictly greater than 0.
542867`\axiom{positive?(pol,{}aRoot)} answers if \axiom{pol} interpreted as \axiom{aRoot} is positive
542964`\spad{positiveRemainder(a,{}b)} (where \spad{b > 1}) yields \spad{r} where \spad{0 <= r < b} and \spad{r == a rem b}.
543663`\spad{possiblyInfinite?(s)} tests if the stream \spad{s} could possibly have an infinite number of elements. Note: for many datatypes,{} \axiom{possiblyInfinite?(\spad{s}) = not explictlyFinite?(\spad{s})}.
543829`\spad{postfix(op,{} a)} creates a form which prints as: a \spad{op}.
543874`\spad{pow()} \undocumented
544164`\spad{powerAssociative?()} tests if all subalgebras generated by a single element are associative.
544647`\spad{powmod(a,{}b,{}p)},{} \spad{0<=a,{}b<p>1},{} means \spad{a^b mod p}.
544764`\axiom{pquo(a,{}\spad{b})} computes the pseudo-quotient of \axiom{a} by \axiom{\spad{b}},{} both viewed as univariate polynomials in the main variable of \axiom{\spad{b}}.
544893`\axiom{pquo(a,{}\spad{b},{}\spad{v})} computes the pseudo-quotient of \axiom{a} by \axiom{\spad{b}},{} both viewed as univariate polynomials in \axiom{\spad{v}}.
545091`\spad{precision()} returns the precision in digits base.
545222`\spad{precision(n)} set the precision in the base to \spad{n} decimal digits.
545327`\spad{predicate(p)} returns the predicate attached to \spad{p},{} the constant function \spad{true} if \spad{p} has no predicates attached to it.
545401`\spad{predicates(p)} returns \spad{[p1,{}...,{}pn]} such that the predicate attached to \spad{p} is \spad{p1} and ... and \spad{pn}.
545505`\spad{prefix?(s,{}t)} tests if the string \spad{s} is the initial substring of \spad{t}. Note: \axiom{prefix?(\spad{s},{}\spad{t}) \spad{==} reduce(and,{}[\spad{s}.\spad{i} = \spad{t}.\spad{i} for \spad{i} in 0..maxIndex \spad{s}])}.
545560`\spad{prefixRagits(rx)} returns the non-cyclic part of the ragits of the fractional part of a radix expansion. For example,{} if \spad{x = 3/28 = 0.10 714285 714285 ...},{} then \spad{prefixRagits(x)=[1,{}0]}.
545626`\spad{prefix(f,{}l)} creates a form depicting the \spad{n}-ary prefix application of \spad{f} to a tuple of arguments given by list \spad{l}.
545737`\axiom{prem(a,{}\spad{b})} computes the pseudo-remainder of \axiom{a} by \axiom{\spad{b}},{} both viewed as univariate polynomials in the main variable of \axiom{\spad{b}}.
545866`\axiom{prem(a,{}\spad{b},{}\spad{v})} computes the pseudo-remainder of \axiom{a} by \axiom{\spad{b}},{} both viewed as univariate polynomials in \axiom{\spad{v}}.
546581`\spad{presub(f,{}n)} creates a form for \spad{f} presubscripted by \spad{n}.
546625`\spad{presuper(f,{}n)} creates a form for \spad{f} presuperscripted by \spad{n}.
546671`\spad{pretendOfType(f)} returns the input form corresponding to \spad{f pretend R}
546745`\spad{prevPrime(n)} returns the largest prime strictly smaller than \spad{n}
546800`\spad{previous(l)} returns the doubly-link list beginning with its previous element. Error: if \spad{l} has no previous element. Note: \axiom{next(previous(\spad{l})) = \spad{l}}.
546938`\axiom{primPartElseUnitCanonical!(\spad{p})} replaces \axiom{\spad{p}} by \axiom{primPartElseUnitCanonical(\spad{p})}.
547124`\axiom{primPartElseUnitCanonical(\spad{p})} returns \axiom{primitivePart(\spad{p})} if \axiom{\spad{R}} is a \spad{gcd}-domain,{} otherwise \axiom{unitCanonical(\spad{p})}.
547873`\spad{prime?(n)} returns \spad{true} if \spad{n} is prime and \spad{false} if not. Note that we ignore sign of \spad{n},{} so \spad{-5} is considered prime. The algorithm used is Rabin\spad{'s} probabilistic primality test (reference: Knuth Volume 2 Semi Numerical Algorithms). If \spad{prime? n} returns \spad{false},{} \spad{n} is proven composite. If \spad{prime? n} returns \spad{true},{} prime? may be in error however,{} the probability of err
547873`--or is very low. and is zero below 25*10^9 (due to a result of Pomerance et al),{} below 10^12 and 10^13 due to results of Pinch,{} and below 341550071728321 due to a result of Jaeschke. Specifically,{} this implementation does at least 10 pseudo prime tests and so the probability of error is \spad{< 4^(-10)}. The running time of this method is cubic in the length of the input \spad{n},{} that is \spad{O( (log n)^3 )},{} for \spad{n<10^20}. beyond that,{} the algorithm is quartic,{} \spad{O( (l
547873`--og n)^4 )}. Two improvements due to Davenport have been incorporated which catches some trivial strong pseudo-primes,{} such as [Jaeschke,{} 1991] 1377161253229053 * 413148375987157,{} which the original algorithm regards as prime
548143`\spad{prime?(x)} tests if \spad{x} can never be written as the product of two non-units of the ring,{} \spadignore{i.e.} \spad{x} is an irreducible element.
548204`\spad{primeFactor(base,{}exponent)} creates a factored object with a single factor whose \spad{base} is asserted to be prime (flag = "prime").
548320`\spad{primeFrobenius(a)} returns \spad{a ^ p} where \spad{p} is the characteristic.
548466`\spad{primeFrobenius(a,{}s)} returns \spad{a^(p^s)} where \spad{p} is the characteristic.
548550`\spad{prime(f)} creates the form \spad{f} followed by a suffix prime (single quote).
548590`\spad{prime(f,{}n)} creates the form \spad{f} followed by \spad{n} primes.
548649`\spad{primes(a,{}b)} returns a list of all primes \spad{p} with \spad{a <= p <= b}
549567`\spad{primitive?(b)} tests whether the element \spad{b} is a generator of the (cyclic) multiplicative group of the field,{} \spadignore{i.e.} is a primitive element. Implementation Note: see \spad{ch}.IX.1.3,{} th.2 in \spad{D}. Lipson.
549626`\spad{primitiveElement()} returns a primitive element stored in a global variable in the domain. At first call,{} the primitive element is computed by calling \spadfun{createPrimitiveElement}.
550686`\spad{primitiveMonomials(p)} gives the list of monomials of the polynomial \spad{p} with their coefficients removed. Note: \spad{primitiveMonomials(sum(a_(i) X^(i))) = [X^(1),{}...,{}X^(n)]}.
550834`\axiom{primitivePart!(\spad{p})} replaces \axiom{\spad{p}} by its primitive part.
551184`\spad{primitivePart(p)} returns the unit normalized form of polynomial \spad{p} divided by the content of \spad{p}.
551266`\spad{primitivePart(f)} removes the content of the denominator and the common content of the numerator of \spad{f}.
551336`\spad{primitivePart(p)} returns the unitCanonical associate of the polynomial \spad{p} with its content divided out.
551420`\spad{primitivePart(l)} returns \spad{l0} such that \spad{l = a * l0} for some a in \spad{R},{} and \spad{content(l0) = 1}.
551580`\spad{primitivePart(p,{} q)} reduces the coefficient of \spad{p} modulo \spad{q},{} takes the primitive part of the result,{} and ensures that the leading coefficient of that result is monic.
551667`\spad{primitivePart(p,{}v)} returns the unitCanonical associate of the polynomial \spad{p} with its content with respect to the variable \spad{v} divided out.
552247`\spad{prinb }\undocumented
552401`\spad{principal?(D)} tests if the argument is the divisor of a function.
552563`\spad{principalIdeal([f1,{}...,{}fn])} returns a record whose generator component is a generator of the ideal generated by \spad{[f1,{}...,{}fn]} whose coef component satisfies \spad{generator = sum (input.i * coef.i)}
552838`\spad{prinpolINFO }\undocumented
552930`\spad{prinshINFO }\undocumented
553015`\spad{printCode(f)} prints out \spad{f} in FORTRAN notation.
553062`\spad{printHeader()} produces the FORTRAN header for the current subprogram in the global symbol table on the current FORTRAN output stream.
553112`\spad{printHeader(f)} produces the FORTRAN header for subprogram \spad{f} in the global symbol table on the current FORTRAN output stream.
553168`\spad{printHeader(f,{}tab)} produces the FORTRAN header for subprogram \spad{f} in symbol table \spad{tab} on the current FORTRAN output stream.
553227`\axiom{printInfo!(\spad{x},{}\spad{y})} initializes the mesages to be printed when manipulating items from the hash-table. If a key is retrieved then \axiom{\spad{x}} is displayed. If an item is stored then \axiom{\spad{y}} is displayed.
553630`\axiom{printStats!()} prints the statistics.
553704`\spad{printTypes(tab)} produces FORTRAN type declarations from \spad{tab},{} on the current FORTRAN output stream
553759`\spad{printTypes(tab)} produces FORTRAN type declarations from \spad{tab},{} on the current FORTRAN output stream
553807`\spad{print(u)} prints the form \spad{u}.
553849`\spad{print(o)} writes the output form \spad{o} on standard output using the two-dimensional formatter.
553901`\axiom{printingInfo?()} returns \spad{true} iff messages are printed when manipulating items from the hash-table.
554064`\spad{processTemplate(tp)} processes the template \spad{tp},{} writing the result to the current FORTRAN output stream.
554131`\spad{processTemplate(tp,{}fn)} processes the template \spad{tp},{} writing the result out to \spad{fn}.
554207`\spad{prod(expr)} creates the form prefixing \spad{expr} by a capital \spad{pi}.
554246`\spad{prod(expr,{}lowerlimit)} creates the form prefixing \spad{expr} by a capital \spad{pi} with a \spad{lowerlimit}.
554288`\spad{prod(expr,{}lowerlimit,{}upperlimit)} creates the form prefixing \spad{expr} by a capital \spad{pi} with both a \spad{lowerlimit} and \spad{upperlimit}.
554333`\spad{product(f(n),{} n = a..b)} returns \spad{f}(a) * ... * \spad{f}(\spad{b}) as a formal product.
554465`\spad{product(f(n),{} n = a..b)} returns \spad{f}(a) * ... * \spad{f}(\spad{b}) as a formal product.
554540`\spad{product(f(n),{} n)} returns the formal product \spad{P}(\spad{n}) which verifies \spad{P}(\spad{n+1})\spad{/P}(\spad{n}) = \spad{f}(\spad{n}).
554603`\spad{product(a,{}b)} is the degree-preserving \spad{R}-linear product: \blankline \indented{2}{\spad{degree product(a,{}b) = degree a + degree b}} \indented{2}{\spad{product(a1+a2,{}b) = product(a1,{}b) + product(a2,{}b)}} \indented{2}{\spad{product(a,{}b1+b2) = product(a,{}b1) + product(a,{}b2)}} \indented{2}{\spad{product(r*a,{}b) = product(a,{}r*b) = r*product(a,{}b)}} \indented{2}{\spad{product(a,{}product(b,{}c)) = product(product(a,{}b),{}c)}}
554656`\spad{product(s,{}t)} is the outer product of the tensors \spad{s} and \spad{t}. For example,{} if \spad{r = product(s,{}t)} for rank 2 tensors \spad{s} and \spad{t},{} then \spad{r} is a rank 4 tensor given by \indented{4}{\spad{r(i,{}j,{}k,{}l) = s(i,{}j)*t(k,{}l)}.}
555039`\spad{property(op,{} s)} returns the value of property \spad{s} if it is attached to \spad{op},{} and "failed" otherwise.
555310`\axiom{pseudoDivide(\spad{P},{}\spad{Q})} computes the pseudoDivide of \axiom{\spad{P}} by \axiom{\spad{Q}}.
555426`\spad{pseudoDivide(p,{}q)} returns \spad{[c,{} q,{} r]},{} when \spad{p' := p*lc(q)^(deg p - deg q + 1) = c * p} is pseudo right-divided by \spad{q},{} \spadignore{i.e.} \spad{p' = s q + r}.
555555`\axiom{pseudoDivide(a,{}\spad{b})} computes \axiom{[pquo(a,{}\spad{b}),{}prem(a,{}\spad{b})]},{} both polynomials viewed as univariate polynomials in the main variable of \axiom{\spad{b}},{} if \axiom{\spad{b}} is not a constant polynomial.
555727`\spad{pseudoQuotient(p,{}q)} returns \spad{r},{} the quotient when \spad{p' := p*lc(q)^(deg p - deg q + 1)} is pseudo right-divided by \spad{q},{} \spadignore{i.e.} \spad{p' = s q + r}.
555821`\spad{pseudoRemainder(p,{}q)} = \spad{r},{} for polynomials \spad{p} and \spad{q},{} returns the remainder when \spad{p' := p*lc(q)^(deg p - deg q + 1)} is pseudo right-divided by \spad{q},{} \spadignore{i.e.} \spad{p' = s q + r}.
558319`\spad{ptree(s)} is a leaf? pendant tree
558362`\spad{ptree(x,{}y)} \undocumented
558409`\spad{puiseux(f)} returns a Puiseux expansion of the expression \spad{f}. Note: \spad{f} should have only one variable; the series will be expanded in powers of that variable.
559354`\spad{purelyAlgebraic?(ts)} returns \spad{true} iff for every algebraic variable \spad{v} of \spad{ts} we have \spad{algebraicCoefficients?(t_v,{}ts_v_-)} where \spad{ts_v} is \axiomOpFrom{select}{TriangularSetCategory}(\spad{ts},{}\spad{v}) and \spad{ts_v_-} is \axiomOpFrom{collectUnder}{TriangularSetCategory}(\spad{ts},{}\spad{v}).
559519`\spad{purelyAlgebraic?(p,{}ts)} returns \spad{true} iff every variable of \spad{p} is algebraic \spad{w}.\spad{r}.\spad{t}. \spad{ts}.
559701`\spad{purelyAlgebraicLeadingMonomial?(p,{}ts)} returns \spad{true} iff the main variable of any non-constant iterarted initial of \spad{p} is algebraic \spad{w}.\spad{r}.\spad{t}. \spad{ts}.
559888`\spad{purelyTranscendental?(p,{}ts)} returns \spad{true} iff every variable of \spad{p} is not algebraic \spad{w}.\spad{r}.\spad{t}. \spad{ts}
559978`\spad{push!(x,{}s)} pushes \spad{x} onto stack \spad{s},{} \spadignore{i.e.} destructively changing \spad{s} so as to have a new first (top) element \spad{x}. Afterwards,{} pop!(\spad{s}) produces \spad{x} and pop!(\spad{s}) produces the original \spad{s}.
560026`\spad{pushFortranOutputStack(f)} pushes \spad{f} onto the Fortran output stack
560106`\spad{pushFortranOutputStack(f)} pushes \spad{f} onto the Fortran output stack
560184`\spad{pushdown(p,{}lv)} \undocumented{}
560253`\spad{pushdown(p,{}v)} \undocumented{}
560316`\spad{pushdown(prf,{}var)} pushes all top level occurences of the variable \spad{var} into the coefficient domain for the polynomial \spad{prf}.
560727`\spad{pushup(p,{}lv)} \undocumented{}
560794`\spad{pushup(p,{}v)} \undocumented{}
560855`\spad{pushup(prf,{}var)} raises all occurences of the variable \spad{var} in the coefficients of the polynomial \spad{prf} back to the polynomial level.
561210`\spad{qShiftAction(q,{} k,{} l,{} g)} gives the coefficient of \spad{x^k} in \spad{z^l} \spad{g}(\spad{x}),{} where \spad{z*}(a+b*x+c*x^2+d*x^3+...) = (a+q*b*x+q^2*c*x^2+q^3*d*x^3+...). In terms of sequences,{} z*u(\spad{n})=q^n*u(\spad{n}).
561314`\spad{qShiftC} gives the coefficients \spad{c_}{\spad{k},{}\spad{k}} in the expansion <x^k> \spad{z} \spad{g}(\spad{x}) = sum_{\spad{i=0}}\spad{^k} \spad{c_}{\spad{k},{}\spad{i}} <x^i> \spad{g}(\spad{x}),{} where \spad{z} acts on \spad{g}(\spad{x}) by shifting. In fact,{} the result is [1,{}\spad{q},{}\spad{q^2},{}...]
561452`\spad{qelt(u,{} x)} applies \axiom{\spad{u}} to \axiom{\spad{x}} without checking whether \axiom{\spad{x}} is in the domain of \axiom{\spad{u}}. If \axiom{\spad{x}} is not in the domain of \axiom{\spad{u}} a memory-access violation may occur. If a check on whether \axiom{\spad{x}} is in the domain of \axiom{\spad{u}} is required,{} use the function \axiom{elt}.
561509`\spad{qelt(m,{}i,{}j)} returns the element in the \spad{i}th row and \spad{j}th column of the matrix \spad{m}. Note: there is NO error check to determine if indices are in the proper ranges.
561593`\spad{qelt(m,{}i,{}j)} returns the element in the \spad{i}th row and \spad{j}th column of the array \spad{m} NO error check to determine if indices are in proper ranges
561818`\spad{qinterval(inf,{}sup)} creates a new interval \axiom{[\spad{inf},{}\spad{sup}]},{} without checking the ordering on the elements.
561872`\spad{qnew(n)} creates a new uninitialized vector of length \spad{n}.
561930`\spad{qnew(n)} creates a new uninitialized vector of length \spad{n}.
561981`\spad{qnew(n,{} m)} creates a new uninitialized \spad{n} by \spad{m} matrix.
562047`\spad{qnew(n,{} m)} creates a new uninitialized \spad{n} by \spad{m} matrix.
562303`\spad{qsetelt!(u,{}x,{}y)} sets the image of \axiom{\spad{x}} to be \axiom{\spad{y}} under \axiom{\spad{u}},{} without checking that \axiom{\spad{x}} is in the domain of \axiom{\spad{u}}. If such a check is required use the function \axiom{setelt}.
562391`\spad{qsetelt!(m,{}i,{}j,{}r)} sets the element in the \spad{i}th row and \spad{j}th column of \spad{m} to \spad{r} NO error check to determine if indices are in proper ranges
562545`\spad{quadraticNorm(f)} returns the \spad{l2} norm of the polynomial \spad{f}.
562889`\spad{quasiAlgebraicSet(pl,{}q)} returns the quasi-algebraic set with defining equations \spad{p} = 0 for \spad{p} belonging to the list \spad{pl},{} and defining inequation \spad{q} \spad{~=} 0.
563080`\axiom{quasiComponent(\spad{ts})} returns \axiom{[\spad{lp},{}\spad{lq}]} where \axiom{\spad{lp}} is the list of the members of \axiom{\spad{ts}} and \axiom{\spad{lq}}is \axiom{initials(\spad{ts})}.
563253`\axiom{quasiMonic?(\spad{p})} returns \spad{false} if \axiom{\spad{p}} belongs to \axiom{\spad{R}},{} otherwise returns \spad{true} iff the initial of \axiom{\spad{p}} lies in the base ring \axiom{\spad{R}}.
563450`\spad{quasiRegular?(x)} return \spad{true} if \spad{constant(p)} is zero.
563513`\spad{quasiRegular?(x)} return \spad{true} if \spad{constant(x)} is zero.
563574`\spad{quasiRegular(x)} return \spad{x} minus its constant term.
563631`\spad{quasiRegular(x)} return \spad{x} minus its constant term.
563686`\spad{quatern(r,{}i,{}j,{}k)} constructs a quaternion from scalars.
563744`\spad{queue([x,{}y,{}...,{}z])} creates a queue with first (top) element \spad{x},{} second element \spad{y},{}...,{}and last (bottom) element \spad{z}.
563859`\spad{quoByVar(a0 + a1 x + a2 x^2 + ...)} returns \spad{a1 + a2 x + a3 x^2 + ...} Thus,{} this function substracts the constant term and divides by the series variable. This function is used when Laurent series are represented by a Taylor series and an order.
564014`\spad{f quo g} creates the equivalent infix form.
564055`\spad{x quo y} is the same as \spad{divide(x,{}y).quotient}. See \spadfunFrom{divide}{EuclideanDomain}.
564101`\spad{a quo b} returns the quotient of \spad{a} and \spad{b},{} forgetting the remainder.
564150`\spad{quote()} provides the string quote character,{} \spad{"}.
564187`\spad{quote(f)} creates the form \spad{f} with a prefix quote.
564269`\spad{quotedOperators(r)} returns the list of operators on the right hand side of \spad{r} that are considered quoted,{} that is they are not evaluated during any rewrite,{} but just applied formally to their arguments.
564340`\spad{quotientByP(x)} returns \spad{b},{} where \spad{x = a + b p}.
565153`\spad{radicalOfLeftTraceForm()} returns basis for null space of \spad{leftTraceMatrix()},{} if the algebra is associative,{} alternative or a Jordan algebra,{} then this space equals the radical (maximal nil ideal) of the algebra.
567101`\spad{radix(x,{}b)} converts \spad{x} to a radix expansion in base \spad{b}.
567286`\spad{ramified?(a)} tests whether \spad{x = a} is ramified.
567356`\spad{ramified?(p)} tests whether \spad{p(x) = 0} is ramified.
567427`\spad{ramifiedAtInfinity?()} tests if infinity is ramified.
567561`\spad{randnum()} is a random number between 0 and size().
567614`\spad{randnum(n)} is a random number between 0 and \spad{n}.
567752`\spad{randomR()} should be local but conditional
567818`\spad{randomR produces} a random element of \spad{R}
567895`\spad{randomR()} produces a random element of \spad{R}
568170`\spad{random()} returns a random element from the set.
568205`\spad{random()} creates a random element. This function is deprecated.
568253`\spad{random()} returns a random fraction. This function is deprecated.
568569`\spad{random(a)} creates a random element from 0 to \spad{n-1}.
568619`\spad{random(n)} returns a random integer from 0 to \spad{n-1}.
568657`\spad{random(n)} returns a random integer from 0 to \spad{n-1}.
569109`\spad{range([i])} provides a user-specified range \spad{i}. This option is expressed in the form \spad{range == [i]}.
569517`\spad{rank()} returns the rank of the algebra.
569580`\spad{rank()} returns the rank of the algebra as \spad{R}-module.
569654`\spad{rank Lv} computes the rank of a list of vectors.
569731`\spad{rank(m)} returns the rank of the matrix \spad{m}.
569821`\spad{rank b} computes the rank of a basis,{} it\spad{'s} the number of vectors in the basis.
569885`\spad{rank(m)} returns the rank of the matrix \spad{m}.
569991`\spad{rank(m)} returns the rank of the matrix \spad{m}.
570082`\spad{rank(m)} returns the rank of the matrix \spad{m}.
570423`\spad{rarrow(f,{}g)} creates a form for the mapping \spad{f -> g}.
570467`\spad{ratDenom(f)} rationalizes the denominators appearing in \spad{f} by moving all the algebraic quantities into the numerators.
570525`\spad{ratDenom(f,{} a)} removes \spad{a} from the denominators in \spad{f} if \spad{a} is an algebraic kernel.
570585`\spad{ratDenom(f,{} [a1,{}...,{}an])} removes the \spad{ai}\spad{'s} which are algebraic kernels from the denominators in \spad{f}.
571394`\spad{ratPoly(f)} returns a polynomial \spad{p} such that \spad{p} has no algebraic coefficients,{} and \spad{p(f) = 0}.
571701`\spad{rational?(x)} returns \spad{true} if \spad{x} is a rational number,{} \spad{false} otherwise.
571761`\spad{rational?(x)} tests if \spad{x} is a rational number.
571844`\spad{rational?(n)} tests if \spad{n} is a rational number (see \spadtype{Fraction Integer}).
571902`\spad{rational?(o)} tests if \spad{o} is rational,{} \spadignore{i.e.} that all seven imaginary parts are 0.
571986`\spad{rational?(q)} returns {\it \spad{true}} if all the imaginary parts of \spad{q} are zero and the real part can be converted into a rational number,{} and {\it \spad{false}} otherwise.
572072`\spad{rational?(u)} tests if \spadvar{\spad{u}} is actually a rational number (see \spadtype{Fraction Integer}).
572148`\spad{rational?(x)} tests if \spad{x} is a finite rational number.
572234`\spad{rational?(x)} tests if \spad{x} is a finite rational number.
572319`\spad{rationalApproximation(f,{} n)} computes a rational approximation \spad{r} to \spad{f} with relative error \spad{< 10^(-n)}.
572490`\spad{rationalApproximation(f,{} n,{} b)} computes a rational approximation \spad{r} to \spad{f} with relative error \spad{< b^(-n)} (that is,{} \spad{|(r-f)/f| < b^(-n)}).
573307`\spad{rationalIfCan(x)} returns \spad{x} as a rational number,{} "failed" if \spad{x} is not a rational number.
573397`\spad{rationalIfCan(x)} returns \spad{x} as a rational number,{} or "failed" if \spad{x} is not a rational number.
573510`\spad{rationalIfCan(n)} creates a rational number,{} or returns "failed" if this is not possible.
573598`\spad{rationalIfCan(o)} returns the real part if all seven imaginary parts are 0,{} and "failed" otherwise.
573712`\spad{rationalIfCan(q)} returns \spad{q} as a rational number,{} or "failed" if this is not possible. Note: if \spad{rational?(q)} is \spad{true},{} the conversion can be done and the rational number will be returned.
573828`\spad{rationalIfCan(u)} returns a rational number if \spad{u} really is one,{} and "failed" otherwise.
573934`\spad{rationalIfCan(x)} returns \spad{x} as a finite rational number if it is one,{} "failed" otherwise.
574050`\spad{rationalIfCan(x)} returns \spad{x} as a finite rational number if it is one and "failed" otherwise.
574165`\spad{rationalPoint?(a,{} b)} tests if \spad{(x=a,{}y=b)} is on the curve.
574245`\spad{rationalPoint?(a,{} b)} tests if \spad{(x=a,{}y=b)} is on the curve.
574322`\spad{rationalPoints()} returns the list of all the affine rational points.
574405`\spad{rationalPoints()} returns the list of all the affine rational points.
575462`\spad{rational(x)} returns \spad{x} as a rational number; error if \spad{x} is not a rational number.
575531`\spad{rational(x)} returns \spad{x} as a rational number. Error: if \spad{x} is not a rational number.
575623`\spad{rational(n)} creates a rational number (see \spadtype{Fraction Integer})..
575690`\spad{rational(o)} returns the real part if all seven imaginary parts are 0. Error: if \spad{o} is not rational.
575783`\spad{rational(q)} tries to convert \spad{q} into a rational number. Error: if this is not possible. If \spad{rational?(q)} is \spad{true},{} the conversion will be done and the rational number returned.
575878`\spad{rational(u)} assumes spadvar{\spad{u}} is actually a rational number and does the conversion to rational number (see \spadtype{Fraction Integer}).
575963`\spad{rational(x)} returns \spad{x} as a finite rational number. Error: if \spad{x} is not a rational number.
576058`\spad{rational(x)} returns \spad{x} as a finite rational number. Error: if \spad{x} cannot be so converted.
576152`\spad{ratpart(ir)} returns the rational part of an integration result
576203`\spad{ravel(t)} produces a list of components from a tensor such that \indented{2}{\spad{unravel(ravel(t)) = t}.}
576266`right contraction inner product
576321`\spad{rdHack1(v,{}u,{}n)} \undocumented
576648`\spad{read!(f)} extracts a value from file \spad{f}. The state of \spad{f} is modified so a subsequent call to \spadfun{read!} will return the next element.
576697`\spad{readIfCan!(f)} returns a value from the file \spad{f},{} if possible. If \spad{f} is not open for reading,{} or if \spad{f} is at the end of file then \spad{"failed"} is the result.
576754`\spad{readIfCan!(f)} returns a value from the file \spad{f},{} if possible. If \spad{f} is not open for reading,{} or if \spad{f} is at the end of file then \spad{"failed"} is the result.
576826`\spad{readIfCan!(f)} returns a string of the contents of a line from file \spad{f},{} if possible. If \spad{f} is not readable or if it is positioned at the end of file,{} then \spad{"failed"} is returned.
576889`\spad{readLine!(f)} returns a string of the contents of a line from the file \spad{f}.
576935`\spad{readLineIfCan!(f)} returns a string of the contents of a line from file \spad{f},{} if possible. If \spad{f} is not readable or if it is positioned at the end of file,{} then \spad{"failed"} is returned.
577002`\spad{readable?(f)} tests if the named file exist and can it be opened for reading.
577057`\spad{real?(t)} tests whether \spad{t} is equivalent to the FORTRAN type REAL.
577109`\spad{real?(f)} returns \spad{true} if \spad{f = real f}.
577181`\spad{real?(f)} returns \spad{true} if \spad{f = real f}.
577498`\spad{realElementary(f)} rewrites \spad{f} in terms of the 4 fundamental real transcendental elementary functions: \spad{log,{} exp,{} tan,{} atan}.
577652`\spad{realLiouvillian(f)} rewrites \spad{f} elementary kernels of \spad{f} in terms 4 fundamental real elementary functions: \spad{log,{} exp,{} tan,{} atan}. Additionally,{} it rewrites Liouvillian functions as indefinite integrals to support better normalization.
579902`\spad{real(f)} returns the real part of \spad{f} where \spad{f} is a complex function.
579960`\spad{real(x)} returns real part of \spad{x}.
580006`\spad{real(o)} extracts real part of octonion \spad{o}.
580053`\spad{real(q)} extracts the real part of quaternion \spad{q}.
580102`\spad{recip()} returns \spad{matrix [[0,{}1],{}[1,{}0]]} representing the map \spad{x -> 1 / x}.
580589`\spad{recip(x)} \undocumented
580689`\spad{recip(x)} \undocumented
580778`\spad{recip(m)} = recip() * \spad{m}
580827`\spad{recip(a)} returns an element,{} which is both a left and a right inverse of \spad{a},{} or \spad{"failed"} if there is no unit element,{} if such an element doesn\spad{'t} exist or cannot be determined (see unitsKnown).
580928`\spad{recip(a)} returns an element,{} which is both a left and a right inverse of \spad{a},{} or \spad{"failed"} if such an element doesn\spad{'t} exist or cannot be determined (see unitsKnown).
580987`\spad{recip(x)} tries to compute the multiplicative inverse for \spad{x} or "failed" if it cannot find the inverse (see unitsKnown).
581039`\spad{recip(x)} computes the multiplicative inverse of \spad{x} or "failed" if \spad{x} is not invertible.
581216`\axiom{recip(\spad{p},{}\spad{ts})} returns the inverse of \axiom{\spad{p}} \spad{w}.\spad{r}.\spad{t} \spad{ts} assuming that \axiom{\spad{p}} is invertible \spad{w}.\spad{r}.\spad{t} \spad{ts}.
581297`\axiom{recip(pol,{}aRoot)} tries to inverse \axiom{pol} interpreted as \axiom{aRoot}
581406`\spad{reciprocalPolynomial(p)} calulates a polynomial which has exactly the inverses of the non-zero roots of \spad{p} as roots,{} and the same number of 0-roots.
582296`\spad{rectangularMatrix(m)} converts a matrix of type \spadtype{Matrix} to a matrix of type \spad{RectangularMatrix}.
582557`\spad{redPo }\undocumented
582670`\spad{redPol }\undocumented
582762`\spad{red()} returns the position of the red hue from total hues.
583445`\spad{reduceBasisAtInfinity(b1,{}...,{}bn)} returns \spad{(x^i * bj)} for all \spad{i},{}\spad{j} such that \spad{x^i*bj} is locally integral at infinity.
583812`\axiom{reduceByQuasiMonic(\spad{p},{}\spad{ts})} returns the same as \axiom{remainder(\spad{p},{}collectQuasiMonic(\spad{ts})).polnum}.
583974`\spad{reduce(f)} produces the equivalence class of \spad{f} in the residue ring
584220`\spad{reduce(x)} \undocumented
584533`\spad{reduce(f)} simplifies all the unreduced algebraic numbers present in \spad{f} by applying their defining relations.
584584`\spad{reduce(frac)} converts the fraction \spad{frac} to an algebra element.
584675`\spad{reduce(up)} converts the univariate polynomial \spad{up} to an algebra element,{} reducing by the \spad{definingPolynomial()} if necessary.
584728`\spad{reduce(D)} converts \spad{D} to some reduced form (the reduced forms can be differents in different implementations).
584793`\spad{reduce(f)} simplifies all the unreduced algebraic numbers present in \spad{f} by applying their defining relations.
584839`\spad{reduce(f)} simplifies all the unreduced algebraic quantities present in \spad{f} by applying their defining relations.
585138`\spad{reduce(r,{}m)} \undocumented
585226`\spad{reduce(r,{}m)} \undocumented
585304`\spad{reduce(r,{}m)} \undocumented
585381`\spad{reduce(f,{}u)} reduces the binary operation \spad{f} across \spad{u}. For example,{} if \spad{u} is \axiom{[\spad{x},{}\spad{y},{}...,{}\spad{z}]} then \axiom{reduce(\spad{f},{}\spad{u})} returns \axiom{\spad{f}(..\spad{f}(\spad{f}(\spad{x},{}\spad{y}),{}...),{}\spad{z})}. Note: if \spad{u} has one element \spad{x},{} \axiom{reduce(\spad{f},{}\spad{u})} returns \spad{x}. Error: if \spad{u} is empty.
586078`\spad{reduce(f,{}m,{}r)} returns a matrix \spad{n} where \spad{n[i,{}j] = f(m[i,{}j],{}r)} for all indices \spad{i} and \spad{j}.
586181`\spad{reduce(f,{}m,{}r)} returns a matrix \spad{n} where \spad{n[i,{}j] = f(m[i,{}j],{}r)} for all indices spad{\spad{i}} and \spad{j}.
586299`\spad{reduce(f,{}a,{}r)} applies function \spad{f} to each successive element of the aggregate \spad{a} and an accumulant initialized to \spad{r}. For example,{} \spad{reduce(_+\$Integer,{}[1,{}2,{}3],{}0)} does \spad{3+(2+(1+0))}. Note: third argument \spad{r} may be regarded as the identity element for the function \spad{f}.
586379`\spad{reduce(f,{}a,{}r)} applies function \spad{f} to each successive element of the aggregate \spad{a} and an accumulant initialised to \spad{r}. For example,{} \spad{reduce(_+\$Integer,{}[1,{}2,{}3],{}0)} does a \spad{3+(2+(1+0))}. Note: third argument \spad{r} may be regarded as an identity element for the function.
586456`\spad{reduce(f,{}u,{}x)} reduces the binary operation \spad{f} across \spad{u},{} where \spad{x} is the identity operation of \spad{f}. Same as \axiom{reduce(\spad{f},{}\spad{u})} if \spad{u} has 2 or more elements. Returns \axiom{\spad{f}(\spad{x},{}\spad{y})} if \spad{u} has one element \spad{y},{} \spad{x} if \spad{u} is empty. For example,{} \axiom{reduce(+,{}\spad{u},{}0)} returns the sum of the elements of \spad{u}.
586930`\spad{reduce(f,{}u,{}x,{}z)} reduces the binary operation \spad{f} across \spad{u},{} stopping when an "absorbing element" \spad{z} is encountered. As for \axiom{reduce(\spad{f},{}\spad{u},{}\spad{x})},{} \spad{x} is the identity operation of \spad{f}. Same as \axiom{reduce(\spad{f},{}\spad{u},{}\spad{x})} when \spad{u} contains no element \spad{z}. Thus the third argument \spad{x} is returned when \s
586930`--pad{u} is empty.
587033`\axiom{reduce(\spad{p},{}\spad{ts},{}redOp,{}redOp?)} returns a polynomial \axiom{\spad{r}} such that \axiom{redOp?(\spad{r},{}\spad{p})} holds for every \axiom{\spad{p}} of \axiom{\spad{ts}} and there exists some product \axiom{\spad{h}} of the initials of the members of \axiom{\spad{ts}} such that \axiom{\spad{h*p} - \spad{r}} lies in the ideal generated by \axiom{\spad{ts}}. The operation \axiom{redOp} must satisfy
587033`-- the following conditions. For every \axiom{\spad{p}} and \axiom{\spad{q}} we have \axiom{redOp?(redOp(\spad{p},{}\spad{q}),{}\spad{q})} and there exists an integer \axiom{\spad{e}} and a polynomial \axiom{\spad{f}} such that \axiom{init(\spad{q})^e*p = \spad{f*q} + redOp(\spad{p},{}\spad{q})}.
587267`\axiom{reduced?(\spad{q},{}\spad{lp})} returns \spad{true} iff \axiom{reduced?(\spad{q},{}\spad{p})} holds for every \axiom{\spad{p}} in \axiom{\spad{lp}}.
587348`\axiom{reduced?(a,{}\spad{b})} returns \spad{true} iff \axiom{degree(a,{}mvar(\spad{b})) < mdeg(\spad{b})}.
587505`\axiom{reduced?(\spad{p},{}\spad{ts},{}redOp?)} returns \spad{true} iff \axiom{\spad{p}} is reduced \spad{w}.\spad{r}.\spad{t}. in the sense of the operation \axiom{redOp?},{} that is if for every \axiom{\spad{t}} in \axiom{\spad{ts}} \axiom{redOp?(\spad{p},{}\spad{t})} holds.
587590`\spad{reducedContinuedFraction(b0,{}b)} constructs a continued fraction in the following way: if \spad{b = [b1,{}b2,{}...]} then the result is the continued fraction \spad{b0 + 1/(b1 + 1/(b2 + ...))}. That is,{} the result is the same as \spad{continuedFraction(b0,{}[1,{}1,{}1,{}...],{}[b1,{}b2,{}b3,{}...])}.
587668`\spad{reducedDiscriminant(up)} \undocumented
587756`\spad{reducedForm(x)} puts the continued fraction \spadvar{\spad{x}} in reduced form,{} \spadignore{i.e.} the function returns an equivalent continued fraction of the form \spad{continuedFraction(b0,{}[1,{}1,{}1,{}...],{}[b1,{}b2,{}b3,{}...])}.
588922`\spad{reducedSystem(A)} returns a matrix \spad{B} such that \spad{A x = 0} and \spad{B x = 0} have the same solutions in \spad{R}.
590442`\spad{reducedSystem(A,{} v)} returns a matrix \spad{B} and a vector \spad{w} such that \spad{A x = v} and \spad{B x = w} have the same solutions in \spad{R}.
590559`\spad{reduction(p,{}prime)} reduces the polynomial \spad{p} modulo \spad{prime} of \spad{R}. Note: this function is exported only because it\spad{'s} conditional.
590612`\spad{reduction(f,{}p)} reduces the coefficients of the polynomial \spad{f} modulo the prime \spad{p}.
590682`\spad{reduction(u,{}pol)} computes the symmetric reduction of \spad{u} mod \spad{pol}
590879`\spad{reductum(p)},{} where \spad{p} is an antisymmetric polynomial,{} returns \spad{p} minus the leading term of \spad{p} if \spad{p} has at least two terms,{} and 0 otherwise.
590928`\spad{reductum(df)},{} where \spad{df} is a differential form,{} returns \spad{df} minus the leading term of \spad{df} if \spad{df} has two or more terms,{} and 0 otherwise.
590995`\spad{reductum(x)} \undocumented
591067`\spad{reductum(x)} \undocumented
591123`\spad{reductum(p)} returns \spad{p} minus its leading term. An error is produced if \spad{p} is zero.
591176`\spad{reductum(u)} returns \spad{u} minus its leading monomial returns zero if handed the zero element.
591231`\spad{reductum(z)} returns a new element created by removing the leading coefficient/support pair from the element \spad{z}. Error: if \spad{z} has no support.
591297`\spad{reductum(l)} is \spad{l - monomial(a(n),{}n)} if \indented{2}{\spad{l = sum(monomial(a(i),{}i),{} i = 0..n)}.}
591356`\spad{reductum(f)} is \spad{f} minus its leading monomial.
591429`\spad{reductum(l)} is \spad{l - monomial(a(n),{}n)} if \indented{2}{\spad{l = sum(monomial(a(i),{}i),{} i = 0..n)}.}
591562`\axiom{reductum(\spad{p},{}\spad{v})} returns the reductum of \axiom{\spad{p}},{} where \axiom{\spad{p}} is viewed as a univariate polynomial in \axiom{\spad{v}}.
591631`\spad{ref(n)} creates a pointer (reference) to the object \spad{n}.
591670`\spad{refine(x)} \undocumented
591707`\spad{refine(p)} performs a refinement on the plot \spad{p}
591742`\axiom{refine(rootChar)} shrinks isolating interval around \axiom{rootChar}
593401`\spad{regularRepresentation(a)} returns the matrix of the linear map defined by left multiplication by \spad{a} with respect to the fixed basis.
593635`\spad{regularRepresentation(a,{}basis)} returns the matrix of the linear map defined by left multiplication by \spad{a} with respect to the \spad{basis} \spad{basis}.
593722`\spad{reindex(t,{}[i1,{}...,{}idim])} permutes the indices of \spad{t}. For example,{} if \spad{r = reindex(t,{} [4,{}1,{}2,{}3])} for a rank 4 tensor \spad{t},{} then \spad{r} is the rank for tensor given by \indented{4}{\spad{r(i,{}j,{}k,{}l) = t(l,{}i,{}j,{}k)}.}
594155`\axiom{approximate(term,{}root,{}prec)} gives an approximation of \axiom{term} over \axiom{root} with precision \axiom{prec}
594394`\spad{f rem g} creates the equivalent infix form.
594435`\spad{x rem y} is the same as \spad{divide(x,{}y).remainder}. See \spadfunFrom{divide}{EuclideanDomain}.
594481`\spad{a rem b} returns the remainder of \spad{a} and \spad{b}.
594711`\axiom{remainder(a,{}\spad{ps})} returns \axiom{[\spad{c},{}\spad{b},{}\spad{r}]} such that \axiom{\spad{b}} is fully reduced in the sense of Groebner bases \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{ps}},{} \axiom{r*a - \spad{c*b}} lies in the ideal generated by \axiom{\spad{ps}}. Furthermore,{} if \axiom{\spad{R}} is a \spad{gcd}-domain,{} \axiom{\spad{b}} is primitive.
594830`remap_variables(\spad{p},{} \spad{lv1},{} \spad{lv2}) replaces corresponding variables from \spad{l1} by variables from \spad{lv2}. Assumes that \spad{lv1} and \spad{lv2} have the same length and that \spad{p} has variables in \spad{lv1}.
595662`\spad{remove!(p,{}d)} destructively changes dictionary \spad{d} by removeing all entries \spad{x} such that \axiom{\spad{p}(\spad{x})} is \spad{true}.
595753`\spad{remove!(p,{}u)} destructively removes all elements \spad{x} of \spad{u} such that \axiom{\spad{p}(\spad{x})} is \spad{true}.
595826`\spad{remove!(k,{}t)} searches the table \spad{t} for the key \spad{k} removing (and return) the entry if there. If \spad{t} has no such key,{} \axiom{remove!(\spad{k},{}\spad{t})} returns "failed".
595907`\spad{remove!(x,{}d)} destructively changes dictionary \spad{d} by removing all entries \spad{y} such that \axiom{\spad{y} = \spad{x}}.
595987`\spad{remove!(x,{}u)} destructively removes all values \spad{x} from \spad{u}.
596067`\spad{remove!(p,{}ms,{}number)} removes destructively at most \spad{number} copies of elements \spad{x} such that \spad{p(x)} is \spadfun{\spad{true}} if \spad{number} is positive,{} all of them if \spad{number} equals zero,{} and all but at most \spad{-number} if \spad{number} is negative.
596131`\spad{remove!(x,{}ms,{}number)} removes destructively at most \spad{number} copies of element \spad{x} if \spad{number} is positive,{} all of them if \spad{number} equals zero,{} and all but at most \spad{-number} if \spad{number} is negative.
596184`remove the specified child node \spad{'c'} below node \spad{'n'}
596320`\spad{removeCosSq(f)} converts every \spad{cos(u)^2} appearing in \spad{f} into \spad{1 - sin(x)^2},{} and also reduces higher powers of \spad{cos(u)} with that formula.
596386`\spad{removeCoshSq(f)} converts every \spad{cosh(u)^2} appearing in \spad{f} into \spad{1 - sinh(x)^2},{} and also reduces higher powers of \spad{cosh(u)} with that formula.
596507`\spad{removeDuplicates!(u)} destructively removes duplicates from \spad{u}.
596595`\spad{removeDuplicates!(d)} destructively removes any duplicate values in dictionary \spad{d}.
596770`\spad{removeDuplicates(u)} returns a copy of \spad{u} with all duplicates removed.
597760`\spad{removeRepeats!(u)} destructively replaces runs of consequtive equal elements of \spad{u} by single elements.
598334`\spad{removeSinSq(f)} converts every \spad{sin(u)^2} appearing in \spad{f} into \spad{1 - cos(x)^2},{} and also reduces higher powers of \spad{sin(u)} with that formula.
598400`\spad{removeSinhSq(f)} converts every \spad{sinh(u)^2} appearing in \spad{f} into \spad{1 - cosh(x)^2},{} and also reduces higher powers of \spad{sinh(u)} with that formula.
599108`\axiom{removeZero(\spad{p},{}\spad{ts})} returns \axiom{0} if \axiom{\spad{p}} reduces to \axiom{0} by pseudo-division \spad{w}.\spad{r}.\spad{t} \axiom{\spad{ts}} otherwise returns a polynomial \axiom{\spad{q}} computed from \axiom{\spad{p}} by removing any coefficient in \axiom{\spad{p}} reducing to \axiom{0}.
599278`\spad{removeZeroes(x)} removes leading zeroes from the representation of the \spad{p}-adic rational \spad{x}. A \spad{p}-adic rational is represented by (1) an exponent and (2) a \spad{p}-adic integer which may have leading zero digits. When the \spad{p}-adic integer has a leading zero digit,{} a 'leading zero' is removed from the \spad{p}-adic rational as follows: the number is rewritten by increasing the exponent by 1 and dividing 
599278`--the \spad{p}-adic integer by \spad{p}. Note: \spad{removeZeroes(f)} removes all leading zeroes from \spad{f}.
599348`\spad{removeZeroes(f(x))} removes leading zeroes from the representation of the Laurent series \spad{f(x)}. A Laurent series is represented by (1) an exponent and (2) a Taylor series which may have leading zero coefficients. When the Taylor series has a leading zero coefficient,{} the 'leading zero' is removed from the Laurent series as follows: the series is rewritten by increasing the exponent by 1 and dividing t
599348`--he Taylor series by its variable. Note: \spad{removeZeroes(f)} removes all leading zeroes from \spad{f}
600301`\spad{remove(p,{}u)} returns a copy of \spad{u} removing all elements \spad{x} such that \axiom{\spad{p}(\spad{x})} is \spad{true}. Note: \axiom{remove(\spad{p},{}\spad{u}) \spad{==} [\spad{x} for \spad{x} in \spad{u} | not \spad{p}(\spad{x})]}.
600381`\spad{remove(f,{}st)} returns a stream consisting of those elements of stream \spad{st} which do not satisfy the predicate \spad{f}. Note: \spad{remove(f,{}st) = [x for x in st | not f(x)]}.
600447`\spad{remove(x,{}u)} returns a copy of \spad{u} with all elements \axiom{\spad{y} = \spad{x}} removed. Note: \axiom{remove(\spad{y},{}\spad{c}) \spad{==} [\spad{x} for \spad{x} in \spad{c} | \spad{x} \spad{~=} \spad{y}]}.
600540`\spad{remove(p,{}ms,{}number)} removes at most \spad{number} copies of elements \spad{x} such that \spad{p(x)} is \spadfun{\spad{true}} if \spad{number} is positive,{} all of them if \spad{number} equals zero,{} and all but at most \spad{-number} if \spad{number} is negative.
600603`\spad{remove(x,{}ms,{}number)} removes at most \spad{number} copies of element \spad{x} if \spad{number} is positive,{} all of them if \spad{number} equals zero,{} and all but at most \spad{-number} if \spad{number} is negative.
600814`\spad{reorder(p,{} perm)} applies the permutation perm to the variables in a polynomial and returns the new correctly ordered polynomial
600899`\spad{reorder(p,{} perm)} applies the permutation perm to the variables in a polynomial and returns the new correctly ordered polynomial
600993`\spad{reorder(p,{} perm)} applies the permutation perm to the variables in a polynomial and returns the new correctly ordered polynomial
601404`\spad{repeatUntilLoop(s,{}c)} creates a repeat ... until loop in FORTRAN.
601771`\spad{replace(s,{}i..j,{}t)} replaces the substring \axiom{\spad{s}(\spad{i}..\spad{j})} of \spad{s} by string \spad{t}.
601847`\spad{representationType()} returns the type of the representation,{} one of: \spad{prime},{} \spad{polynomial},{} \spad{normal},{} or \spad{cyclic}.
602230`\spad{represents([a1,{}..,{}an])} returns \spad{a1*v1 + ... + an*vn},{} where \spad{v1},{}...,{}\spad{vn} are the elements of the fixed basis.
602304`\spad{represents([a1,{}..,{}an])} returns \spad{a1*v1 + ... + an*vn},{} where \spad{v1},{} ...,{} \spad{vn} are the elements of the fixed basis.
602365`\spad{represents([a1,{}...,{}an])} returns \spad{a1*v1 + ... + an*vn},{} where \spad{v1},{} ...,{} \spad{vn} are the elements of the fixed \spad{R}-module basis.
602906`\spad{represents([a1,{}..,{}an],{}[v1,{}..,{}vn])} returns \spad{a1*v1 + ... + an*vn}.
602982`\spad{represents([a1,{}...,{}am],{}[v1,{}...,{}vm])} returns the linear combination \spad{a1*vm + ... + an*vm}.
603069`\spad{represents([A0,{}...,{}A(n-1)],{}D)} returns \spad{(A0 + A1 y +...+ A(n-1)*y^(n-1))/D}.
603147`\spad{reseed(n)} restarts the random number generator at \spad{n}.
603203`\spad{resetBadValues(p)} initializes the list of "bad values" for \spad{p} to \spad{[]}. Note: \spad{p} is not allowed to match any of its "bad values".
603252`\spad{resetNew()} resets the internals counters that new() and new(\spad{s}) use to return distinct symbols every time.
603291`\spad{resetVariableOrder()} cancels any previous use of setVariableOrder and returns to the default system ordering.
603361`\spad{reset(v)} sets the current state of the graph characteristics of the given two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport},{} back to their initial settings.
603415`\spad{reset(v)} sets the current state of the graph characteristics of the given three-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{ThreeDimensionalViewport},{} back to their initial settings.
603471`\spad{reshape(l,{}p)} \undocumented
604015`\axiom{rest(\spad{x})} return \axiom{\spad{x}} without the first entry or error if \axiomOpFrom{retractable?}{Magma}(\axiom{\spad{x}}) is \spad{true}.
604057`\spad{rest(x)} returns \spad{x} except the first letter.
604106`\spad{rest([l1]*[l2]*...[ln])} returns the list \spad{l2,{} .... ln}.
604174`\axiom{rest(\spad{ts})} returns the polynomials of \axiom{\spad{ts}} with smaller main variable than \axiom{mvar(\spad{ts})} if \axiom{\spad{ts}} is not empty,{} otherwise returns "failed"
604249`\spad{rest(u)} returns an aggregate consisting of all but the first element of \spad{u} (equivalently,{} the next node of \spad{u}).
604440`\spad{rest(u,{}n)} returns the \axiom{\spad{n}}th (\spad{n} \spad{>=} 0) node of \spad{u}. Note: \axiom{rest(\spad{u},{}0) = \spad{u}}.
604583`\axiom{resultantEuclidean(\spad{P},{}\spad{Q})} carries out the equality \axiom{coef1*P + coef2*Q = resultant(\spad{P},{}\spad{Q})}
604703`\axiom{resultantEuclidean_naif(\spad{P},{}\spad{Q})} returns the extended resultant of \axiom{\spad{P}} and \axiom{\spad{Q}} computed by means of the naive algorithm.
604827`\axiom{resultantReduitEuclidean(\spad{P},{}\spad{Q})} returns the "reduce resultant" and carries out the equality \axiom{coef1*P + coef2*Q = resultantReduit(\spad{P},{}\spad{Q})}.
604975`\axiom{resultantReduit(\spad{P},{}\spad{Q})} returns the "reduce resultant" of \axiom{\spad{P}} and \axiom{\spad{Q}}.
605068`\axiom{resultant(\spad{P},{} \spad{Q})} returns the resultant of \axiom{\spad{P}} and \axiom{\spad{Q}}
605139`\spad{resultant(p,{}q)} returns the resultant of the polynomials \spad{p} and \spad{q}.
605228`\axiom{resultant(a,{}\spad{b})} computes the resultant of \axiom{a} and \axiom{\spad{b}} where \axiom{a} and \axiom{\spad{b}} are assumed to have the same main variable \axiom{\spad{v}} and are viewed as univariate polynomials in \axiom{\spad{v}}.
605389`\spad{resultant(p,{}q,{}v)} returns the resultant of the polynomials \spad{p} and \spad{q} with respect to the variable \spad{v}.
605485`\axiom{resultantEuclidean_naif(\spad{P},{}\spad{Q})} returns the resultant of \axiom{\spad{P}} and \axiom{\spad{Q}} computed by means of the naive algorithm.
606653`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
606886`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
607379`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
607632`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
607976`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
608081`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
608188`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
608303`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
608420`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
608525`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
608632`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
609077`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
613249`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
613354`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
613461`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
613576`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
613693`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
613798`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
613905`\spad{retractIfCan(a)} tries change \spad{a} into an object of type \spad{S}. If it can,{} then such an object is returned. Otherwise,{} "failed" is returned.
613974`\axiom{retractIfCan(\spad{lp})} returns an element of the domain whose elements are the members of \axiom{\spad{lp}} if such an element exists,{} otherwise \axiom{"failed"} is returned.
614067`\axiom{retractIfCan(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if all its variables belong to \axiom{\spad{V}}.
614248`\axiom{retractIfCan(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if all its variables belong to \axiom{\spad{V}}.
614419`\axiom{retractIfCan(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if all its variables belong to \axiom{\spad{V}}.
614619`\axiom{retractIfCan(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if all its variables belong to \axiom{\spad{V}}.
614827`\axiom{retractIfCan(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if all its variables belong to \axiom{\spad{V}}.
615053`\axiom{retractIfCan(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if all its variables belong to \axiom{\spad{V}}.
615252`\spad{retractIfCan(a)} transforms a into an element of \spad{S} if possible. Returns "failed" if a cannot be made into an element of \spad{S}.
615827`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
616018`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
616305`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
616516`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
616733`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
616817`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
616903`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
616997`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
617093`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
617177`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
617263`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
617624`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
620577`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
620661`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
620747`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
620841`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
620937`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
621021`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
621107`\spad{retract(a)} tries to convert \spad{a} into an object of type \spad{S}. If possible,{} it returns the object. Error: if no such retraction is possible.
621155`\axiom{retract(\spad{lp})} returns an element of the domain whose elements are the members of \axiom{\spad{lp}} if such an element exists,{} otherwise an error is produced.
621227`\axiom{retract(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if \axiom{retractIfCan(\spad{p})} does not return "failed",{} otherwise an error is produced.
621387`\axiom{retract(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if \axiom{retractIfCan(\spad{p})} does not return "failed",{} otherwise an error is produced.
621537`\axiom{retract(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if \axiom{retractIfCan(\spad{p})} does not return "failed",{} otherwise an error is produced.
621716`\axiom{retract(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if \axiom{retractIfCan(\spad{p})} does not return "failed",{} otherwise an error is produced.
621903`\axiom{retract(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if \axiom{retractIfCan(\spad{p})} does not return "failed",{} otherwise an error is produced.
622108`\axiom{retract(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if \axiom{retractIfCan(\spad{p})} does not return "failed",{} otherwise an error is produced.
622286`\spad{retract(a)} transforms a into an element of \spad{S} if possible. Error: if a cannot be made into an element of \spad{S}.
622333`\spad{retractable?(p)} tests if \spad{p} is a 0-form,{} \spadignore{i.e.} if degree(\spad{p}) = 0.
622391`\spad{retractable?(df)} tests if differential form \spad{df} is a 0-form,{} \spadignore{i.e.} if degree(\spad{df}) = 0.
622467`\axiom{retractable?(\spad{x})} tests if \axiom{\spad{x}} is a tree with only one entry.
622527`\axiom{retractable?(\spad{x})} tests if \axiom{\spad{x}} is a tree with only one entry.
622582`\spad{retractable?([l1]*[l2]*...[ln])} returns \spad{true} iff \spad{n} equals \spad{1}.
622663`\spad{retractable?(a)} tests if \spad{a} can be converted into an object of type \spad{S}.
623085`\spad{returns()} creates a representation of a FORTRAN RETURN statement.
623126`\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.
623193`\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.
623251`\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.
623311`\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.
623378`\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.
623443`\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.
623615`\spad{reverse!(l)} reverses the list of monomials forming \spad{l},{} destroying the element \spad{l}.
623669`\spad{reverse!(d)} destructively replaces \spad{d} by its reverse dequeue,{} \spadignore{i.e.} the top (front) element is now the bottom (back) element,{} and so on.
623721`\spad{reverse!(u)} returns \spad{u} with its elements in reverse order.
623930`\spad{reverse(p)} returns the reverse polynomial of \spad{p}.
623998`\spad{reverse(l)} reverses the list of monomials forming \spad{l}. This has some effect if the monoid is non-abelian,{} \spadignore{i.e.} \spad{reverse(a1\^e1 ... an\^en) = an\^en ... a1\^e1} which is different.
624051`\spad{reverse(a)} returns a copy of \axiom{a} with elements in reverse order.
624107`implements reverse for a single term by using: grade: 0 1 2 3... multi: 1 1 \spad{-1} \spad{-1}...
624294`\spad{revert(f(x))} returns a Taylor series \spad{g(x)} such that \spad{f(g(x)) = g(f(x)) = x}. Series \spad{f(x)} should have constant coefficient 0 and invertible 1st order coefficient.
624460`\axiom{rewriteIdealWithHeadRemainder(\spad{lp},{}\spad{cs})} returns \axiom{\spad{lr}} such that the leading monomial of every polynomial in \axiom{\spad{lr}} is reduced in the sense of Groebner bases \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{cs}} and \axiom{(\spad{lp},{}\spad{cs})} and \axiom{(\spad{lr},{}\spad{cs})} generate the same ideal in \axiom{(\spad{R})^(\spad{-1}) \spad{P}
624460`--}.
624806`\axiom{rewriteIdealWithRemainder(\spad{lp},{}\spad{cs})} returns \axiom{\spad{lr}} such that every polynomial in \axiom{\spad{lr}} is fully reduced in the sense of Groebner bases \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{cs}} and \axiom{(\spad{lp},{}\spad{cs})} and \axiom{(\spad{lr},{}\spad{cs})} generate the same ideal in \axiom{(\spad{R})^(\spad{-1}) \spad{P}}.
625186`\axiom{rewriteSetWithReduction(\spad{lp},{}\spad{ts},{}redOp,{}redOp?)} returns a list \axiom{\spad{lq}} of polynomials such that \axiom{[reduce(\spad{p},{}\spad{ts},{}redOp,{}redOp?) for \spad{p} in \spad{lp}]} and \axiom{\spad{lp}} have the same zeros inside the regular zero set of \axiom{\spad{ts}}. Moreover,{} for every polynomial \axiom{\spad{q}} in \axiom{\spad{lq}} and every polynom
625186`--ial \axiom{\spad{t}} in \axiom{\spad{ts}} \axiom{redOp?(\spad{q},{}\spad{t})} holds and there exists a polynomial \axiom{\spad{p}} in the ideal generated by \axiom{\spad{lp}} and a product \axiom{\spad{h}} of \axiom{initials(\spad{ts})} such that \axiom{\spad{h*p} - \spad{r}} lies in the ideal generated by \axiom{\spad{ts}}. The operation \axiom{redOp} must satisfy the following conditions. For every \axiom{\spad{p}} and \axiom{\spad{q}} we have \axiom{redOp?(redOp(\spad{p},{}\spad{q}),{}\spad
625186`--{q})} and there exists an integer \axiom{\spad{e}} and a polynomial \axiom{\spad{f}} such that \axiom{init(\spad{q})^e*p = \spad{f*q} + redOp(\spad{p},{}\spad{q})}.
625301`\spad{rhs(f)} returns the right side of \spad{f}
625344`\spad{rhs(r)} returns the right hand side of the rule \spad{r}.
625392`\spad{rhs(eqn)} returns the right hand side of equation \spad{eqn}.
626586`\spad{ridHack1(i,{}j,{}k,{}l)} \undocumented
626679`\spad{riemannZeta(z)} is the Riemann Zeta function
626743`\spad{riemannZeta(z)} is the Riemann Zeta function.
626802`\spad{rightAlternative?()} tests if \spad{2*associator(a,{}b,{}b) = 0} for all \spad{a},{} \spad{b} in the algebra. Note: we only can test this; in general we don\spad{'t} know whether \spad{2*a=0} implies \spad{a=0}.
626884`\spad{rightAlternative?()} tests if \spad{2*associator(a,{}b,{}b) = 0} for all \spad{a},{} \spad{b} in the algebra. Note: we only can test this; in general we don\spad{'t} know whether \spad{2*a=0} implies \spad{a=0}.
627075`\spad{rightCharacteristicPolynomial(a)} returns the characteristic polynomial of the right regular representation of \spad{a} with respect to any basis.
627190`\spad{rightDiscriminant()} returns the determinant of the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by the right trace of the product \spad{vi*vj},{} where \spad{v1},{}...,{}\spad{vn} are the elements of the fixed \spad{R}-module basis. Note: the same as \spad{determinant(rightTraceMatrix())}.
627262`\spad{rightDiscriminant()} returns the determinant of the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by the right trace of the product \spad{vi*vj},{} where \spad{v1},{}...,{}\spad{vn} are the elements of the fixed \spad{R}-module basis. Note: the same as \spad{determinant(rightTraceMatrix())}.
627487`\spad{rightDiscriminant([v1,{}...,{}vn])} returns the determinant of the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by the right trace of the product \spad{vi*vj}. Note: the same as \spad{determinant(rightTraceMatrix([v1,{}...,{}vn]))}.
627570`\spad{rightDivide(a,{}b)} returns the pair \spad{[q,{}r]} such that \spad{a = q*b + r} and the degree of \spad{r} is less than the degree of \spad{b}. This process is called ``right division\spad{''}.
627906`\spad{rightExactQuotient(a,{}b)} computes the value \spad{q},{} if it exists such that \spad{a = q*b}.
628123`\spad{rightExtendedGcd(a,{}b)} returns \spad{[c,{}d]} such that \spad{g = c * a + d * b = rightGcd(a,{} b)}.
628523`\spad{rightGcd(a,{}b)} computes the value \spad{g} of highest degree such that \indented{3}{\spad{a = aa*g}} \indented{3}{\spad{b = bb*g}} for some values \spad{aa} and \spad{bb}. The value \spad{g} is computed using right-division.
628672`\spad{rightLcm(a,{}b)} computes the value \spad{m} of lowest degree such that \spad{m = a*aa = b*bb} for some values \spad{aa} and \spad{bb}. The value \spad{m} is computed using left-division.
628860`\spad{rightMinimalPolynomial(a)} returns the polynomial determined by the smallest non-trivial linear combination of right powers of \spad{a}. Note: the polynomial never has a constant term as in general the algebra has no unit.
628989`\spad{rightMult(a,{} s)} returns \spad{a * s} where \spad{*} is the monoid operation,{} which is assumed non-commutative.
629110`\spad{rightNorm(a)} returns the determinant of the right regular representation of \spad{a}.
629177`\spad{rightOne(eq)} divides by the right hand side.
629249`\spad{rightOne(eq)} divides by the right hand side.
629500`\spad{rightPower(a,{}n)} returns the \spad{n}\spad{-}th right power of \spad{a},{} \spadignore{i.e.} \spad{rightPower(a,{}n) := rightPower(a,{}n-1) * a} and \spad{rightPower(a,{}0) := 1}.
629567`\spad{rightPower(a,{}n)} returns the \spad{n}\spad{-}th right power of \spad{a},{} \spadignore{i.e.} \spad{rightPower(a,{}n) := rightPower(a,{}n-1) * a} and \spad{rightPower(a,{}1) := a}.
629694`\spad{rightQuotient(a,{}b)} computes the pair \spad{[q,{}r]} such that \spad{a = q*b + r} and the degree of \spad{r} is less than the degree of \spad{b}. The value \spad{q} is returned.
629931`\spad{rightRankPolynomial()} calculates the right minimal polynomial of the generic element in the algebra,{} defined by the same structural constants over the polynomial ring in symbolic coefficients with respect to the fixed basis.
630045`\spad{rightRankPolynomial()} calculates the right minimal polynomial of the generic element in the algebra,{} defined by the same structural constants over the polynomial ring in symbolic coefficients with respect to the fixed basis.
630168`\spad{rightRank(x)} determines the number of linearly independent elements in \spad{b1*x},{}...,{}\spad{bn*x},{} where \spad{b=[b1,{}...,{}bn]} is a basis.
630317`\spad{rightRecip(a)} returns an element,{} which is a right inverse of \spad{a},{} or \spad{"failed"} if there is no unit element,{} if such an element doesn\spad{'t} exist or cannot be determined (see unitsKnown).
630423`\spad{rightRecip(a)} returns an element,{} which is a right inverse of \spad{a},{} or \spad{"failed"} if such an element doesn\spad{'t} exist or cannot be determined (see unitsKnown).
630572`\spad{rightRegularRepresentation(a)} returns the matrix of the linear map defined by right multiplication by \spad{a} with respect to the fixed \spad{R}-module basis.
630854`\spad{rightRegularRepresentation(a,{}[v1,{}...,{}vn])} returns the matrix of the linear map defined by right multiplication by \spad{a} with respect to the \spad{R}-module basis \spad{[v1,{}...,{}vn]}.
631029`\spad{rightRemainder(a,{}b)} computes the pair \spad{[q,{}r]} such that \spad{a = q*b + r} and the degree of \spad{r} is less than the degree of \spad{b}. The value \spad{r} is returned.
631216`\spad{rightTraceMatrix()} is the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by the right trace of the product \spad{vi*vj},{} where \spad{v1},{}...,{}\spad{vn} are the elements of the fixed \spad{R}-module basis.
631295`\spad{rightTraceMatrix()} is the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by the right trace of the product \spad{vi*vj},{} where \spad{v1},{}...,{}\spad{vn} are the elements of the fixed \spad{R}-module basis.
631541`\spad{rightTraceMatrix([v1,{}...,{}vn])} is the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by the right trace of the product \spad{vi*vj}.
631696`\spad{rightTrace(a)} returns the trace of the right regular representation of \spad{a}.
631764`\spad{rightTrim(s,{}c)} returns \spad{s} with all trailing occurrences of \spad{c} deleted. For example,{} \axiom{rightTrim(" abc ",{} char " ")} returns \axiom{" abc"}.
631823`\spad{rightTrim(s,{}cc)} returns \spad{s} with all trailing occurences of characters in \spad{cc} deleted. For example,{} \axiom{rightTrim("(abc)",{} charClass "()")} returns \axiom{"(abc"}.
631962`\spad{rightUnit()} returns a right unit of the algebra (not necessarily unique),{} or \spad{"failed"} if there is none.
632174`\spad{rightUnits()} returns the affine space of all right units of the algebra,{} or \spad{"failed"} if there is none
632299`\spad{rightUnits()} returns the affine space of all right units of the algebra,{} or \spad{"failed"} if there is none.
632437`\spad{rightZero(eq)} subtracts the right hand side.
632501`\axiom{right(\spad{x})} returns right subtree of \axiom{\spad{x}} or error if \axiomOpFrom{retractable?}{LyndonWord}(\axiom{\spad{x}}) is \spad{true}.
632549`\axiom{right(\spad{x})} returns right subtree of \axiom{\spad{x}} or error if \axiomOpFrom{retractable?}{Magma}(\axiom{\spad{x}}) is \spad{true}.
632592`\spad{right(f)} right-justifies form \spad{f} in total space.
632632`\axiom{right(rootChar)} is the right bound of the isolating interval
632725`\spad{right(a)} returns the right child.
632782`\spad{right(f,{}n)} right-justifies form \spad{f} within space of width \spad{n}.
634747`\spad{roman(n)} creates a roman numeral for \spad{n}.
634794`\spad{roman(n)} creates a roman numeral for symbol \spad{n}.
636014`\spad{rootOf(p)} returns \spad{y} such that \spad{p(y) = 0}. Error: if \spad{p} has more than one variable \spad{y}.
636081`\spad{rootOf(p)} returns \spad{y} such that \spad{p(y) = 0}.
636164`\spad{rootOf(p)} returns \spad{y} such that \spad{p(y) = 0}. Error: if \spad{p} has more than one variable \spad{y}.
636883`\spad{rootOf(p,{} y)} returns \spad{y} such that \spad{p(y) = 0}. The object returned displays as \spad{'y}.
637086`\spad{rootOf(p,{}y)} returns \spad{y} such that \spad{p(y) = 0}. The object returned displays as \spad{'y}.
637532`\spad{rootPower(f)} transforms every radical power of the form \spad{(a^(1/n))^m} into a simpler form if \spad{m} and \spad{n} have a common factor.
637684`\spad{rootProduct(f)} combines every product of the form \spad{(a^(1/n))^m * (a^(1/s))^t} into a single power of a root of \spad{a},{} and transforms every radical power of the form \spad{(a^(1/n))^m} into a simpler form.
637838`\spad{rootRadius(p)} calculates the root radius of \spad{p} with a maximal error quotient of {\em 1+globalEps},{} where {\em globalEps} is the internal error bound,{} which can be set by {\em setErrorBound}.
637903`\spad{rootRadius(p,{}errQuot)} calculates the root radius of \spad{p} with a maximal error quotient of {\em errQuot}.
637970`\spad{rootSimp(f)} transforms every radical of the form \spad{(a * b^(q*n+r))^(1/n)} appearing in \spad{f} into \spad{b^q * (a * b^r)^(1/n)}. This transformation is not in general valid for all complex numbers \spad{b}.
638121`\spad{rootSplit(f)} transforms every radical of the form \spad{(a/b)^(1/n)} appearing in \spad{f} into \spad{a^(1/n) / b^(1/n)}. This transformation is not in general valid for all complex numbers \spad{a} and \spad{b}.
638180`\spad{root(f)} creates a form for the square root of form \spad{f}.
638219`\spad{root(f,{}n)} creates a form for the \spad{n}th root of form \spad{f}.
638754`\spad{rootsOf(p)} returns \spad{[y1,{}...,{}yn]} such that \spad{p(\spad{yi}) = 0}. Note: the returned values \spad{y1},{}...,{}\spad{yn} contain new symbols which are bound in the interpreter to the respective values. Error: if \spad{p} has more than one variable \spad{y}.
638828`\spad{rootsOf(p)} returns \spad{[y1,{}...,{}yn]} such that \spad{p(\spad{yi}) = 0}. Note: the returned values \spad{y1},{}...,{}\spad{yn} contain new symbols which are bound in the interpreter to the respective values.
638918`\spad{rootsOf(p,{} y)} returns \spad{[y1,{}...,{}yn]} such that \spad{p(\spad{yi}) = 0}; Note: the returned values \spad{y1},{}...,{}\spad{yn} contain new symbols which are bound in the interpreter to the respective values. Error: if \spad{p} has more than one variable \spad{y}.
639265`\spad{rootsOf(p,{} z)} returns \spad{[y1,{}...,{}yn]} such that \spad{p(\spad{yi}) = 0}; The returned roots contain new symbols \spad{'\%z0},{} \spad{'\%z1} ...; Note: the new symbols are bound in the interpreter to the respective values.
639362`\spad{rootsOf(p,{} y)} returns \spad{[y1,{}...,{}yn]} such that \spad{p(\spad{yi}) = 0}; The returned roots contain new symbols \spad{'\%z0},{} \spad{'\%z1} ...; Note: the new symbols are bound in the interpreter to the respective values.
639442`\spad{rotate! q} rotates queue \spad{q} so that the element at the front of the queue goes to the back of the queue. Note: rotate! \spad{q} is equivalent to enqueue!(dequeue!(\spad{q})).
639623`\spad{rotatex(r)} returns a dhmatrix for rotation about axis \spad{X} for \spad{r} degrees
639680`\spad{rotatey(r)} returns a dhmatrix for rotation about axis \spad{Y} for \spad{r} degrees
639737`\spad{rotatez(r)} returns a dhmatrix for rotation about axis \spad{Z} for \spad{r} degrees
639866`\axiom{roughBase?(\spad{ps})} returns \spad{true} iff for every pair \axiom{{\spad{p},{}\spad{q}}} of polynomials in \axiom{\spad{ps}} their leading monomials are relatively prime.
640186`\axiom{roughEqualIdeals?(\spad{ps1},{}\spad{ps2})} returns \spad{true} iff it can proved that \axiom{\spad{ps1}} and \axiom{\spad{ps2}} generate the same ideal in \axiom{(\spad{R})^(\spad{-1}) \spad{P}} without computing Groebner bases.
640370`\axiom{roughSubIdeal?(\spad{ps1},{}\spad{ps2})} returns \spad{true} iff it can proved that all polynomials in \axiom{\spad{ps1}} lie in the ideal generated by \axiom{\spad{ps2}} in \axiom{\axiom{(\spad{R})^(\spad{-1}) \spad{P}}} without computing Groebner bases.
640550`\axiom{roughUnitIdeal?(\spad{ps})} returns \spad{true} iff \axiom{\spad{ps}} contains some non null element lying in the base ring \axiom{\spad{R}}.
640694`\spad{round x} computes the integer closest to \spad{x}.
640973`\spad{rowEchelon(m)} returns the row echelon form of the matrix \spad{m}. the result will have entries in the quotient field.
641069`\spad{rowEchelon(m)} returns the row echelon form of the matrix \spad{m}.
641148`\spad{rowEchelon(m)} returns the row echelon form of the matrix \spad{m}.
641244`\spad{rowEchelon(m)} returns the row echelon form of the matrix \spad{m}.
641326`\spad{rowEchelon(m)} returns the row echelon form of the matrix \spad{m}.
641572`\spad{row(m,{}i)} returns the \spad{i}th row of the matrix \spad{m}. Error: if the index is outside the proper range.
641649`\spad{row(m,{}i)} returns the \spad{i}th row of \spad{m} error check to determine if index is in proper ranges
641724`\spad{rquo(x,{} s)} returns the exact right quotient of \spad{x} by \spad{s}.
641791`\spad{rquo(x,{} y)} returns the exact right quotient of \spad{x} by \spad{y} \spadignore{i.e.} \spad{q} such that \spad{x = q * y},{} "failed" if \spad{x} is not of the form \spad{q * y}.
641852`\spad{rquo(x,{} y)} returns the exact right quotient of \spad{x} by \spad{y} that is \spad{q} such that \spad{x = q * y},{} "failed" if \spad{x} is not of the form \spad{q * y}.
642091`\spad{rquo(x,{}y)} returns the right simplification of \spad{x} by \spad{y}.
642141`\spad{rquo(x,{}v)} returns the right simplification of \spad{x} by the variable \spad{v}.
642311`\spad{rspace(n,{}m)} creates rectangular white space,{} \spad{n} wide by \spad{m} high.
642365`\spad{rst(s)} returns a pointer to the next node of stream \spad{s}. Caution: this function should only be called after a \spad{empty?} test has been made since there no error check.
642491`\spad{rule(f,{} g)} creates the rewrite rule: \spad{f == eval(g,{} g is f)},{} with left-hand side \spad{f} and right-hand side \spad{g}.
642542`\spad{rule(f,{} g,{} [f1,{}...,{}fn])} creates the rewrite rule \spad{f == eval(eval(g,{} g is f),{} [f1,{}...,{}fn])},{} that is a rule with left-hand side \spad{f} and right-hand side \spad{g}; The symbols \spad{f1},{}...,{}\spad{fn} are the operators that are considered quoted,{} that is they are not evaluated during any rewrite,{} but just applied formally to their arguments.
643963`\spad{safety(d)} specifies the number of values reserved for testing any solutions found. This option is expressed in the form \spad{safety == d}.
644130`\spad{sample yields} a value of type \%
644172`\spad{sample yields} a value of type \%
644210`\spad{sample yields} a value of type \%
644245`\spad{sample()} returns an object of type \%.
644302`\spad{sample()} returns a sample of \%
644337`\spad{satisfy?(v,{} p)} returns \spad{f}(\spad{v}) where \spad{f} is the predicate attached to \spad{p}.
644407`\spad{satisfy?([v1,{}...,{}vn],{} p)} returns \spad{f(v1,{}...,{}vn)} where \spad{f} is the top-level predicate attached to \spad{p}.
644483`\spad{satisfy?(r,{} p)} returns \spad{true} if the matches satisfy the top-level predicate of \spad{p},{} \spad{false} if they don\spad{'t},{} and "failed" if not enough variables of \spad{p} are matched in \spad{r} to decide.
644726`\spad{save()} creates a representation of a SAVE statement.
644966`\spad{scalarMatrix(r)} returns an \spad{n}-by-\spad{n} matrix with \spad{r}\spad{'s} on the diagonal and zeroes elsewhere.
645115`\spad{scalarMatrix(n,{}r)} returns an \spad{n}-by-\spad{n} matrix with \spad{r}\spad{'s} on the diagonal and zeroes elsewhere.
645276`\spad{scaleRoots(p,{}c)} returns the polynomial which has \spad{c} times the roots of \spad{p}.
645349`\spad{scale(k)} returns \spad{matrix [[k,{}0],{}[0,{}1]]} representing the map \spad{x -> k * x}.
645397`\spad{scale(m,{}h)} returns \spad{scale(h) * m} (see \spadfunFrom{shift}{MoebiusTransform}).
645448`\spad{scale(sx,{}sy,{}sz)} returns a dhmatrix for scaling in the \spad{X},{} \spad{Y} and \spad{Z} directions
646147`\spad{scan(f,{}a,{}r)} successively applies \spad{reduce(f,{}x,{}r)} to more and more leading sub-aggregates \spad{x} of aggregrate \spad{a}. More precisely,{} if \spad{a} is \spad{[a1,{}a2,{}...]},{} then \spad{scan(f,{}a,{}r)} returns \spad{[reduce(f,{}[a1],{}r),{}reduce(f,{}[a1,{}a2],{}r),{}...]}.
646225`\spad{scan(f,{}a,{}r)} successively applies \spad{reduce(f,{}x,{}r)} to more and more leading sub-aggregates \spad{x} of aggregate \spad{a}. More precisely,{} if \spad{a} is \spad{[a1,{}a2,{}...]},{} then \spad{scan(f,{}a,{}r)} returns \spad {[reduce(f,{}[a1],{}r),{}reduce(f,{}[a1,{}a2],{}r),{}...]}.
646526`returns the \spad{'x'}(left to right) screen coordinate
646586`returns the \spad{'y'}(bottom to top) screen coordinate
646646`returns the \spad{'z'}(out from screen) screen coordinate
646706`returns screen coordinates as a list
647186`\spad{scripted?(s)} is \spad{true} if \spad{s} has been given any scripts.
647381`\spad{scripts(f,{} [sub,{} super,{} presuper,{} presub])} \indented{1}{creates a form for \spad{f} with scripts on all 4 corners.}
647536`\spad{search(k,{}t)} searches the table \spad{t} for the key \spad{k},{} returning the entry stored in \spad{t} for key \spad{k}. If \spad{t} has no such key,{} \axiom{search(\spad{k},{}\spad{t})} returns "failed".
647616`\spad{sec2cos(f)} converts every \spad{sec(u)} appearing in \spad{f} into \spad{1/cos(u)}.
647678`\spad{secIfCan(z)} returns sec(\spad{z}) if possible,{} and "failed" otherwise.
647758`\spad{sec(x)} applies the secant operator to \spad{x}
648091`\spad{sec(z)} returns the secant of Laurent series \spad{z}.
648177`\spad{sec(z)} returns the secant of a Puiseux series \spad{z}.
648272`\spad{sec(x)} returns the secant of \spad{x}.
648329`\spad{sech2cosh(f)} converts every \spad{sech(u)} appearing in \spad{f} into \spad{1/cosh(u)}.
648393`\spad{sechIfCan(z)} returns sech(\spad{z}) if possible,{} and "failed" otherwise.
648474`\spad{sech(x)} applies the hyperbolic secant operator to \spad{x}
648809`\spad{sech(z)} returns the hyperbolic secant of Laurent series \spad{z}.
648896`\spad{sech(z)} returns the hyperbolic secant of a Puiseux series \spad{z}.
648992`\spad{sech(x)} returns the hyperbolic secant of \spad{x}.
649100`\spad{second(u)} returns the second element of \spad{u}. Note: \axiom{second(\spad{u}) = first(rest(\spad{u}))}.
649156`\spad{seed()} returns the current seed value.
649206`\spad{segment(l)} is an alternate way to construct the segment \spad{l..}.
649256`\spad{segment(segb)} returns the segment from the right hand side of the \spadtype{SegmentBinding}. For example,{} if \spad{segb} is \spad{v=a..b},{} then \spad{segment(segb)} returns \spad{a..b}.
649313`\spad{segment(i,{}j)} is an alternate way to create the segment \spad{i..j}.
649475`\spad{select!(p,{}d)} destructively changes dictionary \spad{d} by removing all entries \spad{x} such that \axiom{\spad{p}(\spad{x})} is not \spad{true}.
649566`\spad{select!(p,{}u)} destructively changes \spad{u} by keeping only values \spad{x} such that \axiom{\spad{p}(\spad{x})}.
650315`\spad{select(x,{}n)} returns the \spad{n}-th element of tuple \spad{x}. tuples are 0-based
650372`\spad{select(p,{}u)} returns a copy of \spad{u} containing only those elements such \axiom{\spad{p}(\spad{x})} is \spad{true}. Note: \axiom{select(\spad{p},{}\spad{u}) \spad{==} [\spad{x} for \spad{x} in \spad{u} | \spad{p}(\spad{x})]}.
650452`\spad{select(f,{}st)} returns a stream consisting of those elements of stream \spad{st} satisfying the predicate \spad{f}. Note: \spad{select(f,{}st) = [x for x in st | f(x)]}.
650518`\spad{select(p,{}t)} returns \spad{[x for x in t | p(x)]}.
650578`\axiom{select(\spad{ts},{}\spad{v})} returns the polynomial of \axiom{\spad{ts}} with \axiom{\spad{v}} as main variable,{} if any.
650656`\spad{selectfirst(x)} \undocumented
650703`\spad{selectsecond(x)} \undocumented
650893`\axiom{discriminantEuclidean(\spad{P})} carries out the equality \axiom{...\spad{P} + \spad{coef2} * \spad{D}(\spad{P}) = discriminant(\spad{P})}. Warning: \axiom{degree(\spad{P}) \spad{>=} degree(\spad{Q})}.
651149`\axiom{semiLastSubResultantEuclidean(\spad{P},{} \spad{Q})} computes the last non zero subresultant \axiom{\spad{S}} and carries out the equality \axiom{...\spad{P} + coef2*Q = \spad{S}}. Warning: \axiom{degree(\spad{P}) \spad{>=} degree(\spad{Q})}.
651275`\axiom{\spad{semiResultantEuclidean1}(\spad{P},{}\spad{Q})} carries out the equality \axiom{\spad{coef1}.\spad{P} + ? \spad{Q} = resultant(\spad{P},{}\spad{Q})}.
651389`\axiom{\spad{semiResultantEuclidean2}(\spad{P},{}\spad{Q})} carries out the equality \axiom{...\spad{P} + coef2*Q = resultant(\spad{P},{}\spad{Q})}. Warning: \axiom{degree(\spad{P}) \spad{>=} degree(\spad{Q})}.
651503`\axiom{resultantEuclidean_naif(\spad{P},{}\spad{Q})} returns the semi-extended resultant of \axiom{\spad{P}} and \axiom{\spad{Q}} computed by means of the naive algorithm.
651620`\axiom{semiResultantReduitEuclidean(\spad{P},{}\spad{Q})} returns the "reduce resultant" and carries out the equality \axiom{...\spad{P} + coef2*Q = resultantReduit(\spad{P},{}\spad{Q})}.
651761`\axiom{\spad{semiSubResultantGcdEuclidean1}(\spad{P},{}\spad{Q})} carries out the equality \axiom{coef1*P + ? \spad{Q} = \spad{+/-} S_i(\spad{P},{}\spad{Q})} where the degree (not the indice) of the subresultant \axiom{S_i(\spad{P},{}\spad{Q})} is the smaller as possible.
651878`\axiom{\spad{semiSubResultantGcdEuclidean2}(\spad{P},{}\spad{Q})} carries out the equality \axiom{...\spad{P} + coef2*Q = \spad{+/-} S_i(\spad{P},{}\spad{Q})} where the degree (not the indice) of the subresultant \axiom{S_i(\spad{P},{}\spad{Q})} is the smaller as possible. Warning: \axiom{degree(\spad{P}) \spad{>=} degree(\spad{Q})}.
651995`\spad{semicolonSeparate(l)} creates the form separating the elements of \spad{l} by semicolons.
652121`\spad{separant(p)} returns the partial derivative of the differential polynomial \spad{p} with respect to its leader.
652640`\spad{separate(s)} makes each of the components of the \spadtype{SubSpace},{} \spad{s},{} into a list of separate and distinct subspaces and returns the list.
652785`\spad{separate(p,{} q)} returns \spad{[a,{} b]} such that polynomial \spad{p = a b} and \spad{a} is relatively prime to \spad{q}.
654496`\spad{series(s)} creates a power series from a stream of \indented{1}{ring elements.} \indented{1}{For univariate series types,{} the stream \spad{s} should be a stream} \indented{1}{of Taylor coefficients. For multivariate series types,{} the} \indented{1}{stream \spad{s} should be a stream of forms the \spad{n}th element} \indented{1}{of which is a} \indented{1}{form of degree \spad{n} in the power series variables.}
654654`\spad{series(f)} returns a series expansion of the expression \spad{f}. Note: \spad{f} should have only one variable; the series will be expanded in powers of that variable.
654725`\spad{series([a0,{}a1,{}a2,{}...])} is the Taylor series \spad{a0 + a1 x + a2 x^2 + ...}.
656417`\spad{setColumn!(m,{}j,{}v)} sets to \spad{j}th column of \spad{m} to \spad{v}
656502`\axiom{setCondition!(\spad{n},{}\spad{t})} returns \spad{n} whose condition has been replaced by \spad{t} if it is not empty,{} else an error is produced.
656560`\spad{setDifference(u1,{}u2)} returns a list of the elements of \spad{u1} that are not also in \spad{u2}. The order of elements in the resulting list is unspecified.
656626`\axiom{setEmpty!(\spad{n})} replaces \spad{n} by \axiom{empty()\$\%}.
656816`\spad{setErrorBound(eps)} changes the internal error bound,{} by default being {\em 10 ^ (-3)} to \spad{eps},{} if \spad{R} is a member in the category \spadtype{QuotientFieldCategory Integer}. The internal {\em globalDigits} is set to {\em ceiling(1/r)^2*10} being {\em 10^7} by default.
657128`\spad{setIntersection(u1,{}u2)} returns a list of the elements that lists \spad{u1} and \spad{u2} have in common. The order of elements in the resulting list is unspecified.
657196`\spad{setLabelValue(i)} resets the counter which produces labels to \spad{i}
657267`\spad{setLegalFortranSourceExtensions(l)} \undocumented{}
657549`allows override of parameters such as orthogonal used for debugging
657698`\spad{setOrder([a1,{}...,{}an])} defines a partial ordering on \spad{S} given \spad{by:} \indented{3}{(1)\space{2}\spad{a1 < a2 < ... < an}.} \indented{3}{(2)\space{2}\spad{b < \spad{ai}\space{3}for i = 1..n} and \spad{b} not among the \spad{ai}\spad{'s}.} \indented{3}{(3)\space{2}undefined on \spad{(b,{} c)} if neither is among the \spad{ai}\spad{'s}.}
657767`\spad{setOrder([b1,{}...,{}bm],{} [a1,{}...,{}an])} defines a partial ordering on \spad{S} given \spad{by:} \indented{3}{(1)\space{2}\spad{b1 < b2 < ... < bm < a1 < a2 < ... < an}.} \indented{3}{(2)\space{2}\spad{bj < c < \spad{ai}}\space{2}for \spad{c} not among the \spad{ai}\spad{'s} and \spad{bj}\spad{'s}.} \indented{3}{(3)\space{2}undefined on \spad{(c,{}d)} if neither is among the \spad{ai}\spad{'s},{}\spad{bj}\spad{'s}.}
657844`\spad{setPoly(x)} \undocumented
657893`\spad{setPosition(x,{} n)} associates the integer \spad{n} to \spad{x}.
657961`\spad{setPredicates(p,{} [p1,{}...,{}pn])} attaches the predicate \spad{p1} and ... and \spad{pn} to \spad{p}.
658232`\spad{setProperty(op,{} s,{} v)} attaches property \spad{s} to \spad{op},{} and sets its value to \spad{v}. Argument \spad{op} is modified "in place",{} \spadignore{i.e.} no copy is made.
658424`\spad{setRow!(m,{}i,{}v)} sets to \spad{i}th row of \spad{m} to \spad{v}
658941`\spad{setUnion(u1,{}u2)} appends the two lists \spad{u1} and \spad{u2},{} then removes all duplicates. The order of elements in the resulting list is unspecified.
659002`\axiom{setValue!(\spad{n},{}\spad{v})} returns \spad{n} whose value has been replaced by \spad{v} if it is not empty,{} else an error is produced.
659056`\spad{setVariableOrder([a1,{}...,{}an])} defines an ordering on the variables given by \spad{a1 > a2 > ... > an > other variables}.
659136`\spad{setVariableOrder([b1,{}...,{}bm],{} [a1,{}...,{}an])} defines an ordering on the variables given by \spad{b1 > b2 > ... > bm >} other variables \spad{> a1 > a2 > ... > an}.
659273`\spad{set()}\$\spad{D} creates an empty set aggregate of type \spad{D}.
659365`\spad{set([x,{}y,{}...,{}z])} creates a set aggregate containing items \spad{x},{}\spad{y},{}...,{}\spad{z}.
659480`\spad{setchildren!(u,{}v)} replaces the current children of node \spad{u} with the members of \spad{v} in left-to-right order.
659681`\spad{setelt(n,{}m)} changes the value of the object \spad{n} to \spad{m}.
660724`\spad{setelt(u,{}"first",{}x)} (also written: \axiom{\spad{u}.first \spad{:=} \spad{x}}) is equivalent to \axiom{setfirst!(\spad{u},{}\spad{x})}.
660814`\spad{setelt(u,{}"last",{}x)} (also written: \axiom{\spad{u}.last \spad{:=} \spad{b}}) is equivalent to \axiom{setlast!(\spad{u},{}\spad{v})}.
660903`\spad{setelt(a,{}"left",{}b)} (also written \axiom{a . left \spad{:=} \spad{b}}) is equivalent to \axiom{setleft!(a,{}\spad{b})}.
660995`\spad{setelt(u,{}"rest",{}v)} (also written: \axiom{\spad{u}.rest \spad{:=} \spad{v}}) is equivalent to \axiom{setrest!(\spad{u},{}\spad{v})}.
661086`\spad{setelt(a,{}"right",{}b)} (also written \axiom{\spad{b} . right \spad{:=} \spad{b}}) is equivalent to \axiom{setright!(a,{}\spad{b})}.
661179`\spad{setelt(a,{}"value",{}x)} (also written \axiom{a . value \spad{:=} \spad{x}}) is equivalent to \axiom{setvalue!(a,{}\spad{x})}
661264`\spad{setelt(u,{}x,{}y)} sets the image of \spad{x} to be \spad{y} under \spad{u},{} assuming \spad{x} is in the domain of \spad{u}. Error: if \spad{x} is not in the domain of \spad{u}.
661350`\spad{setelt(t,{}k,{}e)} (also written \axiom{\spad{t}.\spad{k} \spad{:=} \spad{e}}) is equivalent to \axiom{(insert([\spad{k},{}\spad{e}],{}\spad{t}); \spad{e})}.
661419`\spad{lib.k := v} saves the value \spad{v} in the library \spad{lib}. It can later be extracted using the key \spad{k}.
661469`\spad{setelt(u,{}i..j,{}x)} (also written: \axiom{\spad{u}(\spad{i}..\spad{j}) \spad{:=} \spad{x}}) destructively replaces each element in the segment \axiom{\spad{u}(\spad{i}..\spad{j})} by \spad{x}. The value \spad{x} is returned. Note: \spad{u} is destructively change so that \axiom{\spad{u}.\spad{k} \spad{:=} \spad{x} for \spad{k} in \spad{i}..\spad{j}}; its length remains unchanged.
661721`\spad{setelt(m,{}i,{}j,{}r)} sets the element in the \spad{i}th row and \spad{j}th column of \spad{m} to \spad{r} error check to determine if indices are in proper ranges
661807`\spad{setelt(x,{}rowList,{}colList,{}y)} destructively alters the matrix \spad{x}. If \spad{y} is \spad{m}-by-\spad{n},{} \spad{rowList = [i<1>,{}i<2>,{}...,{}i<m>]} and \spad{colList = [j<1>,{}j<2>,{}...,{}j<n>]},{} then \spad{x(i<k>,{}j<l>)} is set to \spad{y(k,{}l)} for \spad{k = 1,{}...,{}m} and \spad{l = 1,{}...,{}n}.
661894`\spad{setfirst!(u,{}x)} destructively changes the first element of a to \spad{x}.
662036`\spad{setlast!(u,{}x)} destructively changes the last element of \spad{u} to \spad{x}.
662120`\spad{setleaves!(t,{} ls)} sets the leaves of \spad{t} in left-to-right order to the elements of \spad{ls}.
662184`\spad{setleft!(a,{}b)} sets the left child of \axiom{a} to be \spad{b}.
662271`\spad{setnext!(u,{}v)} destructively sets the next node of doubly-linked aggregate \spad{u} to \spad{v},{} returning \spad{v}.
662355`\spad{setprevious!(u,{}v)} destructively sets the previous node of doubly-linked aggregate \spad{u} to \spad{v},{} returning \spad{v}.
662443`\spad{setref(n,{}m)} same as \spad{setelt(n,{}m)}.
662487`\spad{setrest!(u,{}v)} destructively changes the rest of \spad{u} to \spad{v}.
662621`\spad{setright!(a,{}x)} sets the right child of \spad{t} to be \spad{x}.
662786`\spad{setsubMatrix(x,{}i1,{}j1,{}y)} destructively alters the matrix \spad{x}. Here \spad{x(i,{}j)} is set to \spad{y(i-i1+1,{}j-j1+1)} for \spad{i = i1,{}...,{}i1-1+nrows y} and \spad{j = j1,{}...,{}j1-1+ncols y}.
662926`\spad{setvalue!(u,{}x)} sets the value of node \spad{u} to \spad{x}.
663087`\spad{sh(x,{}y)} returns the shuffle-product of \spad{x} by \spad{y}. This multiplication is associative and commutative.
663239`\spad{shallowCopy(x)} \undocumented
663356`\spad{shanksDiscLogAlgorithm(b,{}a,{}p)} computes \spad{s} with \spad{b^s = a} for assuming that \spad{a} and \spad{b} are elements in a 'small' cyclic group of order \spad{p} by Shank\spad{'s} algorithm. Note: this is a subroutine of the function \spadfun{discreteLog}.
676384`\spad{shiftLeft(p,{}n)} returns \spad{p * monomial(1,{}n)}
676549`\spad{shiftRight(p,{}n)} returns \spad{monicDivide(p,{}monomial(1,{}n)).quotient}
676634`\spad{shiftRoots(p,{}c)} returns the polynomial which has for roots \spad{c} added to the roots of \spad{p}.
676707`\spad{shift(k)} returns \spad{matrix [[1,{}k],{}[0,{}1]]} representing the map \spad{x -> x + k}.
676755`\spad{shift(m,{}h)} returns \spad{shift(h) * m} (see \spadfunFrom{shift}{MoebiusTransform}).
676844`\spad{shift(a,{}i)} shift \spad{a} by \spad{i} bits.
676900`\spad{shift(a,{}i)} shift \spad{a} by \spad{i} digits.
677080`\spad{showArrayValues(true)} forces the values of array components to be \indented{1}{displayed rather than just their types.}
677203`\spad{showFortranOutputStack()} returns the Fortran output stack
677347`\spad{showScalarValues(true)} forces the values of scalar components to be \indented{1}{displayed rather than just their types.}
677404`\spad{showTheSymbolTable()} returns the current symbol table.
677459`\spad{showTypeInOutput(bool)} affects the way objects of \spadtype{Any} are displayed. If \spad{bool} is \spad{true} then the type of the original object that was converted to \spadtype{Any} will be printed. If \spad{bool} is \spad{false},{} it will not be printed.
677583`\spad{shrinkable(b)} sets the shrinkable attribute of flexible arrays to \spad{b} and returns the previous value
678296`\spad{sign(r)} \undocumented
678563`\spad{sign(x)} is 1 if \spad{x} is positive,{} \spad{-1} if \spad{x} is negative,{} 0 if \spad{x} equals 0.
678608`\spad{sign(p)} returns the signum of the permutation \spad{p},{} \spad{+1} or \spad{-1}.
679195`\spad{simplifyExp(f)} converts every product \spad{exp(a)*exp(b)} appearing in \spad{f} into \spad{exp(a+b)}.
679261`\spad{simplifyLog(f)} converts every \spad{log(a) - log(b)} appearing in \spad{f} into \spad{log(a/b)},{} every \spad{log(a) + log(b)} into \spad{log(a*b)} and every \spad{n*log(a)} into \spad{log(a^n)}.
679327`\spad{simplifyPower(f,{}n)} \undocumented{}
679407`\spad{simplify(s)} returns a different and presumably simpler representation of \spad{s} with the defining polynomials for the equations forming a groebner basis,{} and the defining polynomial for the inequation reduced with respect to the basis,{} using a heuristic algorithm based on factoring.
679529`\spad{simplify(an)} applies simplifications to \spad{an}
679629`\spad{simplify(f)} performs the following simplifications on \spad{f:}\begin{items} \item 1. rewrites trigs and hyperbolic trigs in terms of \spad{sin} ,{}\spad{cos},{} \spad{sinh},{} \spad{cosh}. \item 2. rewrites \spad{sin^2} and \spad{sinh^2} in terms of \spad{cos} and \spad{cosh},{} \item 3. rewrites \spad{exp(a)*exp(b)} as \spad{exp(a+b)}. \item 4. rewrites \spad{(a^(1/n))^m * (a^(1/s))^t} as a single power of a single radical of \spad
679629`--{a}. \end{items}
680013`\spad{sin2csc(f)} converts every \spad{sin(u)} appearing in \spad{f} into \spad{1/csc(u)}.
680075`\spad{sin?(x)} returns \spad{true} if term is a sin,{} otherwise \spad{false}
680128`\spad{sinIfCan(z)} returns sin(\spad{z}) if possible,{} and "failed" otherwise.
680208`\spad{sin(x)} makes a sin kernel for use in Fourier series
680254`\spad{sin(x)} applies the sine operator to \spad{x}
680575`\spad{sin(z)} returns the sine of Laurent series \spad{z}.
680661`\spad{sin(z)} returns the sine of a Puiseux series \spad{z}.
680756`\spad{sin(x)} represents the Fortran intrinsic function SIN
680836`\spad{sin(x)} returns the sine of \spad{x}.
681360`constuct a ifs representation of a face from an array of points
681598`\spad{singular?(a)} tests whether \spad{x = a} is singular.
681668`\spad{singular?(p)} tests whether \spad{p(x) = 0} is singular.
681739`\spad{singularAtInfinity?()} tests if there is a singularity at infinity.
681818`\spad{sinh2csch(f)} converts every \spad{sinh(u)} appearing in \spad{f} into \spad{1/csch(u)}.
681882`\spad{sinhIfCan(z)} returns sinh(\spad{z}) if possible,{} and "failed" otherwise.
681963`\spad{sinh(x)} applies the hyperbolic sine operator to \spad{x}
682341`\spad{sinh(z)} returns the hyperbolic sine of Laurent series \spad{z}.
682428`\spad{sinh(z)} returns the hyperbolic sine of a Puiseux series \spad{z}.
682524`\spad{sinh(x)} represents the Fortran intrinsic function SINH
682605`\spad{sinh(x)} returns the hyperbolic sine of \spad{x}.
682774`\spad{sipnt(a,{}b)} constructs a 2D point defined by a and \spad{b} of type Integer
682831`\spad{sipnt(a,{}b,{}c)} constructs a 3D point defined by a,{}\spad{b} and \spad{c} of type Integer
682896`\spad{sivec(a,{}b)} constructs a 2D vector defined by a and \spad{b} of type Integer
682953`\spad{sivec(a,{}b,{}c)} constructs a 3D vector defined by a,{}\spad{b} and \spad{c} of type Integer
683318`\spad{size?(u,{}n)} tests if \spad{u} has exactly \spad{n} elements.
683435`\spad{sizeLess?(x,{}y)} tests whether \spad{x} is strictly smaller than \spad{y} with respect to the \spadfunFrom{euclideanSize}{EuclideanDomain}.
684322`\spad{size()} is the base of the random number generator
684372`\spad{size()} returns the number of elements in the set.
684421`\spad{size(x)} returns the number of monomials in \spad{x}.
684531`\spad{size(l)} returns the number of monomials forming \spad{l}.
684657`\spad{size(x)} returns the number of terms in \spad{x}. mapGen(\spad{f},{} \spad{a1}\spad{\^}\spad{e1} ... an\spad{\^}en) returns \spad{f(a1)\^e1 ... f(an)\^en}.
684732`The size of the isolating interval
684933`\spad{slash(f,{}g)} creates a form for the horizontal fraction of \spad{f} over \spad{g}.
685325`\spad{smaller?(x,{} y)} is a strict total ordering on the elements of the set.
685376`constuct a ifs representation of a mesh from a 2 dimentional array of points
685438`\spad{smith(m)} returns the Smith Normal form of the matrix \spad{m}.
686253`\spad{solveLinearPolynomialEquationByFractions([f1,{} ...,{} fn],{} g)} (where the \spad{fi} are relatively prime to each other) returns a list of \spad{ai} such that \spad{g/prod \spad{fi} = sum ai/fi} or returns "failed" if no such exists.
686909`\spad{solveLinearPolynomialEquation([f1,{} ...,{} fn],{} g)} (where the \spad{fi} are relatively prime to each other) returns a list of \spad{ai} such that \spad{g/prod \spad{fi} = sum ai/fi} or returns "failed" if no such list of \spad{ai}\spad{'s} exists.
688556`\spad{solveLinear([v1,{}...,{}vn],{} u)} returns \spad{[c1,{}...,{}cn]} such that \spad{c1*v1 + ... + cn*vn = u},{} "failed" if no such \spad{ci}\spad{'s} exist in \spad{S}.
688776`\spad{solveLinear([v1,{}...,{}vn],{} u)} returns \spad{[c1,{}...,{}cn]} such that \spad{c1*v1 + ... + cn*vn = u},{} "failed" if no such \spad{ci}\spad{'s} exist in \spad{S}.
694044`\spad{someBasis()} returns some \spad{R}-module basis.
694267`\spad{sort!(u)} returns \spad{u} with its elements in ascending order.
694562`\spad{sort!(p,{}u)} returns \spad{u} with its elements ordered by \spad{p}.
694754`\spad{sort(u)} returns an \spad{u} with elements in ascending order. Note: \axiom{sort(\spad{u}) = sort(\spad{<=},{}\spad{u})}.
694985`\spad{sort(p,{}a)} returns a copy of \axiom{a} sorted using total ordering predicate \spad{p}.
695053`\axiom{sort(\spad{v},{}\spad{ps})} returns \axiom{us,{}\spad{vs},{}\spad{ws}} such that \axiom{us} is \axiom{collectUnder(\spad{ps},{}\spad{v})},{} \axiom{\spad{vs}} is \axiom{collect(\spad{ps},{}\spad{v})} and \axiom{\spad{ws}} is \axiom{collectUpper(\spad{ps},{}\spad{v})}.
695329`\spad{sorted?(u)} tests if the elements of \spad{u} are in ascending order.
695625`\spad{sorted?(p,{}a)} tests if \axiom{a} is sorted according to predicate \spad{p}.
695701`\spad{space()} provides the blank character.
696099`\spad{split!(u,{}n)} splits \spad{u} into two aggregates: \axiom{\spad{v} = rest(\spad{u},{}\spad{n})} and \axiom{\spad{w} = first(\spad{u},{}\spad{n})},{} returning \axiom{\spad{v}}. Note: afterwards \axiom{rest(\spad{u},{}\spad{n})} returns \axiom{empty()}.
696277`\spad{splitDenominator([q1,{}...,{}qn])} returns \spad{[[p1,{}...,{}pn],{} d]} such that \spad{\spad{qi} = pi/d} and \spad{d} is a common denominator for the \spad{qi}\spad{'s}.
696365`\spad{splitDenominator([q1,{}...,{}qn])} returns \spad{[[p1,{}...,{}pn],{} d]} such that \spad{\spad{qi} = pi/d} and \spad{d} is a common denominator for the \spad{qi}\spad{'s}.
696446`\spad{splitDenominator(q)} returns \spad{[p,{} d]} such that \spad{q = p/d} and \spad{d} is a common denominator for the elements of \spad{q}.
696547`\spad{splitDenominator(q)} returns \spad{[p,{} d]} such that \spad{q = p/d} and \spad{d} is a common denominator for the coefficients of \spad{q}.
697330`\spad{split(p,{} D)} returns \spad{[n,{}s]} such that \spad{p = n s},{} all the squarefree factors of \spad{n} are normal \spad{w}.\spad{r}.\spad{t}. \spad{D},{} and \spad{s} is special \spad{w}.\spad{r}.\spad{t}. \spad{D}. \spad{D} is the derivation to use.
697639`\spad{split(x,{}b)} splits binary tree \spad{b} into two trees,{} one with elements greater than \spad{x},{} the other with elements less than \spad{x}.
697714`\spad{split(s,{}c)} returns a list of substrings delimited by character \spad{c}.
697775`\spad{split(s,{}cc)} returns a list of substrings delimited by characters in \spad{cc}.
697841`\spad{spnt(a,{}b)} constructs a 2D point defined by a and \spad{b} of type \spad{R}
697905`\spad{spnt(a,{}b,{}c)} constructs a 3D point defined by a,{}\spad{b} and \spad{c} of type \spad{R}
697981`\spad{sqfrFactor(base,{}exponent)} creates a factored object with a single factor whose \spad{base} is asserted to be square-free (flag = "sqfr").
698034`\spad{sqfree(p)} returns the product of square free factors of \spad{p}
698292`\spad{sqrt(x)} represents the Fortran intrinsic function SQRT
698471`\spad{sqrt(x)} returns the square root of \spad{x}. The branch cut lies along the negative real axis,{} continuous with quadrant II.
698515`\axiom{sqrt(\spad{x})} is \axiom{\spad{x} ^ (1/2)}
698859`\spad{square?(m)} returns \spad{true} if \spad{m} is a square matrix (\spadignore{i.e.} if \spad{m} has the same number of rows as columns) and \spad{false} otherwise.
698921`\spad{square?(m)} returns \spad{true} if \spad{m} is a square matrix (\spadignore{i.e.} if \spad{m} has the same number of rows as columns) and \spad{false} otherwise.
699394`\spad{squareFreePart(p)} returns a polynomial which has the same irreducible factors as the univariate polynomial \spad{p},{} but each factor has multiplicity one.
699659`\spad{squareFreePart(p)} returns product of all the irreducible factors of polynomial \spad{p} each taken with multiplicity one.
699744`\spad{squareFreePart(x)} returns a product of prime factors of \spad{x} each taken with multiplicity one.
699808`\spad{squareFreePart(p,{}ts)} returns \spad{lpwt} such that \spad{lpwt.i.val} is a square-free polynomial \spad{w}.\spad{r}.\spad{t}. \spad{lpwt.i.tower},{} this polynomial being associated with \spad{p} modulo \spad{lpwt.i.tower},{} for every \spad{i}. Moreover,{} the list of the \spad{lpwt.i.tower} is a split of Kalkbrener of \spad{ts}. WARNING: This assumes that \spad{p} is a non-constant polynomi
699808`--al such that if \spad{p} is added to \spad{ts},{} then the resulting set is a regular triangular set.
700349`\spad{squareFreePrim(p)} compute the square free decomposition of a primitive multivariate polynomial \spad{p}.
700496`\spad{squareFree(p)} computes the square free decomposition of a multivariate polynomial \spad{p}.
700571`\spad{squareFree(p)} returns the square-free factorization of the polynomial \spad{p}. Each factor has no repeated roots,{} and the factors are pairwise relatively prime.
700649`\spad{squareFree(p)} computes the square-free factorization of the univariate polynomial \spad{p}. Each factor has no repeated roots,{} and the factors are pairwise relatively prime.
701557`\spad{squareFree(p)} returns the square free factorization of the polynomial \spad{p}.
701648`\spad{squareFree(x)} returns the square-free factorization of \spad{x} \spadignore{i.e.} such that the factors are pairwise relatively prime and each has multiple prime factors.
701718`\spad{squareMatrix(m)} converts a matrix of type \spadtype{Matrix} to a matrix of type \spadtype{SquareMatrix}.
701837`\spad{squareTop(m)} returns an \spad{n}-by-\spad{n} matrix consisting of the first \spad{n} rows of the \spad{m}-by-\spad{n} matrix \spad{m}. Error: if \spad{m < n}.
702231`\spad{stack([x,{}y,{}...,{}z])} creates a stack with first (top) element \spad{x},{} second element \spad{y},{}...,{}and last element \spad{z}.
702502`\axiom{startStats!(\spad{x})} initializes the statisitics process and sets the comments to display when statistics are printed
703170`\spad{statistics()} prints out a summary of the outcome of the testcases so far. Use \axiom{clear completely} to reset the statistics.
703350`\spad{stirling1(n,{}m)} returns the Stirling number of the first kind denoted \spad{S[n,{}m]}.
703415`\spad{stirling2(n,{}m)} returns the Stirling number of the second kind denoted \spad{SS[n,{}m]}.
703639`\axiom{stopTableGcd!()} is an internal subroutine,{} exported only for developement.
703707`\axiom{stopTableGcd!()} is an internal subroutine,{} exported only for developement.
703785`\axiom{stopTableGcd!()} is an internal subroutine,{} exported only for developement.
703950`\axiom{stopTableInvSet!()} is an internal subroutine,{} exported only for developement.
704121`\spad{stop()} creates a representation of a STOP statement.
705837`construct transform in general form as a mapping from \spad{PT} to \spad{PT}
706129`\spad{string?(s)} is \spad{true} if \spad{s} is an atom and belong to \spad{Str}.
706207`\spad{string(f)} creates \spad{f} with string quotes.
706248`\spad{string(i)} returns the decimal representation of \spad{i} in a string
706298`\spad{string(s)} returns \spad{s} as an element of \spad{Str}. Error: if \spad{s} is not an atom that also belongs to \spad{Str}.
706371`\spad{string(s)} converts the symbol \spad{s} to a string. Error: if the symbol is subscripted.
706412`\spad{stripCommentsAndBlanks(s)} treats \spad{s} as a piece of FriCAS input,{} and removes comments,{} and leading and trailing blanks.
706624`\axiom{stronglyReduce(\spad{p},{}\spad{ts})} returns a polynomial \axiom{\spad{r}} such that \axiom{stronglyReduced?(\spad{r},{}\spad{ts})} holds and there exists some product \axiom{\spad{h}} of \axiom{initials(\spad{ts})} such that \axiom{\spad{h*p} - \spad{r}} lies in the ideal generated by \axiom{\spad{ts}}.
706767`\axiom{stronglyReduced?(\spad{ts})} returns \spad{true} iff every element of \axiom{\spad{ts}} is reduced \spad{w}.\spad{r}.\spad{t} to any other element of \axiom{\spad{ts}}.
706918`\axiom{stronglyReduced?(\spad{p},{}\spad{ts})} returns \spad{true} iff \axiom{\spad{p}} is reduced \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{ts}}.
706996`\spad{structuralConstants()} calculates the structural constants \spad{[(gammaijk) for k in 1..rank()]} defined by \spad{\spad{vi} * vj = gammaij1 * v1 + ... + gammaijn * vn},{} where \spad{v1},{}...,{}\spad{vn} is the fixed \spad{R}-module basis.
707086`\spad{structuralConstants()} calculates the structural constants \spad{[(gammaijk) for k in 1..rank()]} defined by \spad{\spad{vi} * vj = gammaij1 * v1 + ... + gammaijn * vn},{} where \spad{v1},{}...,{}\spad{vn} is the fixed \spad{R}-module basis.
707461`\spad{structuralConstants([v1,{}v2,{}...,{}vm])} calculates the structural constants \spad{[(gammaijk) for k in 1..m]} defined by \spad{\spad{vi} * vj = gammaij1 * v1 + ... + gammaijm * vm},{} where \spad{[v1,{}...,{}vm]} is an \spad{R}-module basis of a subalgebra.
707836`\spad{struveH(v,{} z)} is the Struve \spad{H} function
707898`\spad{struveH(v,{} z)} is the Struve \spad{H} function.
707956`\spad{struveL(v,{} z)} is the Struve \spad{L} function defined by the formula \spad{struveL(v,{} z) = -\%i^exp(-v*\%pi*\%i/2)*struveH(v,{} \%i*z)}
708018`\spad{struveL(v,{} z)} is the Struve \spad{L} function defined by the formula \spad{struveL(v,{} z) = -\%i^exp(-v*\%pi*\%i/2)*struveH(v,{} \%i*z)}.
708243`\axiom{sturmSequence(\spad{p}) = sylvesterSequence(\spad{p},{}\spad{p'})}
708483`\spad{style(s)} specifies the drawing style in which the graph will be plotted by the indicated string \spad{s}. This option is expressed in the form \spad{style == s}.
708527`\spad{style(l,{}s)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{style}. If the option does not exist the value,{} \spad{s} is returned.
708854`\spad{subHeight(f)} returns the height of form \spad{f} below the base line.
708990`\spad{subMatrix(x,{}i1,{}i2,{}j1,{}j2)} extracts the submatrix \spad{[x(i,{}j)]} where the index \spad{i} ranges from \spad{i1} to \spad{i2} and the index \spad{j} ranges from \spad{j1} to \spad{j2}.
709589`\axiom{subQuasiComponent?(\spad{ts},{}us)} returns \spad{true} iff \axiomOpFrom{internalSubQuasiComponent?}{QuasiComponentPackage} returs \spad{true}.
709673`\axiom{subQuasiComponent?(\spad{ts},{}us)} returns \spad{true} iff \axiomOpFrom{internalSubQuasiComponent?(\spad{ts},{}us)}{QuasiComponentPackage} returs \spad{true}.
709767`\axiom{subResultantChain(a,{}\spad{b})},{} where \axiom{a} and \axiom{\spad{b}} are not contant polynomials with the same main variable,{} returns the subresultant chain of \axiom{a} and \axiom{\spad{b}}.
709873`\axiom{subResultantGcdEuclidean(\spad{P},{}\spad{Q})} carries out the equality \axiom{coef1*P + coef2*Q = \spad{+/-} S_i(\spad{P},{}\spad{Q})} where the degree (not the indice) of the subresultant \axiom{S_i(\spad{P},{}\spad{Q})} is the smaller as possible.
709996`\axiom{subResultantGcd(\spad{P},{} \spad{Q})} returns the \spad{gcd} of two primitive polynomials \axiom{\spad{P}} and \axiom{\spad{Q}}.
710076`\axiom{subResultantGcd(a,{}\spad{b})} computes a \spad{gcd} of \axiom{a} and \axiom{\spad{b}} where \axiom{a} and \axiom{\spad{b}} are assumed to have the same main variable \axiom{\spad{v}} and are viewed as univariate polynomials in \axiom{\spad{v}} with coefficients in the fraction field of the polynomial ring generated by their other variables over \axiom{\spad{R}}.
710174`\spad{subResultantGcd(p,{}q)} computes the \spad{gcd} of the polynomials \spad{p} and \spad{q} using the SubResultant \spad{GCD} algorithm.
710269`\axiom{subResultantsChain(a,{}\spad{b})} returns the list of the non-zero sub-resultants of \axiom{a} and \axiom{\spad{b}} sorted by increasing degree.
710467`\axiom{subTriSet?(\spad{ts},{}us)} returns \spad{true} iff \axiom{\spad{ts}} is a sub-set of \axiom{us}.
710543`\axiom{subTriSet?(\spad{ts},{}us)} returns \spad{true} iff \axiom{\spad{ts}} is a sub-set of \axiom{us}.
710629`\spad{sub(f,{}n)} creates a form for \spad{f} subscripted by \spad{n}.
710670`for each face calculate a new point in the centre of the face,{} then build a new set of faces which include the new point.
710716`\spad{submod(a,{}b,{}p)},{} \spad{0<=a,{}b<p>1},{} means \spad{a-b mod p}.
710914`\spad{subresultantVector(p,{} q)} returns \spad{[p0,{}...,{}pn]} where \spad{pi} is the \spad{i}-th subresultant of \spad{p} and \spad{q}. In particular,{} \spad{p0 = resultant(p,{} q)}.
711110`\spad{subset?(u,{}v)} tests if \spad{u} is a subset of \spad{v}. Note: equivalent to \axiom{reduce(and,{}{member?(\spad{x},{}\spad{v}) for \spad{x} in \spad{u}},{}\spad{true},{}\spad{false})}.
711165`\spad{subspace?(b1,{}b2)} tests if the first space is include in the second.
711226`\spad{subspace()} \undocumented
717842`\spad{subst(f,{} k = g)} replaces the kernel \spad{k} by \spad{g} formally in \spad{f}.
717900`\spad{subst(f,{} [k1 = g1,{}...,{}kn = gn])} replaces the kernels \spad{k1},{}...,{}\spad{kn} by \spad{g1},{}...,{}\spad{gn} formally in \spad{f}.
717964`\spad{subst(eq1,{}eq2)} substitutes \spad{eq2} into both sides of \spad{eq1} the \spad{lhs} of \spad{eq2} should be a kernel
718228`\spad{substitute(x,{}y,{}d)} replace \spad{x}\spad{'s} with \spad{y}\spad{'s} in dictionary \spad{d}.
718289`\spad{substring?(s,{}t,{}i)} tests if \spad{s} is a substring of \spad{t} beginning at index \spad{i}. Note: \axiom{substring?(\spad{s},{}\spad{t},{}0) = prefix?(\spad{s},{}\spad{t})}.
718420`\spad{subtractIfCan(x,{} y)} returns an element \spad{z} such that \spad{z+y=x} or "failed" if no such element exists.
718502`\spad{suchThat(p,{} f)} makes a copy of \spad{p} and adds the predicate \spad{f} to the copy,{} which is returned.
718586`\spad{suchThat(p,{} [f1,{}...,{}fn])} makes a copy of \spad{p} and adds the predicate \spad{f1} and ... and \spad{fn} to the copy,{} which is returned.
718676`\spad{suchThat(x,{} foo)} attaches the predicate foo to \spad{x}; error if \spad{x} is not a symbol.
718756`\spad{suchThat(x,{} [f1,{} f2,{} ...,{} fn])} attaches the predicate \spad{f1} and \spad{f2} and ... and \spad{fn} to \spad{x}. Error: if \spad{x} is not a symbol.
719010`\spad{suchThat(p,{} [a1,{}...,{}an],{} f)} returns a copy of \spad{p} with the top-level predicate set to \spad{f(a1,{}...,{}an)}.
719113`\spad{suchThat(r,{} [a1,{}...,{}an],{} f)} returns the rewrite rule \spad{r} with the predicate \spad{f(a1,{}...,{}an)} attached to it.
719199`\spad{suffix?(s,{}t)} tests if the string \spad{s} is the final substring of \spad{t}. Note: \axiom{suffix?(\spad{s},{}\spad{t}) \spad{==} reduce(and,{}[\spad{s}.\spad{i} = \spad{t}.(\spad{n} - \spad{m} + \spad{i}) for \spad{i} in 0..maxIndex \spad{s}])} where \spad{m} and \spad{n} denote the maxIndex of \spad{s} and \spad{t} respectively.
719254`\spad{sumBasis LLv} computes a basis of the sum of list of subspace. This function is slightly more efficient than \spad{reduce(sumBasis,{}LLv)}
719325`\spad{sumBasis Lb} computes a basis of the sum of list of subspaces. This function is slightly more efficient than \spad{reduce(sumBasis,{}Lb)}
719383`\spad{sumBasis(Lv1,{}Lv2)} computes a basis of the sum of the two subspaces spanned by the vectors of \spad{Lv1} and \spad{Lv2}
719464`\spad{sumBasis(B1,{}B2)} computes a basis of the sum of the two subspaces.
719762`\spad{sum(expr)} creates the form prefixing \spad{expr} by a capital sigma.
719868`\spad{sum(expr,{}lowerlimit)} creates the form prefixing \spad{expr} by a capital sigma with a \spad{lowerlimit}.
719909`\spad{sum(f(n),{} n = a..b)} returns \spad{f}(a) + \spad{f}(\spad{a+1}) + ... + \spad{f}(\spad{b}).
720556`\spad{sum(expr,{}lowerlimit,{}upperlimit)} creates the form prefixing \spad{expr} by a capital sigma with both a \spad{lowerlimit} and \spad{upperlimit}.
720600`\spad{summation(f(n),{} n = a..b)} returns \spad{f}(a) + ... + \spad{f}(\spad{b}) as a formal sum.
720736`\spad{summation(f(n),{} n = a..b)} returns \spad{f}(a) + ... + \spad{f}(\spad{b}) as a formal sum.
720813`\spad{summation(f(n),{} n)} returns the formal sum \spad{S}(\spad{n}) which verifies \spad{S}(\spad{n+1}) - \spad{S}(\spad{n}) = \spad{f}(\spad{n}).
720878`\axiom{supDimElseRittWu(\spad{ts},{}us)} returns \spad{true} iff \axiom{\spad{ts}} has less elements than \axiom{us} otherwise if \axiom{\spad{ts}} has higher rank than \axiom{us} \spad{w}.\spad{r}.\spad{t}. Riit and Wu ordering.
720961`\axiom{supDimElseRittWu(\spad{ts},{}us)} returns \spad{true} iff \axiom{\spad{ts}} has less elements than \axiom{us} otherwise if \axiom{\spad{ts}} has higher rank than \axiom{us} \spad{w}.\spad{r}.\spad{t}. Riit and Wu ordering.
721127`\axiom{supRittWu?(a,{}\spad{b})} returns \spad{true} if \axiom{a} is greater than \axiom{\spad{b}} \spad{w}.\spad{r}.\spad{t}. the Ritt and Wu Wen Tsun ordering using the refinement of Lazard.
721204`\spad{sup(u)} returns the supremum of \axiom{\spad{u}}.
721250`\spad{sup(x,{}y)} returns the least element from which both \spad{x} and \spad{y} can be subtracted.
721304`\spad{superHeight(f)} returns the height of form \spad{f} above the base line.
721355`\spad{super(f,{}n)} creates a form for \spad{f} superscripted by \spad{n}.
721452`\spad{supersub(a,{}[sub1,{}super1,{}sub2,{}super2,{}...])} creates a form with each subscript aligned under each superscript.
721504`\spad{support(x)} returns the list of basis elements with nonzero coefficients.
721564`\spad{surface(c1,{}c2,{}c3)} creates a surface from 3 parametric component functions \spad{c1},{} \spad{c2},{} and \spad{c3}.
721683`\spad{svec(a,{}b)} constructs a 2D vector defined by a and \spad{b} of type \spad{R}
721747`\spad{svec(a,{}b,{}c)} constructs a 3D vector defined by a,{}\spad{b} and \spad{c} of type \spad{R}
721893`\spad{swap!(u,{}i,{}j)} interchanges elements \spad{i} and \spad{j} of aggregate \spad{u}. No meaningful value is returned.
722064`\spad{swapColumns!(m,{}i,{}j)} interchanges the \spad{i}th and \spad{j}th columns of \spad{m}. This destructively alters the matrix.
722213`\spad{swapRows!(m,{}i,{}j)} interchanges the \spad{i}th and \spad{j}th rows of \spad{m}. This destructively alters the matrix.
722288`\spad{swap(p(x,{}y))} returns \spad{p}(\spad{y},{}\spad{x}).
722367`\spad{swap(eq)} interchanges left and right hand side of equation \spad{eq}.
722407`\spad{sylvesterMatrix(p,{}q)} returns the Sylvester matrix for the two polynomials \spad{p} and \spad{q}.
722477`\axiom{sylvesterSequence(\spad{p},{}\spad{q})} is the negated remainder sequence of \spad{p} and \spad{q} divided by the last computed term
722591`\spad{symFunc([r1,{}...,{}rn])} returns the vector of the elementary symmetric functions in the \spad{\spad{ri}'s}: \spad{[r1 + ... + rn,{} r1 r2 + ... + r(n-1) rn,{} ...,{} r1 r2 ... rn]}.
722656`\spad{symFunc(r,{} n)} returns the vector of the elementary symmetric functions in \spad{[r,{}r,{}...,{}r]} \spad{n} times.
722773`\spad{symbol?(s)} is \spad{true} if \spad{s} is an atom and belong to \spad{Sym}.
722851`\spad{symbolIfCan(k)} returns \spad{k} viewed as a symbol if \spad{k} is a symbol,{} and "failed" otherwise.
722916`\spad{symbolTableOf(f,{}tab)} returns the symbol table of \spad{f}
723066`\spad{symbol(s)} returns \spad{s} as an element of \spad{Sym}. Error: if \spad{s} is not an atom that also belongs to \spad{Sym}.
723278`\spad{symmetric?(m)} returns \spad{true} if the matrix \spad{m} is square and symmetric (\spadignore{i.e.} \spad{m[i,{}j] = m[j,{}i]} for all \spad{i} and \spad{j}) and \spad{false} otherwise.
723343`\spad{symmetric?(m)} returns \spad{true} if the matrix \spad{m} is square and symmetric (\spadignore{i.e.} \spad{m[i,{}j] = m[j,{}i]} for all \spad{i} and \spad{j}) and \spad{false} otherwise.
723543`\spad{symmetricDifference(u,{}v)} returns the set aggregate of elements \spad{x} which are members of set aggregate \spad{u} or set aggregate \spad{v} but not both. If \spad{u} and \spad{v} have no elements in common,{} \axiom{symmetricDifference(\spad{u},{}\spad{v})} returns a copy of \spad{u}. Note: \axiom{symmetricDifference(\spad{u},{}\spad{v}) = union(difference(\spad{u},{}\spad{v}),{}difference(\spad{v},{}\spad{u}))}
723791`\spad{symmetricPower(a,{}n)} computes an operator \spad{c} of minimal order such that the nullspace of \spad{c} is generated by all the products of \spad{n} solutions of \spad{a}.
724007`\spad{symmetricProduct(a,{}b)} computes an operator \spad{c} of minimal order such that the nullspace of \spad{c} is generated by all the products of a solution of \spad{a} by a solution of \spad{b}.
724108`\spad{symmetricProduct(a,{}b,{}D)} computes an operator \spad{c} of minimal order such that the nullspace of \spad{c} is generated by all the products of a solution of \spad{a} by a solution of \spad{b}. \spad{D} is the derivation to use.
724260`\spad{symmetricRemainder(a,{}b)} (where \spad{b > 1}) yields \spad{r} where \spad{ -b/2 <= r < b/2 }.
724406`\spad{symmetricSquare(a)} computes \spad{symmetricProduct(a,{}a)} using a more efficient method.
724697`\spad{systemCommand(cmd)} takes the string \spadvar{\spad{cmd}} and passes it to the runtime environment for execution as a system command. Although various things may be printed,{} no usable value is returned.
725109`\spad{tableForDiscreteLogarithm(a,{}n)} returns a table of the discrete logarithms of \spad{a^0} up to \spad{a^(n-1)} which,{} called with key \spad{lookup(a^i)} returns \spad{i} for \spad{i} in \spad{0..n-1}. Error: if not called for prime divisors of order of \indented{7}{multiplicative group.}
725374`\spad{table()}\$\spad{T} creates an empty table of type \spad{T}.
725510`\spad{table([x,{}y,{}...,{}z])} creates a table consisting of entries \axiom{\spad{x},{}\spad{y},{}...,{}\spad{z}}.
725801`\spad{tail(l)} returns the doubly-linked aggregate \spad{l} starting at its second element. Error: if \spad{l} is empty.
725854`\axiom{tail(\spad{p})} returns its reductum,{} where \axiom{\spad{p}} is viewed as a univariate polynomial in its main variable.
725917`\spad{tail(u)} returns the last node of \spad{u}. Note: if \spad{u} is \axiom{shallowlyMutable},{} \axiom{setrest(tail(\spad{u}),{}\spad{v}) = concat(\spad{u},{}\spad{v})}.
725972`\spad{tan2cot(f)} converts every \spad{tan(u)} appearing in \spad{f} into \spad{1/cot(u)}.
726034`\spad{tan2trig(f)} converts every \spad{tan(u)} appearing in \spad{f} into \spad{sin(u)/cos(u)}.
726184`\spad{tanIfCan(z)} returns tan(\spad{z}) if possible,{} and "failed" otherwise.
726394`\spad{tanSum([a1,{}...,{}an])} returns \spad{f(a1,{}...,{}an)} such that if \spad{\spad{ai} = tan(\spad{ui})} then \spad{f(a1,{}...,{}an) = tan(u1 + ... + un)}.
726449`\spad{tan(x)} applies the tangent operator to \spad{x}
726824`\spad{tan(z)} returns the tangent of Laurent series \spad{z}.
726910`\spad{tan(z)} returns the tangent of a Puiseux series \spad{z}.
727005`\spad{tan(x)} represents the Fortran intrinsic function TAN
727085`\spad{tan(x)} returns the tangent of \spad{x}.
727142`\spad{tanh2coth(f)} converts every \spad{tanh(u)} appearing in \spad{f} into \spad{1/coth(u)}.
727206`\spad{tanh2trigh(f)} converts every \spad{tanh(u)} appearing in \spad{f} into \spad{sinh(u)/cosh(u)}.
727271`\spad{tanhIfCan(z)} returns tanh(\spad{z}) if possible,{} and "failed" otherwise.
727352`\spad{tanh(x)} applies the hyperbolic tangent operator to \spad{x}
727730`\spad{tanh(z)} returns the hyperbolic tangent of Laurent series \spad{z}.
727817`\spad{tanh(z)} returns the hyperbolic tangent of a Puiseux series \spad{z}.
727913`\spad{tanh(x)} represents the Fortran intrinsic function TANH
727994`\spad{tanh(x)} returns the hyperbolic tangent of \spad{x}.
728219`\spad{taylorIfCan(f(x))} converts the Laurent series \spad{f(x)} to a Taylor series,{} if possible. If this is not possible,{} "failed" is returned.
728324`\spad{taylorQuoByVar(a0 + a1 x + a2 x^2 + ...)} returns \spad{a1 + a2 x + a3 x^2 + ...}
728401`\spad{taylorRep(f(x))} returns \spad{g(x)},{} where \spad{f = x^n * g(x)} is represented by \spad{[n,{}g(x)]}.
728488`\spad{taylor(f)} returns a Taylor expansion of the expression \spad{f}. Note: \spad{f} should have only one variable; the series will be expanded in powers of that variable.
728629`\spad{taylor(f(x))} converts the Laurent series \spad{f}(\spad{x}) to a Taylor series,{} if possible. Error: if this is not possible.
729385`\spad{tensorMap([s1,{} s2,{} ...],{} f)} returns the stream consisting of all elements of \spad{f}(\spad{s1}) followed by all elements of \spad{f}(\spad{s2}) and so on.
729909`\spad{tensor([x1,{}x2,{}...,{}xn])} constructs the tensor product of \spad{x1,{}x2,{}...,{}xn}.
730026`\spad{tensor(a,{}b)} creates a form for a tensor \spad{b}
730070`\spad{tensor(x,{}y)} constructs the tensor product of the elements \spad{x} and \spad{y}.
730204`\spad{terms(e1 a1 + ... + en an)} returns \spad{[[a1,{} e1],{}...,{}[an,{} en]]}.
730381`\spad{testAbsolutePrecision(eps)} returns the current absolute precision used for floating point comparison,{} and then sets it to \spad{eps}. The default value is 1.0e-15.
730462`testComplexEquals(inp \spad{ex1},{} \spad{ex2}) states that \spad{ex1} and \spad{ex2} should be approximately equal as complex numbers,{} taking into acount \spadfun{testAbsolutePrecision} and \spadfun{testRelativePrecision}.
730572`\spad{testComplexEquals(ex1,{} ex2)} states that \spad{ex1} and \spad{ex2} should be approximately equal as complex numbers,{} taking into acount \spadfun{testAbsolutePrecision} and \spadfun{testRelativePrecision}.
730837`\spad{testEqualsAuxCmp should} be local but is conditional
730898`testEquals(inp,{} \spad{ex1},{} \spad{ex2}) states that \spad{ex1} and \spad{ex2} should be equal. To sidestep the possibility that the equality function of the domain \spad{R} performs some simplifications,{} we convert \spad{ex1} and \spad{ex2} to \axiom{InputForm},{} if possible.
730960`\spad{testEquals(ex1,{} ex2)} states that \spad{ex1} and \spad{ex2} should be equal. To sidestep the possibility that the equality function of the domain \spad{R} performs some simplifications,{} we convert \spad{ex1} and \spad{ex2} to \axiom{InputForm},{} if possible
731016`test output
731060`testLibraryError(inp,{} ex) states that ex should throw an error. Such a test will never count as a fatal error.
731126`\spad{testLibraryError ex} states that ex should throw an error. Such a test will never count as a fatal error.
731242`testNotEquals(inp,{} \spad{ex1},{} \spad{ex2}) states that \spad{ex1} and \spad{ex2} should be different.
731307`\spad{testNotEquals(ex1,{} ex2)} states that \spad{ex1} and \spad{ex2} should be different.
731366`testRealEquals(inp,{} \spad{ex1},{} \spad{ex2}) states that \spad{ex1} and \spad{ex2} should be approximately equal as real numbers,{} taking into acount \spadfun{testAbsolutePrecision} and \spadfun{testRelativePrecision}.
731455`\spad{testRealEquals(ex1,{} ex2)} states that \spad{ex1} and \spad{ex2} should be approximately equal as real numbers,{} taking into acount \spadfun{testAbsolutePrecision} and \spadfun{testRelativePrecision}.
731515`\spad{testRelativePrecision(eps)} returns the current absolute precision used for floating point comparison,{} and then sets it to \spad{eps}. The default value is 1.0e-15.
731596`\spad{testTrueAux(inp,{} ex)} states that ex should be \spad{true}.
731654`\spad{testTrue ex} states that ex should be \spad{true}.
731701`\spad{test(b)} returns \spad{b} and is provided for compatibility with the new compiler.
731742`\spad{testcaseNoClear s},{} starts a new testcase with \spad{s} as title,{} without calling \axiom{clear all}. A testcase is composed of several tests.
731801`\spad{testcase s} starts a new testcase with \spad{s} as title. It also calls \axiom{clear all}. A testcase is composed of several tests.
731853`\spad{testsuiteNoClear s} starts a new testsuite with \spad{s} as title,{} without calling \axiom{clear all}. A testsuite is composed of several testcases.
731913`\spad{testsuite s} starts a new testsuite with \spad{s} as title. It also calls \axiom{clear all}. A testsuite is composed of several testcases.
732110`\spad{third(u)} returns the third element of \spad{u}. Note: \axiom{third(\spad{u}) = first(rest(rest(\spad{u})))}.
732348`\spad{title(s)} specifies a title for a plot by the indicated string \spad{s}. This option is expressed in the form \spad{title == s}.
732392`\spad{title(l,{}s)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{title}. If the option does not exist the value,{} \spad{s} is returned.
732940`\spad{toScale(b)} specifies whether or not a plot is to be drawn to scale; if \spad{b} is \spad{true} it is drawn to scale,{} if \spad{b} is \spad{false} it is not. This option is expressed in the form \spad{toScale == b}.
732987`\spad{toScale(l,{}b)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{toScale}. If the option does not exist the value,{} \spad{b} is returned.
733066`displays table of unary function such as inverse,{} reverse,{} complement,{} or dual basis could have returned type 'List List \%' but matrix displays better
733137`displays multiplication table for binary operation which is represented as a function with two parameters. row number represents first operand in binary order column number represents second operand in binary order could have returned type 'List List \%' but matrix displays better
733481`\spad{top!(d)} returns the element at the top (front) of the dequeue.
733528`\spad{topFortranOutputStack()} returns the top element of the Fortran output stack
733674`\spad{top(s)} returns the top element \spad{x} from \spad{s}; \spad{s} remains unchanged. Note: Use \axiom{pop!(\spad{s})} to obtain \spad{x} and remove it from \spad{s}.
734962`\spad{totalDegree(p)} returns the largest sum over all monomials of all exponents of a monomial.
735244`\spad{totalDegree(p,{} lv)} returns the maximum sum (over all monomials of polynomial \spad{p}) of the variables in the list \spad{lv}.
736056`\spad{traceMatrix()} is the \spad{n}-by-\spad{n} matrix ( \spad{Tr(\spad{vi} * vj)} ),{} where \spad{v1},{} ...,{} \spad{vn} are the elements of the fixed basis.
736119`\spad{traceMatrix()} is the \spad{n}-by-\spad{n} matrix ( \spad{Tr(\spad{vi} * vj)} ),{} where \spad{v1},{} ...,{} \spad{vn} are the elements of the fixed basis.
736317`\spad{traceMatrix([v1,{}..,{}vn])} is the \spad{n}-by-\spad{n} matrix ( \spad{Tr}(\spad{vi} * \spad{vj}) )
736637`\spad{trace(a)} computes the trace of \spad{a} with respect to the field considered as an algebra with 1 over the ground field \spad{F}.
736698`\spad{trace(a)} returns the trace of the regular representation of \spad{a} with respect to any basis.
736750`\spad{trace(m)} returns the trace of the matrix \spad{m}. this is the sum of the elements on the diagonal of the matrix \spad{m}.
736980`\spad{trace(a,{}d)} computes the trace of \spad{a} with respect to the field of extension degree \spad{d} over the ground field of size \spad{q}. Error: if \spad{d} does not divide the extension degree of \spad{a}. Note: \spad{trace(a,{}d) = reduce(+,{}[a^(q^(d*i)) for i in 0..n/d])}.
737071`\spad{trailingCoefficient }\undocumented
737225`\spad{transcendenceDegree()} returns the transcendence degree of the field extension,{} 0 if the extension is algebraic.
737429`\spad{transcendent?(a)} tests whether an element \spad{a} is transcendent with respect to the ground field \spad{F}.
738269`\spad{translate(X,{}Y,{}Z)} returns a dhmatrix for translation by \spad{X},{} \spad{Y},{} and \spad{Z}
738592`\spad{transpose(m)} returns the transpose of the matrix \spad{m}.
738646`\spad{transpose(r)} converts the row \spad{r} to a row matrix.
738706`\spad{transpose(m)} returns the transpose of the matrix \spad{m}.
738765`\spad{transpose(t)} exchanges the first and last indices of \spad{t}. For example,{} if \spad{r = transpose(t)} for a rank 4 tensor \spad{t},{} then \spad{r} is the rank 4 tensor given by \indented{4}{\spad{r(i,{}j,{}k,{}l) = t(l,{}j,{}k,{}i)}.}
738827`\spad{transpose(t,{}i,{}j)} exchanges the \spad{i}\spad{-}th and \spad{j}\spad{-}th indices of \spad{t}. For example,{} if \spad{r = transpose(t,{}2,{}3)} for a rank 4 tensor \spad{t},{} then \spad{r} is the rank 4 tensor given by \indented{4}{\spad{r(i,{}j,{}k,{}l) = t(i,{}k,{}j,{}l)}.}
739300`\spad{tree(ls)} creates a tree from a list of elements of \spad{s}.
739341`\spad{tree(nd)} creates a tree with value \spad{nd},{} and no children
739376`\spad{tree(nd,{}ls)} creates a tree with value \spad{nd},{} and children \spad{ls}.
739838`\axiom{triangular?(\spad{ps})} returns \spad{true} iff \axiom{\spad{ps}} is a triangular set,{} \spadignore{i.e.} two distinct polynomials have distinct main variables and no constant lies in \axiom{\spad{ps}}.
740388`\spad{trigs(f)} rewrites all the complex logs and exponentials appearing in \spad{f} in terms of trigonometric functions.
740454`\spad{trigs(f)} rewrites all the complex logs and exponentials appearing in \spad{f} in terms of trigonometric functions.
740620`\spad{trim(s,{}c)} returns \spad{s} with all characters \spad{c} deleted from right and left ends. For example,{} \axiom{trim(" abc ",{} char " ")} returns \axiom{"abc"}.
740674`\spad{trim(s,{}cc)} returns \spad{s} with all characters in \spad{cc} deleted from right and left ends. For example,{} \axiom{trim("(abc)",{} charClass "()")} returns \axiom{"abc"}.
740808`\axiom{trivialIdeal?(\spad{ps})} returns \spad{true} iff \axiom{\spad{ps}} does not contain non-zero elements.
740886`\spad{trueEqual(x,{}y)} tries to determine if the two numbers are equal
740948`\spad{true} is a logical constant.
740982`\axiom{trunc(\spad{p},{}\spad{n})} returns the polynomial \axiom{\spad{p}} truncated at order \axiom{\spad{n}}.
741055`\spad{trunc(p,{}n)} returns the polynomial \spad{p} truncated at order \spad{n}.
741171`\spad{truncate x} returns the integer between \spad{x} and 0 closest to \spad{x}.
741220`\spad{truncate(f,{}k)} returns a (finite) power series consisting of the sum of all terms of \spad{f} of degree \spad{<= k}.
741300`\spad{truncate(f,{}k1,{}k2)} returns a (finite) power series consisting of the sum of all terms of \spad{f} of degree \spad{d} with \spad{k1 <= d <= k2}.
741610`\spad{tryFunctionalDecomposition?()} returns \spad{true} if factorizers try functional decomposition of polynomials before factoring them.
741690`\spad{tryFunctionalDecomposition(b)} chooses whether factorizers have to look for functional decomposition of polynomials (\spad{true}) or not (\spad{false}). Returns the previous value.
742876`\spad{tubePoints(n)} specifies the number of points,{} \spad{n},{} defining the circle which creates the tube around a 3D curve,{} the default is 6. This option is expressed in the form \spad{tubePoints == n}.
742934`\spad{tubePoints(l,{}n)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{tubePoints}. If the option does not exist the value,{} \spad{n} is returned.
743163`\spad{tubeRadius(r)} specifies a radius,{} \spad{r},{} for a tube plot around a 3D curve; is expressed in the form \spad{tubeRadius == 4}.
743211`\spad{tubeRadius(l,{}n)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{tubeRadius}. If the option does not exist the value,{} \spad{n} is returned.
743452`\spad{twist(f)} is the function \spad{g} \indented{1}{such that \spad{g (a,{}b)= f(b,{}a)}.}
743937`\spad{unary?(op)} tests if \spad{op} is unary.
744303`\spad{uniform01()} \undocumented
744362`\spad{uniform(s)} \undocumented
744443`\spad{uniform(s)} \undocumented
744504`\spad{uniform(f,{}g)} \undocumented
744940`\spad{union(a,{} b)} makes the set-union of two match results.
744996`\spad{union(x,{}u)} returns the set aggregate \spad{u} with the element \spad{x} added. If \spad{u} already contains \spad{x},{} \axiom{union(\spad{x},{}\spad{u})} returns a copy of \spad{u}.
745043`\spad{union(u,{}x)} returns the set aggregate \spad{u} with the element \spad{x} added. If \spad{u} already contains \spad{x},{} \axiom{union(\spad{u},{}\spad{x})} returns a copy of \spad{u}.
745090`\spad{union(u,{}v)} returns the set aggregate of elements which are members of either set aggregate \spad{u} or \spad{v}.
745185`\spad{unit?(x)} tests whether \spad{x} is a unit,{} \spadignore{i.e.} is invertible.
745323`\spad{unitCanonical(x)} returns \spad{unitNormal(x).canonical}.
745618`\spad{unitNormal(x)} tries to choose a canonical element from the associate class of \spad{x}. The attribute canonicalUnitNormal,{} if asserted,{} means that the "canonical" element is the same across all associates of \spad{x} if \spad{unitNormal(x) = [u,{}c,{}a]} then \spad{u*c = x},{} \spad{a*u = 1}.
745706`\spad{unitNormalize(u)} normalizes the unit part of the factorization. For example,{} when working with factored integers,{} this operation will ensure that the bases are all positive integers.
745755`\spad{unitVector(x)} \undocumented
745904`scale until length is 1
745953`\spad{unitVector(n)} produces a vector with 1 in position \spad{n} and zero elsewhere.
746110`\spad{unit()} returns a unit of the algebra (necessarily unique),{} or \spad{"failed"} if there is none.
746251`\spad{unit(u)} extracts the unit part of the factorization.
748454`\spad{univariate(p)} converts the multivariate polynomial \spad{p},{} which should actually involve only one variable,{} into a univariate polynomial in that variable,{} whose coefficients are in the ground ring. Error: if polynomial is genuinely multivariate
748891`\spad{univariate(p,{} x)} converts the polynomial \spad{p} to a one of type \spad{UnivariatePolynomial(x,{}Polynomial(R))},{} ie. as a member of \spad{R[...][x]}.
749137`\spad{univariate(p,{}v)} converts the multivariate polynomial \spad{p} into a univariate polynomial in \spad{v},{} whose coefficients are still multivariate polynomials (in all the other variables).
749542`\spad{univcase should} be local
749667`\spad{universe()}\$\spad{D} returns the universal set for finite set aggregate \spad{D}.
749821`\spad{unmakeSUP(sup)} converts \spad{sup} of type \spadtype{SparseUnivariatePolynomial(R)} to be a member of the given type. Note: converse of makeSUP.
749913`\spad{unparse(f)} returns a string \spad{s} such that the parser would transform \spad{s} to \spad{f}. Error: if \spad{f} is not the parsed form of a string.
750279`\spad{unravel(t)} produces a tensor from a list of components such that \indented{2}{\spad{unravel(ravel(t)) = t}.}
750429`\spad{unvectorise(v)} returns the polynomial which has for coefficients the entries of \spad{v} in the increasing order.
751653`\axiom{updateStatus!(a)} returns a where the status of the vertices are updated to satisfy the "termination condition".
751786`\spad{upperCase!(s)} destructively replaces the alphabetic characters in \spad{s} by upper case characters.
751836`\spad{upperCase?(c)} tests if \spad{c} is an upper case letter,{} \spadignore{i.e.} one of A..\spad{Z}.
751885`\spad{upperCase()} returns the class of all characters for which \spadfunFrom{upperCase?}{Character} is \spad{true}.
751977`\spad{upperCase(s)} returns the string with all characters in upper case.
752026`\spad{upperCase(c)} converts a lower case letter to the corresponding upper case letter. If \spad{c} is not a lower case letter,{} then it is returned unchanged.
752069`\spad{useEisensteinCriterion?()} returns \spad{true} if factorizers check Eisenstein\spad{'s} criterion before factoring.
752145`\spad{useEisensteinCriterion(b)} chooses whether factorizers check Eisenstein\spad{'s} criterion before factoring: \spad{true} for using it,{} \spad{false} else. Returns the previous value.
752227`\spad{useNagFunctions()} indicates whether NAG functions are being used \indented{1}{for mathematical and machine constants.}
752322`\spad{useNagFunctions(v)} sets the flag which controls whether NAG functions \indented{1}{are being used for mathematical and machine constants.\space{2}The previous} \indented{1}{value is returned.}
752424`\spad{useSingleFactorBound?()} returns \spad{true} if algorithm with single factor bound is used for factorization,{} \spad{false} for algorithm with overall bound.
752498`\spad{useSingleFactorBound(b)} chooses the algorithm to be used by the factorizers: \spad{true} for algorithm with single factor bound,{} \spad{false} for algorithm with overall bound. Returns the previous value.
752578`\spad{userOrdered?()} tests if the partial ordering induced by \spadfunFrom{setOrder}{UserDefinedPartialOrdering} is not empty.
752647`\axiom{usingTable?()} returns \spad{true} iff the hash-table is used
752936`\axiom{value(\spad{n})} returns the value of the node \spad{n}.
753023`\spad{value(u)} returns the value of the node \spad{u}.
753073`\spad{value(q)} returns the value (\spadignore{i.e.} right hand side) of \axiom{\spad{q}}.
753120`\spad{values()} returns the number of initial values to display at most.
753188`\spad{values N} sets the number of initial values to display at most.
753421`\spad{var1Steps(n)} indicates the number of subdivisions,{} \spad{n},{} of the first range variable. This option is expressed in the form \spad{var1Steps == n}.
753478`\spad{var1Steps(l,{}n)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{var1Steps}. If the option does not exist the value,{} \spad{n} is returned.
753722`\spad{var2Steps(n)} indicates the number of subdivisions,{} \spad{n},{} of the second range variable. This option is expressed in the form \spad{var2Steps == n}.
753779`\spad{var2Steps(l,{}n)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{var2Steps}. If the option does not exist the value,{} \spad{n} is returned.
753876`\spad{varList(x)} returns the list of variables of \spad{x}.
753933`\axiom{varList(\spad{g})} returns the list of variables of \axiom{\spad{g}}.
754006`\axiom{varList(\spad{x})} returns the list of distinct entries of \axiom{\spad{x}}.
754066`\axiom{varList(\spad{x})} returns the list of distinct entries of \axiom{\spad{x}}.
754121`\spad{varList([l1]*[l2]*...[ln])} returns the list of variables in the word \spad{l1*l2*...*ln}.
754202`\axiom{varList(\spad{x})} returns the list of distinct entries of \axiom{\spad{x}}.
754268`\spad{varList(x)} returns the list of variables which appear in \spad{x}.
754396`\spad{variableName(d)} specifies the variable used in by the algebraic differential equation. This option is expressed in the form \spad{variableName == d}.
754448`\spad{variable()} returns the symbol
754496`\spad{variable(s)} returns a member of the variable set or failed
754582`\spad{variable(v)} returns \spad{s} if \spad{v} is any derivative of the differential indeterminate \spad{s}.
754645`\spad{variable(f)} returns the (unique) power series variable of the power series \spad{f}.
754723`\spad{variable(q)} returns the variable (\spadignore{i.e.} left hand side) of \axiom{\spad{q}}.
754773`\spad{variable(segb)} returns the variable from the left hand side of the \spadtype{SegmentBinding}. For example,{} if \spad{segb} is \spad{v=a..b},{} then \spad{variable(segb)} returns \spad{v}.
754827`\spad{variables(f)} returns the list of variables appearing in the numerator or the denominator of \spad{f}.
755409`\spad{variables(p)} returns the list of matching variables appearing in \spad{p}.
755593`\spad{variables(f)} returns a list of the variables occuring in the power series \spad{f}.
755669`\spad{variables(p)} returns the list of those variables actually appearing in the polynomial \spad{p}.
755743`\axiom{variables(\spad{ps})} returns the decreasingly sorted list of the variables which are variables of some polynomial in \axiom{\spad{ps}}.
756067`\spad{vconcat(u)} vertically concatenates all forms in list \spad{u}.
756115`\spad{vconcat(f,{}g)} vertically concatenates forms \spad{f} and \spad{g}.
756400`\spad{vector(l)} converts the list \spad{l} to a vector.
756479`\spad{vector(l)} converts the list \spad{l} to a vector.
756612`\spad{vectorise(p,{} n)} returns \spad{[a0,{}...,{}a(n-1)]} where \spad{p = a0 + a1*x + ... + a(n-1)*x^(n-1)} + higher order terms. The degree of polynomial \spad{p} can be different from \spad{n-1}.
756761`\spad{vertConcat(x,{}y)} vertically concatenates two matrices with an equal number of columns. The entries of \spad{y} appear below of the entries of \spad{x}. Error: if the matrices do not have the same number of columns.
756824`\spad{viewDefaults()} resets all the default graphics settings.
759339`\spad{viewport2D()} returns an undefined two-dimensional viewport of the domain \spadtype{TwoDimensionalViewport} whose contents are empty.
759394`\spad{viewport3D()} returns an undefined three-dimensional viewport of the domain \spadtype{ThreeDimensionalViewport} whose contents are empty.
759451`\spad{virtualDegree }\undocumented
759553`\spad{void()} produces a void object.
759584`\spad{vspace(n)} creates white space of height \spad{n}.
759630`\spad{weakBiRank(x)} determines the number of linearly independent elements in the \spad{bi*x*bj},{} \spad{i,{}j=1,{}...,{}n},{} where \spad{b=[b1,{}...,{}bn]} is a basis.
759699`\spad{weberE(v,{} z)} is the Weber \spad{E} function
759760`\spad{weberE(v,{} z)} is the Weber \spad{E} function.
759817`\spad{weierstrassPPrime(g2,{} g3,{} x)}
759891`\spad{weierstrassPPrime(g2,{} g3,{} z)} is the derivative of Weierstrass \spad{P} function.
759962`\spad{weierstrassP(g2,{} g3,{} x)}
760031`\spad{weierstrassP(g2,{} g3,{} z)} is the Weierstrass \spad{P} function.
760097`\spad{weierstrassSigma(g2,{} g3,{} x)}
760170`\spad{weierstrassSigma(g2,{} g3,{} z)} is the Weierstrass Sigma function.
760240`\spad{weierstrassZeta(g2,{} g3,{} x)}
760312`\spad{weierstrassZeta(g2,{} g3,{} z)} is the Weierstrass Zeta function.
760634`\spad{weight(p)} returns the maximum weight of all differential monomials appearing in the differential polynomial \spad{p}.
760720`\spad{weight(v)} returns the weight of the derivative \spad{v}.
760798`\spad{weight(op)} returns the weight attached to \spad{op}.
760943`\spad{weight(op,{} n)} attaches the weight \spad{n} to \spad{op}.
761006`\spad{weight(p,{} s)} returns the maximum weight of all differential monomials appearing in the differential polynomial \spad{p} when \spad{p} is viewed as a differential polynomial in the differential indeterminate \spad{s} alone.
761094`\spad{weighted(l)} \undocumented
761276`\spad{weights(p)} returns a list of weights of differential monomials appearing in differential polynomial \spad{p}.
761461`\spad{weights(p,{} s)} returns a list of weights of differential monomials appearing in the differential polynomial \spad{p} when \spad{p} is viewed as a differential polynomial in the differential indeterminate \spad{s} alone.
761556`\spad{whatInfinity(x)} returns 0 if \spad{x} is finite,{} 1 if \spad{x} is +infinity,{} and \spad{-1} if \spad{x} is -infinity.
761624`\spad{whileLoop(s,{}c)} creates a while loop in FORTRAN.
761676`\spad{whittakerM(k,{} m,{} z)} is the Whittaker \spad{M} function
761743`\spad{whittakerM(k,{} m,{} z)} is the Whittaker \spad{M} function.
761807`\spad{whittakerW(k,{} m,{} z)} is the Whittaker \spad{W} function
761874`\spad{whittakerW(k,{} m,{} z)} is the Whittaker \spad{W} function.
761938`\spad{wholePart x} returns the integer part of \spad{x}.
761993`\spad{wholePart(x)} extracts the whole part of \spadvar{\spad{x}}. That is,{} if \spad{x = continuedFraction(b0,{} [a1,{}a2,{}a3,{}...],{} [b1,{}b2,{}b3,{}...])},{} then \spad{wholePart(x) = b0}.
762046`\spad{wholePart(p)} extracts the whole part of the partial fraction \spad{p}.
762097`\spad{wholePart(x)} returns the whole part of the fraction \spad{x} \spadignore{i.e.} the truncated quotient of the numerator by the denominator.
762176`\spad{wholeRadix(l)} creates an integral radix expansion from a list of ragits. For example,{} \spad{wholeRadix([1,{}3,{}4])} will return \spad{134}.
762240`\spad{wholeRagits(rx)} returns the ragits of the integer part of a radix expansion.
762305`\spad{width()} returns the width of the display area (an integer).
762348`\spad{width(f)} returns the width of form \spad{f} (an integer).
762393`\spad{width(u)} returns \axiom{sup(\spad{u}) - inf(\spad{u})}.
762441`\spad{withPredicates(p,{} [p1,{}...,{}pn])} makes a copy of \spad{p} and attaches the predicate \spad{p1} and ... and \spad{pn} to the copy,{} which is returned.
762933`\spad{writable?(f)} tests if the named file be opened for writing. The named file need not already exist.
762988`\spad{write!(f,{}s)} puts the value \spad{s} into the file \spad{f}. The state of \spad{f} is modified so subsequents call to \spad{write!} will append one after another.
763040`\spad{writeLine!(f)} finishes the current line in the file \spad{f}. An empty string is returned. The call \spad{writeLine!(f)} is equivalent to \spad{writeLine!(f,{}"")}.
763087`\spad{writeLine!(f,{}s)} writes the contents of the string \spad{s} and finishes the current line in the file \spad{f}. The value of \spad{s} is returned.
763141`writes 3D SubSpace to a file in Wavefront (.OBJ) format
763212`Write an 'OBJ' (Wavefront) repesentation of node \spad{'n'} to the filename supplied.
763263`Write an 'SVG' repesentation of node \spad{'n'} to the filename supplied.
763323`Write an 'SVG' repesentation of node \spad{'n'} to the filename supplied.
763374`\spad{VRML2} is not \spad{xml} but it has the same semantics and node names as \spad{X3D} but a different syntax so it makes sense to create an \spad{XML} structure and then format the output differently.
763434`Write an 'VRML' repesentation of node \spad{'n'} to the filename supplied.
763486`Write an 'X3D' repesentation of node \spad{'n'} to the filename supplied.
763537`writes an \spad{xml} element and all the elements below it to an \spad{xml} file. This assumes that all elements,{} in a file,{} have a single root which is usually the case.
764570`apply transform to a point producing another point
764617`like testEquals,{} but expects failure.
764729`like testComplexEquals,{} but expects failure.
764794`like testEquals,{} but expects failure.
764858`like testEquals,{} but expects failure.
764916`like testEquals,{} but expects failure.
764984`like testLibraryError,{} but expects failure.
765041`like testEquals,{} but expects failure.
765108`like testNotEquals,{} but expects failure.
765169`like testEquals,{} but expects failure.
765260`like testRealEquals,{} but expects failure.
765322`like testEquals,{} but expects failure.
765382`like testTrue,{} but expects failure.
765431`construct a xmlAttribute from its name and list of values
765497`construct a xmlAttribute from its name and value
765799`\spad{xor(a,{}b)} returns the logical {\em exclusive-or} of bit aggregates \axiom{a} and \axiom{\spad{b}}.
765842`\spad{xor(a,{}b)} returns the logical exclusive {\em or} of Boolean \spad{a} and \spad{b}.
765880`\spad{xor(n,{}m)} returns the bit-by-bit logical {\em xor} of the single integers \spad{n} and \spad{m}.
766063`\spad{yCoordinates(f)} returns \spad{[[A1,{}...,{}An],{} D]} such that \spad{f = (A1 + A2 y +...+ An y^(n-1)) / D}.
766301`\spad{yellow()} returns the position of the yellow hue from total hues.
766624`\spad{zag(f,{}g)} creates a form for the continued fraction form for \spad{f} over \spad{g}.
767051`\spad{zero?(x)} tests if \spad{x} is equal to 0.
767256`\axiom{zero?(pol,{}aRoot)} answers if \axiom{pol} interpreted as \axiom{aRoot} is \axiom{0}
768877`\spad{zeroOf(p)} returns \spad{y} such that \spad{p(y) = 0}. If possible,{} \spad{y} is expressed in terms of radicals. Otherwise it is an implicit algebraic quantity. Error: if \spad{p} has more than one variable \spad{y}.
768944`\spad{zeroOf(p)} returns \spad{y} such that \spad{p(y) = 0}; if possible,{} \spad{y} is expressed in terms of radicals. Otherwise it is an implicit algebraic quantity.
769027`\spad{zeroOf(p)} returns \spad{y} such that \spad{p(y) = 0}. The value \spad{y} is expressed in terms of radicals if possible,{}and otherwise as an implicit algebraic quantity. Error: if \spad{p} has more than one variable.
769346`\spad{zeroOf(p,{} y)} returns \spad{y} such that \spad{p(y) = 0}; if possible,{} \spad{y} is expressed in terms of radicals. Otherwise it is an implicit algebraic quantity which displays as \spad{'y}.
769436`\spad{zeroOf(p,{} y)} returns \spad{y} such that \spad{p(y) = 0}. The value \spad{y} is expressed in terms of radicals if possible,{}and otherwise as an implicit algebraic quantity which displays as \spad{'y}.
769509`\axiom{zeroSetSplitIntoTriangularSystems(\spad{lp})} returns a list of triangular systems \axiom{[[\spad{ts1},{}\spad{qs1}],{}...,{}[\spad{tsn},{}\spad{qsn}]]} such that the zero set of \axiom{\spad{lp}} is the union of the closures of the \axiom{W_i} where \axiom{W_i} consists of the zeros of \axiom{\spad{ts}} which do not cancel any polynomial in \axiom{qsi}.
769635`\axiom{zeroSetSplit(\spad{lp})} returns a list \axiom{\spad{lts}} of triangular sets such that the zero set of \axiom{\spad{lp}} is the union of the closures of the regular zero sets of the members of \axiom{\spad{lts}}.
770242`\spad{zeroSetSplit(lp,{}clos?)} returns \spad{lts} a split of Kalkbrener of the radical ideal associated with \spad{lp}. If \spad{clos?} is \spad{false},{} it is also a decomposition of the variety associated with \spad{lp} into the regular zero set of the \spad{ts} in \spad{lts} (or,{} in other words,{} a split of Lazard of this variety). See the example illustrating the \spadtype{RegularTriangularSet} constru
770242`--ctor for more explanations about decompositions by means of regular triangular sets.
771102`\spad{zero(n)} creates a zero vector of length \spad{n}.
771270`\spad{zero(m,{}n)} returns an \spad{m}-by-\spad{n} zero matrix.
771764`\spad{zerosOf(p)} returns \spad{[y1,{}...,{}yn]} such that \spad{p(\spad{yi}) = 0}. The \spad{yi}\spad{'s} are expressed in radicals if possible. Otherwise they are implicit algebraic quantities containing new symbols. The new symbols are bound in the interpreter to the respective values. Error: if \spad{p} has more than one variable \spad{y}.
771838`\spad{zerosOf(p)} returns \spad{[y1,{}...,{}yn]} such that \spad{p(\spad{yi}) = 0}. The \spad{yi}\spad{'s} are expressed in radicals if possible. Otherwise they are implicit algebraic quantities containing new symbols. The new symbols are bound in the interpreter to the respective values.
771928`\spad{zerosOf(p)} returns \spad{[y1,{}...,{}yn]} such that \spad{p(\spad{yi}) = 0}. The \spad{yi}\spad{'s} are expressed in radicals if possible. Note: the returned values \spad{y1},{}...,{}\spad{yn} contain new symbols which are bound in the interpreter to the respective values. Error: if \spad{p} has more than one variable.
772275`\spad{zerosOf(p,{} y)} returns \spad{[y1,{}...,{}yn]} such that \spad{p(\spad{yi}) = 0}. The \spad{yi}\spad{'s} are expressed in radicals if possible,{} and otherwise as implicit algebraic quantities containing new symbols which display as \spad{'\%z0},{} \spad{'\%z1},{} ...; The new symbols are bound in the interpreter to respective values.
772372`\spad{zerosOf(p,{} y)} returns \spad{[y1,{}...,{}yn]} such that \spad{p(\spad{yi}) = 0}. The \spad{yi}\spad{'s} are expressed in radicals if possible,{} and otherwise as implicit algebraic quantities containing new symbols which display as \spad{'\%z0},{} \spad{'\%z1},{} ...; The new symbols are bound in the interpreter to the respective values.
772452`\spad{zeta()} returns the function which is constantly one
772875`\spad{f ~= g} creates the equivalent infix form.
772915`\spad{x~=y} tests if \spad{x} and \spad{y} are not equal.
772994`\spad{~(x)} returns the logical complement of \spad{x}.
773025`reverse,{} complement,{}canonical dual basis
773076`\spad{~ n} returns the bit-by-bit logical {\em not } of the single integer \spad{n}.
773115`Factorization of univariate polynomials with coefficients in \spadtype{AlgebraicNumber}.
773203`AlgebraPackage assembles a variety of useful functions for general algebras.
773300`This package provides algebraic functions over an integral domain.
773388`algebraic Hermite redution.
773584`This package provides functions for integrating a function on an algebraic curve.
773825`This package provides functions for the integration of algebraic integrands over transcendental functions.
773937`AlgebraicManipulations provides functions to simplify and expand expressions involving algebraic operators.
774029`This package factors multivariate polynomials over the domain of \spadtype{AlgebraicNumber} by allowing the user to specify a list of algebraic numbers generating the particular extension to factor over.
774167`\spadtype{AnyFunctions1} implements several utility functions for working with \spadtype{Any}. These functions are used to go back and forth between objects of \spadtype{Any} and objects of other types.
774214`This package apply rewrite rules to expressions,{} calling the pattern matcher.
774326`\spad{ApplyUnivariateSkewPolynomial} (internal) allows univariate skew polynomials to be applied to appropriate modules.
774453`\spadtype{AssociatedEquations} provides functions to compute the associated equations needed for factoring operators
774571`Attaching predicates to symbols for pattern matching. Date Created: 21 Mar 1989 Date Last Updated: 23 May 1990
774623`This package provides balanced factorisations of polynomials.
774736`This package exports functions to set some commonly used properties of operators,{} including properties which contain functions.
774800`\spadtype{BezoutMatrix} contains functions for computing resultants and discriminants using Bezout matrices.
774980`\spadtype{BoundIntegerRoots} provides functions to find lower bounds on the integer roots of a polynomial.
775086`\indented{1}{Author: Frederic Lehobey,{} James \spad{H}. Davenport} Date Created: 28 June 1994 Date Last Updated: 11 July 1997 Basic Operations: brillhartIrreducible? Related Domains: Also See: AMS Classifications: Keywords: factorization Examples: References: [1] John Brillhart,{} Note on Irreducibility Testing,{} Mathematics of Computation,{} vol. 35,{} num. 35,{} Oct. 1980,{} 1379-1381 [2] James Davenport,{} On Brillh
775086`--art Irreducibility. To appear. [3] John Brillhart,{} On the Euler and Bernoulli polynomials,{} \spad{J}. Reine Angew. Math.,{} \spad{v}. 234,{} (1969),{} \spad{pp}. 45-64
775169`This package implements asymptotically efficient Chinese reconstruction
775227`This package provides functions to enable conversion of tensors given conversion of the components.
775363`Tools to send a point to infinity on an algebraic curve.
775528`This package implements characteristicPolynomials for monogenic algebras using resultants
775691`This package provides a characteristicPolynomial function for any matrix over a commutative ring.
775770`\indented{1}{Author: Clifton Williamson} Date Created: 9 August 1993 Date Last Updated: 3 December 1993 Basic Operations: chineseRemainder,{} factorList Related Domains: PAdicWildFunctionFieldIntegralBasis(\spad{K},{}\spad{R},{}UP,{}\spad{F}) Also See: WildFunctionFieldIntegralBasis,{} FunctionFieldIntegralBasis AMS Classifications: Keyword
775770`--s: function field,{} finite field,{} integral basis Examples: References: Description:
775935`CoerceVectorMatrixPackage: an unexposed,{} technical package for data conversions
776005`Provides combinatorial functions over an integral domain.
776100`CommonDenominator provides functions to compute the common denominator of a finite linear aggregate of elements of the quotient field of an integral domain.
776221`This package exports the elementary operators,{} with some semantics already attached to them. The semantics that is attached here is not dependent on the set in which the operators will be applied.
776267`A package for swapping the order of two variables in a tower of two UnivariatePolynomialCategory extensions.
776422`\indented{1}{Author:} Date Created: Date Last Updated: Basic Functions: Related Constructors: Complex,{} UnivariatePolynomial Also See: AMS Classifications: Keywords: complex,{} polynomial factorization,{} factor References:
776543`This package extends maps from underlying rings to maps between complex over those rings.
776629`This package provides the generalized euclidean algorithm which is needed as the basic step for factoring polynomials.
776752`This package supports matching patterns involving complex expressions
776876`This package supports converting complex expressions to patterns
777002`\spadtype{ComplexRootFindingPackage} provides functions to find all roots of a polynomial \spad{p} over the complex number by using Plesken\spad{'s} idea to calculate in the polynomial ring modulo \spad{f} and employing the Chinese Remainder Theorem. In this first version,{} the precision (see \spadfunFrom{digits}{Float}) is not increased when this is necessary to avoid rounding errors
777002`--. Hence it is the user\spad{'s} responsibility to increase the precision if necessary. Note also,{} if this package is called with \spadignore{e.g.} \spadtype{Fraction Integer},{} the precise calculations could require a lot of time. Also note that evaluating the zeros is not necessarily a good check whether the result is correct: already evaluation can cause rounding errors.
777238`\spadtype{ComplexTrigonometricManipulations} provides function that compute the real and imaginary parts of complex functions.
777370`Solution of linear ordinary differential equations,{} constant coefficient case.
777538`CoordinateSystems provides coordinate transformation functions for plotting. Functions in this package return conversion functions which take points expressed in other coordinate systems and return points with the corresponding Cartesian coordinates.
777604`Enumeration by cycle indices.
777648`This package provides tools for working with cyclic streams.
777734`This package \undocumented{}
777792`\spadtype{DefiniteIntegrationTools} provides common tools used by the definite integration of both rational and elementary functions.
777916`This package \undocumented{}
778002`any solution of a homogeneous linear Diophantine equation can be represented as a sum of minimal solutions,{} which form a "basis" (a minimal solution cannot be represented as a nontrivial sum of solutions) in the case of an inhomogeneous linear Diophantine equation,{} each solution is the sum of a inhomogeneous solution and any number of homogeneous solutions therefore,{} it suffices to compute two sets: \indented{3}{1. all minimal inhomogeneous so
778002`--lutions} \indented{3}{2. all minimal homogeneous solutions} the algorithm implemented is a completion procedure,{} which enumerates all solutions in a recursive depth-first-search it can be seen as finding monotone paths in a graph for more details see Reference
778056`\indented{2}{This package provides operations which all take as arguments} direct products of elements of some type \spad{A} and functions from \spad{A} to another type \spad{B}. The operations all iterate over their vector argument and either return a value of type \spad{B} or a direct product over \spad{B}.
778151`DiscreteLogarithmPackage implements help functions for discrete logarithms in monoids using small cyclic groups.
778220`DisplayPackage allows one to print strings in a nice manner,{} including highlighting substrings.
778264`Package for the factorization of a univariate polynomial with coefficients in a finite field. The algorithm used is the "distinct degree" algorithm of Cantor-Zassenhaus,{} modified to use trace instead of the norm and a table for computing Frobenius as suggested by Naudin and Quitte .
778378`This package provides special functions for double precision real and complex floating point.
778434`This package provides functions for computing the residues of a function on an algebraic curve.
778659`Hack for the draw interface. DrawNumericHack provides a "coercion" from something of the form \spad{x = a..b} where \spad{a} and \spad{b} are formal expressions to a binding of the form \spad{x = c..d} where \spad{c} and \spad{d} are the numerical values of \spad{a} and \spad{b}. This "coercion" fails if \spad{a} and \spad{b} contains symbolic variables,{} but is meant for expressions involving \%\spad{pi}.
778728`This package \undocumented{}
778777`This package \undocumented{}
778833`This is a package for the exact computation of eigenvalues and eigenvectors. This package can be made to work for matrices with coefficients which are rational functions over a ring where we can factor polynomials. Rational eigenvalues are always explicitly computed while the non-rational ones are expressed in terms of their minimal polynomial.
778882`\spadtype{ElementaryFunctionDefiniteIntegration} provides functions to compute definite integrals of elementary functions.
779025`\spad{ElementaryFunctionLODESolver} provides the top-level functions for finding closed form solutions of linear ordinary differential equations and initial value problems.
779207`\spad{ElementaryFunctionODESolver} provides the top-level functions for finding closed form solutions of ordinary differential equations and initial value problems.
779337`This package provides functions to determine the sign of an elementary function around a point or infinity.
779456`ElementaryFunctionStructurePackage provides functions to test the algebraic independence of various elementary functions,{} using the Risch structure theorem (real and complex versions). It also provides transformations on elementary functions which are not considered simplifications.
779588`Provides elementary functions over an integral domain.
779687`This package provides functions for integration,{} limited integration,{} extended integration and the risch differential equation for elemntary functions.
779799`\indented{1}{Risch differential equation,{} elementary case.} Author: Manuel Bronstein Date Created: 12 August 1992 Date Last Updated: 17 August 1992 Keywords: elementary,{} function,{} integration.
779920`\indented{1}{Risch differential equation,{} elementary case.} Author: Manuel Bronstein Date Created: 1 February 1988 Date Last Updated: 2 November 1995 Keywords: elementary,{} function,{} integration.
780034`The elliptic functions \spad{sn},{} \spad{sc} and \spad{dn} are expanded as Taylor series.
780163`This package provides operations for mapping the sides of equations.
780223`ErrorFunctions implements error functions callable from the system interpreter. Typically,{} these functions would be called in user functions. The simple forms of the functions take one argument which is either a string (an error message) or a list of strings which all together make up a message. The list can contain formatting codes (see below). The more sophisticated versions takes two arguments where the first argument is the name of the function from which
780223`-- the error was invoked and the second argument is either a string or a list of strings,{} as above. When you use the one argument version in an interpreter function,{} the system will automatically insert the name of the function as the new first argument. Thus in the user interpreter function \indented{2}{\spad{f x == if x < 0 then error "negative argument" else x}} the call to error will actually be of the form \indented{2}{\spad{error("f",{}"negative argument")}} because the interpreter wil
780223`--l have created a new first argument. \blankline Formatting codes: error messages may contain the following formatting codes (they should either start or end a string or else have blanks around them): \indented{3}{\spad{\%l}\space{6}start a new line} \indented{3}{\spad{\%b}\space{6}start printing in a bold font (where available)} \indented{3}{\spad{\%d}\space{6}stop\space{2}printing in a bold font (where available)} \indented{3}{\spad{ \%ceon}\space{2}start centering message lines} \indented{3}
780223`--{\spad{\%ceoff}\space{2}stop\space{2}centering message lines} \indented{3}{\spad{\%rjon}\space{3}start displaying lines "ragged left"} \indented{3}{\spad{\%rjoff}\space{2}stop\space{2}displaying lines "ragged left"} \indented{3}{\spad{\%i}\space{6}indent\space{3}following lines 3 additional spaces} \indented{3}{\spad{\%u}\space{6}unindent following lines 3 additional spaces} \indented{3}{\spad{\%xN}\space{5}insert \spad{N} blanks (eg,{} \spad{\%x10} inserts 10 blanks)} \blankline
780265`\spadtype{EuclideanGroebnerBasisPackage} computes groebner bases for polynomial ideals over euclidean domains. The basic computation provides a distinguished set of generators for these ideals. This basis allows an easy test for membership: the operation \spadfun{euclideanNormalForm} returns zero on ideal members.
780265`-- The string "info" and "redcrit" can be given as additional args to provide incremental information during the computation. If "info" is given,{} \indented{1}{a computational summary is given for each \spad{s}-polynomial. If "redcrit"} is given,{} the reduced critical pairs are printed. The term ordering is determined by the polynomial type used. Suggested types include \spadtype{DistributedMultivariatePolynomial},{} \spadtype{HomogeneousDistributedMultivariatePolynomial},{} \spadtype{GeneralD
780265`--istributedMultivariatePolynomial}.
780457`This package is to be used in conjuction with \indented{12}{the CycleIndicators package. It provides an evaluation} \indented{12}{function for SymmetricPolynomials.}
780570`This package provides support for \spad{XML} files
780605`Lifting of maps to Expressions. Date Created: 16 Jan 1989 Date Last Updated: 22 Jan 1990
780894`This package allows a map from any expression space into any object to be lifted to a kernel over the expression set,{} using a given property of the operator of the kernel.
780972`This package allows a mapping \spad{E} \spad{->} \spad{F} to be lifted to a kernel over \spad{E}; This lifting can fail if the operator of the kernel cannot be applied in \spad{F}; Do not use this package with \spad{E} = \spad{F},{} since this may drop some properties of the operators.
781061`Taylor series solutions of explicit ODE\spad{'s}.
781161`\spadtype{ExpressionToOpenMath} provides support for converting objects of type \spadtype{Expression} into OpenMath.
781231`This package provides functions to convert functional expressions to power series.
781360`\indented{1}{Author: Clifton \spad{J}. Williamson} Date Created: Bastille Day 1989 Date Last Updated: 5 June 1990 Keywords: Examples: Package for constructing tubes around 3-dimensional parametric curves.
781409`This is just an interface between several packages and domains. The goal is to compute lexicographical Groebner bases of sets of polynomial with type \spadtype{Polynomial R} by the {\em FGLM} algorithm if this is possible (\spadignore{i.e.} if the input system generates a zero-dimensional ideal).
781484`\spadtype{FactoredFunctionUtilities} implements some utility functions for manipulating factored objects.
781555`\spadtype{FactoredFunctions2} contains functions that involve factored objects whose underlying domains may not be the same. For example,{} \spadfun{map} might be used to coerce an object of type \spadtype{Factored(Integer)} to \spadtype{Factored(Complex(Integer))}.
781635`computes various functions on factored arguments.
781700`This package provides utilities used by the factorizers which operate on polynomials represented as univariate polynomials with multivariate coefficients.
781833`\indented{1}{Finds the order of a divisor over a finite field} Author: Manuel Bronstein Date Created: 1988 Date Last Updated: 11 Jul 1990
782026`This package provides a mapping function for \spadtype{FiniteAbelianMonoidRing}
782206`\indented{1}{Lift a map to finite divisors.} Author: Manuel Bronstein Date Created: 1988 Date Last Updated: 19 May 1993
782559`FiniteFieldFunctions(\spad{GF}) is a package with functions concerning finite extension fields of the finite ground field {\em GF},{} \spadignore{e.g.} Zech logarithms.
782628`FiniteFieldHomomorphisms(\spad{F1},{}\spad{GF},{}\spad{F2}) exports coercion functions of elements between the fields {\em F1} and {\em F2},{} which both must be finite simple algebraic extensions of the finite ground field {\em GF}.
782783`\spad{FiniteFieldPolynomialPackage2}(\spad{F},{}\spad{GF}) exports some functions concerning finite fields,{} which depend on a finite field {\em GF} and an algebraic extension \spad{F} of {\em GF},{} \spadignore{e.g.} a zero of a polynomial over {\em GF} in \spad{F}.
782907`This package provides a number of functions for generating,{} counting and testing irreducible,{} normal,{} primitive,{} random polynomials over finite fields.
782987`This package solves linear diophantine equations for Bivariate polynomials over finite fields
783159`\spad{FiniteLinearAggregateFunctions2} provides functions involving two FiniteLinearAggregates where the underlying domains might be different. An example of this might be creating a list of rational numbers by mapping a function across a list of integers where the function divides each integer by 1000.
783295`This package exports 3 sorting algorithms which work over FiniteLinearAggregates.
783398`\spad{FiniteSetAggregateFunctions2} provides functions involving two finite set aggregates where the underlying domains might be different. An example of this is to create a set of rational numbers by mapping a function across a set of integers,{} where the function divides each integer by 1000.
783583`\indented{3}{This is a package for the approximation of complex solutions for} systems of equations of rational functions with complex rational coefficients. The results are expressed as either complex rational numbers or complex floats depending on the type of the precision parameter which can be either a rational number or a floating point number.
783655`\indented{3}{This is a package for the approximation of real solutions for} systems of polynomial equations over the rational numbers. The results are expressed as either rational numbers or floats depending on the type of the precision parameter which can be either a rational number or a floating point number.
783730`\spadtype{FortranCodePackage1} provides some utilities for producing useful objects in FortranCode domain. The Package may be used with the FortranCode domain and its \spad{printCode} or possibly via an outputAsFortran. (The package provides items of use in connection with ASPs in the AXIOM-NAG link and,{} where appropriate,{} naming accords with that in IRENA.) The easy-to-use functions use Fortran loop variables \spad{I1},{} \spad{I2},{} and it is users
783730`--' responsibility to check that this is sensible. The advanced functions use SegmentBinding to allow users control over Fortran loop variable names.
783778`Code to manipulate Fortran Output Stack
783829`provides an interface to the boot code for calling Fortran
783870`This package lifts the interpolation functions from \spadtype{FractionFreeFastGaussian} to fractions.
784072`This package implements the interpolation algorithm proposed in Beckermann,{} Bernhard and Labahn,{} George,{} Fraction-free computation of matrix rational interpolants and matrix GCDs,{} SIAM Journal on Matrix Analysis and Applications 22.
784196`This package extends a map between integral domains to a map between Fractions over those domains by applying the map to the numerators and denominators.
784278`\indented{1}{Lifting of morphisms to fractional ideals.} Author: Manuel Bronstein Date Created: 1 Feb 1989 Date Last Updated: 27 Feb 1990 Keywords: ideal,{} algebra,{} module.
784594`\spad{FramedNonAssociativeAlgebraFunctions2} implements functions between two framed non associative algebra domains defined over different rings. The function map is used to coerce between algebras over different domains having the same structural constants.
784772`linear extensions of maps on the basis
784899`Lifts a map from rings to function fields over them.
785301`In this package \spad{R} is a Euclidean domain and \spad{F} is a framed algebra over \spad{R}. The package provides functions to compute the integral closure of \spad{R} in the quotient field of \spad{F}. It is assumed that \spad{char(R/P) = char(R)} for any prime \spad{P} of \spad{R}. A typical instance of this is when \spad{R = K[x]} and \spad{F} is a func
785301`--tion field over \spad{R}.
785448`Attaching assertions to symbols for pattern matching; Date Created: 21 Mar 1989 Date Last Updated: 23 May 1990
785537`Attaching predicates to symbols for pattern matching. Date Created: 21 Mar 1989 Date Last Updated: 23 May 1990
785642`\spadtype{FunctionSpaceComplexIntegration} provides functions for the indefinite integration of complex-valued functions.
785777`This package allows a mapping \spad{R} \spad{->} \spad{S} to be lifted to a mapping from a function space over \spad{R} to a function space over \spad{S}.
785906`\spadtype{FunctionSpaceIntegration} provides functions for the indefinite integration of real-valued functions.
786033`FunctionsSpacePrimitiveElement provides functions to compute primitive elements in functions spaces.
786143`This package provides function which replaces transcendental kernels in a function space by random integers. The correspondence between the kernels and the integers is fixed between calls to new().
786236`computes sums of top-level expressions.
786340`This package converts expressions in some function space to exponential expansions.
786493`This package converts expressions in some function space to power series in a variable \spad{x} with coefficients in that function space. The function \spadfun{exprToUPS} converts expressions to power series whose coefficients do not contain the
786493`-- variable \spad{x}. The function \spadfun{exprToGenUPS} converts functional expressions to power series whose coefficients may involve functions of \spad{log(x)}.
786755`\indented{1}{Used internally by IR2F} Author: Manuel Bronstein Date Created: 12 May 1988 Date Last Updated: 22 September 1993 Keywords: function,{} space,{} polynomial,{} factoring
786913`Provides some special functions over an integral domain.
787013`\spadtype{GaloisGroupFactorizationUtilities} provides functions that will be used by the factorizer.
787158`\spadtype{GaloisGroupFactorizer} provides functions to factor resolvents.
787250`\spadtype{GaloisGroupPolynomialUtilities} provides useful functions for univariate polynomials which should be added to \spadtype{UnivariatePolynomialCategory} or to \spadtype{Factored} (July 1994).
787358`\spadtype{GaloisGroupUtilities} provides several useful functions.
787415`Package for the factorization of complex or gaussian integers.
787474`\spad{Gcd} basis provides functions to find structure of multiplicative group (and semigroup) generated by elements of \spad{Gcd} domain.
787533`\indented{1}{Author : \spad{P}.Gianni.} January 1990 The equation \spad{Af+Bg=h} and its generalization to \spad{n} polynomials is solved for solutions over the \spad{R},{} euclidean domain. A table containing the solutions of \spad{Af+Bg=x^k} is used. The operations are performed modulus a prime which are in principle big enough,{} but the solutions are tested and,{} in case of failure,{} a hensel lifting
787533`-- process is used to get to the right solutions. It will be used in the factorization of multivariate polynomials over finite field,{} with \spad{R=F[x]}.
787631`\indented{1}{Description} This package provides operations for the factorization of univariate polynomials with integer coefficients. The factorization is done by "lifting" the finite "berlekamp's" factorization
787693`\indented{1}{Author : \spad{P}.Gianni} General Hensel Lifting Used for Factorization of bivariate polynomials over a finite field.
787804`This package provides operations for \spad{GCD} computations on polynomials
787973`\indented{2}{This is the top level package for doing multivariate factorization} over basic domains like \spadtype{Integer} or \spadtype{Fraction Integer}.
788158`\spadtype{GenerateUnivariatePowerSeries} provides functions that create power series from explicit formulas for their \spad{n}th coefficient.
788286`This internal package rationalises integrands on curves of the form: \indented{2}{\spad{y\^2 = a x\^2 + b x + c}} \indented{2}{\spad{y\^2 = (a x + b) / (c x + d)}} \indented{2}{\spad{f(x,{} y) = 0} where \spad{f} has degree 1 in \spad{x}} The rationalization is done for integration,{} limited integration,{} extended integration and the risch differential equation.
788405`Gosper\spad{'s} summation algorithm.
788594`TwoDimensionalPlotSettings sets global flags and constants for 2-dimensional plotting.
788638`GrayCode provides a function for efficiently running through all subsets of a finite set,{} only changing one element by another one.
788673`\spadtype{GroebnerFactorizationPackage} provides the function groebnerFactor" which uses the factorization routines of \Language{} to factor each polynomial under consideration while doing the groebner basis algorithm. Then it writes the ideal as an intersection of ideals determined by the irreducible factors.
788673`-- Note that the whole ring may occur as well as other redundancies. We also use the fact,{} that from the second factor on we can assume that the preceding factors are not equal to 0 and we divide all polynomials under considerations by the elements of this list of "nonZeroRestrictions". The result is a list of groebner bases,{} whose union of solutions of the corresponding systems of equations is the solution of the system of equation corresponding to the input list. The term ordering is deter
788673`--mined by the polynomial type used. Suggested types include \spadtype{DistributedMultivariatePolynomial},{} \spadtype{HomogeneousDistributedMultivariatePolynomial},{} \spadtype{GeneralDistributedMultivariatePolynomial}.
788869`\indented{1}{Author:} Date Created: Date Last Updated: Keywords: Description This package provides low level tools for Groebner basis computations
789049`\spadtype{GroebnerPackage} computes groebner bases for polynomial ideals. The basic computation provides a distinguished set of generators for polynomial ideals over fields. This basis allows an easy test for membership: the operation \spadfun{normalForm} returns zero on ideal members. When the provided coefficient domain,{} Dom,{} is not 
789049`--a field,{} the result is equivalent to considering the extended ideal with \spadtype{Fraction(Dom)} as coefficients,{} but considerably more efficient since all calculations are performed in Dom. Additional argument "info" and "redcrit" can be given to provide incremental information during computation. Argument "info" produces a computational summary for each \spad{s}-polynomial. Argument "redcrit" prints out the reduced critical pairs. The term ordering is determined by the polynomial type u
789049`--sed. Suggested types include \spadtype{DistributedMultivariatePolynomial},{} \spadtype{HomogeneousDistributedMultivariatePolynomial},{} \spadtype{GeneralDistributedMultivariatePolynomial}.
789215`\indented{1}{Author : \spad{P}.Gianni,{} Summer \spad{'88},{} revised November \spad{'89}} Solve systems of polynomial equations using Groebner bases Total order Groebner bases are computed and then converted to lex ones This package is mostly intended for internal use.
789301`This package exports guessing of sequences of rational functions
789351`This package exports coercion and retraction from EXPR INT to finite fields
789433`This package exports guessing of sequences of numbers in a finite field
789505`This package exports guessing of sequences of rational numbers
789548`This package exports coercion and retraction from EXPR INT to polynomials
789609`This package exports guessing of sequences of rational functions over the integers
789661`This package exports guessing of sequences of rational functions
789712`This package implements guessing of sequences. Packages for the most common cases are provided as \spadtype{GuessInteger},{} \spadtype{GuessPolynomial},{} etc.
789851`\indented{1}{Author : Larry Lambe} Date Created : August 1988 Date Last Updated : March 9 1990 Related Constructors: OrderedSetInts,{} Commutator,{} FreeNilpotentLie AMS Classification: Primary 17B05,{} 17B30; Secondary 17A50 Keywords: free Lie algebra,{} Hall basis,{} basic commutators Description : Generate a basis for the free Lie algebra on \spad{n} generators over a ring \spad{R} with identity up to basic commutators of length \spad{c} using the algorithm of \spad
789851`--{P}. Hall as given in Serre\spad{'s} book Lie Groups \spad{
789851`--} Lie Algebras
789885`This package provides the functions for the heuristic integer \spad{gcd}. Geddes\spad{'s} algorithm,{}for univariate polynomials with integer coefficients
789961`\indented{2}{This package provides functions for the primary decomposition of} polynomial ideals over the rational numbers. The ideals are members of the \spadtype{PolynomialIdeal} domain,{} and the polynomial generators are required to be from the \spadtype{DistributedMultivariatePolynomial} domain.
790057`This package provides operations to create incrementing functions.
790123`This package computes infinite products of univariate Taylor series over an integral domain of characteristic 0.
790267`This package computes infinite products of univariate Taylor series over an arbitrary finite field.
790460`This package computes infinite products of univariate Taylor series over a field of prime order.
790587`Functions defined on streams with entries in two sets.
790655`Functions defined on streams with entries in two sets.
790732`Default infinity signatures for the interpreter; Date Created: 4 Oct 1989 Date Last Updated: 4 Oct 1989
790771`Factorization of univariate polynomials with coefficients in an algebraic extension of a field over which we can factor UP\spad{'s}.
790940`InnerCommonDenominator provides functions to compute the common denominator of a finite linear aggregate of elements of the quotient field of an integral domain.
791096`\spadtype{InnerMatrixLinearAlgebraFunctions} is an internal package which provides standard linear algebra functions on domains in \spad{MatrixCategory}
791265`\spadtype{InnerMatrixQuotientFieldFunctions} provides functions on matrices over an integral domain which involve the quotient field of that integral domain. The functions rowEchelon and inver
791265`--se return matrices with entries in the quotient field.
791767`\indented{2}{This is an inner package for factoring multivariate polynomials} over various coefficient domains in characteristic 0. The univariate factor operation is passed as a parameter. Multivariate hensel lifting is used to lift the univariate factorization
791917`InnerNormalBasisFieldFunctions(\spad{GF}) (unexposed): This package has functions used by every normal basis finite field extension domain.
791998`This package is the inner package to be used by NumericRealEigenPackage and NumericComplexEigenPackage for the computation of numeric eigenvalues and eigenvectors.
792091`This is an internal package for computing approximate solutions to systems of polynomial equations. The parameter \spad{K} specifies the coefficient field of the input polynomials and must be either \spad{Fraction(Integer)} or \spad{Complex(Fraction Integer)}. The parameter \spad{F} specifies where the solutions must lie and can be one of the following: \spad{Float},{} \spad{Fraction(Integer)},{} \spa
792091`--d{Complex(Float)},{} \spad{Complex(Fraction Integer)}. The last parameter specifies the type of the precision operand and must be either \spad{Fraction(Integer)} or \spad{Float}.
792194`Find the sign of a polynomial around a point or infinity.
792284`tools for the summation packages.
792416`This package provides transformations from trigonometric functions to exponentials and logarithms,{} and back. \spad{F} and \spad{FG} should be the same type of function space.
792580`Tools for manipulating input forms.
792636`This package provides functions to lookup bits in integers
792752`\indented{1}{This Package contains basic methods for integer factorization.} The factor operation employs trial division up to 10,{}000. It then tests to see if \spad{n} is a perfect power before using Pollards rho method. Because Pollards method may fail,{} the result of factor may contain composite factors. We should also employ Lenstra\spad{'s} eliptic curve method.
792831`Test for linear dependence over the integers.
792920`This package provides various number theoretic functions on the integers.
792979`The \spadtype{IntegerPrimesPackage} implements a modification of Rabin\spad{'s} probabilistic primality test and the utility functions \spadfun{nextPrime},{} \spadfun{prevPrime} and \spadfun{primes}.
793050`Provides integer testing and retraction functions. Date Created: March 1990 Date Last Updated: 9 April 1991
793122`The \spadtype{IntegerRoots} package computes square roots and \indented{2}{\spad{n}th roots of integers efficiently.}
793184`This package provides the implementation for the \spadfun{solveLinearPolynomialEquation} operation over the integers. It uses a lifting technique from the package GenExEuclid
793250`IntegralBasisPolynomialTools provides functions for \indented{1}{mapping functions on the coefficients of univariate and bivariate} \indented{1}{polynomials.}
793401`This package contains functions used in the packages FunctionFieldIntegralBasis and NumberFieldIntegralBasis.
793539`\indented{1}{Internally used by the integration packages} Author: Manuel Bronstein Date Created: 1987 Date Last Updated: 12 August 1992 Keywords: integration.
793610`\indented{2}{This package allows a sum of logs over the roots of a polynomial} \indented{2}{to be expressed as explicit logarithms and arc tangents,{} provided} \indented{2}{that the indexing polynomial can be factored into quadratics.} Date Created: 21 August 1988 Date Last Updated: 4 October 1993
793690`This package allows a sum of logs over the roots of a polynomial to be expressed as explicit logarithms and arc tangents,{} provided that the indexing polynomial can be factored into quadratics.
793818`\indented{1}{Tools for the integrator} Author: Manuel Bronstein Date Created: 25 April 1990 Date Last Updated: 9 June 1993 Keywords: elementary,{} function,{} integration.
793901`A package to print strings without line-feed nor carriage-return.
793951`\indented{1}{An internal package for computing the rational univariate representation} \indented{1}{of a zero-dimensional algebraic variety given by a square-free} \indented{1}{triangular set.} \indented{1}{The main operation is \axiomOpFrom{rur}{InternalRationalUn
793951`--ivariateRepresentationPackage}.} \indented{1}{It is based on the {\em generic} algorithm description in [1]. \newline References:} [1] \spad{D}. LAZARD "Solving Zero-dimensional Algebraic Systems" \indented{4}{Journal of Symbolic Computation,{} 1992,{} 13,{} 117-131}
794193`This package computes the inverse Laplace Transform.
794322`IrrRepSymNatPackage contains functions for computing the ordinary irreducible representations of symmetric groups on \spad{n} letters {\em {1,{}2,{}...,{}n}} in Young\spad{'s} natural form and their dimensions. These representations can be labelled by number partitions of \spad{n},{} \spadignore{i.e.} a weakly decreasing sequence of integers summing up to \spad{n},{} \spadignore{e.g.} {\em [3,{}3,{}3,{}1]} labels an irreducible representation for \spad{n} e
794322`--quals 10. Note: whenever a \spadtype{List Integer} appears in a signature,{} a partition required.
794368`This package exports the function generateIrredPoly that computes a monic irreducible polynomial of degree \spad{n} over a finite field.
794446`This package exports some auxiliary functions on kernels
794520`\spadtype{Kovacic} provides a modified Kovacic\spad{'s} algorithm for solving explicitely irreducible 2nd order linear ordinary differential equations.
794628`This package computes the forward Laplace Transform.
794749`A package for solving polynomial systems by means of Lazard triangular sets [1]. This package provides two operations. One for solving in the sense of the regular zeros,{} and the other for solving in the sense of the Zariski closure. Both produce square-free reg
794749`--ular sets. Moreover,{} the decompositions do not contain any redundant component. However,{} only zero-dimensional regular sets are normalized,{} since normalization may be time consumming in positive dimension. The decomposition process is that of [2].\newline References : \indented{1}{[1] \spad{D}. LAZARD "A new method for solving algebraic systems of} \indented{5}{positive dimension" Discr. App. Math. 33:147-160,{}1991} \indented{1}{[2] \spad{M}. MORENO MAZA "A new algorithm for computing t
794749`--riangular} \indented{5}{decomposition of algebraic varieties" NAG Tech. Rep. 4/98.}
794993`Package for leading coefficient determination in the lifting step. Package working for every \spad{R} euclidean with property \spad{"F"}. Uses algorithm given in section 3 of \spad{P}. \spad{S}. Wang "An improved multivariate polynomial factoring algorithm",{} Math. of Comput. 32,{} 1978,{} \spad{p}. 1215-\spad{-1231},{} generalized to arbitrary euclidea
794993`--n domain.
795144`A package for solving polynomial systems with finitely many solutions. The decompositions are given by means of regular triangular sets. The computations use lexicographical Groebner bases. The main operations are \axiomOpFrom{lexTriangular}{LexTriangularPackage} and \axiomOpFrom{squareFreeLexTriangular}{LexTriangularPackage}. The second one provide decompositions by means of square-free regular triangular sets. Both are 
795144`--based on the {\em lexTriangular} method described in [1]. They differ from the algorithm described in [2] by the fact that multiciplities of the roots are not kept. With the \axiomOpFrom{squareFreeLexTriangular}{LexTriangularPackage} operation all multiciplities are removed. With the other operation some multiciplities may remain. Both operations admit an optional argument to produce normalized triangular sets. \newline
795226`\indented{1}{Given a Groebner basis \spad{B} with respect to the total degree ordering for a} zero-dimensional ideal \spad{I},{} compute a Groebner basis with respect to the lexicographical ordering by using linear algebra. This is an internal package,{} operations in this package may crash the system on invalid input.
795303`Test for linear dependence.
795399`\spadtype{LinearOrdinaryDifferentialOperatorFactorizer} provides a factorizer for linear ordinary differential operators whose coefficients are rational functions.
795529`\spad{LinearOrdinaryDifferentialOperatorsOps} provides symmetric products and sums for linear ordinary differential operators.
795657`Given a PolynomialFactorizationExplicit ring,{} this package provides a defaulting rule for the \spad{solveLinearPolynomialEquation} operation,{} by moving into the field of fractions,{} and solving it there via the \spad{multiEuclidean} operation.
795756`This package solves linear system in the matrix form \spad{AX = B}. It is essentially a particular instantiation of the package \spadtype{LinearSystemMatrixPackage} for Matrix and Vector. This package\spad{'s} existence makes it easier to use \spadfun{solve} in the FriCAS interpreter.
795818`This package solves linear system in the matrix form \spad{AX = B}.
795996`this package finds the solutions of linear systems presented as a list of polynomials.
796147`This package provides liouvillian functions over an integral domain.
796247`\spadtype{ListFunctions2} implements utility functions that operate on two kinds of lists,{} each with a possibly different type of element.
796305`\spadtype{ListFunctions3} implements utility functions that operate on three kinds of lists,{} each with a possibly different type of element.
796372`\spadtype{ListToMap} allows mappings to be described by a pair of lists of equal lengths. The image of an element \spad{x},{} which appears in position \spad{n} in the first list,{} is then the \spad{n}th element of the second list. A default value or default function can be specified to be used when \spad{x} does not appear in the first list. In the absence of defaults,{} an error will occur in that case.
796435`\indented{1}{Utilities for MPolyCat} Author: Manuel Bronstein Date Created: 1987 Date Last Updated: 28 March 1990 (\spad{PG})
796659`This package \undocumented
796897`\indented{3}{This package exports a factor operation for multivariate polynomials} with coefficients which are polynomials over some ring \spad{R} over which we can factor. It is used internally by packages such as the solve package which need to work with polynomials in a specific set of variables with coefficients which are polynom
796897`--ials in all the other variables.
797069`\indented{3}{This package exports a factor operation for multivariate polynomials} with coefficients which are rational functions over some ring \spad{R} over which we can factor. It is used internally by packages such as primary decomposition which need to work with polynomials with rational function coefficient
797069`--s,{} \spadignore{i.e.} themselves fractions of polynomials.
797262`\indented{1}{MRationalFactorize contains the factor function for multivariate} polynomials over the quotient field of a ring \spad{R} such that the package MultivariateFactorize can factor multivariate polynomials over \spad{R}.
797426`transforms top-level objects into compiled functions.
797541`MakeFloatCompiledFunction transforms top-level objects into compiled Lisp functions whose arguments are Lisp floats. This by-passes the \Language{} compiler and interpreter,{} thereby gaining several orders of magnitude.
797623`transforms top-level objects into interpreter functions.
797691`MakeRecord is used internally by the interpreter to create record types which are used for doing parallel iterations on streams.
797748`transforms top-level objects into compiled functions.
797849`various Currying operations.
797908`various Currying operations.
797983`various Currying operations.
798074`various Currying operations.
798147`various Currying operations.
798236`various Currying operations.
798341`\spadtype{MatrixCategoryFunctions2} provides functions between two matrix domains. The functions provided are \spadfun{map} and \spadfun{reduce}.
798631`MatrixCommonDenominator provides functions to compute the common denominator of a matrix of elements of the quotient field of an integral domain.
798730`\spadtype{MatrixLinearAlgebraFunctions} provides functions to compute inverses and canonical forms.
798903`This package exports tools for merging lists
798955`\indented{1}{<description of package>} Author: Jim Wen Date Created: \spad{??} Date Last Updated: October 1991 by Jon Steinbach Keywords: Examples: References:
799020`This package supports factorization and gcds of univariate polynomials over the integers modulo different primes. The inputs are given as polynomials over the integers with the prime passed explicitly as an extra argument.
799121`This package provides generic interface to modular \indented{1}{Hermite-Pade solver.}
799253`\indented{1}{Modular hermitian row reduction.} Author: Manuel Bronstein Date Created: 22 February 1989 Date Last Updated: 24 November 1993 Keywords: matrix,{} reduction.
799330`\spad{MonoidRingFunctions2} implements functions between two monoid rings defined with the same monoid over different rings.
799404`Tools for handling monomial extensions.
799505`\spadtype{MoreSystemCommands} implements an interface with the system command facility. These are the commands that are issued from source files or the system interpreter and they start with a close parenthesis,{} \spadignore{e.g.} \spadsyscom{what} commands.
799553`Package for factorization of multivariate polynomials over finite fields.
799705`\indented{1}{\spadtype{MultiVariableCalculusFunctions} Package provides several} \indented{1}{functions for multivariable calculus.} These include gradient,{} hessian and jacobian,{} divergence and laplacian. Various forms for banded and sparse storage of matrices are included.
799882`Lifting of a map through 2 levels of polynomials.
800204`\indented{2}{This is the top level package for doing multivariate factorization} over basic domains like \spadtype{Integer} or \spadtype{Fraction Integer}.
800362`This package provides the functions for the multivariate "lifting",{} using an algorithm of Paul Wang. This package will work for every euclidean domain \spad{R} which has property \spad{F},{} \spadignore{i.e.} there exists a factor operation in \spad{R[x]}.
800505`Author : \spad{P}.Gianni This package provides the functions for the computation of the square free decomposition of a multivariate polynomial. It uses the package GenExEuclid for the resolution of the equation \spad{Af + Bg = h} and its generalization to \spad{n} polynomials over an integral domain and the package \spad{MultivariateLifting} for the "multi
800505`--variate" lifting.
800654`Package for the determination of the coefficients in the lifting process. Used by \spadtype{MultivariateLifting}. This package will work for every euclidean domain \spad{R} which has property \spad{F},{} \spadignore{i.e.} there exists a factor operation in \spad{R[x]}.
800870`This package lifts a mapping from coefficient rings \spad{R} to \spad{S} to a mapping from sparse univariate polynomial over \spad{R} to a sparse univariate polynomial over \spad{S}. Note that the mapping is assumed to send zero to zero,{} since it will only be applied to the non-zero coefficients of the polynomial.
800953`This package exports Newton interpolation for the special case where the result is known to be in the original integral domain
801018`This package provides a division and related operations for \spadtype{MonogenicLinearOperator}\spad{s} over a \spadtype{Field}. Since the multiplication is in general non-commutative,{} these operations all have left- and right-hand versions. This package provides the operations based on left-division.
801114`NonLinearFirstOrderODESolver provides a function for finding closed form first integrals of nonlinear ordinary differential equations of order 1.
801245`NonLinearSolvePackage is an interface to \spadtype{SystemSolvePackage} that attempts to retract the coefficients of the equations before solving. The solutions are given in the algebraic closure of \spad{R} whenever possible.
801313`\spadtype{NoneFunctions1} implements functions on \spadtype{None}. It particular it includes a particulary dangerous coercion from any other type to \spadtype{None}.
801362`This package implements the norm of a polynomial with coefficients in a monogenic algebra (using resultants)
801540`This package \undocumented
801764`A package for computing normalized assocites of univariate polynomials with coefficients in a tower of simple extensions of a field.\newline References : \indented{1}{[1] \spad{D}. LAZARD "A new method for solving algebraic systems of} \indented{5}{positive dimension" Discr. App. Math. 33:147-160,{}1991} \indented{1}
801764`--{[2] \spad{M}. MORENO MAZA and \spad{R}. RIOBOO "Computations of \spad{gcd} over} \indented{5}{algebraic towers of simple extensions" In proceedings of \spad{AAECC11}} \indented{5}{Paris,{} 1995.} \indented{1}{[3] \spad{M}. MORENO MAZA "Calculs de pgcd au-dessus des tours} \indented{5}{d'extensions simples et resolution des systemes d'equations} \indented{5}{algebriques" These,{} Universite \spad{P}.etM. Curie,{} Paris,{} 1997.}
801953`In this package \spad{F} is a framed algebra over the integers (typically \spad{F = Z[a]} for some algebraic integer a). The package provides functions to compute the integral closure of \spad{Z} in the quotient quotient field of \spad{F}.
802080`NumberFormats provides function to format and read arabic and roman numbers,{} to convert numbers to strings and to read floating-point numbers.
802122`This package provides polynomials as functions on a ring.
802204`This package computes explicitly eigenvalues and eigenvectors of matrices with entries over the complex rational numbers. The results are expressed either as complex floating numbers or as complex rational numbers depending on the type of the precision parameter.
802277`\spadtype{NumericContinuedFraction} provides functions \indented{2}{for converting floating point numbers to continued fractions.}
802354`This package computes explicitly eigenvalues and eigenvectors of matrices with entries over the Rational Numbers. The results are expressed as floating numbers or as rational numbers depending on the type of the parameter Par.
802424`\indented{1}{Author: Clifton \spad{J}. Williamson} Date Created: Bastille Day 1989 Date Last Updated: 5 June 1990 Keywords: Examples: Package for constructing tubes around 3-dimensional parametric curves.
802503`Numeric provides real and complex numerical evaluation functions for various symbolic types.
802563`This package is a suite of functions for the numerical integration of an ordinary differential equation of \spad{n} variables: \blankline \indented{8}{\center{dy/dx = \spad{f}(\spad{y},{}\spad{x})\space{5}\spad{y} is an \spad{n}-vector}} \blankline \par All the routines are based on a 4-th order Runge-Kutta kernel. These routines generally have as arguments: \spad{n},{} the number of dependent variables; \spad{x1},{} the initial point; 
802563`--\spad{h},{} the step size; \spad{y},{} a vector of initial conditions of length \spad{n} which upon exit contains the solution at \spad{x1 + h}; \spad{derivs},{} a function which computes the right hand side of the ordinary differential equation: \spad{derivs(dydx,{}y,{}x)} computes \spad{dydx},{} a vector which contains the derivative information. \blankline \par In order of increasing complexity:\begin{items} \blankline \item \spad{rk4(y,{}n,{}x1,{}h,{}derivs)} advances the solution vector t
802563`--o \spad{x1 + h} and return the values in \spad{y}. \blankline \item \spad{rk4(y,{}n,{}x1,{}h,{}derivs,{}t1,{}t2,{}t3,{}t4)} is the same as \spad{rk4(y,{}n,{}x1,{}h,{}derivs)} except that you must provide 4 scratch arrays \spad{t1}-\spad{t4} of size \spad{n}. \blankline \item Starting with \spad{y} at \spad{x1},{} \spad{rk4f(y,{}n,{}x1,{}x2,{}ns,{}derivs)} uses \spad{ns} fixed steps of a 4-th order Runge-Kutta integrator to advance the solution vector to \spad{x2} and return the values in \spad
802563`--{y}. Argument \spad{x2},{} is the final point,{} and \spad{ns},{} the number of steps to take. \blankline \item \spad{rk4qc(y,{}n,{}x1,{}step,{}eps,{}yscal,{}derivs)} takes a 5-th order Runge-Kutta step with monitoring of local truncation to ensure accuracy and adjust stepsize. The function takes two half steps and one full step and scales the difference in solutions at the final point. If the error is within \spad{eps},{} the step is taken and the result is returned. If the error is not withi
802563`--n \spad{eps},{} the stepsize if decreased and the procedure is tried again until the desired accuracy is reached. Upon input,{} an trial step size must be given and upon return,{} an estimate of the next step size to use is returned as well as the step size which produced the desired accuracy. The scaled error is computed as \center{\spad{error = MAX(ABS((y2steps(i) - y1step(i))/yscal(i)))}} and this is compared against \spad{eps}. If this is greater than \spad{eps},{} the step size is reduced
802563`-- accordingly to \center{\spad{hnew = 0.9 * hdid * (error/eps)^(-1/4)}} If the error criterion is satisfied,{} then we check if the step size was too fine and return a more efficient one. If \spad{error > \spad{eps} * (6.0E-04)} then the next step size should be \center{\spad{hnext = 0.9 * hdid * (error/\spad{eps})^(\spad{-1/5})}} Otherwise \spad{hnext = 4.0 * hdid} is returned. A more detailed discussion of this and related topics can be found in the book "Numerical Recipies" by \spad{W}.Press
802563`--,{} \spad{B}.\spad{P}. Flannery,{} \spad{S}.A. Teukolsky,{} \spad{W}.\spad{T}. Vetterling published by Cambridge University Press. Argument \spad{step} is a record of 3 floating point numbers \spad{(try ,{} did ,{} next)},{} \spad{eps} is the required accuracy,{} \spad{yscal} is the scaling vector for the difference in solutions. On input,{} \spad{step.try} should be the guess at a step size to achieve the accuracy. On output,{} \spad{step.did} contains the step size which achieved the accurac
802563`--y and \spad{step.next} is the next step size to use. \blankline \item \spad{rk4qc(y,{}n,{}x1,{}step,{}eps,{}yscal,{}derivs,{}t1,{}t2,{}t3,{}t4,{}t5,{}t6,{}t7)} is the same as \spad{rk4qc(y,{}n,{}x1,{}step,{}eps,{}yscal,{}derivs)} except that the user must provide the 7 scratch arrays \spad{t1-t7} of size \spad{n}. \blankline \item \spad{rk4a(y,{}n,{}x1,{}x2,{}eps,{}h,{}ns,{}derivs)} is a driver program which uses \spad{rk4qc} to integrate \spad{n} ordinary differential equations starting at \s
802563`--pad{x1} to \spad{x2},{} keeping the local truncation error to within \spad{eps} by changing the local step size. The scaling vector is defined as \center{\spad{yscal(i) = abs(y(i)) + abs(h*dydx(i)) + tiny}} where \spad{y(i)} is the solution at location \spad{x},{} \spad{dydx} is the ordinary differential equation\spad{'s} right hand side,{} \spad{h} is the current step size and \spad{tiny} is 10 times the smallest positive number representable. The user must supply an estimate for a trial step
802563`-- size and the maximum number of calls to \spad{rk4qc} to use. Argument \spad{x2} is the final point,{} \spad{eps} is local truncation,{} \spad{ns} is the maximum number of call to \spad{rk4qc} to use. \end{items}
802630`This suite of routines performs numerical quadrature using algorithms derived from the basic trapezoidal rule. Because the error term of this rule contains only even powers of the step size (for open and closed versions),{} fast convergence can be obtained if the integrand is sufficiently smooth. \blankline Each routine returns a Record of type TrapAns,{} which contains\indent{3} \newline value (\spadtype{Float}):\tab{20} estimate of the integral \newlin
802630`--e error (\spadtype{Float}):\tab{20} estimate of the error in the computation \newline totalpts (\spadtype{Integer}):\tab{20} total number of function evaluations \newline success (\spadtype{Boolean}):\tab{20} if the integral was computed within the user specified error criterion \indent{0}\indent{0} To produce this estimate,{} each routine generates an internal sequence of sub-estimates,{} denoted by {\em S(i)},{} depending on the routine,{} to which the various convergence criteria are applie
802630`--d. The user must supply a relative accuracy,{} \spad{eps_r},{} and an absolute accuracy,{} \spad{eps_a}. Convergence is obtained when either \center{\spad{ABS(S(i) - S(i-1)) < eps_r * ABS(S(i-1))}} \center{or \spad{ABS(S(i) - S(i-1)) < eps_a}} are \spad{true} statements. \blankline The routines come in three families and three flavors: \newline\tab{3} closed:\tab{20}romberg,{}\tab{30}simpson,{}\tab{42}trapezoidal \newline\tab{3} open: \tab{20}rombergo,{}\tab{30}simpsono,{}\tab{42}trapezoidalo 
802630`--\newline\tab{3} adaptive closed:\tab{20}aromberg,{}\tab{30}asimpson,{}\tab{42}atrapezoidal \par The {\em S(i)} for the trapezoidal family is the value of the integral using an equally spaced absicca trapezoidal rule for that level of refinement. \par The {\em S(i)} for the simpson family is the value of the integral using an equally spaced absicca simpson rule for that level of refinement. \par The {\em S(i)} for the romberg family is the estimate of the integral using an equally spaced absicc
802630`--a romberg method. For the \spad{i}\spad{-}th level,{} this is an appropriate combination of all the previous trapezodial estimates so that the error term starts with the \spad{2*(i+1)} power only. \par The three families come in a closed version,{} where the formulas include the endpoints,{} an open version where the formulas do not include the endpoints and an adaptive version,{} where the user is required to input the number of subintervals over which the appropriate closed family integrator
802630`-- will apply with the usual convergence parmeters for each subinterval. This is useful where a large number of points are needed only in a small fraction of the entire domain. \par Each routine takes as arguments: \newline \spad{f}\tab{10} integrand \newline a\tab{10} starting point \newline \spad{b}\tab{10} ending point \newline \spad{eps_r}\tab{10} relative error \newline \spad{eps_a}\tab{10} absolute error \newline \spad{nmin} \tab{10} refinement level when to start checking for convergence 
802630`--(> 1) \newline \spad{nmax} \tab{10} maximum level of refinement \par The adaptive routines take as an additional parameter \newline \spad{nint}\tab{10} the number of independent intervals to apply a closed \indented{1}{family integrator of the same name.} \par Notes: \newline Closed family level \spad{i} uses \spad{1 + 2^i} points. \newline Open family level \spad{i} uses \spad{1 + 3^i} points.
802679`\spadtype{ODEIntegration} provides an interface to the integrator. This package is intended for use by the differential equations solver but not at top-level.
802797`\spad{ODETools} provides tools for the linear ODE solver.
802902`\spad{OctonionCategoryFunctions2} implements functions between two octonion domains defined over different rings. The function map is used to coerce between octonion types.
803046`\indented{1}{This package provides tools for operating on one-dimensional arrays} with unary and binary functions involving different underlying types
803121`Lifting of maps to one-point completions. Date Created: 4 Oct 1989 Date Last Updated: 4 Oct 1989
803210`\spadtype{OpenMathPackage} provides some simple utilities to make reading OpenMath objects easier.
803253`\spadtype{OpenMathServerPackage} provides the necessary operations to run AXIOM as an OpenMath server,{} reading/writing objects to/from a port. Please note the facilities available here are very basic. The idea is that a user calls \spadignore{e.g.} \axiom{Omserve(4000,{}60)} and then another process sends OpenMath objects to port 4000 and reads the result.
803305`This package exports tools to create FriCAS Library information databases.
803350`Lifting of maps to ordered completions. Date Created: 4 Oct 1989 Date Last Updated: 4 Oct 1989
803438`\indented{3}{This package provides ordering functions on vectors which} are suitable parameters for OrderedDirectProduct.
803533`This package provides orthogonal polynomials as functions on a ring.
803610`OutPackage allows pretty-printing from programs.
803649`In this package \spad{K} is a finite field,{} \spad{R} is a ring of univariate polynomials over \spad{K},{} and \spad{F} is a monogenic algebra over \spad{R}. We require that \spad{F} is monogenic,{} \spadignore{i.e.} that \spad{F = K[x,{}y]/(f(x,{}y))},{} because the integral basis algorithm used will factor the pol
803649`--ynomial \spad{f(x,{}y)}. The package provides a function to compute the integral closure of \spad{R} in the quotient field of \spad{F} as well as a function to compute a "local integral basis" at a specific prime.
803838`\indented{1}{This package computes reliable Pad&ea. approximants using} a generalized Viskovatov continued fraction algorithm. Authors: Trager,{}Burge,{} Hassner & Watt. Date Created: April 1987 Date Last Updated: 12 April 1990 Keywords: Pade,{} series Examples: References: \indented{2}{"Pade Approximants,{} Part I: Basic Theory",{} Baker & Graves-Morris.}
803919`\indented{1}{This package computes reliable Pad&ea. approximants using} a generalized Viskovatov continued fraction algorithm. Authors: Burge,{} Hassner & Watt. Date Created: April 1987 Date Last Updated: 12 April 1990 Keywords: Pade,{} series Examples: References: \indented{2}{"Pade Approximants,{} Part I: Basic Theory",{} Baker & Graves-Morris.}
804050`This package implements fixed-point computations on streams.
804119`Author: William Sit,{} spring 89
804305`This package \undocumented
804386`This package \undocumented
804467`This package \undocumented
804545`The package \spadtype{PartialFractionPackage} gives an easier to use interfact the domain \spadtype{PartialFraction}. The user gives a fraction of polynomials,{} and a variable and the package converts it to the proper datatype for the \spadtype{PartialFraction} domain.
804624`PartitionsAndPermutations contains functions for generating streams of integer partitions,{} and streams of sequences of integers composed from a multi-set.
804680`Tools for patterns.
804751`Lifts maps to patterns.
804829`Attaching assertions to symbols for pattern matching. Date Created: 21 Mar 1989 Date Last Updated: 23 May 1990
804879`This package provides pattern matching functions on function spaces.
805007`This package provides pattern matching functions on integers.
805088`\spadtype{PatternMatchIntegration} provides functions that use the pattern matcher to find some indefinite and definite integrals involving special functions and found in the litterature.
805203`This package provides pattern matching functions on kernels.
805293`This package provides pattern matching functions on lists.
805409`This package provides pattern matching functions on polynomials.
805587`This packages provides tools for matching recursively in type towers.
805702`This package provides pattern matching functions on quotients.
805839`Lifts maps to pattern matching results.
805943`This package provides pattern matching functions on symbols.
806003`This package provides tools for the pattern matcher.
806110`This package provides the top-level pattern macthing functions.
806251`Permanent implements the functions {\em permanent},{} the permanent for square matrices.
806326`PermutationGroupExamples provides permutation groups for some classes of groups: symmetric,{} alternating,{} dihedral,{} cyclic,{} direct products of cyclic,{} which are in fact the finite abelian groups of symmetric groups called Young subgroups. Furthermore,{} Rubik\spad{'s} group as permutation group of 48 integers and a list of sporadic simple groups derived from the atlas of finite groups.
806376`\indented{1}{Provides a coercion from the symbolic fractions in \%\spad{pi} with} integer coefficients to any Expression type. Date Created: 21 Feb 1990 Date Last Updated: 21 Feb 1990
806441`\spad{PlotFunctions1} provides facilities for plotting curves where functions \spad{SF} \spad{->} \spad{SF} are specified by giving an expression
806510`This package exports plotting tools
806550`This package \undocumented
806614`This package \undocumented
806662`This package provides function for testing whether a divisor on a curve is a torsion divisor.
806876`\indented{1}{Utilities for PFOQ and PFO} Author: Manuel Bronstein Date Created: 25 Aug 1988 Date Last Updated: 11 Jul 1990
807038`This package provides function for testing whether a divisor on a curve is a torsion divisor.
807260`Package with the conversion functions among different kind of polynomials
807324`Groebner functions for \spad{P} \spad{F} \indented{2}{This package is an interface package to the groebner basis} package which allows you to compute groebner bases for polynomials in either lexicographic ordering or total degree ordering refined by reverse lex. The input is the ordinary polynomial type which is internally converted to a type with the required ordering. The resulting grobner basis is converted back to ordinary polynomials. The orderi
807324`--ng among the variables is controlled by an explicit list of variables which is passed as a second argument. The coefficient domain is allowed to be any \spad{gcd} domain,{} but the groebner basis is computed as if the polynomials were over a field.
807377`This package provides a coerce from polynomials over algebraic numbers to \spadtype{Expression AlgebraicNumber}.
807431`This package provides a very general map function,{} which given a set \spad{S} and polynomials over \spad{R} with maps from the variables into \spad{S} and the coefficients into \spad{S},{} maps polynomials into \spad{S}. \spad{S} is assumed to support \spad{+},{} \spad{*} and \spad{^}.
807604`This package transforms multivariate polynomials or fractions into univariate polynomials or fractions,{} and back.
807772`This package \undocumented
807863`This package \undocumented
807959`\indented{1}{PolynomialFactorizationByRecursionUnivariate} \spad{R} is a \spadfun{PolynomialFactorizationExplicit} domain,{} \spad{S} is univariate polynomials over \spad{R} We are interested in handling SparseUnivariatePolynomials over \spad{S},{} is a variable we shall call \spad{z}
808103`PolynomialFactorizationByRecursion(\spad{R},{}\spad{E},{}\spad{VarSet},{}\spad{S}) is used for factorization of sparse univariate polynomials over a domain \spad{S} of multivariate polynomials over \spad{R}.
808288`\indented{2}{This package takes a mapping between coefficient rings,{} and lifts} it to a mapping between polynomials over those rings.
808490`This package exports interpolation algorithms
808600`This package exports interpolation algorithms
808673`This package provides various polynomial number theoretic functions over the integers.
808735`computes \spad{n}-th roots of quotients of multivariate polynomials
808893`This package provides modest routines for polynomial system solving. The aim of many of the operations of this package is to remove certain factors in some polynomials in order to avoid unnecessary computations in algorithms involving splitting techniques by partial factorization.
809052`This package factors the formulas out of the general solve code,{} allowing their recursive use over different domains. Care is taken to introduce few radicals so that radical extension domains can more easily simplify the results.
809163`This package computes square-free decomposition of multivariate polynomials over a coefficient ring which is an arbitrary \spad{gcd} domain. The requirement on the coefficient domain guarantees that the \spadfun{content} can be removed so that factors will be primitive as well as square-free. Over an infinite ring of finite characteristic,{}it may not 
809163`--be possible to guarantee that the factors are square-free.
809316`This package is primarily to help the interpreter do coercions. It allows you to view a polynomial as a univariate polynomial in one of its variables with coefficients which are again a polynomial in all the other variables.
809396`PowerSeriesLimitPackage implements limits of expressions in one or more variables as one of the variables approaches a limiting value. Included are two-sided limits,{} left- and right- hand limits,{} and limits at plus or minus infinity.
809514`\spadtype{PrecomputedAssociatedEquations} stores some generic precomputations which speed up the computations of the associated equations needed for factoring operators.
809644`\indented{1}{This package provides tools for operating on primitive arrays} with unary and binary functions involving different underlying types
809715`PrimitiveElement provides functions to compute primitive elements in algebraic extensions.
809779`\spad{PrimitiveRatDE} provides functions for in-field solutions of linear \indented{1}{ordinary differential equations,{} in the transcendental case.} \indented{1}{The derivation to use is given by the parameter \spad{L}.}
809995`In-field solution of Riccati equations,{} primitive case.
810212`PrintPackage provides a print function for output forms.
810252`PseudoLinearNormalForm provides a function for computing a block-companion form for pseudo-linear operators.
810313`This package contains some functions: \axiomOpFrom{discriminant}{PseudoRemainderSequence},{} \axiomOpFrom{resultant}{PseudoRemainderSequence},{} \axiomOpFrom{subResultantGcd}{PseudoRemainderSequence},{} \axiomOpFrom{chainSubResultants}{PseudoRemainderSequence},{} \axiomOpFrom{degreeSubResultant}{PseudoRemainderSequence},{} \axiomOpFrom{lastSubResultant}{PseudoRemainderSequence},{} \axiomOpFrom{
810313`--resultantEuclidean}{PseudoRemainderSequence},{} \axiomOpFrom{subResultantGcdEuclidean}{PseudoRemainderSequence},{} \axiomOpFrom{\spad{semiSubResultantGcdEuclidean1}}{PseudoRemainderSequence},{} \axiomOpFrom{\spad{semiSubResultantGcdEuclidean2}}{PseudoRemainderSequence},{} etc. This procedures are coming from improvements of the subresultants algorithm. \indented{2}{Version : 7} \indented{2}{References : Lionel Ducos "Optimizations of the subresultant algorithm"} \indented{2}{to appear in the J
810313`--ournal of Pure and Applied Algebra.} \indented{2}{Author : Ducos Lionel \axiom{Lionel.Ducos@mathlabo.univ-poitiers.\spad{fr}}}
810423`This package provides functions for integration,{} limited integration,{} extended integration and the risch differential equation for pure algebraic integrands.
810547`In-field solution of an linear ordinary differential equation,{} pure algebraic case.
810741`This package \undocumented{}
810895`\spadtype{QuasiAlgebraicSet2} adds a function \spadfun{radicalSimplify} which uses \spadtype{IdealDecompositionPackage} to simplify the representation of a quasi-algebraic set. A quasi-algebraic set is the intersection of a Zariski closed set,{} defined as the common zeros of a given list of polynomials (the defining polynomials for equations),{} and a principal Zariski open set,{} defined as the complement of the
810895`-- common zeros of a polynomial \spad{f} (the defining polynomial for the inequation). Quasi-algebraic sets are implemented in the domain \spadtype{QuasiAlgebraicSet},{} where two simplification routines are provided: \spadfun{idealSimplify} and \spadfun{simplify}. The function \spadfun{radicalSimplify} is added for comparison study only. Because the domain \spadtype{IdealDecompositionPackage} provides facilities for computing with radical ideals,{} it is necessary to restrict the ground ring to
810895`-- the domain \spadtype{Fraction Integer},{} and the polynomial ring to be of type \spadtype{DistributedMultivariatePolynomial}. The routine \spadfun{radicalSimplify} uses these to compute groebner basis of radical ideals and is inefficient and restricted when compared to the two in \spadtype{QuasiAlgebraicSet}.
810985`A package for removing redundant quasi-components and redundant branches when decomposing a variety by means of quasi-components of regular triangular sets. \newline References : \indented{1}{[1] \spad{D}. LAZARD "A new method for solving algebraic systems of} \indented{5}{positive dimension" Discr. App. Math. 33:1
810985`--47-160,{}1991} \indented{1}{[2] \spad{M}. MORENO MAZA "Calculs de pgcd au-dessus des tours} \indented{5}{d'extensions simples et resolution des systemes d'equations} \indented{5}{algebriques" These,{} Universite \spad{P}.etM. Curie,{} Paris,{} 1997.} \indented{1}{[3] \spad{M}. MORENO MAZA "A new algorithm for computing triangular} \indented{5}{decomposition of algebraic varieties" NAG Tech. Rep. 4/98.}
811176`\spadtype{QuaternionCategoryFunctions2} implements functions between two quaternion domains. The function \spadfun{map} is used by the system interpreter to coerce between quaternion types.
811327`This package extends a function between integral domains to a mapping between their quotient fields.
811483`Package for the computation of eigenvalues and eigenvectors. This package works for matrices with coefficients which are rational functions over the integers. (see \spadtype{Fraction Polynomial Integer}). The eigenvalues and eigenvectors are expressed in terms of radicals.
811528`This package tries to find solutions expressed in terms of radicals for systems of equations of rational functions with coefficients in an integral domain \spad{R}.
811606`This package provides tools for creating radix expansions.
811650`This package exports random distributions
811759`This package exports integer distributions
811814`Random number generators \indented{2}{All random numbers used in the system should originate from} \indented{2}{the same generator.\space{2}This package is intended to be the source.}
811955`\spadtype{RationalFunctionDefiniteIntegration} provides functions to compute definite integrals of rational functions.
812049`Factorization of univariate polynomials with coefficients which are rational functions with integer coefficients.
812163`\spadtype{RationalFunctionFactorizer} contains the factor function (called factorFraction) which factors fractions of polynomials by factoring the numerator and denominator. Since any non zero fraction is a unit the usual factor operation will just return the original fraction.
812238`This package provides functions for the integration of rational functions.
812320`Computation of limits for rational functions.
812390`Find the sign of a rational function around a point or infinity.
812451`Computes sums of rational functions.
812525`Utilities that provide the same top-level manipulations on fractions than on polynomials.
812583`This package provides functions for the base case of the Risch algorithm.
812689`This package exports rational interpolation algorithms
812760`\spad{RationalLODE} provides functions for in-field solutions of linear \indented{1}{ordinary differential equations,{} in the rational case.}
812859`rational number testing and retraction functions. Date Created: March 1990 Date Last Updated: 9 April 1991
812942`In-field solution of Riccati equations,{} rational case.
813044`\indented{1}{A package for computing the rational univariate representation} \indented{1}{of a zero-dimensional algebraic variety given by a regular} \indented{1}{triangular set. This package is essentially an interface for the} \spadtype{InternalRationalUnivariateRepresentationPackage} constructor. It is used in the \spadtype{ZeroDimensionalSolvePackage} for solving polynomial systems with fi
813044`--nitely many solutions.
813155`\axiomType{RealPolynomialUtilitiesPackage} provides common functions used by interval coding.
813294`\indented{1}{This package provides numerical solutions of systems of polynomial} equations for use in ACPLOT.
813341`\indented{2}{This package provides functions for finding the real zeros} of univariate polynomials over the rational numbers to arbitrary user-specified precision. The results are returned as a list of isolating intervals,{} expressed as records with "left" and "right" rational number components.
813438`\indented{2}{This package provides functions for finding the real zeros} of univariate polynomials over the integers to arbitrary user-specified precision. The results are returned as a list of isolating intervals which are expressed as records with "left" and "right" rational number components.
813523`\spadtype{RectangularMatrixCategoryFunctions2} provides functions between two matrix domains. The functions provided are \
813523`--spadfun{map} and \spadfun{reduce}.
813908`This package provides an operator for the \spad{n}-th term of a recurrence and an operator for the coefficient of \spad{x^n} in a function specified by a functional equation.
814010`Elimination of an algebraic from the coefficentss of a linear ordinary differential equation.
814226`\indented{1}{Finds the order of a divisor over a finite field} Author: Manuel Bronstein Date Created: 1988 Date Last Updated: 8 November 1994
814432`\spadtype{ReductionOfOrder} provides functions for reducing the order of linear ordinary differential equations once some solutions are known.
814539`A package providing a new algorithm for solving polynomial systems by means of regular chains. Two ways of solving are proposed: in the sense of Zariski closure (like in Kalkbrener\spad{'s} algorithm) or in the sense of the regular zeros (like in Wu,{} Wang or Lazard methods). This algorithm is valid for n
814539`--ay type of regular set. It does not care about the way a polynomial is added in an regular set,{} or how two quasi-components are compared (by an inclusion-test),{} or how the invertibility test is made in the tower of simple extensions associated with a regular set. These operations are realized respectively by the domain \spad{TS} and the packages \axiomType{QCMPACK}(\spad{R},{}\spad{E},{}\spad{V},{}\spad{P},{}\spad{TS}) and \axiomType{RSETGCD}(\spad{R},{}\spad{E},{}\spad{V},{}\spad{P},{}\sp
814539`--ad{TS}). The same way it does not care about the way univariate polynomial \spad{gcd} (with coefficients in the tower of simple extensions associated with a regular set) are computed. The only requirement is that these \spad{gcd} need to have invertible initials (normalized or not). WARNING. There is no need for a user to call diectly any operation of this package since they can be accessed by the domain \axiom{\spad{TS}}. Thus,{} the operations of this package are not documented.\newline Refe
814539`--rences : \indented{1}{[1] \spad{M}. MORENO MAZA "A new algorithm for computing triangular} \indented{5}{decomposition of algebraic varieties" NAG Tech. Rep. 4/98.}
814739`An internal package for computing gcds and resultants of univariate polynomials with coefficients in a tower of simple extensions of a field.\newline References : \indented{1}{[1] \spad{M}. MORENO MAZA and \spad{R}. RIOBOO "Computations of \spad{gcd} over} \indented{5}{algebraic towers of simple extensions
814739`--" In proceedings of \spad{AAECC11}} \indented{5}{Paris,{} 1995.} \indented{1}{[2] \spad{M}. MORENO MAZA "Calculs de pgcd au-dessus des tours} \indented{5}{d'extensions simples et resolution des systemes d'equations} \indented{5}{algebriques" These,{} Universite \spad{P}.etM. Curie,{} Paris,{} 1997.} \indented{1}{[3] \spad{M}. MORENO MAZA "A new algorithm for computing triangular} \indented{5}{decomposition of algebraic varieties" NAG Tech. Rep. 4/98.}
814939`Implements multiplication by repeated addition
815007`Implements exponentiation by repeated squaring
815075`\spad{RepresentationPackage1} provides functions for representation theory for finite groups and algebras. The package creates permutation representations and uses tensor products and its symmetric and antisymmetric components to create new representations of larger degree from given ones. Note: instead of having parameters from \spadtype{Permutation} this package allows list notation of permutations as well: \spadignore{e.g.} \spad{[1,{}4,{}3,{}2
815075`--]} denotes permutes 2 and 4 and fixes 1 and 3.
815131`\spad{RepresentationPackage2} provides functions for working with modular representations of finite groups and algebra. The routines in this package are created,{} using ideas of \spad{R}. Parker,{} (the meat-Axe) to get smaller representations from bigger ones,{} \spadignore{i.e.} finding sub- and factormodules,{} or to show,{} that such the representations are irreducible. Note: most functions are randomized functions of Las Vegas type \spadigno
815131`--re{i.e.} every answer is correct,{} but with small probability the algorithm fails to get an answer.
815187`This package provides coercions for the special types \spadtype{Exit} and \spadtype{Void}.
815248`RetractSolvePackage is an interface to \spadtype{SystemSolvePackage} that attempts to retract the coefficients of the equations before solving.
815344`Factorization of univariate polynomials with coefficients in an algebraic extension of \spadtype{Fraction Polynomial Integer}.
815534`\spadtype{ScriptFormulaFormat1} provides a utility coercion for changing to SCRIPT formula format anything that has a coercion to the standard output format.
815599`This package provides operations for mapping functions onto \spadtype{SegmentBinding}\spad{s}.
815670`This package provides operations for mapping functions onto segments.
815730`Factorization of univariate polynomials with coefficients in an algebraic extension of the rational numbers (\spadtype{Fraction Integer}).
815913`\indented{1}{Package to allow simplify to be called on AlgebraicNumbers} by converting to EXPR(INT)
815979`\spadtype{SmithNormalForm} is a package which provides some standard canonical forms for matrices.
816138`This package exports sorting algorithnms
816230`\indented{1}{Cache of elements in a set} Author: Manuel Bronstein Date Created: 31 Oct 1988 Date Last Updated: 14 May 1991 \indented{2}{A sorted cache of a cachable set \spad{S} is a dynamic structure that} \indented{2}{keeps the elements of \spad{S} sorted and assigns an integer to each} \indented{2}{element of \spad{S} once it is in the cache. This way,{} equality and ordering} \indented{2}{on \spad{S} are tested directly on the integers associate
816230`--d with the elements} \indented{2}{of \spad{S},{} once they have been entered in the cache.}
816401`This package lifts a mapping from coefficient rings \spad{R} to \spad{S} to a mapping from sparse univariate polynomial over \spad{R} to a sparse univariate polynomial over \spad{S}. Note that the mapping is assumed to send zero to zero,{} since it will only be applied to the non-zero coefficients of the polynomial.
816480`SpecialOutputPackage allows FORTRAN,{} Tex and \indented{2}{Script Formula Formatter output from programs.}
816530`\indented{2}{A internal package for removing redundant quasi-components and redundant} \indented{2}{branches when decomposing a variety by means of quasi-components} \indented{2}{of regular triangular sets. \newline} References : \indented{1}{[1] \spad{D}. LAZARD "A new method for solving algebraic system
816530`--s of} \indented{5}{positive dimension" Discr. App. Math. 33:147-160,{}1991} \indented{5}{Tech. Report (PoSSo project)} \indented{1}{[2] \spad{M}. MORENO MAZA "Calculs de pgcd au-dessus des tours} \indented{5}{d'extensions simples et resolution des systemes d'equations} \indented{5}{algebriques" These,{} Universite \spad{P}.etM. Curie,{} Paris,{} 1997.} \indented{1}{[3] \spad{M}. MORENO MAZA "A new algorithm for computing triangular} \indented{5}{decomposition of algebraic varieties" NAG Tech. 
816530`--Rep. 4/98.}
816731`A package providing a new algorithm for solving polynomial systems by means of regular chains. Two ways of solving are provided: in the sense of Zariski closure (like in Kalkbrener\spad{'s} algorithm) or in the sense of the regular zeros (like in Wu,{} Wang or Lazard- Moreno methods). T
816731`--his algorithm is valid for nay type of regular set. It does not care about the way a polynomial is added in an regular set,{} or how two quasi-components are compared (by an inclusion-test),{} or how the invertibility test is made in the tower of simple extensions associated with a regular set. These operations are realized respectively by the domain \spad{TS} and the packages \spad{QCMPPK(R,{}E,{}V,{}P,{}TS)} and \spad{RSETGCD(R,{}E,{}V,{}P,{}TS)}. The same way it does not care about the way 
816731`--univariate polynomial gcds (with coefficients in the tower of simple extensions associated with a regular set) are computed. The only requirement is that these gcds need to have invertible initials (normalized or not). WARNING. There is no need for a user to call diectly any operation of this package since they can be accessed by the domain \axiomType{\spad{TS}}. Thus,{} the operations of this package are not documented.\newline References : \indented{1}{[1] \spad{M}. MORENO MAZA "A new algori
816731`--thm for computing triangular} \indented{5}{decomposition of algebraic varieties" NAG Tech. Rep. 4/98.}
816951`A internal package for computing gcds and resultants of univariate polynomials with coefficients in a tower of simple extensions of a field. There is no need to use directly this package since its main operations are available from \spad{TS}. \newline References : \indented{1}{[1] \spad{M}. MORENO
816951`-- MAZA and \spad{R}. RIOBOO "Computations of \spad{gcd} over} \indented{5}{algebraic towers of simple extensions" In proceedings of \spad{AAECC11}} \indented{5}{Paris,{} 1995.} \indented{1}{[2] \spad{M}. MORENO MAZA "Calculs de pgcd au-dessus des tours} \indented{5}{d'extensions simples et resolution des systemes d'equations} \indented{5}{algebriques" These,{} Universite \spad{P}.etM. Curie,{} Paris,{} 1997.} \indented{1}{[3] \spad{M}. MORENO MAZA "A new algorithm for computing triangular} \ind
816951`--ented{5}{decomposition of algebraic varieties" NAG Tech. Rep. 4/98.}
817160`This package provides standard arithmetic operations on matrices. The functions in this package store the results of computations in existing matrices,{} rather than creating new matrices. This package works only for matrices of type Matrix and uses the internal representation of this type.
817229`Functions defined on streams with entries in one set.
817282`Functions defined on streams with entries in two sets.
817344`Functions defined on streams with entries in three sets.
817415`This package computes infinite products of Taylor series over an integral domain of characteristic 0. Here Taylor series are represented by streams of Taylor coefficients.
817497`StreamTaylorSeriesOperations implements Taylor series arithmetic,{} where a Taylor series is represented by a stream of its coefficients.
817605`StreamTranscendentalFunctionsNonCommutative implements transcendental functions on Taylor series over a non-commutative ring,{} where a Taylor series is represented by a stream of its coefficients.
817709`StreamTranscendentalFunctions implements transcendental functions on Taylor series,{} where a Taylor series is represented by a stream of its coefficients.
817797`StructuralConstantsPackage provides functions creating structural constants from a multiplication tables or a basis of a matrix algebra and other useful functions in this context.
817859`This package produces functions for counting etc. real roots of univariate polynomials in \spad{x} over \spad{R},{} which must be an OrderedIntegralDomain
817939`This package computes the subresultants of two polynomials which is needed for the `Lazard Rioboo' enhancement to Tragers integrations formula For efficiency reasons this has been rewritten to call Lionel Ducos package which is currently the best one. \blankline
818045`\indented{1}{SupFractionFactorize} contains the factor function for univariate polynomials over the quotient field of a ring \spad{S} such that the package MultivariateFactorize works for \spad{S}
818187`Computes all the symmetric functions in \spad{n} variables.
818242`SymmetricGroupCombinatoricFunctions contains combinatoric functions concerning symmetric groups and representation theory: list young tableaus,{} improper partitions,{} subsets bijection of Coleman.
818304`SystemODESolver provides tools for triangulating and solving some systems of linear ordinary differential equations.
818470`TableauBumpers implements the Schenstead-Knuth correspondence between sequences and pairs of Young tableaux. The 2 Young tableaux are represented as a single tableau with pairs as components.
818529`\axiom{TabulatedComputationPackage(Key ,{}Entry)} provides some modest support for dealing with operations with type \axiom{Key \spad{->} Entry}. The result of such operations can be stored and retrieved with this package by using a hash-table. The user does not need to worry about the management of this hash-table. However,{} onnly one hash-table is built by calling \axiom{TabulatedComputationPackage(Key ,
818529`--{}Entry)}.
818626`Expands tangents of sums and scalar products.
818860`This package provides functions for template manipulation
818906`linear extensions of maps on the basis
819059`\spadtype{TexFormat1} provides a utility coercion for changing to TeX format anything that has a coercion to the standard output format.
819110`Tools for the sign finding utilities.
819160`TopLevelDrawFunctionsForAlgebraicCurves provides top level functions for drawing non-singular algebraic curves.
819282`TopLevelDrawFunctionsForCompiledFunctions provides top level functions for drawing graphics of expressions.
819354`TopLevelDrawFunctionsForPoints provides top level functions for drawing curves and surfaces described by sets of points.
819413`TopLevelDrawFunctions provides top level functions for drawing graphics of expressions.
819499`This package exports a function for making a \spadtype{ThreeSpace}
819545`This package finds the function \spad{func3} where \spad{func1} and \spad{func2} \indented{1}{are given and\space{2}\spad{func1} = \spad{func3}(\spad{func2}) .\space{2}If there is no solution then} \indented{1}{function \spad{func1} will be returned.} \indented{1}{An example would be\space{2}\spad{func1:= 8*X^3+32*X^2-14*X ::EXPR INT} and} \indented{1}{\spad{func2:=2*X ::EXPR INT} convert them via univariate} \indented{1}{to 
819545`--FRAC SUP EXPR INT and then the solution is \spad{func3:=X^3+X^2-X}} \indented{1}{of type FRAC SUP EXPR INT}
819623`This package tries to find solutions of equations of type Expression(\spad{R}). This means expressions involving transcendental,{} exponential,{} logarithmic and nthRoot functions. After trying to transform different kernels to one kernel by applying several rules,{} it calls zerosOf for the SparseUnivariatePolynomial in the remaining kernel. For example the expression \spad{sin(x)*cos(x)-2} will be transformed to \indented{3}{\spad
819623`--{-2 tan(x/2)^4 -2 tan(x/2)^3 -4 tan(x/2)^2 +2 tan(x/2) -2}} by using the function normalize and then to \indented{3}{\spad{-2 tan(x)^2 + tan(x) -2}} with help of subsTan. This function tries to express the given function in terms of \spad{tan(x/2)} to express in terms of \spad{tan(x)} . Other examples are the expressions \spad{sqrt(x+1)+sqrt(x+7)+1} or \indented{1}{\spad{sqrt(sin(x))+1} .}
819694`Hermite integration,{} transcendental case.
819805`This package provides functions for the transcendental case of the Risch algorithm.
819906`TranscendentalManipulations provides functions to simplify and expand expressions involving transcendental operators.
820019`\indented{1}{Risch differential equation system,{} transcendental case.} Author: Manuel Bronstein Date Created: 17 August 1992 Date Last Updated: 3 February 1994
820133`\indented{1}{Risch differential equation,{} transcendental case.} Author: Manuel Bronstein Date Created: Jan 1988 Date Last Updated: 2 November 1995
820240`This package provides functions that compute "fraction-free" inverses of upper and lower triangular matrices over a integral domain. By "fraction-free inverses" we mean the following: given a matrix \spad{B} with entries in \spad{R} and an element \spad{d} of \spad{R} such that \spad{d} * inv(\spad{B}) also has entries in \spad{R},{} w
820240`--e return \spad{d} * inv(\spad{B}). Thus,{} it is not necessary to pass to the quotient field in any of our computations.
820410`\spadtype{TrigonometricManipulations} provides transformations from trigonometric functions to complex exponentials and logarithms,{} and back.
820530`Tools for constructing tubes around 3-dimensional parametric curves.
820574`\indented{1}{The purpose of this package is to provide reasonable plots of} functions with singularities.
820627`A basic package for the factorization of bivariate polynomials over a finite field. The functions here represent the base step for the multivariate factorizer.
820691`This is a low-level package which implements operations \indented{1}{on vectors treated as univariate modular polynomials.\space{2}Most} \indented{1}{operations takes modulus as parameter.\space{2}Modulus is machine} \indented{1}{sized prime which should be small enough to avoid overflow} \indented{1}{in intermediate calculations.}
820750`\spad{RUTSodetools} provides tools to interface with the series \indented{1}{ODE solver when presented with linear ODEs.}
820932`UnittestAux is an auxiliary package to Unittest,{} that performs \indented{13}{checking of tests,{} dealing also with fatal errors.}
820985`UnittestCount is a package that keeps track of statistics \indented{13}{collected during testing.\space{2}It is separate from} \indented{13}{\spadtype{UnittestAux} because the latter takes an argument,{}} \indented{13}{and we want the state to be common for all arguments.}
821028`Unittest is a package that provides functions for regression \indented{13}{testing.\space{2}Note that it calls the interpreter,{} thus,{} all} \indented{13}{functions require that there arguments be passed as strings.}
821067`Package for the factorization of univariate polynomials with integer coefficients. The factorization is done by "lifting" (HENSEL) the factorization over a finite field.
821226`Mapping package for univariate Laurent series \indented{2}{This package allows one to apply a function to the coefficients of} \indented{2}{a univariate Laurent series.}
821386`Mapping from polynomials over \spad{R} to polynomials over \spad{S} given a map from \spad{R} to \spad{S} assumed to send zero to zero.
821548`UnivariatePolynomialCommonDenominator provides functions to compute the common denominator of the coefficients of univariate polynomials over the quotient field of a \spad{gcd} domain.
821700`UnivariatePolynomialDecompositionPackage implements functional decomposition of univariate polynomial with coefficients in an \spad{IntegralDomain} of \spad{CharacteristicZero}.
821838`UnivariatePolynomialDivisionPackage provides a division for non monic univarite polynomials with coefficients in an \spad{IntegralDomain}.
821959`This package lifts a mapping from coefficient rings \spad{R} to \spad{S} to a mapping from \spadtype{UnivariatePolynomial}(\spad{x},{}\spad{R}) to \spadtype{UnivariatePolynomial}(\spad{y},{}\spad{S}). Note that the mapping is assumed to send zero to zero,{} since it will only be applied to the non-zero coefficients of the polynomial.
822053`This package implements Karatsuba\spad{'s} trick for multiplying (large) univariate polynomials. It could be improved with a version doing the work on place and also with a special case for squares. We've done this in Basicmath,{} but we believe that this out of the scope of FriCAS.
822166`This package provides for square-free decomposition of univariate polynomials over arbitrary rings,{} \spadignore{i.e.} a partial factorization such that each factor is a product of irreducibles with multiplicity one and the factors are pairwise relatively prime. If the ring has characteristic zero,{} the result is guaranteed to satisfy this condition. If the ring is an infin
822166`--ite ring of finite characteristic,{} then it may not be possible to decide when polynomials contain factors which are \spad{p}th powers. In this case,{} the flag associated with that polynomial is set to "nil" (meaning that that polynomials are not guaranteed to be square-free).
822295`Mapping package for univariate Puiseux series. This package allows one to apply a function to the coefficients of a univariate Puiseux series.
822456`\spad{UnivariateSkewPolynomialCategoryOps} provides products and \indented{1}{divisions of univariate skew polynomials.}
822570`Mapping package for univariate Taylor series. \indented{2}{This package allows one to apply a function to the coefficients of} \indented{2}{a univariate Taylor series.}
822759`\indented{1}{This package provides Taylor series solutions to regular} linear or non-linear ordinary differential equations of arbitrary order.
822901`This package provides operations for mapping functions onto segments.
822973`Provides functions to force a partial ordering on any set.
823040`This packages provides functions to allow the user to select the ordering on the variables and operators for displaying polynomials,{} fractions and expressions. The ordering affects the display only and not the computations.
823094`\indented{2}{This package provides operations which all take as arguments} vectors of elements of some type \spad{A} and functions from \spad{A} to another of type \spad{B}. The operations all iterate over their vector argument and either return a value of type \spad{B} or a vector over \spad{B}.
823156`ViewportDefaultsPackage describes default and user definable values for graphics
823205`ViewportPackage provides functions for creating GraphImages and TwoDimensionalViewports from lists of lists of points.
823247`This package implements the Weierstrass preparation theorem \spad{f} for multivariate power series. weierstrass(\spad{v},{}\spad{p}) where \spad{v} is a variable,{} and \spad{p} is a TaylorSeries(\spad{R}) in which there is term of form c*v^s where \spad{c} is a constant. Let \spad{s} be minimal as above. The result is a list of TaylorSeries coefficients A[\spad{i}] of the equivalent polynomial A = A[0] + A[1]\spad{*v} + A[2]\spad{*v^2} + ... + 
823247`--A[\spad{s}-1]\spad{*v^}(\spad{s}-1) + \spad{v^s} such that p=A*B ,{} \spad{B} being a TaylorSeries of minimum degree 0
823305`In this package \spad{K} is a finite field,{} \spad{R} is a ring of univariate polynomials over \spad{K},{} and \spad{F} is a framed algebra over \spad{R}. The package provides a function to compute the integral closure of \spad{R} in the quotient field of \spad{F} as well as a function to compute a "local integral basis" at
823305`-- a specific prime.
823486`This package provides computations of logarithms and exponentials for polynomials in non-commutative variables. \newline Author: Michel Petitot (petitot@lifl.\spad{fr}).
823617`A package for computing symbolically the complex and real roots of zero-dimensional algebraic systems over the integer or rational numbers. Complex roots are given by means of univariate representations of irreducible regular chains. Real roots are given by means of tuples of coordinates lying in the \spadtype{RealClosure} of the coefficient ring. This constructor takes three argumen
823617`--ts. The first one \spad{R} is the coefficient ring. The second one \spad{ls} is the list of variables involved in the systems to solve. The third one must be \spad{concat(ls,{}s)} where \spad{s} is an additional symbol used for the univariate representations. WARNING: The third argument is not checked. All operations are based on triangular decompositions. The default is to compute these decompositions directly from the input system by using the \spadtype{RegularChain} domain constructor. The 
823617`--lexTriangular algorithm can also be used for computing these decompositions (see the \spadtype{LexTriangularPackage} package constructor). For that purpose,{} the operations \axiomOpFrom{univariateSolve}{ZeroDimensionalSolvePackage},{} \axiomOpFrom{realSolve}{ZeroDimensionalSolvePackage} and \axiomOpFrom{positiveSolve}{ZeroDimensionalSolvePackage} admit an optional argument. \newline Author: Marc Moreno Maza.
823738`The class of abelian groups,{} \spadignore{i.e.} additive monoids where each element has an additive inverse. \blankline
823797`The class of multiplicative monoids,{} \spadignore{i.e.} semigroups with an additive identity element. \blankline
823858`Abelian monoid ring elements (not necessarily of finite support) of this ring are of the form formal SUM (r_i * e_i) where the r_i are coefficents and the e_i,{} elements of the ordered abelian monoid,{} are thought of as exponents or monomials. The monomials commute with each other,{} and with the coefficients (which themselves may or may not be commutative). See \spadtype{FiniteAbelianMonoidRing} f
823858`--or the case of finite support. A useful common model for polynomials and power series. Conceptually at least,{} only the non-zero terms are ever operated on.
823962`the class of all additive (commutative) semigroups,{} \spadignore{i.e.} a set with a commutative and associative operation \spadop{+}. \blankline
824028`The notion of aggregate serves to model any data structure aggregate,{} designating any collection of objects,{} with heterogenous or homogeneous members,{} with a finite or infinite number of members,{} explicitly or implicitly represented. An aggregate can in principle represent everything from a string of characters to abstract sets such as "the set of \spad{x} satisfying relation {\em r(x)}" An attribute \spadatt{finiteAggregate} is used to assert th
824028`--at a domain has a finite number of elements.
824077`The category of associative algebras (modules which are themselves rings). \blankline
824149`Model for algebraically closed fields.
824228`Model for algebraically closed function spaces.
824350`Category for the inverse trigonometric functions.
824447`A bag aggregate is an aggregate for which one can insert and extract objects,{} and where the order in which objects are inserted determines the order of extraction. Examples of bags are stacks,{} queues,{} and dequeues.
824516`\spadtype{BasicType} is the basic category for describing a collection of elements with \spadop{=} (equality).
824569`A binary-recursive aggregate has 0,{} 1 or 2 children and serves as a model for a binary tree or a doubly-linked aggregate structure
824662`\spadtype{BinaryTreeCategory(S)} is the category of binary trees: a tree which is either empty or else is a \spadfun{node} consisting of a value and a \spadfun{left} and \spadfun{right},{} both binary trees.
824750`The bit aggregate category models aggregates representing large quantities of Boolean data.
824807`A collection is a homogeneous aggregate which can built from list of members. The operation used to build the aggregate is generically named \spadfun{construct}. However,{} each collection provides its own special function with the same name as the data type,{} except with an initial lower case letter,{} \spadignore{e.g.} \spadfun{list} for \spadtype{List},{} \spadfun{flexibleArray} for \spadtype{FlexibleArray},{} and so on.
824872`This category represents the extension of a ring by a square root of \spad{-1}.
824960`A dictionary is an aggregate in which entries can be inserted,{} searched for and removed. Duplicates are thrown away on insertion. This category models the usual notion of dictionary which involves large amounts of data where copying is impractical. Principal operations are thus destructive (non-copying) ones.
825032`This category is a collection of operations common to both categories \spadtype{Dictionary} and \spadtype{MultiDictionary}
825124`Differential extensions of a ring \spad{R}. Given a differentiation on \spad{R},{} extend it to a differentiation on \%.
825212`\spadtype{DifferentialPolynomialCategory} is a category constructor specifying basic functions in an ordinary differential polynomial ring with a given ordered set of differential indeterminates. In addition,{} it implements defaults for the basic functions. The functions \spadfun{order} and \spadfun{weight} are e
825212`--xtended from the set of derivatives of differential indeterminates to the set of differential polynomials. Other operations provided on differential polynomials are \spadfun{leader},{} \spadfun{initial},{} \spadfun{separant},{} \spadfun{differentialVariables},{} and \spadfun{isobaric?}. Furthermore,{} if the ground ring is a differential ring,{} then evaluation (substitution of differential indeterminates by elements of the ground ring or by differential polynomials) is provided by \spadfun{ev
825212`--al}. A convenient way of referencing derivatives is provided by the functions \spadfun{makeVariable}. \blankline To construct a domain using this constructor,{} one needs to provide a ground ring \spad{R},{} an ordered set \spad{S} of differential indeterminates,{} a ranking \spad{V} on the set of derivatives of the differential indeterminates,{} and a set \spad{E} of exponents in bijection with the set of differential monomials in the given differential indeterminates. \blankline
825404`An ordinary differential ring,{} that is,{} a ring with an operation \spadfun{differentiate}. \blankline
825471`\spadtype{DifferentialVariableCategory} constructs the set of derivatives of a given set of (ordinary) differential indeterminates. If \spad{x},{}...,{}\spad{y} is an ordered set of differential indeterminates,{} and the prime notation is used for differentiation,{} then the set of derivatives (including zero-th order) of the differential indeterminates is \spad{x},{}\spad{x'},{}\spad{x''},{}...
825471`--,{} \spad{y},{}\spad{y'},{}\spad{y''},{}... (Note: in the interpreter,{} the \spad{n}-th derivative of \spad{y} is displayed as \spad{y} with a subscript \spad{n}.) This set is viewed as a set of algebraic indeterminates,{} totally ordered in a way compatible with differentiation and the given order on the differential indeterminates. Such a total order is called a ranking of the differential indeterminates. \blankline A domain in this category is needed to construct a differential polynomial 
825471`--domain. Differential polynomials are ordered by a ranking on the derivatives,{} and by an order (extending the ranking) on on the set of differential monomials. One may thus associate a domain in this category with a ranking of the differential indeterminates,{} just as one associates a domain in the category \spadtype{OrderedAbelianMonoidSup} with an ordering of the set of monomials in a set of algebraic indeterminates. The ranking is specified through the binary relation \spadfun{<}. For exa
825471`--mple,{} one may define one derivative to be less than another by lexicographically comparing first the \spadfun{order},{} then the given order of the differential indeterminates appearing in the derivatives. This is the default implementation. \blankline The notion of weight generalizes that of degree. A polynomial domain may be made into a graded ring if a weight function is given on the set of indeterminates,{} Very often,{} a grading is the first step in ordering the set of monomials. For d
825471`--ifferential polynomial domains,{} this constructor provides a function \spadfun{weight},{} which allows the assignment of a non-negative number to each derivative of a differential indeterminate. For example,{} one may define the weight of a derivative to be simply its \spadfun{order} (this is the default assignment). This weight function can then be extended to the set of all differential polynomials,{} providing a graded ring structure.
825580`attribute to indicate an aggregate of finite size
825696`A division ring (sometimes called a skew field),{} \spadignore{i.e.} a not necessarily commutative ring where all non-zero elements have multiplicative inverses.
825755`Category for the elementary functions.
825842`An eltable aggregate is one which can be viewed as a function. For example,{} the list \axiom{[1,{}7,{}4]} can applied to 0,{}1,{} and 2 respectively will return the integers 1,{}7,{} and 4; thus this list may be viewed as mapping 0 to 1,{} 1 to 7 and 2 to 4. In general,{} an aggregate can map members of a domain {\em Dom} to an image domain {\em Im}.
825947`A constructive euclidean domain,{} \spadignore{i.e.} one can divide producing a quotient and a remainder where the remainder is either zero or is smaller (\spadfun{euclideanSize}) than the divisor. \blankline Conditional attributes: \indented{2}{multiplicativeValuation\tab{25}\spad{Size(a*b)=Size(a)*Size(b)}} \indented{2}{additiveValuation\tab{25}\spad{Size(a*b)=Size(a)+Size(b)}}
826011`This category provides \spadfun{eval} operations. A domain may belong to this category if it is possible to make ``evaluation\spad{''} substitutions.
826080`An expression space is a set which is closed under certain operators.
826140`An extensible aggregate is one which allows insertion and deletion of entries. These aggregates are models of lists and streams which are represented by linked structures so as to make insertion,{} deletion,{} and concatenation efficient. However,{} access to elements of these extensible aggregates is generally slow since access is made from the end. See \spadtype{FlexibleArray} for an exception.
826235`ExtensionField {\em F} is the category of fields which extend the field \spad{F}
826304`The category of commutative fields,{} \spadignore{i.e.} commutative rings where all non-zero elements have multiplicative inverses. The \spadfun{factor} operation while trivial is useful to have defined. \blankline
826347`FieldOfPrimeCharacteristic is the category of fields of prime characteristic,{} \spadignore{e.g.} finite fields,{} algebraic closures of fields of prime characteristic,{} transcendental extensions of of fields of prime characteristic.
826430`The category of domains composed of a finite set of elements. We include the functions \spadfun{lookup} and \spadfun{index} to give a bijection between the finite set and an initial segment of positive integers. \blankline
826476`This category is similar to AbelianMonoidRing,{} except that the sum is assumed to be finite. It is a useful model for polynomials,{} but is somewhat more general.
826593`FiniteAlgebraicExtensionField {\em F} is the category of fields which are finite algebraic extensions of the field {\em F}. If {\em F} is finite then any finite algebraic extension of {\em F} is finite,{} too. Let {\em K} be a finite algebraic extension of the finite field {\em F}. The exponentiation of elements of {\em K} defines a \spad{Z}-module structure on the multiplicative group of {\em K}. The
826593`-- additive group of {\em K} becomes a module over the ring of polynomials over {\em F} via the operation \spadfun{linearAssociatedExp}(a:K,{}f:SparseUnivariatePolynomial \spad{F}) which is linear over {\em F},{} \spadignore{i.e.} for elements {\em a} from {\em K},{} {\em c,{}d} from {\em F} and {\em f,{}g} univariate polynomials over {\em F} we have \spadfun{linearAssociatedExp}(a,{}cf+dg) equals {\em c} times \spadfun{linearAssociatedExp}(a,{}\spad{f}) plus {\em d} times \spadfun{linearAssocia
826593`--tedExp}(a,{}\spad{g}). Therefore \spadfun{linearAssociatedExp} is defined completely by its action on monomials from {\em F[X]}: \spadfun{linearAssociatedExp}(a,{}monomial(1,{}\spad{k})\spad{\$}SUP(\spad{F})) is defined to be \spadfun{Frobenius}(a,{}\spad{k}) which is {\em a^(q^k)} where {\em q=size()\$F}. The operations order and discreteLog associated with the multiplicative exponentiation have additive analogues associated to the operation \spadfun{linearAssociatedExp}. These are the functi
826593`--ons \spadfun{linearAssociatedOrder} and \spadfun{linearAssociatedLog},{} respectively.
826696`This category describes finite rational divisors on a curve,{} that is finite formal sums SUM(\spad{n} * \spad{P}) where the \spad{n}\spad{'s} are integers and the \spad{P}\spad{'s} are finite rational points on the curve.
826924`FiniteFieldCategory is the category of finite fields
826997`A finite linear aggregate is a linear aggregate of finite length. The finite property of the aggregate adds several exports to the list of exports from \spadtype{LinearAggregate} such as \spadfun{reverse},{} \spadfun{sort},{} and so on.
827084`A FiniteRankAlgebra is an algebra over a commutative ring \spad{R} which is a free \spad{R}-module of finite rank.
827217`A FiniteRankNonAssociativeAlgebra is a non associative algebra over a commutative ring \spad{R} which is a free \spad{R}-module of finite rank.
827337`A finite-set aggregate models the notion of a finite set,{} that is,{} a collection of elements characterized by membership,{} but not by order or multiplicity. See \spadtype{Set} for an example.
827425`This category is intended as a model for floating point systems. A floating point system is a model for the real numbers. In fact,{} it is an approximation in the sense that not all real numbers are exactly representable by floating point numbers. A floating point system is characterized by the following: \blankline \indented{2}{1: \spadfunFrom{base}{FloatingPointSystem} of the \spadfunFrom{exponent}{FloatingPointSystem}.} \indented{9
827425`--}{(actual implemenations are usually binary or decimal)} \indented{2}{2: \spadfunFrom{precision}{FloatingPointSystem} of the \spadfunFrom{mantissa}{FloatingPointSystem} (arbitrary or fixed)} \indented{2}{3: rounding error for operations} \blankline Because a Float is an approximation to the real numbers,{} even though it is defined to be a join of a Field and OrderedRing,{} some of the attributes do not hold. In particular associative(\spad{"+"}) does not hold. Algorithms defined over a field 
827425`--need special considerations when the field is a floating point system.
827494`A \spadtype{FramedAlgebra} is a \spadtype{FiniteRankAlgebra} together with a fixed \spad{R}-module basis.
827619`FramedNonAssociativeAlgebra(\spad{R}) is a \spadtype{FiniteRankNonAssociativeAlgebra} (\spadignore{i.e.} a non associative algebra over \spad{R} which is a free \spad{R}-module of finite rank) over a commutative ring \spad{R} together with a fixed \spad{R}-module basis.
827731`A domain of this category implements formal linear combinations of elements from a domain \spad{Basis} with coefficients in a domain \spad{R}. The domain \spad{Basis} needs only to belong to the category \spadtype{SetCategory} and \spad{R} to the category \spadtype{Ring}. Thus the coefficient ring may be non-commutative. See the \spadtype{XDistributedPolynomial} constructor for examples of domains built with
827731`-- the \spadtype{FreeModuleCategory} category constructor. Author: Michel Petitot (petitot@lifl.\spad{fr}) \blankline Note (Franz Lehner,{} June 2009): Since \spad{leadingTerm} makes no sense for unordered base sets,{} and at the time of this writing this domain was never used for such,{} it was changed to \spad{OrderedSet}. \spad{FreeModule} originally was not of FreeModuleCategory. Some functions (like \spad{support},{} \spad{coefficients},{} \spad{monomials},{} ...) from here could be moved t
827731`--o \spad{IndexedDirectProductCategory} but at the moment there is no need for this.
827827`This category provides a selection of evaluation operations depending on what the argument type \spad{R} provides.
827915`\spad{S} is \spadtype{FullyLinearlyExplicitRingOver R} means that \spad{S} is a \spadtype{LinearlyExplicitRingOver R} and,{} in addition,{} if \spad{R} is a \spadtype{LinearlyExplicitRingOver Integer},{} then so is \spad{S}
828020`\indented{2}{A is fully retractable to \spad{B} means that A is retractable to \spad{B},{} and,{}} \indented{2}{in addition,{} if \spad{B} is retractable to the integers or rational} \indented{2}{numbers then so is A.} \indented{2}{In particular,{} what we are asserting is that there are no integers} \indented{2}{(rationals) in A which don\spad{'t} retract into \spad{B}.} Date Created: March 1990 Date Last Updated: 9 Apr
828020`--il 1991
828103`This category is a model for the function field of a plane algebraic curve.
828310`A space of formal functions with arguments in an arbitrary ordered set.
828384`This category describes domains where \spadfun{\spad{gcd}} can be computed but where there is no guarantee of the existence of \spadfun{factor} operation for factorisation into irreducibles. However,{} if such a \spadfun{factor} operation exist,{} factorization will be unique up to order and units.
828436`GradedAlgebra(\spad{R},{}\spad{E}) denotes ``E-graded \spad{R}-algebra\spad{''}. A graded algebra is a graded module together with a degree preserving \spad{R}-linear map,{} called the {\em product}. \blankline The name ``product\spad{''} is written out in full so inner and outer products with the same mapping type can be distinguished by name.
828538`GradedModule(\spad{R},{}\spad{E}) denotes ``E-graded \spad{R}-module\spad{''},{} \spadignore{i.e.} collection of \spad{R}-modules indexed by an abelian monoid \spad{E}. An element \spad{g} of \spad{G[s]} for some specific \spad{s} in \spad{E} is said to be an element of \spad{G} with {\em degree} \spad{s}. Sums are defined in each module \spad{G[s]} so two elements of \spad{G} have a sum if they have the s
828538`--ame degree. \blankline Morphisms can be defined and composed by degree to give the mathematical category of graded modules.
828636`The class of multiplicative groups,{} \spadignore{i.e.} monoids with multiplicative inverses. \blankline
828679`A homogeneous aggregate is an aggregate of elements all of the same type. In the current system,{} all aggregates are homogeneous. Two attributes characterize classes of aggregates. Aggregates from domains with attribute \spadatt{finiteAggregate} have a finite number of members. Of course,{} such a domain may have an infinite number of elements,{} like,{} for example \spadtype{List}. Those domains with attribute \spada
828679`--tt{shallowlyMutable} allow an element to be modified or updated without changing its overall value.
828764`Category for the hyperbolic trigonometric functions.
828850`An indexed aggregate is a many-to-one mapping of indices to entries. For example,{} a one-dimensional-array is an indexed aggregate where the index is an integer. Also,{} a table is an indexed aggregate where the indices and entries may have any type.
828969`This category provides \spadfun{eval} operations. A domain may belong to this category if it is possible to make ``evaluation\spad{''} substitutions. The difference between this and \spadtype{Evalable} is that the operations in this category specify the substitution as a pair of arguments rather than as an equation.
829060`An \spad{IntegerNumberSystem} is a model for the integers.
829129`The category of commutative integral domains,{} \spadignore{i.e.} commutative rings with no zero divisors. \blankline Conditional attributes: \indented{2}{canonicalUnitNormal\tab{20}the canonical field is the same for all associates} \indented{2}{canonicalsClosed\tab{20}the product of two canonicals is itself canonical}
829191`A keyed dictionary is a dictionary of key-entry pairs for which there is a unique entry for each key.
829309`LazyStreamAggregate is the category of streams with lazy evaluation. It is understood that the function 'empty?' will cause lazy evaluation if necessary to determine if there are entries. Functions which call 'empty?',{} \spadignore{e.g.} 'first' and 'rest',{} will also cause lazy evaluation if necessary.
829394`The category of all left algebras over an arbitrary ring.
829460`The category of Lie Algebras. It is used by the following domains of non-commutative algebra: \axiomType{LiePolynomial} and \axiomType{XPBWPolynomial}. \newline Author : Michel Petitot (petitot@lifl.\spad{fr}).
829537`A linear aggregate is an aggregate whose elements are indexed by integers. Examples of linear aggregates are strings,{} lists,{} and arrays. Most of the exported operations for linear aggregates are non-destructive but are not always efficient for a particular aggregate. For example,{} \spadfun{concat} of two lists needs only to copy its first argument,{} whereas \spadfun{concat} of two arrays needs to copy both arguments. Most 
829537`--of the operations exported here apply to infinite objects (\spadignore{e.g.} streams) as well to finite ones. For finite linear aggregates,{} see \spadtype{FiniteLinearAggregate}.
829612`\spad{LinearOrdinaryDifferentialOperatorCategory} is the category of differential operators with coefficients in a ring A with a given derivation. Multiplication of operators corresponds to functional composition: \indented{4}{\spad{(L1 * L2).(f) = L1 L2 f}}
829741`A list aggregate is a model for a linked list data structure. A linked list is a versatile data structure. Insertion and deletion are efficient and searching is a linear operation.
829812``Logic' provides the basic operations for lattices,{} \spadignore{e.g.} boolean algebra.
829855`\spadtype{MatrixCategory} is a general matrix category which allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and colums returned as objects of type Col. A domain belonging to this category will be shallowly mutable. The index of the 'first' row may be obtained by calling the 
829855`--function \spadfun{minRowIndex}. The index of the 'first' column may be obtained by calling the function \spadfun{minColIndex}. The index of the first element of a Row is the same as the index of the first column in a matrix and vice versa.
830003`The category of modules over a commutative ring. \blankline
830072`Monad is the class of all multiplicative monads,{} \spadignore{i.e.} sets with a binary operation.
830115`\indented{1}{MonadWithUnit is the class of multiplicative monads with unit,{}} \indented{1}{\spadignore{i.e.} sets with a binary operation and a unit element.} Axioms \indented{3}{leftIdentity("*":(\%,{}\%)\spad{->}\%,{}1)\space{3}\tab{30} 1*x=x} \indented{3}{rightIdentity("*":(\%,{}\%)\spad{->}\%,{}1)\space{2}\tab{30} x*1=x} Common Additional Axioms \indented{3}{unitsKnown
830115`---if "recip" says "failed",{} that PROVES input wasn\spad{'t} a unit}
830176`A \spadtype{MonogenicAlgebra} is an algebra of finite rank which can be generated by a single element.
830307`The class of multiplicative monoids,{} \spadignore{i.e.} semigroups with a multiplicative identity element. \blankline
830353`\spadtype{MultivariateTaylorSeriesCategory} is the most general multivariate Taylor series category.
830495`NonAssociativeAlgebra is the category of non associative algebras (modules which are themselves non associative rngs). Axioms \indented{3}{\spad{r*}(a*b) = (r*a)\spad{*b} = a*(\spad{r*b})}
830591`A NonAssociativeRing is a non associative \spad{rng} which has a unit,{} the multiplication is not necessarily commutative or associative.
830662`NonAssociativeRng is a basic ring-type structure,{} not necessarily commutative or associative,{} and not necessarily with unit. Axioms \indented{2}{\spad{x*}(\spad{y+z}) = x*y + \spad{x*z}} \indented{2}{(x+y)\spad{*z} = \spad{x*z} + \spad{y*z}} Common Additional Axioms \indented{2}{noZeroDivisors\space{2}ab = 0 \spad{=>} \spad{a=0} or \spad{b=0}}
830729`OctonionCategory gives the categorial frame for the octonions,{} and eight-dimensional non-associative algebra,{} doubling the the quaternions in the same way as doubling the Complex numbers to get the quaternions.
830814`One-dimensional-array aggregates serves as models for one-dimensional arrays. Categorically,{} these aggregates are finite linear aggregates with the \spadatt{shallowlyMutable} property,{} that is,{} any component of the array may be changed without affecting the identity of the overall array. Array data structures are typically represented by a fixed area in storage and therefore cannot efficiently gro
830814`--w or shrink on demand as can list structures (see however \spadtype{FlexibleArray} for a data structure which is a cross between a list and an array). Iteration over,{} and access to,{} elements of arrays is extremely fast (and often can be optimized to open-code). Insertion and deletion however is generally slow since an entirely new data structure must be created for the result.
830915`Ordered sets which are also rings,{} that is,{} domains where the ring operations are compatible with the ordering. \blankline
830972`The class of totally ordered sets,{} that is,{} sets such that for each pair of elements \spad{(a,{}b)} exactly one of the following relations holds \spad{a<b or a=b or b<a} and the relation is transitive,{} \spadignore{i.e.} \spad{a<b and b<c => a<c}. This order should be the natural order on given structure.
831026`A partial differential ring with differentiations indexed by a parameter type \spad{S}. \blankline
831125`The category for general multi-variate polynomials over a ring \spad{R},{} in variables from VarSet,{} with exponents from the \spadtype{OrderedAbelianMonoidSup}.
831270`This is the category of domains that know "enough" about themselves in order to factor univariate polynomials over themselves. This will be used in future releases for supporting factorization over finitely generated coefficient fields,{} it is not yet available in the current release of axiom.
831366`A category for finite subsets of a polynomial ring. Such a set is only regarded as a set of polynomials and not identified to the ideal it generates. So two distinct sets may generate the same the ideal. Furthermore,{} for \spad{R} being an integral domain,{} a set of polynomials may be viewed as a representa
831366`--tion of the ideal it generates in the polynomial ring \spad{(R)^(-1) P},{} or the set of its zeros (described for instance by the radical of the previous ideal,{} or a split of the associated affine variety) and so on. So this category provides operations about those different notions.
831563`\spadtype{PowerSeriesCategory} is the most general power series category with exponents in an ordered abelian monoid.
831717`\spadtype{QuaternionCategory} describes the category of quaternions and implements functions that are not representation specific.
831811`QuotientField(\spad{S}) is the category of fractions of an Integral Domain \spad{S}.
831908`The \spad{RadicalCategory} is a model for the rational numbers.
831972`\axiomType{RealClosedField} provides common acces functions for all real closed fields.
832037`The real number system category is intended as a model for the real numbers. The real numbers form an ordered normed field. Note that we have purposely not included \spadtype{DifferentialRing} or the elementary functions (see \spadtype{TranscendentalFunctionCategory}) in the definition.
832100`\axiomType{RealRootCharacterizationCategory} provides common acces functions for all real root codings.
832309`\spadtype{RectangularMatrixCategory} is a category of matrices of fixed dimensions. The dimensions of the matrix will be parameters of the domain. Domains in this category will be \spad{R}-modules and will be non-mutable.
832534`A recursive aggregate over a type \spad{S} is a model for a a directed graph containing values of type \spad{S}. Recursively,{} a recursive aggregate is a {\em node} consisting of a \spadfun{value} from \spad{S} and 0 or more \spadfun{children} which are recursive aggregates. A node with no children is called a \spadfun{leaf} node. A recursive aggregate may be cyclic for which some operations as noted may go into an infini
832534`--te loop.
832615`A category for general multi-variate polynomials with coefficients in a ring,{} variables in an ordered set,{} and exponents from an ordered abelian monoid,{} with a \axiomOp{sup} operation. When not constant,{} such a polynomial is viewed as a univariate polynomial in its main variable \spad{w}. \spad{r}. \spad{t}. to the total ordering on the elements in 
832615`--the ordered set,{} so that some operations usually defined for univariate polynomials make sense here.
832763`The category of regular triangular sets,{} introduced under the name regular chains in [1] (and other papers). In [3] it is proved that regular triangular sets and towers of simple extensions of a field are equivalent notions. In the following definitions,{} all polynomials and ideals are taken from the polyno
832763`--mial ring \spad{k[x1,{}...,{}xn]} where \spad{k} is the fraction field of \spad{R}. The triangular set \spad{[t1,{}...,{}tm]} is regular iff for every \spad{i} the initial of \spad{ti+1} is invertible in the tower of simple extensions associated with \spad{[t1,{}...,{}\spad{ti}]}. A family \spad{[T1,{}...,{}Ts]} of regular triangular sets is a split of Kalkbrener of a given ideal \spad{I} iff the radical of \spad{I} is equal to the intersection of the radical ideals generated by the saturated 
832763`--ideals of the \spad{[T1,{}...,{}\spad{Ti}]}. A family \spad{[T1,{}...,{}Ts]} of regular triangular sets is a split of Kalkbrener of a given triangular set \spad{T} iff it is a split of Kalkbrener of the saturated ideal of \spad{T}. Let \spad{K} be an algebraic closure of \spad{k}. Assume that \spad{V} is finite with cardinality \spad{n} and let \spad{A} be the affine space \spad{K^n}. For a regular triangular set \spad{T} let denote by \spad{W(T)} the set of regular zeros of \spad{T}. A family
832763`-- \spad{[T1,{}...,{}Ts]} of regular triangular sets is a split of Lazard of a given subset \spad{S} of \spad{A} iff the union of the \spad{W(\spad{Ti})} contains \spad{S} and is contained in the closure of \spad{S} (\spad{w}.\spad{r}.\spad{t}. Zariski topology). A family \spad{[T1,{}...,{}Ts]} of regular triangular sets is a split of Lazard of a given triangular set \spad{T} if it is a split of Lazard of \spad{W(T)}. Note that if \spad{[T1,{}...,{}Ts]} is a split of Lazard of \spad{T} then it i
832763`--s also a split of Kalkbrener of \spad{T}. The converse is \spad{false}. This category provides operations related to both kinds of splits,{} the former being related to ideals decomposition whereas the latter deals with varieties decomposition. See the example illustrating the \spadtype{RegularTriangularSet} constructor for more explanations about decompositions by means of regular triangular sets. \newline References : \indented{1}{[1] \spad{M}. KALKBRENER "Three contributions to elimination 
832763`--theory"} \indented{5}{\spad{Phd} Thesis,{} University of Linz,{} Austria,{} 1991.} \indented{1}{[2] \spad{M}. KALKBRENER "Algorithmic properties of polynomial rings"} \indented{5}{Journal of Symbol. Comp. 1998} \indented{1}{[3] \spad{P}. AUBRY,{} \spad{D}. LAZARD and \spad{M}. MORENO MAZA "On the Theories} \indented{5}{of Triangular Sets" Journal of Symbol. Comp. (to appear)} \indented{1}{[4] \spad{M}. MORENO MAZA "A new algorithm for computing triangular} \indented{5}{decomposition of algebra
832763`--ic varieties" NAG Tech. Rep. 4/98.}
832959`A is retractable to \spad{B} means that some elementsif A can be converted into elements of \spad{B} and any element of \spad{B} can be converted into an element of A.
833032`The category of rings with unity,{} always associative,{} but not necessarily commutative.
833072`the class of all multiplicative semigroups,{} \spadignore{i.e.} a set with an associative operation \spadop{*}. \blankline
833124`A set category lists a collection of set-theoretic operations useful for both finite sets and multisets. Note however that finite sets are distinct from multisets. Although the operations defined for set categories are common to both,{} the relationship between the two cannot be described by inclusion or inheritance.
833201`\spadtype{SetCategory} is the basic category for describing a collection of elements with \spadop{=} (equality) and \spadfun{coerce} to output form. \blankline Conditional Attributes: \indented{3}{canonical\tab{15}data structure equality is the same as \spadop{=}}
833257`\spadtype{SquareMatrixCategory} is a general square matrix category which allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and colums returned as objects of type Col.
833462`A stream aggregate is a linear aggregate which possibly has an infinite number of elements. A basic domain constructor which builds stream aggregates is \spadtype{Stream}. From streams,{} a number of infinite structures such power series can be built. A stream aggregate may also be infinite since it may be cyclic. For example,{} see \spadtype{DecimalExpansion}.
833537`A string aggregate is a category for strings,{} that is,{} one dimensional arrays of characters.
833600`A table aggregate is a model of a table,{} \spadignore{i.e.} a discrete many-to-one mapping from keys to entries.
833716`Category of tensor powers of modules over commutative rings.
833850`Category for the transcendental elementary functions.
833945`The category of triangular sets of multivariate polynomials with coefficients in an integral domain. Let \axiom{\spad{R}} be an integral domain and \axiom{\spad{V}} a finite ordered set of variables,{} say \axiom{\spad{X1} < \spad{X2} < ... < \spad{Xn}}. A set \axiom{\spad{S}} of polynomials in \axiom{\spad{R}[\spad{X1
833945`--},{}\spad{X2},{}...,{}\spad{Xn}]} is triangular if no elements of \axiom{\spad{S}} lies in \axiom{\spad{R}},{} and if two distinct elements of \axiom{\spad{S}} have distinct main variables. Note that the empty set is a triangular set. A triangular set is not necessarily a (lexicographical) Groebner basis and the notion of reduction related to triangular sets is based on the recursive view of polynomials. We recall this notion here and refer to [1] for more details. A polynomial \axiom{\spad{P}
833945`--} is reduced \spad{w}.\spad{r}.\spad{t} a non-constant polynomial \axiom{\spad{Q}} if the degree of \axiom{\spad{P}} in the main variable of \axiom{\spad{Q}} is less than the main degree of \axiom{\spad{Q}}. A polynomial \axiom{\spad{P}} is reduced \spad{w}.\spad{r}.\spad{t} a triangular set \axiom{\spad{T}} if it is reduced \spad{w}.\spad{r}.\spad{t}. every polynomial of \axiom{\spad{T}}. \newline References : \indented{1}{[1] \spad{P}. AUBRY,{} \spad{D}. LAZARD and \spad{M}. MORENO MAZA "On 
833945`--the Theories} \indented{5}{of Triangular Sets" Journal of Symbol. Comp. (to appear)}
834132`Category for the trigonometric functions.
834225`\indented{1}{TwoDimensionalArrayCategory is a general array category which} allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and columns returned as objects of type Col. The index of the 'first' row may be obtained by calling the function 'minRowInde
834225`--x'. The index of the 'first' column may be obtained by calling the function 'minColIndex'. The index of the first element of a 'Row' is the same as the index of the first column in an array and vice versa.
834400`A unary-recursive aggregate is a one where nodes may have either 0 or 1 children. This aggregate models,{} though not precisely,{} a linked list possibly with a single cycle. A node with one children models a non-empty list,{} with the \spadfun{value} of the list designating the head,{} or \spadfun{first},{} of the list,{} and the child designating the tail,{} or \spadfun{rest},{} of the list. A node with no chil
834400`--d then designates the empty list. Since these aggregates are recursive aggregates,{} they may be cyclic.
834491`A constructive unique factorization domain,{} \spadignore{i.e.} where we can constructively factor members into a product of a finite number of irreducible elements.
834572`This is a category of univariate Laurent series constructed from univariate Taylor series. A Laurent series is represented by a pair \spad{[n,{}f(x)]},{} where \spad{n} is an arbitrary integer and \spad{f(x)} is a Taylor series. This pair represents the Laurent series \spad{x^n * f(x)}.
834761`The category of univariate polynomials over a ring \spad{R}. No particular model is assumed - implementations can be either sparse or dense.
834863`\spadtype{UnivariatePowerSeriesCategory} is the most general univariate power series category with exponents in an ordered abelian monoid. Note: this category exports a substitution function if it is possible to multiply exponents. Note: this category exports a derivative operation if it is possible to multiply coefficients by exponents.
835015`This is a category of univariate Puiseux series constructed from univariate Laurent series. A Puiseux series is represented by a pair \spad{[r,{}f(x)]},{} where \spad{r} is a positive rational number and \spad{f(x)} is a Laurent series. This pair represents the Puiseux series \spad{f(x^r)}.
835205`This is the category of univariate skew polynomials over an Ore coefficient ring. The multiplication is given by \spad{x a = \sigma(a) x + \delta a}. This category is an evolution of the types \indented{2}{MonogenicLinearOperator,{} OppositeMonogenicLinearOperator,{} and} \indented{2}{NonCommutativeOperatorDivision} developped by Jean Della Dora and Stephen \spad{M}. Watt.
835316`\spadtype{UnivariateTaylorSeriesCategory} is the category of Taylor series in one variable.
835431`\spadtype{VectorCategory} represents the type of vector like objects,{} \spadignore{i.e.} finite sequences indexed by some finite segment of the integers. The operations available on vectors depend on the structure of the underlying components. Many operations from the component domain are defined for vectors componentwise. It can by assumed that extraction or updating components can be done in constant time.
835506`Vector Spaces (not necessarily finite dimensional) over a field.
