This is ginac.info, produced by makeinfo version 4.2 from ginac.texi.

START-INFO-DIR-ENTRY
* ginac: (ginac).                   C++ library for symbolic computation.
END-INFO-DIR-ENTRY

This is a tutorial that documents GiNaC 1.0.14, an open framework for
symbolic computation within the C++ programming language.

Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany

Permission is granted to make and distribute verbatim copies of this
manual provided the copyright notice and this permission notice are
preserved on all copies.

Permission is granted to copy and distribute modified versions of this
manual under the conditions for verbatim copying, provided that the
entire resulting derived work is distributed under the terms of a
permission notice identical to this one.


File: ginac.info,  Node: Top,  Next: Introduction,  Prev: (dir),  Up: (dir)

GiNaC
*****

This is a tutorial that documents GiNaC 1.0.14, an open framework for
symbolic computation within the C++ programming language.

* Menu:

* Introduction::                 GiNaC's purpose.
* A Tour of GiNaC::              A quick tour of the library.
* Installation::                 How to install the package.
* Basic Concepts::               Description of fundamental classes.
* Methods and Functions::        Algorithms for symbolic manipulations.
* Extending GiNaC::              How to extend the library.
* A Comparison With Other CAS::  Compares GiNaC to traditional CAS.
* Internal Structures::          Description of some internal structures.
* Package Tools::                Configuring packages to work with GiNaC.
* Bibliography::
* Concept Index::


File: ginac.info,  Node: Introduction,  Next: A Tour of GiNaC,  Prev: Top,  Up: Top

Introduction
************

The motivation behind GiNaC derives from the observation that most
present day computer algebra systems (CAS) are linguistically and
semantically impoverished.  Although they are quite powerful tools for
learning math and solving particular problems they lack modern
linguistic structures that allow for the creation of large-scale
projects.  GiNaC is an attempt to overcome this situation by extending a
well established and standardized computer language (C++) by some
fundamental symbolic capabilities, thus allowing for integrated systems
that embed symbolic manipulations together with more established areas
of computer science (like computation-intense numeric applications,
graphical interfaces, etc.) under one roof.

The particular problem that led to the writing of the GiNaC framework is
still a very active field of research, namely the calculation of higher
order corrections to elementary particle interactions.  There,
theoretical physicists are interested in matching present day theories
against experiments taking place at particle accelerators.  The
computations involved are so complex they call for a combined symbolical
and numerical approach.  This turned out to be quite difficult to
accomplish with the present day CAS we have worked with so far and so we
tried to fill the gap by writing GiNaC.  But of course its applications
are in no way restricted to theoretical physics.

This tutorial is intended for the novice user who is new to GiNaC but
already has some background in C++ programming.  However, since a
hand-made documentation like this one is difficult to keep in sync with
the development, the actual documentation is inside the sources in the
form of comments.  That documentation may be parsed by one of the many
Javadoc-like documentation systems.  If you fail at generating it you
may access it from the GiNaC home page
(http://www.ginac.de/reference/).  It is an invaluable resource not
only for the advanced user who wishes to extend the system (or chase
bugs) but for everybody who wants to comprehend the inner workings of
GiNaC.  This little tutorial on the other hand only covers the basic
things that are unlikely to change in the near future.

License
=======

The GiNaC framework for symbolic computation within the C++ programming
language is Copyright (C) 1999-2003 Johannes Gutenberg University
Mainz, Germany.

This program is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by the
Free Software Foundation; either version 2 of the License, or (at your
option) any later version.

This program is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
General Public License for more details.

You should have received a copy of the GNU General Public License along
with this program; see the file COPYING.  If not, write to the Free
Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA
02111-1307, USA.


File: ginac.info,  Node: A Tour of GiNaC,  Next: How to use it from within C++,  Prev: Introduction,  Up: Top

A Tour of GiNaC
***************

This quick tour of GiNaC wants to arise your interest in the subsequent
chapters by showing off a bit.  Please excuse us if it leaves many open
questions.

* Menu:

* How to use it from within C++::  Two simple examples.
* What it can do for you::         A Tour of GiNaC's features.


File: ginac.info,  Node: How to use it from within C++,  Next: What it can do for you,  Prev: A Tour of GiNaC,  Up: A Tour of GiNaC

How to use it from within C++
=============================

The GiNaC open framework for symbolic computation within the C++
programming language does not try to define a language of its own as
conventional CAS do.  Instead, it extends the capabilities of C++ by
symbolic manipulations.  Here is how to generate and print a simple
(and rather pointless) bivariate polynomial with some large
coefficients:

     #include <iostream>
     #include <ginac/ginac.h>
     using namespace std;
     using namespace GiNaC;
     
     int main()
     {
         symbol x("x"), y("y");
         ex poly;
     
         for (int i=0; i<3; ++i)
             poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
     
         cout << poly << endl;
         return 0;
     }

Assuming the file is called `hello.cc', on our system we can compile
and run it like this:

     $ c++ hello.cc -o hello -lcln -lginac
     $ ./hello
     355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2

(*Note Package Tools::, for tools that help you when creating a software
package that uses GiNaC.)

Next, there is a more meaningful C++ program that calls a function which
generates Hermite polynomials in a specified free variable.

     #include <iostream>
     #include <ginac/ginac.h>
     using namespace std;
     using namespace GiNaC;
     
     ex HermitePoly(const symbol & x, int n)
     {
         ex HKer=exp(-pow(x, 2));
         // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
         return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
     }
     
     int main()
     {
         symbol z("z");
     
         for (int i=0; i<6; ++i)
             cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
     
         return 0;
     }

When run, this will type out

     H_0(z) == 1
     H_1(z) == 2*z
     H_2(z) == 4*z^2-2
     H_3(z) == -12*z+8*z^3
     H_4(z) == -48*z^2+16*z^4+12
     H_5(z) == 120*z-160*z^3+32*z^5

This method of generating the coefficients is of course far from optimal
for production purposes.

In order to show some more examples of what GiNaC can do we will now use
the `ginsh', a simple GiNaC interactive shell that provides a
convenient window into GiNaC's capabilities.


File: ginac.info,  Node: What it can do for you,  Next: Installation,  Prev: How to use it from within C++,  Up: A Tour of GiNaC

What it can do for you
======================

After invoking `ginsh' one can test and experiment with GiNaC's
features much like in other Computer Algebra Systems except that it does
not provide programming constructs like loops or conditionals.  For a
concise description of the `ginsh' syntax we refer to its accompanied
man page. Suffice to say that assignments and comparisons in `ginsh'
are written as they are in C, i.e. `=' assigns and `==' compares.

It can manipulate arbitrary precision integers in a very fast way.
Rational numbers are automatically converted to fractions of coprime
integers:

     > x=3^150;
     369988485035126972924700782451696644186473100389722973815184405301748249
     > y=3^149;
     123329495011708990974900260817232214728824366796574324605061468433916083
     > x/y;
     3
     > y/x;
     1/3

Exact numbers are always retained as exact numbers and only evaluated as
floating point numbers if requested.  For instance, with numeric
radicals is dealt pretty much as with symbols.  Products of sums of them
can be expanded:

     > expand((1+a^(1/5)-a^(2/5))^3);
     1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
     > expand((1+3^(1/5)-3^(2/5))^3);
     10-5*3^(3/5)
     > evalf((1+3^(1/5)-3^(2/5))^3);
     0.33408977534118624228

The function `evalf' that was used above converts any number in GiNaC's
expressions into floating point numbers.  This can be done to arbitrary
predefined accuracy:

     > evalf(1/7);
     0.14285714285714285714
     > Digits=150;
     150
     > evalf(1/7);
     0.1428571428571428571428571428571428571428571428571428571428571428571428
     5714285714285714285714285714285714285

Exact numbers other than rationals that can be manipulated in GiNaC
include predefined constants like Archimedes' `Pi'.  They can both be
used in symbolic manipulations (as an exact number) as well as in
numeric expressions (as an inexact number):

     > a=Pi^2+x;
     x+Pi^2
     > evalf(a);
     9.869604401089358619+x
     > x=2;
     2
     > evalf(a);
     11.869604401089358619

Built-in functions evaluate immediately to exact numbers if this is
possible.  Conversions that can be safely performed are done
immediately; conversions that are not generally valid are not done:

     > cos(42*Pi);
     1
     > cos(acos(x));
     x
     > acos(cos(x));
     acos(cos(x))

(Note that converting the last input to `x' would allow one to conclude
that `42*Pi' is equal to `0'.)

Linear equation systems can be solved along with basic linear algebra
manipulations over symbolic expressions.  In C++ GiNaC offers a matrix
class for this purpose but we can see what it can do using `ginsh''s
bracket notation to type them in:

     > lsolve(a+x*y==z,x);
     y^(-1)*(z-a);
     > lsolve({3*x+5*y == 7, -2*x+10*y == -5}, {x, y});
     {x==19/8,y==-1/40}
     > M = [ [1, 3], [-3, 2] ];
     [[1,3],[-3,2]]
     > determinant(M);
     11
     > charpoly(M,lambda);
     lambda^2-3*lambda+11
     > A = [ [1, 1], [2, -1] ];
     [[1,1],[2,-1]]
     > A+2*M;
     [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
     > evalm(%);
     [[3,7],[-4,3]]
     > B = [ [0, 0, a], [b, 1, -b], [-1/a, 0, 0] ];
     > evalm(B^(2^12345));
     [[1,0,0],[0,1,0],[0,0,1]]

Multivariate polynomials and rational functions may be expanded,
collected and normalized (i.e. converted to a ratio of two coprime
polynomials):

     > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
     12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
     > b = x^2 + 4*x*y - y^2;
     4*x*y-y^2+x^2
     > expand(a*b);
     8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
     > collect(a+b,x);
     4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
     > collect(a+b,y);
     12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
     > normal(a/b);
     3*y^2+x^2

You can differentiate functions and expand them as Taylor or Laurent
series in a very natural syntax (the second argument of `series' is a
relation defining the evaluation point, the third specifies the order):

     > diff(tan(x),x);
     tan(x)^2+1
     > series(sin(x),x==0,4);
     x-1/6*x^3+Order(x^4)
     > series(1/tan(x),x==0,4);
     x^(-1)-1/3*x+Order(x^2)
     > series(tgamma(x),x==0,3);
     x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
     (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
     > evalf(%);
     x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
     -(0.90747907608088628905)*x^2+Order(x^3)
     > series(tgamma(2*sin(x)-2),x==Pi/2,6);
     -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
     -Euler-1/12+Order((x-1/2*Pi)^3)

Here we have made use of the `ginsh'-command `%' to pop the previously
evaluated element from `ginsh''s internal stack.

If you ever wanted to convert units in C or C++ and found this is
cumbersome, here is the solution.  Symbolic types can always be used as
tags for different types of objects.  Converting from wrong units to the
metric system is now easy:

     > in=.0254*m;
     0.0254*m
     > lb=.45359237*kg;
     0.45359237*kg
     > 200*lb/in^2;
     140613.91592783185568*kg*m^(-2)


File: ginac.info,  Node: Installation,  Next: Prerequisites,  Prev: What it can do for you,  Up: Top

Installation
************

GiNaC's installation follows the spirit of most GNU software. It is
easily installed on your system by three steps: configuration, build,
installation.

* Menu:

* Prerequisites::                Packages upon which GiNaC depends.
* Configuration::                How to configure GiNaC.
* Building GiNaC::               How to compile GiNaC.
* Installing GiNaC::             How to install GiNaC on your system.


File: ginac.info,  Node: Prerequisites,  Next: Configuration,  Prev: Installation,  Up: Installation

Prerequisites
=============

In order to install GiNaC on your system, some prerequisites need to be
met.  First of all, you need to have a C++-compiler adhering to the
ANSI-standard `ISO/IEC 14882:1998(E)'.  We used GCC for development so
if you have a different compiler you are on your own.  For the
configuration to succeed you need a Posix compliant shell installed in
`/bin/sh', GNU `bash' is fine.  Perl is needed by the built process as
well, since some of the source files are automatically generated by
Perl scripts.  Last but not least, Bruno Haible's library CLN is
extensively used and needs to be installed on your system.  Please get
it either from `ftp://ftp.santafe.edu/pub/gnu/', from GiNaC's FTP site
(ftp://ftpthep.physik.uni-mainz.de/pub/gnu/) or from Bruno Haible's FTP
site (ftp://ftp.ilog.fr/pub/Users/haible/gnu/) (it is covered by GPL)
and install it prior to trying to install GiNaC.  The configure script
checks if it can find it and if it cannot it will refuse to continue.


File: ginac.info,  Node: Configuration,  Next: Building GiNaC,  Prev: Prerequisites,  Up: Installation

Configuration
=============

To configure GiNaC means to prepare the source distribution for
building.  It is done via a shell script called `configure' that is
shipped with the sources and was originally generated by GNU Autoconf.
Since a configure script generated by GNU Autoconf never prompts, all
customization must be done either via command line parameters or
environment variables.  It accepts a list of parameters, the complete
set of which can be listed by calling it with the `--help' option.  The
most important ones will be shortly described in what follows:

   * `--disable-shared': When given, this option switches off the build
     of a shared library, i.e. a `.so' file.  This may be convenient
     when developing because it considerably speeds up compilation.

   * `--prefix=PREFIX': The directory where the compiled library and
     headers are installed. It defaults to `/usr/local' which means
     that the library is installed in the directory `/usr/local/lib',
     the header files in `/usr/local/include/ginac' and the
     documentation (like this one) into `/usr/local/share/doc/GiNaC'.

   * `--libdir=LIBDIR': Use this option in case you want to have the
     library installed in some other directory than `PREFIX/lib/'.

   * `--includedir=INCLUDEDIR': Use this option in case you want to
     have the header files installed in some other directory than
     `PREFIX/include/ginac/'. For instance, if you specify
     `--includedir=/usr/include' you will end up with the header files
     sitting in the directory `/usr/include/ginac/'. Note that the
     subdirectory `ginac' is enforced by this process in order to keep
     the header files separated from others.  This avoids some clashes
     and allows for an easier deinstallation of GiNaC. This ought to be
     considered A Good Thing (tm).

   * `--datadir=DATADIR': This option may be given in case you want to
     have the documentation installed in some other directory than
     `PREFIX/share/doc/GiNaC/'.


In addition, you may specify some environment variables.  `CXX' holds
the path and the name of the C++ compiler in case you want to override
the default in your path.  (The `configure' script searches your path
for `c++', `g++', `gcc', `CC', `cxx' and `cc++' in that order.)  It may
be very useful to define some compiler flags with the `CXXFLAGS'
environment variable, like optimization, debugging information and
warning levels.  If omitted, it defaults to `-g -O2'.(1)

The whole process is illustrated in the following two examples.
(Substitute `setenv VARIABLE VALUE' for `export VARIABLE=VALUE' if the
Berkeley C shell is your login shell.)

Here is a simple configuration for a site-wide GiNaC library assuming
everything is in default paths:

     $ export CXXFLAGS="-Wall -O2"
     $ ./configure

And here is a configuration for a private static GiNaC library with
several components sitting in custom places (site-wide GCC and private
CLN).  The compiler is persuaded to be picky and full assertions and
debugging information are switched on:

     $ export CXX=/usr/local/gnu/bin/c++
     $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
     $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
     $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
     $ ./configure --disable-shared --prefix=$(HOME)

---------- Footnotes ----------

(1) The `configure' script is itself generated from the file
`configure.ac'.  It is only distributed in packaged releases of GiNaC.
If you got the naked sources, e.g. from CVS, you must generate
`configure' along with the various `Makefile.in' by using the
`autogen.sh' script.  This will require a fair amount of support from
your local toolchain, though.


File: ginac.info,  Node: Building GiNaC,  Next: Installing GiNaC,  Prev: Configuration,  Up: Installation

Building GiNaC
==============

After proper configuration you should just build the whole library by
typing
     $ make
at the command prompt and go for a cup of coffee.  The exact time it
takes to compile GiNaC depends not only on the speed of your machines
but also on other parameters, for instance what value for `CXXFLAGS'
you entered.  Optimization may be very time-consuming.

Just to make sure GiNaC works properly you may run a collection of
regression tests by typing

     $ make check

This will compile some sample programs, run them and check the output
for correctness.  The regression tests fall in three categories.  First,
the so called _exams_ are performed, simple tests where some predefined
input is evaluated (like a pupils' exam).  Second, the _checks_ test
the coherence of results among each other with possible random input.
Third, some _timings_ are performed, which benchmark some predefined
problems with different sizes and display the CPU time used in seconds.
Each individual test should return a message `passed'.  This is mostly
intended to be a QA-check if something was broken during development,
not a sanity check of your system.  Some of the tests in sections
_checks_ and _timings_ may require insane amounts of memory and CPU
time.  Feel free to kill them if your machine catches fire.  Another
quite important intent is to allow people to fiddle around with
optimization.

Generally, the top-level Makefile runs recursively to the
subdirectories.  It is therefore safe to go into any subdirectory
(`doc/', `ginsh/', ...) and simply type `make' TARGET there in case
something went wrong.


File: ginac.info,  Node: Installing GiNaC,  Next: Basic Concepts,  Prev: Building GiNaC,  Up: Installation

Installing GiNaC
================

To install GiNaC on your system, simply type

     $ make install

As described in the section about configuration the files will be
installed in the following directories (the directories will be created
if they don't already exist):

   * `libginac.a' will go into `PREFIX/lib/' (or `LIBDIR') which
     defaults to `/usr/local/lib/'.  So will `libginac.so' unless the
     configure script was given the option `--disable-shared'.  The
     proper symlinks will be established as well.

   * All the header files will be installed into `PREFIX/include/ginac/'
     (or `INCLUDEDIR/ginac/', if specified).

   * All documentation (HTML and Postscript) will be stuffed into
     `PREFIX/share/doc/GiNaC/' (or `DATADIR/doc/GiNaC/', if DATADIR was
     specified).


For the sake of completeness we will list some other useful make
targets: `make clean' deletes all files generated by `make', i.e. all
the object files.  In addition `make distclean' removes all files
generated by the configuration and `make maintainer-clean' goes one
step further and deletes files that may require special tools to
rebuild (like the `libtool' for instance).  Finally `make uninstall'
removes the installed library, header files and documentation(1).

---------- Footnotes ----------

(1) Uninstallation does not work after you have called `make distclean'
since the `Makefile' is itself generated by the configuration from
`Makefile.in' and hence deleted by `make distclean'.  There are two
obvious ways out of this dilemma.  First, you can run the configuration
again with the same PREFIX thus creating a `Makefile' with a working
`uninstall' target.  Second, you can do it by hand since you now know
where all the files went during installation.


File: ginac.info,  Node: Basic Concepts,  Next: Expressions,  Prev: Installing GiNaC,  Up: Top

Basic Concepts
**************

This chapter will describe the different fundamental objects that can be
handled by GiNaC.  But before doing so, it is worthwhile introducing you
to the more commonly used class of expressions, representing a flexible
meta-class for storing all mathematical objects.

* Menu:

* Expressions::                  The fundamental GiNaC class.
* Automatic evaluation::         Evaluation and canonicalization.
* Error handling::               How the library reports errors.
* The Class Hierarchy::          Overview of GiNaC's classes.
* Symbols::                      Symbolic objects.
* Numbers::                      Numerical objects.
* Constants::                    Pre-defined constants.
* Fundamental containers::       Sums, products and powers.
* Lists::                        Lists of expressions.
* Mathematical functions::       Mathematical functions.
* Relations::                    Equality, Inequality and all that.
* Matrices::                     Matrices.
* Indexed objects::              Handling indexed quantities.
* Non-commutative objects::      Algebras with non-commutative products.


File: ginac.info,  Node: Expressions,  Next: Automatic evaluation,  Prev: Basic Concepts,  Up: Basic Concepts

Expressions
===========

The most common class of objects a user deals with is the expression
`ex', representing a mathematical object like a variable, number,
function, sum, product, etc...  Expressions may be put together to form
new expressions, passed as arguments to functions, and so on.  Here is a
little collection of valid expressions:

     ex MyEx1 = 5;                       // simple number
     ex MyEx2 = x + 2*y;                 // polynomial in x and y
     ex MyEx3 = (x + 1)/(x - 1);         // rational expression
     ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
     ex MyEx5 = MyEx4 + 1;               // similar to above

Expressions are handles to other more fundamental objects, that often
contain other expressions thus creating a tree of expressions (*Note
Internal Structures::, for particular examples).  Most methods on `ex'
therefore run top-down through such an expression tree.  For example,
the method `has()' scans recursively for occurrences of something
inside an expression.  Thus, if you have declared `MyEx4' as in the
example above `MyEx4.has(y)' will find `y' inside the argument of `sin'
and hence return `true'.

The next sections will outline the general picture of GiNaC's class
hierarchy and describe the classes of objects that are handled by `ex'.


File: ginac.info,  Node: Automatic evaluation,  Next: Error handling,  Prev: Expressions,  Up: Basic Concepts

Automatic evaluation and canonicalization of expressions
========================================================

GiNaC performs some automatic transformations on expressions, to
simplify them and put them into a canonical form. Some examples:

     ex MyEx1 = 2*x - 1 + x;  // 3*x-1
     ex MyEx2 = x - x;        // 0
     ex MyEx3 = cos(2*Pi);    // 1
     ex MyEx4 = x*y/x;        // y

This behavior is usually referred to as "automatic" or "anonymous
evaluation". GiNaC only performs transformations that are

   * at most of complexity O(n log n)

   * algebraically correct, possibly except for a set of measure zero
     (e.g.  x/x is transformed to 1 although this is incorrect for x=0)

There are two types of automatic transformations in GiNaC that may not
behave in an entirely obvious way at first glance:

   * The terms of sums and products (and some other things like the
     arguments of symmetric functions, the indices of symmetric tensors
     etc.) are re-ordered into a canonical form that is deterministic,
     but not lexicographical or in any other way easily guessable (it
     almost always depends on the number and order of the symbols you
     define). However, constructing the same expression twice, either
     implicitly or explicitly, will always result in the same canonical
     form.

   * Expressions of the form 'number times sum' are automatically
     expanded (this has to do with GiNaC's internal representation of
     sums and products). For example
          ex MyEx5 = 2*(x + y);   // 2*x+2*y
          ex MyEx6 = z*(x + y);   // z*(x+y)

The general rule is that when you construct expressions, GiNaC
automatically creates them in canonical form, which might differ from
the form you typed in your program. This may create some awkward
looking output (`-y+x' instead of `y-x') but allows for more efficient
operation and usually yields some immediate simplifications.

Internally, the anonymous evaluator in GiNaC is implemented by the
methods

     ex ex::eval(int level = 0) const;
     ex basic::eval(int level = 0) const;

but unless you are extending GiNaC with your own classes or functions,
there should never be any reason to call them explicitly. All GiNaC
methods that transform expressions, like `subs()' or `normal()',
automatically re-evaluate their results.


File: ginac.info,  Node: Error handling,  Next: The Class Hierarchy,  Prev: Automatic evaluation,  Up: Basic Concepts

Error handling
==============

GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
generated by GiNaC are subclassed from the standard `exception' class
defined in the `<stdexcept>' header. In addition to the predefined
`logic_error', `domain_error', `out_of_range', `invalid_argument',
`runtime_error', `range_error' and `overflow_error' types, GiNaC also
defines a `pole_error' exception that gets thrown when trying to
evaluate a mathematical function at a singularity.

The `pole_error' class has a member function

     int pole_error::degree(void) const;

that returns the order of the singularity (or 0 when the pole is
logarithmic or the order is undefined).

When using GiNaC it is useful to arrange for exceptions to be catched in
the main program even if you don't want to do any special error
handling.  Otherwise whenever an error occurs in GiNaC, it will be
delegated to the default exception handler of your C++ compiler's
run-time system which usually only aborts the program without giving
any information what went wrong.

Here is an example for a `main()' function that catches and prints
exceptions generated by GiNaC:

     #include <iostream>
     #include <stdexcept>
     #include <ginac/ginac.h>
     using namespace std;
     using namespace GiNaC;
     
     int main(void)
     {
         try {
             ...
             // code using GiNaC
             ...
         } catch (exception &p) {
             cerr << p.what() << endl;
             return 1;
         }
         return 0;
     }


File: ginac.info,  Node: The Class Hierarchy,  Next: Symbols,  Prev: Error handling,  Up: Basic Concepts

The Class Hierarchy
===================

GiNaC's class hierarchy consists of several classes representing
mathematical objects, all of which (except for `ex' and some helpers)
are internally derived from one abstract base class called `basic'.
You do not have to deal with objects of class `basic', instead you'll
be dealing with symbols, numbers, containers of expressions and so on.

To get an idea about what kinds of symbolic composites may be built we
have a look at the most important classes in the class hierarchy and
some of the relations among the classes:

<picture missing>

The abstract classes shown here (the ones without drop-shadow) are of no
interest for the user.  They are used internally in order to avoid code
duplication if two or more classes derived from them share certain
features.  An example is `expairseq', a container for a sequence of
pairs each consisting of one expression and a number (`numeric').  What
_is_ visible to the user are the derived classes `add' and `mul',
representing sums and products.  *Note Internal Structures::, where
these two classes are described in more detail.  The following table
shortly summarizes what kinds of mathematical objects are stored in the
different classes:

`symbol'         Algebraic symbols a, x, y...
`constant'       Constants like Pi
`numeric'        All kinds of numbers, 42, 7/3*I, 3.14159...
`add'            Sums like x+y or a-(2*b)+3
`mul'            Products like x*y or 2*a^2*(x+y+z)/b
`ncmul'          Products of non-commutative objects
`power'          Exponentials such as x^2, a^b, `sqrt('2`)' ...
`pseries'        Power Series, e.g. x-1/6*x^3+1/120*x^5+O(x^7)
`function'       A symbolic function like sin(2*x)
`lst'            Lists of expressions {x, 2*y, 3+z}
`matrix'         mxn matrices of expressions
`relational'     A relation like the identity x`=='y
`indexed'        Indexed object like A_ij
`tensor'         Special tensor like the delta and metric tensors
`idx'            Index of an indexed object
`varidx'         Index with variance
`spinidx'        Index with variance and dot (used in
                 Weyl-van-der-Waerden spinor formalism)
`wildcard'       Wildcard for pattern matching


File: ginac.info,  Node: Symbols,  Next: Numbers,  Prev: The Class Hierarchy,  Up: Basic Concepts

Symbols
=======

Symbols are for symbolic manipulation what atoms are for chemistry.  You
can declare objects of class `symbol' as any other object simply by
saying `symbol x,y;'.  There is, however, a catch in here having to do
with the fact that C++ is a compiled language.  The information about
the symbol's name is thrown away by the compiler but at a later stage
you may want to print expressions holding your symbols.  In order to
avoid confusion GiNaC's symbols are able to know their own name.  This
is accomplished by declaring its name for output at construction time in
the fashion `symbol x("x");'.  If you declare a symbol using the
default constructor (i.e. without string argument) the system will deal
out a unique name.  That name may not be suitable for printing but for
internal routines when no output is desired it is often enough.  We'll
come across examples of such symbols later in this tutorial.

This implies that the strings passed to symbols at construction time may
not be used for comparing two of them.  It is perfectly legitimate to
write `symbol x("x"),y("x");' but it is likely to lead into trouble.
Here, `x' and `y' are different symbols and statements like `x-y' will
not be simplified to zero although the output `x-x' looks funny.  Such
output may also occur when there are two different symbols in two
scopes, for instance when you call a function that declares a symbol
with a name already existent in a symbol in the calling function.
Again, comparing them (using `operator==' for instance) will always
reveal their difference.  Watch out, please.

Although symbols can be assigned expressions for internal reasons, you
should not do it (and we are not going to tell you how it is done).  If
you want to replace a symbol with something else in an expression, you
can use the expression's `.subs()' method (*note Substituting
Expressions::).


File: ginac.info,  Node: Numbers,  Next: Constants,  Prev: Symbols,  Up: Basic Concepts

Numbers
=======

For storing numerical things, GiNaC uses Bruno Haible's library CLN.
The classes therein serve as foundation classes for GiNaC.  CLN stands
for Class Library for Numbers or alternatively for Common Lisp Numbers.
In order to find out more about CLN's internals, the reader is referred
to the documentation of that library.  *note (cln)Introduction::, for
more information. Suffice to say that it is by itself build on top of
another library, the GNU Multiple Precision library GMP, which is an
extremely fast library for arbitrary long integers and rationals as well
as arbitrary precision floating point numbers.  It is very commonly used
by several popular cryptographic applications.  CLN extends GMP by
several useful things: First, it introduces the complex number field
over either reals (i.e. floating point numbers with arbitrary precision)
or rationals.  Second, it automatically converts rationals to integers
if the denominator is unity and complex numbers to real numbers if the
imaginary part vanishes and also correctly treats algebraic functions.
Third it provides good implementations of state-of-the-art algorithms
for all trigonometric and hyperbolic functions as well as for
calculation of some useful constants.

The user can construct an object of class `numeric' in several ways.
The following example shows the four most important constructors.  It
uses construction from C-integer, construction of fractions from two
integers, construction from C-float and construction from a string:

     #include <iostream>
     #include <ginac/ginac.h>
     using namespace GiNaC;
     
     int main()
     {
         numeric two = 2;                      // exact integer 2
         numeric r(2,3);                       // exact fraction 2/3
         numeric e(2.71828);                   // floating point number
         numeric p = "3.14159265358979323846"; // constructor from string
         // Trott's constant in scientific notation:
         numeric trott("1.0841015122311136151E-2");
     
         std::cout << two*p << std::endl;  // floating point 6.283...
         ...

The imaginary unit in GiNaC is a predefined `numeric' object with the
name `I':

         ...
         numeric z1 = 2-3*I;                    // exact complex number 2-3i
         numeric z2 = 5.9+1.6*I;                // complex floating point number
     }

It may be tempting to construct fractions by writing `numeric r(3/2)'.
This would, however, call C's built-in operator `/' for integers first
and result in a numeric holding a plain integer 1.  *Never use the
operator `/' on integers* unless you know exactly what you are doing!
Use the constructor from two integers instead, as shown in the example
above.  Writing `numeric(1)/2' may look funny but works also.

We have seen now the distinction between exact numbers and floating
point numbers.  Clearly, the user should never have to worry about
dynamically created exact numbers, since their `exactness' always
determines how they ought to be handled, i.e. how `long' they are.  The
situation is different for floating point numbers.  Their accuracy is
controlled by one _global_ variable, called `Digits'.  (For those
readers who know about Maple: it behaves very much like Maple's
`Digits').  All objects of class numeric that are constructed from then
on will be stored with a precision matching that number of decimal
digits:

     #include <iostream>
     #include <ginac/ginac.h>
     using namespace std;
     using namespace GiNaC;
     
     void foo()
     {
         numeric three(3.0), one(1.0);
         numeric x = one/three;
     
         cout << "in " << Digits << " digits:" << endl;
         cout << x << endl;
         cout << Pi.evalf() << endl;
     }
     
     int main()
     {
         foo();
         Digits = 60;
         foo();
         return 0;
     }

The above example prints the following output to screen:

     in 17 digits:
     0.33333333333333333334
     3.1415926535897932385
     in 60 digits:
     0.33333333333333333333333333333333333333333333333333333333333333333334
     3.1415926535897932384626433832795028841971693993751058209749445923078

Note that the last number is not necessarily rounded as you would
naively expect it to be rounded in the decimal system.  But note also,
that in both cases you got a couple of extra digits.  This is because
numbers are internally stored by CLN as chunks of binary digits in order
to match your machine's word size and to not waste precision.  Thus, on
architectures with different word size, the above output might even
differ with regard to actually computed digits.

It should be clear that objects of class `numeric' should be used for
constructing numbers or for doing arithmetic with them.  The objects
one deals with most of the time are the polymorphic expressions `ex'.

Tests on numbers
----------------

Once you have declared some numbers, assigned them to expressions and
done some arithmetic with them it is frequently desired to retrieve some
kind of information from them like asking whether that number is
integer, rational, real or complex.  For those cases GiNaC provides
several useful methods.  (Internally, they fall back to invocations of
certain CLN functions.)

As an example, let's construct some rational number, multiply it with
some multiple of its denominator and test what comes out:

     #include <iostream>
     #include <ginac/ginac.h>
     using namespace std;
     using namespace GiNaC;
     
     // some very important constants:
     const numeric twentyone(21);
     const numeric ten(10);
     const numeric five(5);
     
     int main()
     {
         numeric answer = twentyone;
     
         answer /= five;
         cout << answer.is_integer() << endl;  // false, it's 21/5
         answer *= ten;
         cout << answer.is_integer() << endl;  // true, it's 42 now!
     }

Note that the variable `answer' is constructed here as an integer by
`numeric''s copy constructor but in an intermediate step it holds a
rational number represented as integer numerator and integer
denominator.  When multiplied by 10, the denominator becomes unity and
the result is automatically converted to a pure integer again.
Internally, the underlying CLN is responsible for this behavior and we
refer the reader to CLN's documentation.  Suffice to say that the same
behavior applies to complex numbers as well as return values of certain
functions.  Complex numbers are automatically converted to real numbers
if the imaginary part becomes zero.  The full set of tests that can be
applied is listed in the following table.

*Method*               *Returns true if the object is...*
`.is_zero()'           ...equal to zero
`.is_positive()'       ...not complex and greater than 0
`.is_integer()'        ...a (non-complex) integer
`.is_pos_integer()'    ...an integer and greater than 0
`.is_nonneg_integer()' ...an integer and greater equal 0
`.is_even()'           ...an even integer
`.is_odd()'            ...an odd integer
`.is_prime()'          ...a prime integer (probabilistic primality test)
`.is_rational()'       ...an exact rational number (integers are
                       rational, too)
`.is_real()'           ...a real integer, rational or float (i.e. is not
                       complex)
`.is_cinteger()'       ...a (complex) integer (such as 2-3*I)
`.is_crational()'      ...an exact (complex) rational number (such as
                       2/3+7/2*I)


File: ginac.info,  Node: Constants,  Next: Fundamental containers,  Prev: Numbers,  Up: Basic Concepts

Constants
=========

Constants behave pretty much like symbols except that they return some
specific number when the method `.evalf()' is called.

The predefined known constants are:

*Name*     *Common Name*          *Numerical Value (to 35 digits)*
`Pi'       Archimedes' constant   3.14159265358979323846264338327950288
`Catalan'  Catalan's constant     0.91596559417721901505460351493238411
`Euler'    Euler's (or            0.57721566490153286060651209008240243
           Euler-Mascheroni)      
           constant               


File: ginac.info,  Node: Fundamental containers,  Next: Lists,  Prev: Constants,  Up: Basic Concepts

Sums, products and powers
=========================

Simple rational expressions are written down in GiNaC pretty much like
in other CAS or like expressions involving numerical variables in C.
The necessary operators `+', `-', `*' and `/' have been overloaded to
achieve this goal.  When you run the following code snippet, the
constructor for an object of type `mul' is automatically called to hold
the product of `a' and `b' and then the constructor for an object of
type `add' is called to hold the sum of that `mul' object and the
number one:

         ...
         symbol a("a"), b("b");
         ex MyTerm = 1+a*b;
         ...

For exponentiation, you have already seen the somewhat clumsy (though
C-ish) statement `pow(x,2);' to represent `x' squared.  This direct
construction is necessary since we cannot safely overload the
constructor `^' in C++ to construct a `power' object.  If we did, it
would have several counterintuitive and undesired effects:

   * Due to C's operator precedence, `2*x^2' would be parsed as
     `(2*x)^2'.

   * Due to the binding of the operator `^', `x^a^b' would result in
     `(x^a)^b'. This would be confusing since most (though not all)
     other CAS interpret this as `x^(a^b)'.

   * Also, expressions involving integer exponents are very frequently
     used, which makes it even more dangerous to overload `^' since it
     is then hard to distinguish between the semantics as
     exponentiation and the one for exclusive or.  (It would be
     embarrassing to return `1' where one has requested `2^3'.)

All effects are contrary to mathematical notation and differ from the
way most other CAS handle exponentiation, therefore overloading `^' is
ruled out for GiNaC's C++ part.  The situation is different in `ginsh',
there the exponentiation-`^' exists.  (Also note that the other
frequently used exponentiation operator `**' does not exist at all in
C++).

To be somewhat more precise, objects of the three classes described
here, are all containers for other expressions.  An object of class
`power' is best viewed as a container with two slots, one for the
basis, one for the exponent.  All valid GiNaC expressions can be
inserted.  However, basic transformations like simplifying
`pow(pow(x,2),3)' to `x^6' automatically are only performed when this
is mathematically possible.  If we replace the outer exponent three in
the example by some symbols `a', the simplification is not safe and
will not be performed, since `a' might be `1/2' and `x' negative.

Objects of type `add' and `mul' are containers with an arbitrary number
of slots for expressions to be inserted.  Again, simple and safe
simplifications are carried out like transforming `3*x+4-x' to `2*x+4'.


File: ginac.info,  Node: Lists,  Next: Mathematical functions,  Prev: Fundamental containers,  Up: Basic Concepts

Lists of expressions
====================

The GiNaC class `lst' serves for holding a "list" of arbitrary
expressions. They are not as ubiquitous as in many other computer
algebra packages, but are sometimes used to supply a variable number of
arguments of the same type to GiNaC methods such as `subs()' and
`to_rational()', so you should have a basic understanding of them.

Lists of up to 16 expressions can be directly constructed from single
expressions:

     {
         symbol x("x"), y("y");
         lst l(x, 2, y, x+y);
         // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
         ...

Use the `nops()' method to determine the size (number of expressions) of
a list and the `op()' method or the `[]' operator to access individual
elements:

         ...
         cout << l.nops() << endl;                // prints '4'
         cout << l.op(2) << " " << l[0] << endl;  // prints 'y x'
         ...

`lst' is one of the few GiNaC classes that allow in-place modifications
(the only other one is `matrix'). You can modify single elements:

         ...
         l.let_op(1) = 7; // l is now {x, 7, y, x+y}
         ...

You can append or prepend an expression to a list with the `append()'
and `prepend()' methods:

         ...
         l.append(4*x);   // l is now {x, 7, y, x+y, 4*x}
         l.prepend(0);    // l is now {0, x, 7, y, x+y, 4*x}
         ...

You can remove the first or last element of a list with `remove_first()'
and `remove_last()':

         ...
         l.remove_first();   // l is now {x, 7, y, x+y, 4*x}
         l.remove_last();    // l is now {x, 7, y, x+y}
         ...

You can bring the elements of a list into a canonical order with
`sort()':

         ...
         lst l1(x, 2, y, x+y);
         lst l2(2, x+y, x, y);
         l1.sort();
         l2.sort();
         // l1 and l2 are now equal
         ...

Finally, you can remove all but the first element of consecutive groups
of elements with `unique()':

         ...
         lst l3(x, 2, 2, 2, y, x+y, y+x);
         l3.unique();        // l3 is now {x, 2, y, x+y}
     }


File: ginac.info,  Node: Mathematical functions,  Next: Relations,  Prev: Lists,  Up: Basic Concepts

Mathematical functions
======================

There are quite a number of useful functions hard-wired into GiNaC.  For
instance, all trigonometric and hyperbolic functions are implemented
(*Note Built-in Functions::, for a complete list).

These functions (better called _pseudofunctions_) are all objects of
class `function'.  They accept one or more expressions as arguments and
return one expression.  If the arguments are not numerical, the
evaluation of the function may be halted, as it does in the next
example, showing how a function returns itself twice and finally an
expression that may be really useful:

         ...
         symbol x("x"), y("y");
         ex foo = x+y/2;
         cout << tgamma(foo) << endl;
          // -> tgamma(x+(1/2)*y)
         ex bar = foo.subs(y==1);
         cout << tgamma(bar) << endl;
          // -> tgamma(x+1/2)
         ex foobar = bar.subs(x==7);
         cout << tgamma(foobar) << endl;
          // -> (135135/128)*Pi^(1/2)
         ...

Besides evaluation most of these functions allow differentiation, series
expansion and so on.  Read the next chapter in order to learn more about
this.

It must be noted that these pseudofunctions are created by inline
functions, where the argument list is templated.  This means that
whenever you call `GiNaC::sin(1)' it is equivalent to `sin(ex(1))' and
will therefore not result in a floating point number.  Unless of course
the function prototype is explicitly overridden - which is the case for
arguments of type `numeric' (not wrapped inside an `ex').  Hence, in
order to obtain a floating point number of class `numeric' you should
call `sin(numeric(1))'.  This is almost the same as calling
`sin(1).evalf()' except that the latter will return a numeric wrapped
inside an `ex'.


File: ginac.info,  Node: Relations,  Next: Matrices,  Prev: Mathematical functions,  Up: Basic Concepts

Relations
=========

Sometimes, a relation holding between two expressions must be stored
somehow.  The class `relational' is a convenient container for such
purposes.  A relation is by definition a container for two `ex' and a
relation between them that signals equality, inequality and so on.
They are created by simply using the C++ operators `==', `!=', `<',
`<=', `>' and `>=' between two expressions.

*Note Mathematical functions::, for examples where various applications
of the `.subs()' method show how objects of class relational are used
as arguments.  There they provide an intuitive syntax for
substitutions.  They are also used as arguments to the `ex::series'
method, where the left hand side of the relation specifies the variable
to expand in and the right hand side the expansion point.  They can also
be used for creating systems of equations that are to be solved for
unknown variables.  But the most common usage of objects of this class
is rather inconspicuous in statements of the form `if
(expand(pow(a+b,2))==a*a+2*a*b+b*b) {...}'.  Here, an implicit
conversion from `relational' to `bool' takes place.  Note, however,
that `==' here does not perform any simplifications, hence `expand()'
must be called explicitly.


File: ginac.info,  Node: Matrices,  Next: Indexed objects,  Prev: Relations,  Up: Basic Concepts

Matrices
========

A "matrix" is a two-dimensional array of expressions. The elements of a
matrix with m rows and n columns are accessed with two `unsigned'
indices, the first one in the range 0...m-1, the second one in the
range 0...n-1.

There are a couple of ways to construct matrices, with or without preset
elements:

     matrix::matrix(unsigned r, unsigned c);
     matrix::matrix(unsigned r, unsigned c, const lst & l);
     ex lst_to_matrix(const lst & l);
     ex diag_matrix(const lst & l);
     ex unit_matrix(unsigned x);
     ex unit_matrix(unsigned r, unsigned c);
     ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
     ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);

The first two functions are `matrix' constructors which create a matrix
with `r' rows and `c' columns. The matrix elements can be initialized
from a (flat) list of expressions `l'. Otherwise they are all set to
zero. The `lst_to_matrix()' function constructs a matrix from a list of
lists, each list representing a matrix row. `diag_matrix()' constructs
a diagonal matrix given the list of diagonal elements.  `unit_matrix()'
creates an `x' by `x' (or `r' by `c') unit matrix. And finally,
`symbolic_matrix' constructs a matrix filled with newly generated
symbols made of the specified base name and the position of each
element in the matrix.

Matrix elements can be accessed and set using the parenthesis (function
call) operator:

     const ex & matrix::operator()(unsigned r, unsigned c) const;
     ex & matrix::operator()(unsigned r, unsigned c);

It is also possible to access the matrix elements in a linear fashion
with the `op()' method. But C++-style subscripting with square brackets
`[]' is not available.

Here are a couple of examples of constructing matrices:

     {
         symbol a("a"), b("b");
     
         matrix M(2, 2);
         M(0, 0) = a;
         M(1, 1) = b;
         cout << M << endl;
          // -> [[a,0],[0,b]]
     
         cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
          // -> [[a,0],[0,b]]
     
         cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
          // -> [[a,0],[0,b]]
     
         cout << diag_matrix(lst(a, b)) << endl;
          // -> [[a,0],[0,b]]
     
         cout << unit_matrix(3) << endl;
          // -> [[1,0,0],[0,1,0],[0,0,1]]
     
         cout << symbolic_matrix(2, 3, "x") << endl;
          // -> [[x00,x01,x02],[x10,x11,x12]]
     }

There are three ways to do arithmetic with matrices. The first (and most
efficient one) is to use the methods provided by the `matrix' class:

     matrix matrix::add(const matrix & other) const;
     matrix matrix::sub(const matrix & other) const;
     matrix matrix::mul(const matrix & other) const;
     matrix matrix::mul_scalar(const ex & other) const;
     matrix matrix::pow(const ex & expn) const;
     matrix matrix::transpose(void) const;
     matrix matrix::inverse(void) const;

All of these methods return the result as a new matrix object. Here is
an example that calculates A*B-2*C for three matrices A, B and C:

     {
         matrix A(2, 2, lst(1, 2, 3, 4));
         matrix B(2, 2, lst(-1, 0, 2, 1));
         matrix C(2, 2, lst(8, 4, 2, 1));
     
         matrix result = A.mul(B).sub(C.mul_scalar(2));
         cout << result << endl;
          // -> [[-13,-6],[1,2]]
         ...
     }

The second (and probably the most natural) way is to construct an
expression containing matrices with the usual arithmetic operators and
`pow()'.  For efficiency reasons, expressions with sums, products and
powers of matrices are not automatically evaluated in GiNaC. You have
to call the method

     ex ex::evalm() const;

to obtain the result:

     {
         ...
         ex e = A*B - 2*C;
         cout << e << endl;
          // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
         cout << e.evalm() << endl;
          // -> [[-13,-6],[1,2]]
         ...
     }

The non-commutativity of the product `A*B' in this example is
automatically recognized by GiNaC. There is no need to use a special
operator here. *Note Non-commutative objects::, for more information
about dealing with non-commutative expressions.

Finally, you can work with indexed matrices and call
`simplify_indexed()' to perform the arithmetic:

     {
         ...
         idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
         e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
         cout << e << endl;
          // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
         cout << e.simplify_indexed() << endl;
          // -> [[-13,-6],[1,2]].i.j
     }

Using indices is most useful when working with rectangular matrices and
one-dimensional vectors because you don't have to worry about having to
transpose matrices before multiplying them. *Note Indexed objects::, for
more information about using matrices with indices, and about indices in
general.

The `matrix' class provides a couple of additional methods for
computing determinants, traces, and characteristic polynomials:

     ex matrix::determinant(unsigned algo = determinant_algo::automatic) const;
     ex matrix::trace(void) const;
     ex matrix::charpoly(const symbol & lambda) const;

The `algo' argument of `determinant()' allows to select between
different algorithms for calculating the determinant. The possible
values are defined in the `flags.h' header file. By default, GiNaC uses
a heuristic to automatically select an algorithm that is likely to give
the result most quickly.


File: ginac.info,  Node: Indexed objects,  Next: Non-commutative objects,  Prev: Matrices,  Up: Basic Concepts

Indexed objects
===============

GiNaC allows you to handle expressions containing general indexed
objects in arbitrary spaces. It is also able to canonicalize and
simplify such expressions and perform symbolic dummy index summations.
There are a number of predefined indexed objects provided, like delta
and metric tensors.

There are few restrictions placed on indexed objects and their indices
and it is easy to construct nonsense expressions, but our intention is
to provide a general framework that allows you to implement algorithms
with indexed quantities, getting in the way as little as possible.

Indexed quantities and their indices
------------------------------------

Indexed expressions in GiNaC are constructed of two special types of
objects, "index objects" and "indexed objects".

   * Index objects are of class `idx' or a subclass. Every index has a
     "value" and a "dimension" (which is the dimension of the space the
     index lives in) which can both be arbitrary expressions but are
     usually a number or a simple symbol. In addition, indices of class
     `varidx' have a "variance" (they can be co- or contravariant), and
     indices of class `spinidx' have a variance and can be "dotted" or
     "undotted".

   * Indexed objects are of class `indexed' or a subclass. They contain
     a "base expression" (which is the expression being indexed), and
     one or more indices.


*Note:* when printing expressions, covariant indices and indices
without variance are denoted `.i' while contravariant indices are
denoted `~i'. Dotted indices have a `*' in front of the index value. In
the following, we are going to use that notation in the text so instead
of A^i_jk we will write `A~i.j.k'. Index dimensions are not visible in
the output.

A simple example shall illustrate the concepts:

     #include <iostream>
     #include <ginac/ginac.h>
     using namespace std;
     using namespace GiNaC;
     
     int main()
     {
         symbol i_sym("i"), j_sym("j");
         idx i(i_sym, 3), j(j_sym, 3);
     
         symbol A("A");
         cout << indexed(A, i, j) << endl;
          // -> A.i.j
         ...

The `idx' constructor takes two arguments, the index value and the
index dimension. First we define two index objects, `i' and `j', both
with the numeric dimension 3. The value of the index `i' is the symbol
`i_sym' (which prints as `i') and the value of the index `j' is the
symbol `j_sym' (which prints as `j'). Next we construct an expression
containing one indexed object, `A.i.j'. It has the symbol `A' as its
base expression and the two indices `i' and `j'.

Note the difference between the indices `i' and `j' which are of class
`idx', and the index values which are the symbols `i_sym' and `j_sym'.
The indices of indexed objects cannot directly be symbols or numbers
but must be index objects. For example, the following is not correct
and will raise an exception:

     symbol i("i"), j("j");
     e = indexed(A, i, j); // ERROR: indices must be of type idx

You can have multiple indexed objects in an expression, index values can
be numeric, and index dimensions symbolic:

         ...
         symbol B("B"), dim("dim");
         cout << 4 * indexed(A, i)
               + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
          // -> B.j.2.i+4*A.i
         ...

`B' has a 4-dimensional symbolic index `k', a 3-dimensional numeric
index of value 2, and a symbolic index `i' with the symbolic dimension
`dim'. Note that GiNaC doesn't automatically notify you that the free
indices of `A' and `B' in the sum don't match (you have to call
`simplify_indexed()' for that, see below).

In fact, base expressions, index values and index dimensions can be
arbitrary expressions:

         ...
         cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
          // -> (B+A).(1+2*i)
         ...

It's also possible to construct nonsense like `Pi.sin(x)'. You will not
get an error message from this but you will probably not be able to do
anything useful with it.

The methods

     ex idx::get_value(void);
     ex idx::get_dimension(void);

return the value and dimension of an `idx' object. If you have an index
in an expression, such as returned by calling `.op()' on an indexed
object, you can get a reference to the `idx' object with the function
`ex_to<idx>()' on the expression.

There are also the methods

     bool idx::is_numeric(void);
     bool idx::is_symbolic(void);
     bool idx::is_dim_numeric(void);
     bool idx::is_dim_symbolic(void);

for checking whether the value and dimension are numeric or symbolic
(non-numeric). Using the `info()' method of an index (see *Note
Information About Expressions::) returns information about the index
value.

If you need co- and contravariant indices, use the `varidx' class:

         ...
         symbol mu_sym("mu"), nu_sym("nu");
         varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
         varidx mu_co(mu_sym, 4, true);       // covariant index .mu
     
         cout << indexed(A, mu, nu) << endl;
          // -> A~mu~nu
         cout << indexed(A, mu_co, nu) << endl;
          // -> A.mu~nu
         cout << indexed(A, mu.toggle_variance(), nu) << endl;
          // -> A.mu~nu
         ...

A `varidx' is an `idx' with an additional flag that marks it as co- or
contravariant. The default is a contravariant (upper) index, but this
can be overridden by supplying a third argument to the `varidx'
constructor. The two methods

     bool varidx::is_covariant(void);
     bool varidx::is_contravariant(void);

allow you to check the variance of a `varidx' object (use
`ex_to<varidx>()' to get the object reference from an expression).
There's also the very useful method

     ex varidx::toggle_variance(void);

which makes a new index with the same value and dimension but the
opposite variance. By using it you only have to define the index once.

The `spinidx' class provides dotted and undotted variant indices, as
used in the Weyl-van-der-Waerden spinor formalism:

         ...
         symbol K("K"), C_sym("C"), D_sym("D");
         spinidx C(C_sym, 2), D(D_sym);          // default is 2-dimensional,
                                                 // contravariant, undotted
         spinidx C_co(C_sym, 2, true);           // covariant index
         spinidx D_dot(D_sym, 2, false, true);   // contravariant, dotted
         spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
     
         cout << indexed(K, C, D) << endl;
          // -> K~C~D
         cout << indexed(K, C_co, D_dot) << endl;
          // -> K.C~*D
         cout << indexed(K, D_co_dot, D) << endl;
          // -> K.*D~D
         ...

A `spinidx' is a `varidx' with an additional flag that marks it as
dotted or undotted. The default is undotted but this can be overridden
by supplying a fourth argument to the `spinidx' constructor. The two
methods

     bool spinidx::is_dotted(void);
     bool spinidx::is_undotted(void);

allow you to check whether or not a `spinidx' object is dotted (use
`ex_to<spinidx>()' to get the object reference from an expression).
Finally, the two methods

     ex spinidx::toggle_dot(void);
     ex spinidx::toggle_variance_dot(void);

create a new index with the same value and dimension but opposite
dottedness and the same or opposite variance.

Substituting indices
--------------------

Sometimes you will want to substitute one symbolic index with another
symbolic or numeric index, for example when calculating one specific
element of a tensor expression. This is done with the `.subs()' method,
as it is done for symbols (see *Note Substituting Expressions::).

You have two possibilities here. You can either substitute the whole
index by another index or expression:

         ...
         ex e = indexed(A, mu_co);
         cout << e << " becomes " << e.subs(mu_co == nu) << endl;
          // -> A.mu becomes A~nu
         cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
          // -> A.mu becomes A~0
         cout << e << " becomes " << e.subs(mu_co == 0) << endl;
          // -> A.mu becomes A.0
         ...

The third example shows that trying to replace an index with something
that is not an index will substitute the index value instead.

Alternatively, you can substitute the _symbol_ of a symbolic index by
another expression:

         ...
         ex e = indexed(A, mu_co);
         cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
          // -> A.mu becomes A.nu
         cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
          // -> A.mu becomes A.0
         ...

As you see, with the second method only the value of the index will get
substituted. Its other properties, including its dimension, remain
unchanged.  If you want to change the dimension of an index you have to
substitute the whole index by another one with the new dimension.

Finally, substituting the base expression of an indexed object works as
expected:

         ...
         ex e = indexed(A, mu_co);
         cout << e << " becomes " << e.subs(A == A+B) << endl;
          // -> A.mu becomes (B+A).mu
         ...

Symmetries
----------

Indexed objects can have certain symmetry properties with respect to
their indices. Symmetries are specified as a tree of objects of class
`symmetry' that is constructed with the helper functions

     symmetry sy_none(...);
     symmetry sy_symm(...);
     symmetry sy_anti(...);
     symmetry sy_cycl(...);

`sy_none()' stands for no symmetry, `sy_symm()' and `sy_anti()' specify
fully symmetric or antisymmetric, respectively, and `sy_cycl()'
represents a cyclic symmetry. Each of these functions accepts up to four
arguments which can be either symmetry objects themselves or unsigned
integer numbers that represent an index position (counting from 0). A
symmetry specification that consists of only a single `sy_symm()',
`sy_anti()' or `sy_cycl()' with no arguments specifies the respective
symmetry for all indices.

Here are some examples of symmetry definitions:

         ...
         // No symmetry:
         e = indexed(A, i, j);
         e = indexed(A, sy_none(), i, j);     // equivalent
         e = indexed(A, sy_none(0, 1), i, j); // equivalent
     
         // Symmetric in all three indices:
         e = indexed(A, sy_symm(), i, j, k);
         e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
         e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
                                                    // different canonical order
     
         // Symmetric in the first two indices only:
         e = indexed(A, sy_symm(0, 1), i, j, k);
         e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
     
         // Antisymmetric in the first and last index only (index ranges need not
         // be contiguous):
         e = indexed(A, sy_anti(0, 2), i, j, k);
         e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
     
         // An example of a mixed symmetry: antisymmetric in the first two and
         // last two indices, symmetric when swapping the first and last index
         // pairs (like the Riemann curvature tensor):
         e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
     
         // Cyclic symmetry in all three indices:
         e = indexed(A, sy_cycl(), i, j, k);
         e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
     
         // The following examples are invalid constructions that will throw
         // an exception at run time.
     
         // An index may not appear multiple times:
         e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
         e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
     
         // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
         // same number of indices:
         e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
     
         // And of course, you cannot specify indices which are not there:
         e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
         ...

If you need to specify more than four indices, you have to use the
`.add()' method of the `symmetry' class. For example, to specify full
symmetry in the first six indices you would write `sy_symm(0, 1, 2,
3).add(4).add(5)'.

If an indexed object has a symmetry, GiNaC will automatically bring the
indices into a canonical order which allows for some immediate
simplifications:

         ...
         cout << indexed(A, sy_symm(), i, j)
               + indexed(A, sy_symm(), j, i) << endl;
          // -> 2*A.j.i
         cout << indexed(B, sy_anti(), i, j)
               + indexed(B, sy_anti(), j, i) << endl;
          // -> 0
         cout << indexed(B, sy_anti(), i, j, k)
               - indexed(B, sy_anti(), j, k, i) << endl;
          // -> 0
         ...

Dummy indices
-------------

GiNaC treats certain symbolic index pairs as "dummy indices" meaning
that a summation over the index range is implied. Symbolic indices
which are not dummy indices are called "free indices". Numeric indices
are neither dummy nor free indices.

To be recognized as a dummy index pair, the two indices must be of the
same class and their value must be the same single symbol (an index like
`2*n+1' is never a dummy index). If the indices are of class `varidx'
they must also be of opposite variance; if they are of class `spinidx'
they must be both dotted or both undotted.

The method `.get_free_indices()' returns a vector containing the free
indices of an expression. It also checks that the free indices of the
terms of a sum are consistent:

     {
         symbol A("A"), B("B"), C("C");
     
         symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
         idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
     
         ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
         cout << exprseq(e.get_free_indices()) << endl;
          // -> (.i,.k)
          // 'j' and 'l' are dummy indices
     
         symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
         varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
     
         e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
           + indexed(C, mu, sigma, rho, sigma.toggle_variance());
         cout << exprseq(e.get_free_indices()) << endl;
          // -> (~mu,~rho)
          // 'nu' is a dummy index, but 'sigma' is not
     
         e = indexed(A, mu, mu);
         cout << exprseq(e.get_free_indices()) << endl;
          // -> (~mu)
          // 'mu' is not a dummy index because it appears twice with the same
          // variance
     
         e = indexed(A, mu, nu) + 42;
         cout << exprseq(e.get_free_indices()) << endl; // ERROR
          // this will throw an exception:
          // "add::get_free_indices: inconsistent indices in sum"
     }

Simplifying indexed expressions
-------------------------------

In addition to the few automatic simplifications that GiNaC performs on
indexed expressions (such as re-ordering the indices of symmetric
tensors and calculating traces and convolutions of matrices and
predefined tensors) there is the method

     ex ex::simplify_indexed(void);
     ex ex::simplify_indexed(const scalar_products & sp);

that performs some more expensive operations:

   * it checks the consistency of free indices in sums in the same way
     `get_free_indices()' does

   * it tries to give dummy indices that appear in different terms of a
     sum   the same name to allow simplifications like a_i*b_i-a_j*b_j=0

   * it (symbolically) calculates all possible dummy index
     summations/contractions   with the predefined tensors (this will
     be explained in more detail in the   next section)

   * it detects contractions that vanish for symmetry reasons, for
     example   the contraction of a symmetric and a totally
     antisymmetric tensor

   * as a special case of dummy index summation, it can replace scalar
     products   of two tensors with a user-defined value

The last point is done with the help of the `scalar_products' class
which is used to store scalar products with known values (this is not an
arithmetic class, you just pass it to `simplify_indexed()'):

     {
         symbol A("A"), B("B"), C("C"), i_sym("i");
         idx i(i_sym, 3);
     
         scalar_products sp;
         sp.add(A, B, 0); // A and B are orthogonal
         sp.add(A, C, 0); // A and C are orthogonal
         sp.add(A, A, 4); // A^2 = 4 (A has length 2)
     
         e = indexed(A + B, i) * indexed(A + C, i);
         cout << e << endl;
          // -> (B+A).i*(A+C).i
     
         cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
              << endl;
          // -> 4+C.i*B.i
     }

The `scalar_products' object `sp' acts as a storage for the scalar
products added to it with the `.add()' method. This method takes three
arguments: the two expressions of which the scalar product is taken,
and the expression to replace it with. After `sp.add(A, B, 0)',
`simplify_indexed()' will replace all scalar products of indexed
objects that have the symbols `A' and `B' as base expressions with the
single value 0. The number, type and dimension of the indices don't
matter; `A~mu~nu*B.mu.nu' would also be replaced by 0.

The example above also illustrates a feature of the `expand()' method:
if passed the `expand_indexed' option it will distribute indices over
sums, so `(A+B).i' becomes `A.i+B.i'.

Predefined tensors
------------------

Some frequently used special tensors such as the delta, epsilon and
metric tensors are predefined in GiNaC. They have special properties
when contracted with other tensor expressions and some of them have
constant matrix representations (they will evaluate to a number when
numeric indices are specified).

Delta tensor
............

The delta tensor takes two indices, is symmetric and has the matrix
representation `diag(1, 1, 1, ...)'. It is constructed by the function
`delta_tensor()':

     {
         symbol A("A"), B("B");
     
         idx i(symbol("i"), 3), j(symbol("j"), 3),
             k(symbol("k"), 3), l(symbol("l"), 3);
     
         ex e = indexed(A, i, j) * indexed(B, k, l)
              * delta_tensor(i, k) * delta_tensor(j, l) << endl;
         cout << e.simplify_indexed() << endl;
          // -> B.i.j*A.i.j
     
         cout << delta_tensor(i, i) << endl;
          // -> 3
     }

General metric tensor
.....................

The function `metric_tensor()' creates a general symmetric metric
tensor with two indices that can be used to raise/lower tensor indices.
The metric tensor is denoted as `g' in the output and if its indices
are of mixed variance it is automatically replaced by a delta tensor:

     {
         symbol A("A");
     
         varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
     
         ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
         cout << e.simplify_indexed() << endl;
          // -> A~mu~rho
     
         e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
         cout << e.simplify_indexed() << endl;
          // -> g~mu~rho
     
         e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
           * metric_tensor(nu, rho);
         cout << e.simplify_indexed() << endl;
          // -> delta.mu~rho
     
         e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
           * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
             + indexed(A, mu.toggle_variance(), rho));
         cout << e.simplify_indexed() << endl;
          // -> 4+A.rho~rho
     }

Minkowski metric tensor
.......................

The Minkowski metric tensor is a special metric tensor with a constant
matrix representation which is either `diag(1, -1, -1, ...)' (negative
signature, the default) or `diag(-1, 1, 1, ...)' (positive signature).
It is created with the function `lorentz_g()' (although it is output as
`eta'):

     {
         varidx mu(symbol("mu"), 4);
     
         e = delta_tensor(varidx(0, 4), mu.toggle_variance())
           * lorentz_g(mu, varidx(0, 4));       // negative signature
         cout << e.simplify_indexed() << endl;
          // -> 1
     
         e = delta_tensor(varidx(0, 4), mu.toggle_variance())
           * lorentz_g(mu, varidx(0, 4), true); // positive signature
         cout << e.simplify_indexed() << endl;
          // -> -1
     }

Spinor metric tensor
....................

The function `spinor_metric()' creates an antisymmetric tensor with two
indices that is used to raise/lower indices of 2-component spinors.  It
is output as `eps':

     {
         symbol psi("psi");
     
         spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
         ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
     
         e = spinor_metric(A, B) * indexed(psi, B_co);
         cout << e.simplify_indexed() << endl;
          // -> psi~A
     
         e = spinor_metric(A, B) * indexed(psi, A_co);
         cout << e.simplify_indexed() << endl;
          // -> -psi~B
     
         e = spinor_metric(A_co, B_co) * indexed(psi, B);
         cout << e.simplify_indexed() << endl;
          // -> -psi.A
     
         e = spinor_metric(A_co, B_co) * indexed(psi, A);
         cout << e.simplify_indexed() << endl;
          // -> psi.B
     
         e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
         cout << e.simplify_indexed() << endl;
          // -> 2
     
         e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
         cout << e.simplify_indexed() << endl;
          // -> -delta.A~C
     }

The matrix representation of the spinor metric is `[[0, 1], [-1, 0]]'.

Epsilon tensor
..............

The epsilon tensor is totally antisymmetric, its number of indices is
equal to the dimension of the index space (the indices must all be of
the same numeric dimension), and `eps.1.2.3...' (resp. `eps~0~1~2...')
is defined to be 1. Its behavior with indices that have a variance also
depends on the signature of the metric. Epsilon tensors are output as
`eps'.

There are three functions defined to create epsilon tensors in 2, 3 and
4 dimensions:

     ex epsilon_tensor(const ex & i1, const ex & i2);
     ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
     ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);

The first two functions create an epsilon tensor in 2 or 3 Euclidean
dimensions, the last function creates an epsilon tensor in a
4-dimensional Minkowski space (the last `bool' argument specifies
whether the metric has negative or positive signature, as in the case
of the Minkowski metric tensor):

     {
         varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
                sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
         e = lorentz_eps(mu, nu, rho, sig) *
             lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
         cout << simplify_indexed(e) << endl;
          // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
     
         idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
         symbol A("A"), B("B");
         e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
         cout << simplify_indexed(e) << endl;
          // -> -B.k*A.j*eps.i.k.j
         e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
         cout << simplify_indexed(e) << endl;
          // -> 0
     }

Linear algebra
--------------

The `matrix' class can be used with indices to do some simple linear
algebra (linear combinations and products of vectors and matrices,
traces and scalar products):

     {
         idx i(symbol("i"), 2), j(symbol("j"), 2);
         symbol x("x"), y("y");
     
         // A is a 2x2 matrix, X is a 2x1 vector
         matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
     
         cout << indexed(A, i, i) << endl;
          // -> 5
     
         ex e = indexed(A, i, j) * indexed(X, j);
         cout << e.simplify_indexed() << endl;
          // -> [[2*y+x],[4*y+3*x]].i
     
         e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
         cout << e.simplify_indexed() << endl;
          // -> [[3*y+3*x,6*y+2*x]].j
     }

You can of course obtain the same results with the `matrix::add()',
`matrix::mul()' and `matrix::trace()' methods (*note Matrices::) but
with indices you don't have to worry about transposing matrices.

Matrix indices always start at 0 and their dimension must match the
number of rows/columns of the matrix. Matrices with one row or one
column are vectors and can have one or two indices (it doesn't matter
whether it's a row or a column vector). Other matrices must have two
indices.

You should be careful when using indices with variance on matrices.
GiNaC doesn't look at the variance and doesn't know that `F~mu~nu' and
`F.mu.nu' are different matrices. In this case you should use only one
form for `F' and explicitly multiply it with a matrix representation of
the metric tensor.


File: ginac.info,  Node: Non-commutative objects,  Next: Methods and Functions,  Prev: Indexed objects,  Up: Basic Concepts

Non-commutative objects
=======================

GiNaC is equipped to handle certain non-commutative algebras. Three
classes of non-commutative objects are built-in which are mostly of use
in high energy physics:

   * Clifford (Dirac) algebra (class `clifford')

   * su(3) Lie algebra (class `color')

   * Matrices (unindexed) (class `matrix')

The `clifford' and `color' classes are subclasses of `indexed' because
the elements of these algebras usually carry indices. The `matrix'
class is described in more detail in *Note Matrices::.

Unlike most computer algebra systems, GiNaC does not primarily provide
an operator (often denoted `&*') for representing inert products of
arbitrary objects. Rather, non-commutativity in GiNaC is a property of
the classes of objects involved, and non-commutative products are
formed with the usual `*' operator, as are ordinary products. GiNaC is
capable of figuring out by itself which objects commute and will group
the factors by their class. Consider this example:

         ...
         varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
         idx a(symbol("a"), 8), b(symbol("b"), 8);
         ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
         cout << e << endl;
          // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
         ...

As can be seen, GiNaC pulls out the overall commutative factor `-16' and
groups the non-commutative factors (the gammas and the su(3) generators)
together while preserving the order of factors within each class
(because Clifford objects commute with color objects). The resulting
expression is a _commutative_ product with two factors that are
themselves non-commutative products (`gamma~mu*gamma~nu' and
`T.a*T.b'). For clarification, parentheses are placed around the
non-commutative products in the output.

Non-commutative products are internally represented by objects of the
class `ncmul', as opposed to commutative products which are handled by
the `mul' class. You will normally not have to worry about this
distinction, though.

The advantage of this approach is that you never have to worry about
using (or forgetting to use) a special operator when constructing
non-commutative expressions. Also, non-commutative products in GiNaC
are more intelligent than in other computer algebra systems; they can,
for example, automatically canonicalize themselves according to rules
specified in the implementation of the non-commutative classes. The
drawback is that to work with other than the built-in algebras you have
to implement new classes yourself. Symbols always commute and it's not
possible to construct non-commutative products using symbols to
represent the algebra elements or generators. User-defined functions
can, however, be specified as being non-commutative.

Information about the commutativity of an object or expression can be
obtained with the two member functions

     unsigned ex::return_type(void) const;
     unsigned ex::return_type_tinfo(void) const;

The `return_type()' function returns one of three values (defined in
the header file `flags.h'), corresponding to three categories of
expressions in GiNaC:

   * `return_types::commutative': Commutes with everything. Most GiNaC
     classes are of this kind.

   * `return_types::noncommutative': Non-commutative, belonging to a
     certain class of non-commutative objects which can be determined
     with the   `return_type_tinfo()' method. Expressions of this
     category commute   with everything except `noncommutative'
     expressions of the same   class.

   * `return_types::noncommutative_composite': Non-commutative, composed
      of non-commutative objects of different classes. Expressions of
     this   category don't commute with any other `noncommutative' or
     `noncommutative_composite' expressions.

The value returned by the `return_type_tinfo()' method is valid only
when the return type of the expression is `noncommutative'. It is a
value that is unique to the class of the object and usually one of the
constants in `tinfos.h', or derived therefrom.

Here are a couple of examples:

*Expression*             *`return_type()'*        *`return_type_tinfo()'*
`42'                     `commutative'            -
`2*x-y'                  `commutative'            -
`dirac_ONE()'            `noncommutative'         `TINFO_clifford'
`dirac_gamma(mu)*dirac_gamma(nu)'`noncommutative'         `TINFO_clifford'
`2*color_T(a)'           `noncommutative'         `TINFO_color'
`dirac_ONE()*color_T(a)' `noncommutative_composite'-

Note: the `return_type_tinfo()' of Clifford objects is only equal to
`TINFO_clifford' for objects with a representation label of zero.
Other representation labels yield a different `return_type_tinfo()',
but it's the same for any two objects with the same label. This is also
true for color objects.

A last note: With the exception of matrices, positive integer powers of
non-commutative objects are automatically expanded in GiNaC. For
example, `pow(a*b, 2)' becomes `a*b*a*b' if `a' and `b' are
non-commutative expressions).

Clifford algebra
----------------

Clifford algebra elements (also called Dirac gamma matrices, although
GiNaC doesn't treat them as matrices) are designated as `gamma~mu' and
satisfy `gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu' where
`eta~mu~nu' is the Minkowski metric tensor. Dirac gammas are
constructed by the function

     ex dirac_gamma(const ex & mu, unsigned char rl = 0);

which takes two arguments: the index and a "representation label" in the
range 0 to 255 which is used to distinguish elements of different
Clifford algebras (this is also called a "spin line index"). Gammas
with different labels commute with each other. The dimension of the
index can be 4 or (in the framework of dimensional regularization) any
symbolic value. Spinor indices on Dirac gammas are not supported in
GiNaC.

The unity element of a Clifford algebra is constructed by

     ex dirac_ONE(unsigned char rl = 0);

*Note:* You must always use `dirac_ONE()' when referring to multiples
of the unity element, even though it's customary to omit it.  E.g.
instead of `dirac_gamma(mu)*(dirac_slash(q,4)+m)' you have to write
`dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())'. Otherwise, GiNaC
will complain and/or produce incorrect results.

There is a special element `gamma5' that commutes with all other
gammas, has a unit square, and in 4 dimensions equals `gamma~0 gamma~1
gamma~2 gamma~3', provided by

     ex dirac_gamma5(unsigned char rl = 0);

The chiral projectors `(1+/-gamma5)/2' are also available as proper
objects, constructed by

     ex dirac_gammaL(unsigned char rl = 0);
     ex dirac_gammaR(unsigned char rl = 0);

They observe the relations `gammaL^2 = gammaL', `gammaR^2 = gammaR',
and `gammaL gammaR = gammaR gammaL = 0'.

Finally, the function

     ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);

creates a term that represents a contraction of `e' with the Dirac
Lorentz vector (it behaves like a term of the form `e.mu gamma~mu' with
a unique index whose dimension is given by the `dim' argument).  Such
slashed expressions are printed with a trailing backslash, e.g. `e\'.

In products of dirac gammas, superfluous unity elements are
automatically removed, squares are replaced by their values, and
`gamma5', `gammaL' and `gammaR' are moved to the front.

The `simplify_indexed()' function performs contractions in gamma
strings, for example

     {
         ...
         symbol a("a"), b("b"), D("D");
         varidx mu(symbol("mu"), D);
         ex e = dirac_gamma(mu) * dirac_slash(a, D)
              * dirac_gamma(mu.toggle_variance());
         cout << e << endl;
          // -> gamma~mu*a\*gamma.mu
         e = e.simplify_indexed();
         cout << e << endl;
          // -> -D*a\+2*a\
         cout << e.subs(D == 4) << endl;
          // -> -2*a\
         ...
     }

To calculate the trace of an expression containing strings of Dirac
gammas you use the function

     ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);

This function takes the trace of all gammas with the specified
representation label; gammas with other labels are left standing. The
last argument to `dirac_trace()' is the value to be returned for the
trace of the unity element, which defaults to 4. The `dirac_trace()'
function is a linear functional that is equal to the usual trace only
in D = 4 dimensions.  In particular, the functional is not cyclic in D
!= 4 dimensions when acting on expressions containing `gamma5', so it's
not a proper trace.  This `gamma5' scheme is described in greater
detail in `The Role of gamma5 in Dimensional Regularization'.

The value of the trace itself is also usually different in 4 and in D
!= 4 dimensions:

     {
         // 4 dimensions
         varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
         ex e = dirac_gamma(mu) * dirac_gamma(nu) *
                dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
         cout << dirac_trace(e).simplify_indexed() << endl;
          // -> -8*eta~rho~nu
     }
     ...
     {
         // D dimensions
         symbol D("D");
         varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
         ex e = dirac_gamma(mu) * dirac_gamma(nu) *
                dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
         cout << dirac_trace(e).simplify_indexed() << endl;
          // -> 8*eta~rho~nu-4*eta~rho~nu*D
     }

Here is an example for using `dirac_trace()' to compute a value that
appears in the calculation of the one-loop vacuum polarization
amplitude in QED:

     {
         symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
         varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
     
         scalar_products sp;
         sp.add(l, l, pow(l, 2));
         sp.add(l, q, ldotq);
     
         ex e = dirac_gamma(mu) *
                (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
                dirac_gamma(mu.toggle_variance()) *
                (dirac_slash(l, D) + m * dirac_ONE());
         e = dirac_trace(e).simplify_indexed(sp);
         e = e.collect(lst(l, ldotq, m));
         cout << e << endl;
          // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
     }

The `canonicalize_clifford()' function reorders all gamma products that
appear in an expression to a canonical (but not necessarily simple)
form.  You can use this to compare two expressions or for further
simplifications:

     {
         varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
         ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
         cout << e << endl;
          // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
     
         e = canonicalize_clifford(e);
         cout << e << endl;
          // -> 2*eta~mu~nu
     }

Color algebra
-------------

For computations in quantum chromodynamics, GiNaC implements the base
elements and structure constants of the su(3) Lie algebra (color
algebra). The base elements T_a are constructed by the function

     ex color_T(const ex & a, unsigned char rl = 0);

which takes two arguments: the index and a "representation label" in the
range 0 to 255 which is used to distinguish elements of different color
algebras. Objects with different labels commute with each other. The
dimension of the index must be exactly 8 and it should be of class
`idx', not `varidx'.

The unity element of a color algebra is constructed by

     ex color_ONE(unsigned char rl = 0);

*Note:* You must always use `color_ONE()' when referring to multiples
of the unity element, even though it's customary to omit it.  E.g.
instead of `color_T(a)*(color_T(b)*indexed(X,b)+1)' you have to write
`color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())'. Otherwise, GiNaC
may produce incorrect results.

The functions

     ex color_d(const ex & a, const ex & b, const ex & c);
     ex color_f(const ex & a, const ex & b, const ex & c);

create the symmetric and antisymmetric structure constants d_abc and
f_abc which satisfy {T_a, T_b} = 1/3 delta_ab + d_abc T_c and [T_a,
T_b] = i f_abc T_c.

There's an additional function

     ex color_h(const ex & a, const ex & b, const ex & c);

which returns the linear combination `color_d(a, b, c)+I*color_f(a, b,
c)'.

The function `simplify_indexed()' performs some simplifications on
expressions containing color objects:

     {
         ...
         idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
             k(symbol("k"), 8), l(symbol("l"), 8);
     
         e = color_d(a, b, l) * color_f(a, b, k);
         cout << e.simplify_indexed() << endl;
          // -> 0
     
         e = color_d(a, b, l) * color_d(a, b, k);
         cout << e.simplify_indexed() << endl;
          // -> 5/3*delta.k.l
     
         e = color_f(l, a, b) * color_f(a, b, k);
         cout << e.simplify_indexed() << endl;
          // -> 3*delta.k.l
     
         e = color_h(a, b, c) * color_h(a, b, c);
         cout << e.simplify_indexed() << endl;
          // -> -32/3
     
         e = color_h(a, b, c) * color_T(b) * color_T(c);
         cout << e.simplify_indexed() << endl;
          // -> -2/3*T.a
     
         e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
         cout << e.simplify_indexed() << endl;
          // -> -8/9*ONE
     
         e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
         cout << e.simplify_indexed() << endl;
          // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
         ...

To calculate the trace of an expression containing color objects you
use the function

     ex color_trace(const ex & e, unsigned char rl = 0);

This function takes the trace of all color `T' objects with the
specified representation label; `T's with other labels are left
standing. For example:

         ...
         e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
         cout << e << endl;
          // -> -I*f.a.c.b+d.a.c.b
     }


File: ginac.info,  Node: Methods and Functions,  Next: Information About Expressions,  Prev: Non-commutative objects,  Up: Top

Methods and Functions
*********************

In this chapter the most important algorithms provided by GiNaC will be
described.  Some of them are implemented as functions on expressions,
others are implemented as methods provided by expression objects.  If
they are methods, there exists a wrapper function around it, so you can
alternatively call it in a functional way as shown in the simple
example:

         ...
         cout << "As method:   " << sin(1).evalf() << endl;
         cout << "As function: " << evalf(sin(1)) << endl;
         ...

The general rule is that wherever methods accept one or more parameters
(ARG1, ARG2, ...) the order of arguments the function wrapper accepts
is the same but preceded by the object to act on (OBJECT, ARG1, ARG2,
...).  This approach is the most natural one in an OO model but it may
lead to confusion for MapleV users because where they would type
`A:=x+1; subs(x=2,A);' GiNaC would require `A=x+1; subs(A,x==2);'
(after proper declaration of `A' and `x').  On the other hand, since
MapleV returns 3 on `A:=x^2+3; coeff(A,x,0);' (GiNaC: `A=pow(x,2)+3;
coeff(A,x,0);') it is clear that MapleV is not trying to be consistent
here.  Also, users of MuPAD will in most cases feel more comfortable
with GiNaC's convention.  All function wrappers are implemented as
simple inline functions which just call the corresponding method and
are only provided for users uncomfortable with OO who are dead set to
avoid method invocations.  Generally, nested function wrappers are much
harder to read than a sequence of methods and should therefore be
avoided if possible.  On the other hand, not everything in GiNaC is a
method on class `ex' and sometimes calling a function cannot be avoided.

* Menu:

* Information About Expressions::
* Substituting Expressions::
* Pattern Matching and Advanced Substitutions::
* Applying a Function on Subexpressions::
* Polynomial Arithmetic::           Working with polynomials.
* Rational Expressions::            Working with rational functions.
* Symbolic Differentiation::
* Series Expansion::                Taylor and Laurent expansion.
* Symmetrization::
* Built-in Functions::              List of predefined mathematical functions.
* Input/Output::                    Input and output of expressions.


File: ginac.info,  Node: Information About Expressions,  Next: Substituting Expressions,  Prev: Methods and Functions,  Up: Methods and Functions

Getting information about expressions
=====================================

Checking expression types
-------------------------

Sometimes it's useful to check whether a given expression is a plain
number, a sum, a polynomial with integer coefficients, or of some other
specific type.  GiNaC provides a couple of functions for this:

     bool is_a<T>(const ex & e);
     bool is_exactly_a<T>(const ex & e);
     bool ex::info(unsigned flag);
     unsigned ex::return_type(void) const;
     unsigned ex::return_type_tinfo(void) const;

When the test made by `is_a<T>()' returns true, it is safe to call one
of the functions `ex_to<T>()', where `T' is one of the class names
(*Note The Class Hierarchy::, for a list of all classes). For example,
assuming `e' is an `ex':

     {
         ...
         if (is_a<numeric>(e))
             numeric n = ex_to<numeric>(e);
         ...
     }

`is_a<T>(e)' allows you to check whether the top-level object of an
expression `e' is an instance of the GiNaC class `T' (*Note The Class
Hierarchy::, for a list of all classes). This is most useful, e.g., for
checking whether an expression is a number, a sum, or a product:

     {
         symbol x("x");
         ex e1 = 42;
         ex e2 = 4*x - 3;
         is_a<numeric>(e1);  // true
         is_a<numeric>(e2);  // false
         is_a<add>(e1);      // false
         is_a<add>(e2);      // true
         is_a<mul>(e1);      // false
         is_a<mul>(e2);      // false
     }

In contrast, `is_exactly_a<T>(e)' allows you to check whether the
top-level object of an expression `e' is an instance of the GiNaC class
`T', not including parent classes.

The `info()' method is used for checking certain attributes of
expressions. The possible values for the `flag' argument are defined in
`ginac/flags.h', the most important being explained in the following
table:

*Flag*                 *Returns true if the object is...*
`numeric'              ...a number (same as `is_<numeric>(...)')
`real'                 ...a real integer, rational or float (i.e. is not
                       complex)
`rational'             ...an exact rational number (integers are
                       rational, too)
`integer'              ...a (non-complex) integer
`crational'            ...an exact (complex) rational number (such as
                       2/3+7/2*I)
`cinteger'             ...a (complex) integer (such as 2-3*I)
`positive'             ...not complex and greater than 0
`negative'             ...not complex and less than 0
`nonnegative'          ...not complex and greater than or equal to 0
`posint'               ...an integer greater than 0
`negint'               ...an integer less than 0
`nonnegint'            ...an integer greater than or equal to 0
`even'                 ...an even integer
`odd'                  ...an odd integer
`prime'                ...a prime integer (probabilistic primality test)
`relation'             ...a relation (same as `is_a<relational>(...)')
`relation_equal'       ...a `==' relation
`relation_not_equal'   ...a `!=' relation
`relation_less'        ...a `<' relation
`relation_less_or_equal'...a `<=' relation
`relation_greater'     ...a `>' relation
`relation_greater_or_equal'...a `>=' relation
`symbol'               ...a symbol (same as `is_a<symbol>(...)')
`list'                 ...a list (same as `is_a<lst>(...)')
`polynomial'           ...a polynomial (i.e. only consists of sums and
                       products of numbers and symbols with positive
                       integer powers)
`integer_polynomial'   ...a polynomial with (non-complex) integer
                       coefficients
`cinteger_polynomial'  ...a polynomial with (possibly complex) integer
                       coefficients (such as 2-3*I)
`rational_polynomial'  ...a polynomial with (non-complex) rational
                       coefficients
`crational_polynomial' ...a polynomial with (possibly complex) rational
                       coefficients (such as 2/3+7/2*I)
`rational_function'    ...a rational function (x+y, z/(x+y))
`algebraic'            ...an algebraic object (sqrt(2), sqrt(x)-1)

To determine whether an expression is commutative or non-commutative
and if so, with which other expressions it would commute, you use the
methods `return_type()' and `return_type_tinfo()'. *Note
Non-commutative objects::, for an explanation of these.

Accessing subexpressions
------------------------

GiNaC provides the two methods

     unsigned ex::nops();
     ex ex::op(unsigned i);

for accessing the subexpressions in the container-like GiNaC classes
like `add', `mul', `lst', and `function'. `nops()' determines the
number of subexpressions (`operands') contained, while `op()' returns
the `i'-th (0..`nops()-1') subexpression.  In the case of a `power'
object, `op(0)' will return the basis and `op(1)' the exponent. For
`indexed' objects, `op(0)' is the base expression and `op(i)', i>0 are
the indices.

The left-hand and right-hand side expressions of objects of class
`relational' (and only of these) can also be accessed with the methods

     ex ex::lhs();
     ex ex::rhs();

Comparing expressions
---------------------

Expressions can be compared with the usual C++ relational operators like
`==', `>', and `<' but if the expressions contain symbols, the result
is usually not determinable and the result will be `false', except in
the case of the `!=' operator. You should also be aware that GiNaC will
only do the most trivial test for equality (subtracting both
expressions), so something like `(pow(x,2)+x)/x==x+1' will return
`false'.

Actually, if you construct an expression like `a == b', this will be
represented by an object of the `relational' class (*note Relations::)
which is not evaluated until (explicitly or implicitly) cast to a
`bool'.

There are also two methods

     bool ex::is_equal(const ex & other);
     bool ex::is_zero();

for checking whether one expression is equal to another, or equal to
zero, respectively.

*Warning:* You will also find an `ex::compare()' method in the GiNaC
header files. This method is however only to be used internally by
GiNaC to establish a canonical sort order for terms, and using it to
compare expressions will give very surprising results.


File: ginac.info,  Node: Substituting Expressions,  Next: Pattern Matching and Advanced Substitutions,  Prev: Information About Expressions,  Up: Methods and Functions

Substituting expressions
========================

Algebraic objects inside expressions can be replaced with arbitrary
expressions via the `.subs()' method:

     ex ex::subs(const ex & e);
     ex ex::subs(const lst & syms, const lst & repls);

In the first form, `subs()' accepts a relational of the form `object ==
expression' or a `lst' of such relationals:

     {
         symbol x("x"), y("y");
     
         ex e1 = 2*x^2-4*x+3;
         cout << "e1(7) = " << e1.subs(x == 7) << endl;
          // -> 73
     
         ex e2 = x*y + x;
         cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
          // -> -10
     }

If you specify multiple substitutions, they are performed in parallel,
so e.g.  `subs(lst(x == y, y == x))' exchanges `x' and `y'.

The second form of `subs()' takes two lists, one for the objects to be
replaced and one for the expressions to be substituted (both lists must
contain the same number of elements). Using this form, you would write
`subs(lst(x, y), lst(y, x))' to exchange `x' and `y'.

`subs()' performs syntactic substitution of any complete algebraic
object; it does not try to match sub-expressions as is demonstrated by
the following example:

     {
         symbol x("x"), y("y"), z("z");
     
         ex e1 = pow(x+y, 2);
         cout << e1.subs(x+y == 4) << endl;
          // -> 16
     
         ex e2 = sin(x)*sin(y)*cos(x);
         cout << e2.subs(sin(x) == cos(x)) << endl;
          // -> cos(x)^2*sin(y)
     
         ex e3 = x+y+z;
         cout << e3.subs(x+y == 4) << endl;
          // -> x+y+z
          // (and not 4+z as one might expect)
     }

A more powerful form of substitution using wildcards is described in the
next section.


File: ginac.info,  Node: Pattern Matching and Advanced Substitutions,  Next: Applying a Function on Subexpressions,  Prev: Substituting Expressions,  Up: Methods and Functions

Pattern matching and advanced substitutions
===========================================

GiNaC allows the use of patterns for checking whether an expression is
of a certain form or contains subexpressions of a certain form, and for
substituting expressions in a more general way.

A "pattern" is an algebraic expression that optionally contains
wildcards.  A "wildcard" is a special kind of object (of class
`wildcard') that represents an arbitrary expression. Every wildcard has
a "label" which is an unsigned integer number to allow having multiple
different wildcards in a pattern. Wildcards are printed as `$label'
(this is also the way they are specified in `ginsh'). In C++ code,
wildcard objects are created with the call

     ex wild(unsigned label = 0);

which is simply a wrapper for the `wildcard()' constructor with a
shorter name.

Some examples for patterns:

*Constructed as*                     *Output as*
`wild()'                             `$0'
`pow(x,wild())'                      `x^$0'
`atan2(wild(1),wild(2))'             `atan2($1,$2)'
`indexed(A,idx(wild(),3))'           `A.$0'

Notes:

   * Wildcards behave like symbols and are subject to the same algebraic
      rules. E.g., `$0+2*$0' is automatically transformed to `3*$0'.

   * As shown in the last example, to use wildcards for indices you
     have to   use them as the value of an `idx' object. This is
     because indices must   always be of class `idx' (or a subclass).

   * Wildcards only represent expressions or subexpressions. It is not
     possible to use them as placeholders for other properties like
     index   dimension or variance, representation labels, symmetry of
     indexed objects   etc.

   * Because wildcards are commutative, it is not possible to use
     wildcards   as part of noncommutative products.

   * A pattern does not have to contain wildcards. `x' and `x+y'   are
     also valid patterns.

Matching expressions
--------------------

The most basic application of patterns is to check whether an expression
matches a given pattern. This is done by the function

     bool ex::match(const ex & pattern);
     bool ex::match(const ex & pattern, lst & repls);

This function returns `true' when the expression matches the pattern
and `false' if it doesn't. If used in the second form, the actual
subexpressions matched by the wildcards get returned in the `repls'
object as a list of relations of the form `wildcard == expression'.  If
`match()' returns false, the state of `repls' is undefined.  For
reproducible results, the list should be empty when passed to
`match()', but it is also possible to find similarities in multiple
expressions by passing in the result of a previous match.

The matching algorithm works as follows:

   * A single wildcard matches any expression. If one wildcard appears
     multiple times in a pattern, it must match the same expression in
     all   places (e.g. `$0' matches anything, and `$0*($0+1)' matches
     `x*(x+1)' but not `x*(y+1)').

   * If the expression is not of the same class as the pattern, the
     match   fails (i.e. a sum only matches a sum, a function only
     matches a function,   etc.).

   * If the pattern is a function, it only matches the same function
     (i.e. `sin($0)' matches `sin(x)' but doesn't match `exp(x)').

   * Except for sums and products, the match fails if the number of
     subexpressions (`nops()') is not equal to the number of
     subexpressions   of the pattern.

   * If there are no subexpressions, the expressions and the pattern
     must   be equal (in the sense of `is_equal()').

   * Except for sums and products, each subexpression (`op()') must
     match the corresponding subexpression of the pattern.

Sums (`add') and products (`mul') are treated in a special way to
account for their commutativity and associativity:

   * If the pattern contains a term or factor that is a single wildcard,
      this one is used as the "global wildcard". If there is more than
     one   such wildcard, one of them is chosen as the global wildcard
     in a random   way.

   * Every term/factor of the pattern, except the global wildcard, is
     matched against every term of the expression in sequence. If no
     match is   found, the whole match fails. Terms that did match are
     not considered in   further matches.

   * If there are no unmatched terms left, the match succeeds. Otherwise
      the match fails unless there is a global wildcard in the
     pattern, in   which case this wildcard matches the remaining terms.

In general, having more than one single wildcard as a term of a sum or a
factor of a product (such as `a+$0+$1') will lead to unpredictable or
ambiguous results.

Here are some examples in `ginsh' to demonstrate how it works (the
`match()' function in `ginsh' returns `FAIL' if the match fails, and
the list of wildcard replacements otherwise):

     > match((x+y)^a,(x+y)^a);
     {}
     > match((x+y)^a,(x+y)^b);
     FAIL
     > match((x+y)^a,$1^$2);
     {$1==x+y,$2==a}
     > match((x+y)^a,$1^$1);
     FAIL
     > match((x+y)^(x+y),$1^$1);
     {$1==x+y}
     > match((x+y)^(x+y),$1^$2);
     {$1==x+y,$2==x+y}
     > match((a+b)*(a+c),($1+b)*($1+c));
     {$1==a}
     > match((a+b)*(a+c),(a+$1)*(a+$2));
     {$1==c,$2==b}
       (Unpredictable. The result might also be [$1==c,$2==b].)
     > match((a+b)*(a+c),($1+$2)*($1+$3));
       (The result is undefined. Due to the sequential nature of the algorithm
        and the re-ordering of terms in GiNaC, the match for the first factor
        may be {$1==a,$2==b} in which case the match for the second factor
        succeeds, or it may be {$1==b,$2==a} which causes the second match to
        fail.)
     > match(a*(x+y)+a*z+b,a*$1+$2);
       (This is also ambiguous and may return either {$1==z,$2==a*(x+y)+b} or
        {$1=x+y,$2=a*z+b}.)
     > match(a+b+c+d+e+f,c);
     FAIL
     > match(a+b+c+d+e+f,c+$0);
     {$0==a+e+b+f+d}
     > match(a+b+c+d+e+f,c+e+$0);
     {$0==a+b+f+d}
     > match(a+b,a+b+$0);
     {$0==0}
     > match(a*b^2,a^$1*b^$2);
     FAIL
       (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
        even though a==a^1.)
     > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
     {$0==x}
     > match(atan2(y,x^2),atan2(y,$0));
     {$0==x^2}

Matching parts of expressions
-----------------------------

A more general way to look for patterns in expressions is provided by
the member function

     bool ex::has(const ex & pattern);

This function checks whether a pattern is matched by an expression
itself or by any of its subexpressions.

Again some examples in `ginsh' for illustration (in `ginsh', `has()'
returns `1' for `true' and `0' for `false'):

     > has(x*sin(x+y+2*a),y);
     1
     > has(x*sin(x+y+2*a),x+y);
     0
       (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
        has the subexpressions "x", "y" and "2*a".)
     > has(x*sin(x+y+2*a),x+y+$1);
     1
       (But this is possible.)
     > has(x*sin(2*(x+y)+2*a),x+y);
     0
       (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
        which "x+y" is not a subexpression.)
     > has(x+1,x^$1);
     0
       (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
        "x^something".)
     > has(4*x^2-x+3,$1*x);
     1
     > has(4*x^2+x+3,$1*x);
     0
       (Another possible pitfall. The first expression matches because the term
        "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
        contains a linear term you should use the coeff() function instead.)

The method

     bool ex::find(const ex & pattern, lst & found);

works a bit like `has()' but it doesn't stop upon finding the first
match. Instead, it inserts all found matches into the specified list. If
there are multiple occurrences of the same expression, it is entered
only once to the list. `find()' returns false if no matches were found
(in `ginsh', it returns an empty list):

     > find(1+x+x^2+x^3,x);
     {x}
     > find(1+x+x^2+x^3,y);
     {}
     > find(1+x+x^2+x^3,x^$1);
     {x^3,x^2}
       (Note the absence of "x".)
     > expand((sin(x)+sin(y))*(a+b));
     sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
     > find(%,sin($1));
     {sin(y),sin(x)}

Substituting expressions
------------------------

Probably the most useful application of patterns is to use them for
substituting expressions with the `subs()' method. Wildcards can be
used in the search patterns as well as in the replacement expressions,
where they get replaced by the expressions matched by them. `subs()'
doesn't know anything about algebra; it performs purely syntactic
substitutions.

Some examples:

     > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
     b^3+a^3+(x+y)^3
     > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
     b^4+a^4+(x+y)^4
     > subs((a+b+c)^2,a+b==x);
     (a+b+c)^2
     > subs((a+b+c)^2,a+b+$1==x+$1);
     (x+c)^2
     > subs(a+2*b,a+b==x);
     a+2*b
     > subs(4*x^3-2*x^2+5*x-1,x==a);
     -1+5*a-2*a^2+4*a^3
     > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
     -1+5*x-2*a^2+4*a^3
     > subs(sin(1+sin(x)),sin($1)==cos($1));
     cos(1+cos(x))
     > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
     a+b

The last example would be written in C++ in this way:

     {
         symbol a("a"), b("b"), x("x"), y("y");
         e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
         e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
         cout << e.expand() << endl;
          // -> a+b
     }


File: ginac.info,  Node: Applying a Function on Subexpressions,  Next: Polynomial Arithmetic,  Prev: Pattern Matching and Advanced Substitutions,  Up: Methods and Functions

Applying a Function on Subexpressions
=====================================

Sometimes you may want to perform an operation on specific parts of an
expression while leaving the general structure of it intact. An example
of this would be a matrix trace operation: the trace of a sum is the sum
of the traces of the individual terms. That is, the trace should "map"
on the sum, by applying itself to each of the sum's operands. It is
possible to do this manually which usually results in code like this:

     ex calc_trace(ex e)
     {
         if (is_a<matrix>(e))
             return ex_to<matrix>(e).trace();
         else if (is_a<add>(e)) {
             ex sum = 0;
             for (unsigned i=0; i<e.nops(); i++)
                 sum += calc_trace(e.op(i));
             return sum;
         } else if (is_a<mul>)(e)) {
             ...
         } else {
             ...
         }
     }

This is, however, slightly inefficient (if the sum is very large it can
take a long time to add the terms one-by-one), and its applicability is
limited to a rather small class of expressions. If `calc_trace()' is
called with a relation or a list as its argument, you will probably
want the trace to be taken on both sides of the relation or of all
elements of the list.

GiNaC offers the `map()' method to aid in the implementation of such
operations:

     ex ex::map(map_function & f) const;
     ex ex::map(ex (*f)(const ex & e)) const;

In the first (preferred) form, `map()' takes a function object that is
subclassed from the `map_function' class. In the second form, it takes
a pointer to a function that accepts and returns an expression.
`map()' constructs a new expression of the same type, applying the
specified function on all subexpressions (in the sense of `op()'),
non-recursively.

The use of a function object makes it possible to supply more arguments
to the function that is being mapped, or to keep local state
information.  The `map_function' class declares a virtual function call
operator that you can overload. Here is a sample implementation of
`calc_trace()' that uses `map()' in a recursive fashion:

     struct calc_trace : public map_function {
         ex operator()(const ex &e)
         {
             if (is_a<matrix>(e))
                 return ex_to<matrix>(e).trace();
             else if (is_a<mul>(e)) {
                 ...
             } else
                 return e.map(*this);
         }
     };

This function object could then be used like this:

     {
         ex M = ... // expression with matrices
         calc_trace do_trace;
         ex tr = do_trace(M);
     }

Here is another example for you to meditate over.  It removes quadratic
terms in a variable from an expanded polynomial:

     struct map_rem_quad : public map_function {
         ex var;
         map_rem_quad(const ex & var_) : var(var_) {}
     
         ex operator()(const ex & e)
         {
             if (is_a<add>(e) || is_a<mul>(e))
          	    return e.map(*this);
             else if (is_a<power>(e) &&
                      e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
                 return 0;
             else
                 return e;
         }
     };
     
     ...
     
     {
         symbol x("x"), y("y");
     
         ex e;
         for (int i=0; i<8; i++)
             e += pow(x, i) * pow(y, 8-i) * (i+1);
         cout << e << endl;
          // -> 4*y^5*x^3+5*y^4*x^4+8*y*x^7+7*y^2*x^6+2*y^7*x+6*y^3*x^5+3*y^6*x^2+y^8
     
         map_rem_quad rem_quad(x);
         cout << rem_quad(e) << endl;
          // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
     }

`ginsh' offers a slightly different implementation of `map()' that
allows applying algebraic functions to operands. The second argument to
`map()' is an expression containing the wildcard `$0' which acts as the
placeholder for the operands:

     > map(a*b,sin($0));
     sin(a)*sin(b)
     > map(a+2*b,sin($0));
     sin(a)+sin(2*b)
     > map({a,b,c},$0^2+$0);
     {a^2+a,b^2+b,c^2+c}

Note that it is only possible to use algebraic functions in the second
argument. You can not use functions like `diff()', `op()', `subs()'
etc. because these are evaluated immediately:

     > map({a,b,c},diff($0,a));
     {0,0,0}
       This is because "diff($0,a)" evaluates to "0", so the command is equivalent
       to "map({a,b,c},0)".


File: ginac.info,  Node: Polynomial Arithmetic,  Next: Rational Expressions,  Prev: Applying a Function on Subexpressions,  Up: Methods and Functions

Polynomial arithmetic
=====================

Expanding and collecting
------------------------

A polynomial in one or more variables has many equivalent
representations.  Some useful ones serve a specific purpose.  Consider
for example the trivariate polynomial 4*x*y + x*z + 20*y^2 + 21*y*z +
4*z^2 (written down here in output-style).  It is equivalent to the
factorized polynomial (x + 5*y + 4*z)*(4*y + z).  Other representations
are the recursive ones where one collects for exponents in one of the
three variable.  Since the factors are themselves polynomials in the
remaining two variables the procedure can be repeated.  In our example,
two possibilities would be (4*y + z)*x + 20*y^2 + 21*y*z + 4*z^2 and
20*y^2 + (21*z + 4*x)*y + 4*z^2 + x*z.

To bring an expression into expanded form, its method

     ex ex::expand();

may be called.  In our example above, this corresponds to 4*x*y + x*z +
20*y^2 + 21*y*z + 4*z^2.  Again, since the canonical form in GiNaC is
not easily guessable you should be prepared to see different orderings
of terms in such sums!

Another useful representation of multivariate polynomials is as a
univariate polynomial in one of the variables with the coefficients
being polynomials in the remaining variables.  The method `collect()'
accomplishes this task:

     ex ex::collect(const ex & s, bool distributed = false);

The first argument to `collect()' can also be a list of objects in which
case the result is either a recursively collected polynomial, or a
polynomial in a distributed form with terms like c*x1^e1*...*xn^en, as
specified by the `distributed' flag.

Note that the original polynomial needs to be in expanded form (for the
variables concerned) in order for `collect()' to be able to find the
coefficients properly.

The following `ginsh' transcript shows an application of `collect()'
together with `find()':

     > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
     d*p*sin(x)+p*sin(x)+q*d*sin(x)+q*sin(y)+d*sin(x)+q*d*sin(y)+sin(y)+d*sin(y)+q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
     > collect(a,{p,q});
     d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
     > collect(a,find(a,sin($1)));
     (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
     > collect(a,{find(a,sin($1)),p,q});
     (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
     > collect(a,{find(a,sin($1)),d});
     (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)

Polynomials can often be brought into a more compact form by collecting
common factors from the terms of sums. This is accomplished by the
function

     ex collect_common_factors(const ex & e);

This function doesn't perform a full factorization but only looks for
factors which are already explicitly present:

     > collect_common_factors(a*x+a*y);
     (x+y)*a
     > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
     a*(2*x*y+y^2+x^2)
     > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
     (c+a)*a*(x*y+y^2+x)*b

Degree and coefficients
-----------------------

The degree and low degree of a polynomial can be obtained using the two
methods

     int ex::degree(const ex & s);
     int ex::ldegree(const ex & s);

These functions only work reliably if the input polynomial is collected
in terms of the object `s'. Otherwise, they are only guaranteed to
return the upper/lower bounds of the exponents. If you need accurate
results, you have to call `expand()' and/or `collect()' on the input
polynomial.  For example

     > a=(x+1)^2-x^2;
     (1+x)^2-x^2;
     > degree(a,x);
     2
     > degree(expand(a),x);
     1

`degree()' also works on rational functions, returning the asymptotic
degree:

     > degree((x+1)/(x^3+1),x);
     -2

If the input is not a polynomial or rational function in the variable
`s', the behavior of `degree()' and `ldegree()' is undefined.

To extract a coefficient with a certain power from an expanded
polynomial you use

     ex ex::coeff(const ex & s, int n);

You can also obtain the leading and trailing coefficients with the
methods

     ex ex::lcoeff(const ex & s);
     ex ex::tcoeff(const ex & s);

which are equivalent to `coeff(s, degree(s))' and `coeff(s,
ldegree(s))', respectively.

An application is illustrated in the next example, where a multivariate
polynomial is analyzed:

     {
         symbol x("x"), y("y");
         ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
                      - pow(x+y,2) + 2*pow(y+2,2) - 8;
         ex Poly = PolyInp.expand();
     
         for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) {
             cout << "The x^" << i << "-coefficient is "
                  << Poly.coeff(x,i) << endl;
         }
         cout << "As polynomial in y: "
              << Poly.collect(y) << endl;
     }

When run, it returns an output in the following fashion:

     The x^0-coefficient is y^2+11*y
     The x^1-coefficient is 5*y^2-2*y
     The x^2-coefficient is -1
     The x^3-coefficient is 4*y
     As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y

As always, the exact output may vary between different versions of GiNaC
or even from run to run since the internal canonical ordering is not
within the user's sphere of influence.

`degree()', `ldegree()', `coeff()', `lcoeff()', `tcoeff()' and
`collect()' can also be used to a certain degree with non-polynomial
expressions as they not only work with symbols but with constants,
functions and indexed objects as well:

     {
         symbol a("a"), b("b"), c("c");
         idx i(symbol("i"), 3);
     
         ex e = pow(sin(x) - cos(x), 4);
         cout << e.degree(cos(x)) << endl;
          // -> 4
         cout << e.expand().coeff(sin(x), 3) << endl;
          // -> -4*cos(x)
     
         e = indexed(a+b, i) * indexed(b+c, i);
         e = e.expand(expand_options::expand_indexed);
         cout << e.collect(indexed(b, i)) << endl;
          // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
     }

Polynomial division
-------------------

The two functions

     ex quo(const ex & a, const ex & b, const symbol & x);
     ex rem(const ex & a, const ex & b, const symbol & x);

compute the quotient and remainder of univariate polynomials in the
variable `x'. The results satisfy a = b*quo(a, b, x) + rem(a, b, x).

The additional function

     ex prem(const ex & a, const ex & b, const symbol & x);

computes the pseudo-remainder of `a' and `b' which satisfies c*a = b*q
+ prem(a, b, x), where c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) +
1).

Exact division of multivariate polynomials is performed by the function

     bool divide(const ex & a, const ex & b, ex & q);

If `b' divides `a' over the rationals, this function returns `true' and
returns the quotient in the variable `q'. Otherwise it returns `false'
in which case the value of `q' is undefined.

Unit, content and primitive part
--------------------------------

The methods

     ex ex::unit(const symbol & x);
     ex ex::content(const symbol & x);
     ex ex::primpart(const symbol & x);

return the unit part, content part, and primitive polynomial of a
multivariate polynomial with respect to the variable `x' (the unit part
being the sign of the leading coefficient, the content part being the
GCD of the coefficients, and the primitive polynomial being the input
polynomial divided by the unit and content parts). The product of unit,
content, and primitive part is the original polynomial.

GCD and LCM
-----------

The functions for polynomial greatest common divisor and least common
multiple have the synopsis

     ex gcd(const ex & a, const ex & b);
     ex lcm(const ex & a, const ex & b);

The functions `gcd()' and `lcm()' accept two expressions `a' and `b' as
arguments and return a new expression, their greatest common divisor or
least common multiple, respectively.  If the polynomials `a' and `b'
are coprime `gcd(a,b)' returns 1 and `lcm(a,b)' returns the product of
`a' and `b'.

     #include <ginac/ginac.h>
     using namespace GiNaC;
     
     int main()
     {
         symbol x("x"), y("y"), z("z");
         ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
         ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
     
         ex P_gcd = gcd(P_a, P_b);
         // x + 5*y + 4*z
         ex P_lcm = lcm(P_a, P_b);
         // 4*x*y^2 + 13*y*x*z + 20*y^3 + 81*y^2*z + 67*y*z^2 + 3*x*z^2 + 12*z^3
     }

Square-free decomposition
-------------------------

GiNaC still lacks proper factorization support.  Some form of
factorization is, however, easily implemented by noting that factors
appearing in a polynomial with power two or more also appear in the
derivative and hence can easily be found by computing the GCD of the
original polynomial and its derivatives.  Any decent system has an
interface for this so called square-free factorization.  So we provide
one, too:
     ex sqrfree(const ex & a, const lst & l = lst());
Here is an example that by the way illustrates how the exact form of the
result may slightly depend on the order of differentiation, calling for
some care with subsequent processing of the result:
         ...
         symbol x("x"), y("y");
         ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
     
         cout << sqrfree(BiVarPol, lst(x,y)) << endl;
          // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
     
         cout << sqrfree(BiVarPol, lst(y,x)) << endl;
          // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
     
         cout << sqrfree(BiVarPol) << endl;
          // -> depending on luck, any of the above
         ...
Note also, how factors with the same exponents are not fully factorized
with this method.


File: ginac.info,  Node: Rational Expressions,  Next: Symbolic Differentiation,  Prev: Polynomial Arithmetic,  Up: Methods and Functions

Rational expressions
====================

The `normal' method
-------------------

Some basic form of simplification of expressions is called for
frequently.  GiNaC provides the method `.normal()', which converts a
rational function into an equivalent rational function of the form
`numerator/denominator' where numerator and denominator are coprime.
If the input expression is already a fraction, it just finds the GCD of
numerator and denominator and cancels it, otherwise it performs
fraction addition and multiplication.

`.normal()' can also be used on expressions which are not rational
functions as it will replace all non-rational objects (like functions
or non-integer powers) by temporary symbols to bring the expression to
the domain of rational functions before performing the normalization,
and re-substituting these symbols afterwards. This algorithm is also
available as a separate method `.to_rational()', described below.

This means that both expressions `t1' and `t2' are indeed simplified in
this little code snippet:

     {
         symbol x("x");
         ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
         ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
         std::cout << "t1 is " << t1.normal() << std::endl;
         std::cout << "t2 is " << t2.normal() << std::endl;
     }

Of course this works for multivariate polynomials too, so the ratio of
the sample-polynomials from the section about GCD and LCM above would be
normalized to `P_a/P_b' = `(4*y+z)/(y+3*z)'.

Numerator and denominator
-------------------------

The numerator and denominator of an expression can be obtained with

     ex ex::numer();
     ex ex::denom();
     ex ex::numer_denom();

These functions will first normalize the expression as described above
and then return the numerator, denominator, or both as a list,
respectively.  If you need both numerator and denominator, calling
`numer_denom()' is faster than using `numer()' and `denom()' separately.

Converting to a rational expression
-----------------------------------

Some of the methods described so far only work on polynomials or
rational functions. GiNaC provides a way to extend the domain of these
functions to general expressions by using the temporary replacement
algorithm described above. You do this by calling

     ex ex::to_rational(lst &l);

on the expression to be converted. The supplied `lst' will be filled
with the generated temporary symbols and their replacement expressions
in a format that can be used directly for the `subs()' method. It can
also already contain a list of replacements from an earlier application
of `.to_rational()', so it's possible to use it on multiple expressions
and get consistent results.

For example,

     {
         symbol x("x");
         ex a = pow(sin(x), 2) - pow(cos(x), 2);
         ex b = sin(x) + cos(x);
         ex q;
         lst l;
         divide(a.to_rational(l), b.to_rational(l), q);
         cout << q.subs(l) << endl;
     }

will print `sin(x)-cos(x)'.


File: ginac.info,  Node: Symbolic Differentiation,  Next: Series Expansion,  Prev: Rational Expressions,  Up: Methods and Functions

Symbolic differentiation
========================

GiNaC's objects know how to differentiate themselves.  Thus, a
polynomial (class `add') knows that its derivative is the sum of the
derivatives of all the monomials:

     {
         symbol x("x"), y("y"), z("z");
         ex P = pow(x, 5) + pow(x, 2) + y;
     
         cout << P.diff(x,2) << endl;
          // -> 20*x^3 + 2
         cout << P.diff(y) << endl;    // 1
          // -> 1
         cout << P.diff(z) << endl;    // 0
          // -> 0
     }

If a second integer parameter N is given, the `diff' method returns the
Nth derivative.

If _every_ object and every function is told what its derivative is,
all derivatives of composed objects can be calculated using the chain
rule and the product rule.  Consider, for instance the expression
`1/cosh(x)'.  Since the derivative of `cosh(x)' is `sinh(x)' and the
derivative of `pow(x,-1)' is `-pow(x,-2)', GiNaC can readily compute
the composition.  It turns out that the composition is the generating
function for Euler Numbers, i.e. the so called Nth Euler number is the
coefficient of `x^n/n!' in the expansion of `1/cosh(x)'.  We may use
this identity to code a function that generates Euler numbers in just
three lines:

     #include <ginac/ginac.h>
     using namespace GiNaC;
     
     ex EulerNumber(unsigned n)
     {
         symbol x;
         const ex generator = pow(cosh(x),-1);
         return generator.diff(x,n).subs(x==0);
     }
     
     int main()
     {
         for (unsigned i=0; i<11; i+=2)
             std::cout << EulerNumber(i) << std::endl;
         return 0;
     }

When you run it, it produces the sequence `1', `-1', `5', `-61',
`1385', `-50521'.  We increment the loop variable `i' by two since all
odd Euler numbers vanish anyways.


File: ginac.info,  Node: Series Expansion,  Next: Symmetrization,  Prev: Symbolic Differentiation,  Up: Methods and Functions

Series expansion
================

Expressions know how to expand themselves as a Taylor series or (more
generally) a Laurent series.  As in most conventional Computer Algebra
Systems, no distinction is made between those two.  There is a class of
its own for storing such series (`class pseries') and a built-in
function (called `Order') for storing the order term of the series.  As
a consequence, if you want to work with series, i.e. multiply two
series, you need to call the method `ex::series' again to convert it to
a series object with the usual structure (expansion plus order term).
A sample application from special relativity could read:

     #include <ginac/ginac.h>
     using namespace std;
     using namespace GiNaC;
     
     int main()
     {
         symbol v("v"), c("c");
     
         ex gamma = 1/sqrt(1 - pow(v/c,2));
         ex mass_nonrel = gamma.series(v==0, 10);
     
         cout << "the relativistic mass increase with v is " << endl
              << mass_nonrel << endl;
     
         cout << "the inverse square of this series is " << endl
              << pow(mass_nonrel,-2).series(v==0, 10) << endl;
     }

Only calling the series method makes the last output simplify to
1-v^2/c^2+O(v^10), without that call we would just have a long series
raised to the power -2.

As another instructive application, let us calculate the numerical
value of Archimedes' constant (for which there already exists the
built-in constant `Pi') using Machin's amazing formula
Pi==16*atan(1/5)-4*atan(1/239).  We may expand the arcus tangent around
`0' and insert the fractions `1/5' and `1/239'.  But, as we have seen,
a series in GiNaC carries an order term with it and the question arises
what the system is supposed to do when the fractions are plugged into
that order term.  The solution is to use the function
`series_to_poly()' to simply strip the order term off:

     #include <ginac/ginac.h>
     using namespace GiNaC;
     
     ex machin_pi(int degr)
     {
         symbol x;
         ex pi_expansion = series_to_poly(atan(x).series(x,degr));
         ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
                        -4*pi_expansion.subs(x==numeric(1,239));
         return pi_approx;
     }
     
     int main()
     {
         using std::cout;  // just for fun, another way of...
         using std::endl;  // ...dealing with this namespace std.
         ex pi_frac;
         for (int i=2; i<12; i+=2) {
             pi_frac = machin_pi(i);
             cout << i << ":\t" << pi_frac << endl
                  << "\t" << pi_frac.evalf() << endl;
         }
         return 0;
     }

Note how we just called `.series(x,degr)' instead of
`.series(x==0,degr)'.  This is a simple shortcut for `ex''s method
`series()': if the first argument is a symbol the expression is
expanded in that symbol around point `0'.  When you run this program,
it will type out:

     2:      3804/1195
             3.1832635983263598326
     4:      5359397032/1706489875
             3.1405970293260603143
     6:      38279241713339684/12184551018734375
             3.141621029325034425
     8:      76528487109180192540976/24359780855939418203125
             3.141591772182177295
     10:     327853873402258685803048818236/104359128170408663038552734375
             3.1415926824043995174


File: ginac.info,  Node: Symmetrization,  Next: Built-in Functions,  Prev: Series Expansion,  Up: Methods and Functions

Symmetrization
==============

The three methods

     ex ex::symmetrize(const lst & l);
     ex ex::antisymmetrize(const lst & l);
     ex ex::symmetrize_cyclic(const lst & l);

symmetrize an expression by returning the sum over all symmetric,
antisymmetric or cyclic permutations of the specified list of objects,
weighted by the number of permutations.

The three additional methods

     ex ex::symmetrize();
     ex ex::antisymmetrize();
     ex ex::symmetrize_cyclic();

symmetrize or antisymmetrize an expression over its free indices.

Symmetrization is most useful with indexed expressions but can be used
with almost any kind of object (anything that is `subs()'able):

     {
         idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
         symbol A("A"), B("B"), a("a"), b("b"), c("c");
     
         cout << indexed(A, i, j).symmetrize() << endl;
          // -> 1/2*A.j.i+1/2*A.i.j
         cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
          // -> -1/2*A.j.i.k+1/2*A.i.j.k
         cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
          // -> 1/3*{a,b,c}+1/3*{b,c,a}+1/3*{c,a,b}
     }


File: ginac.info,  Node: Built-in Functions,  Next: Input/Output,  Prev: Symmetrization,  Up: Methods and Functions

Predefined mathematical functions
=================================

GiNaC contains the following predefined mathematical functions:

*Name*                 *Function*
`abs(x)'               absolute value
`csgn(x)'              complex sign
`sqrt(x)'              square root (not a GiNaC function, rather an
                       alias for `pow(x, numeric(1, 2))')
`sin(x)'               sine
`cos(x)'               cosine
`tan(x)'               tangent
`asin(x)'              inverse sine
`acos(x)'              inverse cosine
`atan(x)'              inverse tangent
`atan2(y, x)'          inverse tangent with two arguments
`sinh(x)'              hyperbolic sine
`cosh(x)'              hyperbolic cosine
`tanh(x)'              hyperbolic tangent
`asinh(x)'             inverse hyperbolic sine
`acosh(x)'             inverse hyperbolic cosine
`atanh(x)'             inverse hyperbolic tangent
`exp(x)'               exponential function
`log(x)'               natural logarithm
`Li2(x)'               Dilogarithm
`zeta(x)'              Riemann's zeta function
`zeta(n, x)'           derivatives of Riemann's zeta function
`tgamma(x)'            Gamma function
`lgamma(x)'            logarithm of Gamma function
`beta(x, y)'           Beta function (`tgamma(x)*tgamma(y)/tgamma(x+y)')
`psi(x)'               psi (digamma) function
`psi(n, x)'            derivatives of psi function (polygamma functions)
`factorial(n)'         factorial function
`binomial(n, m)'       binomial coefficients
`Order(x)'             order term function in truncated power series

For functions that have a branch cut in the complex plane GiNaC follows
the conventions for C++ as defined in the ANSI standard as far as
possible.  In particular: the natural logarithm (`log') and the square
root (`sqrt') both have their branch cuts running along the negative
real axis where the points on the axis itself belong to the upper part
(i.e. continuous with quadrant II).  The inverse trigonometric and
hyperbolic functions are not defined for complex arguments by the C++
standard, however.  In GiNaC we follow the conventions used by CLN,
which in turn follow the carefully designed definitions in the Common
Lisp standard.  It should be noted that this convention is identical to
the one used by the C99 standard and by most serious CAS.  It is to be
expected that future revisions of the C++ standard incorporate these
functions in the complex domain in a manner compatible with C99.


File: ginac.info,  Node: Input/Output,  Next: Extending GiNaC,  Prev: Built-in Functions,  Up: Methods and Functions

Input and output of expressions
===============================

Expression output
-----------------

The easiest way to print an expression is to write it to a stream:

     {
         symbol x("x");
         ex e = 4.5+pow(x,2)*3/2;
         cout << e << endl;    // prints '(4.5)+3/2*x^2'
         // ...

The output format is identical to the `ginsh' input syntax and to that
used by most computer algebra systems, but not directly pastable into a
GiNaC C++ program (note that in the above example, `pow(x,2)' is
printed as `x^2').

It is possible to print expressions in a number of different formats
with the method

     void ex::print(const print_context & c, unsigned level = 0);

The type of `print_context' object passed in determines the format of
the output. The possible types are defined in `ginac/print.h'.  All
constructors of `print_context' and derived classes take an `ostream &'
as their first argument.

To print an expression in a way that can be directly used in a C or C++
program, you pass a `print_csrc' object like this:

         // ...
         cout << "float f = ";
         e.print(print_csrc_float(cout));
         cout << ";\n";
     
         cout << "double d = ";
         e.print(print_csrc_double(cout));
         cout << ";\n";
     
         cout << "cl_N n = ";
         e.print(print_csrc_cl_N(cout));
         cout << ";\n";
         // ...

The three possible types mostly affect the way in which floating point
numbers are written.

The above example will produce (note the `x^2' being converted to
`x*x'):

     float f = (3.0/2.0)*(x*x)+4.500000e+00;
     double d = (3.0/2.0)*(x*x)+4.5000000000000000e+00;
     cl_N n = cln::cl_RA("3/2")*(x*x)+cln::cl_F("4.5_17");

The `print_context' type `print_tree' provides a dump of the internal
structure of an expression for debugging purposes:

         // ...
         e.print(print_tree(cout));
     }

produces

     add, hash=0x0, flags=0x3, nops=2
         power, hash=0x9, flags=0x3, nops=2
             x (symbol), serial=3, hash=0x44a113a6, flags=0xf
             2 (numeric), hash=0x80000042, flags=0xf
         3/2 (numeric), hash=0x80000061, flags=0xf
         -----
         overall_coeff
         4.5L0 (numeric), hash=0x8000004b, flags=0xf
         =====

This kind of output is also available in `ginsh' as the `print()'
function.

Another useful output format is for LaTeX parsing in mathematical mode.
It is rather similar to the default `print_context' but provides some
braces needed by LaTeX for delimiting boxes and also converts some
common objects to conventional LaTeX names. It is possible to give
symbols a special name for LaTeX output by supplying it as a second
argument to the `symbol' constructor.

For example, the code snippet

         // ...
         symbol x("x");
         ex foo = lgamma(x).series(x==0,3);
         foo.print(print_latex(std::cout));

will print out:

         {(-\ln(x))}+{(-\gamma_E)} x+{(1/12 \pi^2)} x^{2}+\mathcal{O}(x^3)

If you need any fancy special output format, e.g. for interfacing GiNaC
with other algebra systems or for producing code for different
programming languages, you can always traverse the expression tree
yourself:

     static void my_print(const ex & e)
     {
         if (is_a<function>(e))
             cout << ex_to<function>(e).get_name();
         else
             cout << e.bp->class_name();
         cout << "(";
         unsigned n = e.nops();
         if (n)
             for (unsigned i=0; i<n; i++) {
                 my_print(e.op(i));
                 if (i != n-1)
                     cout << ",";
             }
         else
             cout << e;
         cout << ")";
     }
     
     int main(void)
     {
         my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
         return 0;
     }

This will produce

     add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
     symbol(y))),numeric(-2)))

If you need an output format that makes it possible to accurately
reconstruct an expression by feeding the output to a suitable parser or
object factory, you should consider storing the expression in an
`archive' object and reading the object properties from there.  See the
section on archiving for more information.

Expression input
----------------

GiNaC provides no way to directly read an expression from a stream
because you will usually want the user to be able to enter something
like `2*x+sin(y)' and have the `x' and `y' correspond to the symbols
`x' and `y' you defined in your program and there is no way to specify
the desired symbols to the `>>' stream input operator.

Instead, GiNaC lets you construct an expression from a string,
specifying the list of symbols and indices to be used:

     {
         symbol x("x"), y("y"), p("p");
         idx i(symbol("i"), 3);
         ex e("2*x+sin(y)+p.i", lst(x, y, p, i));
     }

The input syntax is the same as that used by `ginsh' and the stream
output operator `<<'. The symbols and indices in the string are matched
by name to the symbols and indices in the list and if GiNaC encounters a
symbol or index not specified in the list it will throw an exception.
Only indices whose values are single symbols can be used (i.e. numeric
indices or compound indices as in "A.(2*n+1)" are not allowed).

With this constructor, it's also easy to implement interactive GiNaC
programs:

     #include <iostream>
     #include <string>
     #include <stdexcept>
     #include <ginac/ginac.h>
     using namespace std;
     using namespace GiNaC;
     
     int main()
     {
         symbol x("x");
         string s;
     
         cout << "Enter an expression containing 'x': ";
         getline(cin, s);
     
         try {
             ex e(s, lst(x));
             cout << "The derivative of " << e << " with respect to x is ";
             cout << e.diff(x) << ".\n";
         } catch (exception &p) {
             cerr << p.what() << endl;
         }
     }

Archiving
---------

GiNaC allows creating "archives" of expressions which can be stored to
or retrieved from files. To create an archive, you declare an object of
class `archive' and archive expressions in it, giving each expression a
unique name:

     #include <fstream>
     using namespace std;
     #include <ginac/ginac.h>
     using namespace GiNaC;
     
     int main()
     {
         symbol x("x"), y("y"), z("z");
     
         ex foo = sin(x + 2*y) + 3*z + 41;
         ex bar = foo + 1;
     
         archive a;
         a.archive_ex(foo, "foo");
         a.archive_ex(bar, "the second one");
         // ...

The archive can then be written to a file:

         // ...
         ofstream out("foobar.gar");
         out << a;
         out.close();
         // ...

The file `foobar.gar' contains all information that is needed to
reconstruct the expressions `foo' and `bar'.

The tool `viewgar' that comes with GiNaC can be used to view the
contents of GiNaC archive files:

     $ viewgar foobar.gar
     foo = 41+sin(x+2*y)+3*z
     the second one = 42+sin(x+2*y)+3*z

The point of writing archive files is of course that they can later be
read in again:

         // ...
         archive a2;
         ifstream in("foobar.gar");
         in >> a2;
         // ...

And the stored expressions can be retrieved by their name:

         // ...
         lst syms(x, y);
     
         ex ex1 = a2.unarchive_ex(syms, "foo");
         ex ex2 = a2.unarchive_ex(syms, "the second one");
     
         cout << ex1 << endl;              // prints "41+sin(x+2*y)+3*z"
         cout << ex2 << endl;              // prints "42+sin(x+2*y)+3*z"
         cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
     }

Note that you have to supply a list of the symbols which are to be
inserted in the expressions. Symbols in archives are stored by their
name only and if you don't specify which symbols you have, unarchiving
the expression will create new symbols with that name. E.g. if you
hadn't included `x' in the `syms' list above, the `ex1.subs(x == 2)'
statement would have had no effect because the `x' in `ex1' would have
been a different symbol than the `x' which was defined at the beginning
of the program, although both would appear as `x' when printed.

You can also use the information stored in an `archive' object to
output expressions in a format suitable for exact reconstruction. The
`archive' and `archive_node' classes have a couple of member functions
that let you access the stored properties:

     static void my_print2(const archive_node & n)
     {
         string class_name;
         n.find_string("class", class_name);
         cout << class_name << "(";
     
         archive_node::propinfovector p;
         n.get_properties(p);
     
         unsigned num = p.size();
         for (unsigned i=0; i<num; i++) {
             const string &name = p[i].name;
             if (name == "class")
                 continue;
             cout << name << "=";
     
             unsigned count = p[i].count;
             if (count > 1)
                 cout << "{";
     
             for (unsigned j=0; j<count; j++) {
                 switch (p[i].type) {
                     case archive_node::PTYPE_BOOL: {
                         bool x;
                         n.find_bool(name, x, j);
                         cout << (x ? "true" : "false");
                         break;
                     }
                     case archive_node::PTYPE_UNSIGNED: {
                         unsigned x;
                         n.find_unsigned(name, x, j);
                         cout << x;
                         break;
                     }
                     case archive_node::PTYPE_STRING: {
                         string x;
                         n.find_string(name, x, j);
                         cout << '\"' << x << '\"';
                         break;
                     }
                     case archive_node::PTYPE_NODE: {
                         const archive_node &x = n.find_ex_node(name, j);
                         my_print2(x);
                         break;
                     }
                 }
     
                 if (j != count-1)
                     cout << ",";
             }
     
             if (count > 1)
                 cout << "}";
     
             if (i != num-1)
                 cout << ",";
         }
     
         cout << ")";
     }
     
     int main(void)
     {
         ex e = pow(2, x) - y;
         archive ar(e, "e");
         my_print2(ar.get_top_node(0)); cout << endl;
         return 0;
     }

This will produce:

     add(rest={power(basis=numeric(number="2"),exponent=symbol(name="x")),
     symbol(name="y")},coeff={numeric(number="1"),numeric(number="-1")},
     overall_coeff=numeric(number="0"))

Be warned, however, that the set of properties and their meaning for
each class may change between GiNaC versions.


File: ginac.info,  Node: Extending GiNaC,  Next: What does not belong into GiNaC,  Prev: Input/Output,  Up: Top

Extending GiNaC
***************

By reading so far you should have gotten a fairly good understanding of
GiNaC's design-patterns.  From here on you should start reading the
sources.  All we can do now is issue some recommendations how to tackle
GiNaC's many loose ends in order to fulfill everybody's dreams.  If you
develop some useful extension please don't hesitate to contact the GiNaC
authors--they will happily incorporate them into future versions.

* Menu:

* What does not belong into GiNaC::  What to avoid.
* Symbolic functions::               Implementing symbolic functions.
* Adding classes::                   Defining new algebraic classes.


File: ginac.info,  Node: What does not belong into GiNaC,  Next: Symbolic functions,  Prev: Extending GiNaC,  Up: Extending GiNaC

What doesn't belong into GiNaC
==============================

First of all, GiNaC's name must be read literally.  It is designed to be
a library for use within C++.  The tiny `ginsh' accompanying GiNaC
makes this even more clear: it doesn't even attempt to provide a
language.  There are no loops or conditional expressions in `ginsh', it
is merely a window into the library for the programmer to test stuff
(or to show off).  Still, the design of a complete CAS with a language
of its own, graphical capabilities and all this on top of GiNaC is
possible and is without doubt a nice project for the future.

There are many built-in functions in GiNaC that do not know how to
evaluate themselves numerically to a precision declared at runtime
(using `Digits').  Some may be evaluated at certain points, but not
generally.  This ought to be fixed.  However, doing numerical
computations with GiNaC's quite abstract classes is doomed to be
inefficient.  For this purpose, the underlying foundation classes
provided by CLN are much better suited.


File: ginac.info,  Node: Symbolic functions,  Next: Adding classes,  Prev: What does not belong into GiNaC,  Up: Extending GiNaC

Symbolic functions
==================

The easiest and most instructive way to start extending GiNaC is
probably to create your own symbolic functions. These are implemented
with the help of two preprocessor macros:

     DECLARE_FUNCTION_<n>P(<name>)
     REGISTER_FUNCTION(<name>, <options>)

The `DECLARE_FUNCTION' macro will usually appear in a header file. It
declares a C++ function with the given `name' that takes exactly `n'
parameters of type `ex' and returns a newly constructed GiNaC
`function' object that represents your function.

The `REGISTER_FUNCTION' macro implements the function. It must be passed
the same `name' as the respective `DECLARE_FUNCTION' macro, and a set
of options that associate the symbolic function with C++ functions you
provide to implement the various methods such as evaluation, derivative,
series expansion etc. They also describe additional attributes the
function might have, such as symmetry and commutation properties, and a
name for LaTeX output. Multiple options are separated by the member
access operator `.' and can be given in an arbitrary order.

(By the way: in case you are worrying about all the macros above we can
assure you that functions are GiNaC's most macro-intense classes. We
have done our best to avoid macros where we can.)

A minimal example
-----------------

Here is an example for the implementation of a function with two
arguments that is not further evaluated:

     DECLARE_FUNCTION_2P(myfcn)
     
     static ex myfcn_eval(const ex & x, const ex & y)
     {
         return myfcn(x, y).hold();
     }
     
     REGISTER_FUNCTION(myfcn, eval_func(myfcn_eval))

Any code that has seen the `DECLARE_FUNCTION' line can use `myfcn()' in
algebraic expressions:

     {
         ...
         symbol x("x");
         ex e = 2*myfcn(42, 3*x+1) - x;
          // this calls myfcn_eval(42, 3*x+1), and inserts its return value into
          // the actual expression
         cout << e << endl;
          // prints '2*myfcn(42,1+3*x)-x'
         ...
     }

The `eval_func()' option specifies the C++ function that implements the
`eval()' method, GiNaC's anonymous evaluator. This function takes the
same number of arguments as the associated symbolic function (two in
this case) and returns the (possibly transformed or in some way
simplified) symbolically evaluated function (*Note Automatic
evaluation::, for a description of the automatic evaluation process).
If no (further) evaluation is to take place, the `eval_func()' function
must return the original function with `.hold()', to avoid a potential
infinite recursion. If your symbolic functions produce a segmentation
fault or stack overflow when using them in expressions, you are
probably missing a `.hold()' somewhere.

There is not much you can do with the `myfcn' function. It merely acts
as a kind of container for its arguments (which is, however, sometimes
perfectly sufficient). Let's have a look at the implementation of
GiNaC's cosine function.

The cosine function
-------------------

The GiNaC header file `inifcns.h' contains the line

     DECLARE_FUNCTION_1P(cos)

which declares to all programs using GiNaC that there is a function
`cos' that takes one `ex' as an argument. This is all they need to know
to use this function in expressions.

The implementation of the cosine function is in `inifcns_trans.cpp'. The
`eval_func()' function looks something like this (actually, it doesn't
look like this at all, but it should give you an idea what is going on):

     static ex cos_eval(const ex & x)
     {
         if (<x is a multiple of 2*Pi>)
             return 1;
         else if (<x is a multiple of Pi>)
             return -1;
         else if (<x is a multiple of Pi/2>)
             return 0;
         // more rules...
     
         else if (<x has the form 'acos(y)'>)
             return y;
         else if (<x has the form 'asin(y)'>)
             return sqrt(1-y^2);
         // more rules...
     
         else
             return cos(x).hold();
     }

In this way, `cos(4*Pi)' automatically becomes 1, `cos(asin(a+b))'
becomes `sqrt(1-(a+b)^2)', etc. If no reasonable symbolic
transformation can be done, the unmodified function is returned with
`.hold()'.

GiNaC doesn't automatically transform `cos(2)' to `-0.416146...'.  The
user has to call `evalf()' for that. This is implemented in a different
function:

     static ex cos_evalf(const ex & x)
     {
         if (is_a<numeric>(x))
             return cos(ex_to<numeric>(x));
         else
             return cos(x).hold();
     }

Since we are lazy we defer the problem of numeric evaluation to
somebody else, in this case the `cos()' function for `numeric' objects,
which in turn hands it over to the `cos()' function in CLN. The
`.hold()' isn't really needed here, but reminds us that the
corresponding `eval()' function would require it in this place.

Differentiation will surely turn up and so we need to tell `cos' what
its first derivative is (higher derivatives, `.diff(x,3)' for instance,
are then handled automatically by `basic::diff' and `ex::diff'):

     static ex cos_deriv(const ex & x, unsigned diff_param)
     {
         return -sin(x);
     }

The second parameter is obligatory but uninteresting at this point.  It
specifies which parameter to differentiate in a partial derivative in
case the function has more than one parameter, and its main application
is for correct handling of the chain rule.

An implementation of the series expansion is not needed for `cos()' as
it doesn't have any poles and GiNaC can do Taylor expansion by itself
(as long as it knows what the derivative of `cos()' is). `tan()', on
the other hand, does have poles and may need to do Laurent expansion:

     static ex tan_series(const ex & x, const relational & rel,
                          int order, unsigned options)
     {
         // Find the actual expansion point
         const ex x_pt = x.subs(rel);
     
         if (<x_pt is not an odd multiple of Pi/2>)
             throw do_taylor();  // tell function::series() to do Taylor expansion
     
         // On a pole, expand sin()/cos()
         return (sin(x)/cos(x)).series(rel, order+2, options);
     }

The `series()' implementation of a function _must_ return a `pseries'
object, otherwise your code will crash.

Now that all the ingredients have been set up, the `REGISTER_FUNCTION'
macro is used to tell the system how the `cos()' function behaves:

     REGISTER_FUNCTION(cos, eval_func(cos_eval).
                            evalf_func(cos_evalf).
                            derivative_func(cos_deriv).
                            latex_name("\\cos"));

This registers the `cos_eval()', `cos_evalf()' and `cos_deriv()' C++
functions with the `cos()' function, and also gives it a proper LaTeX
name.

Function options
----------------

GiNaC functions understand several more options which are always
specified as `.option(params)'. None of them are required, but you need
to specify at least one option to `REGISTER_FUNCTION()' (usually the
`eval()' method).

     eval_func(<C++ function>)
     evalf_func(<C++ function>)
     derivative_func(<C++ function>)
     series_func(<C++ function>)

These specify the C++ functions that implement symbolic evaluation,
numeric evaluation, partial derivatives, and series expansion,
respectively.  They correspond to the GiNaC methods `eval()', `evalf()',
`diff()' and `series()'.

The `eval_func()' function needs to use `.hold()' if no further
automatic evaluation is desired or possible.

If no `series_func()' is given, GiNaC defaults to simple Taylor
expansion, which is correct if there are no poles involved. If the
function has poles in the complex plane, the `series_func()' needs to
check whether the expansion point is on a pole and fall back to Taylor
expansion if it isn't. Otherwise, the pole usually needs to be
regularized by some suitable transformation.

     latex_name(const string & n)

specifies the LaTeX code that represents the name of the function in
LaTeX output. The default is to put the function name in an `\mbox{}'.

     do_not_evalf_params()

This tells `evalf()' to not recursively evaluate the parameters of the
function before calling the `evalf_func()'.

     set_return_type(unsigned return_type, unsigned return_type_tinfo)

This allows you to explicitly specify the commutation properties of the
function (*Note Non-commutative objects::, for an explanation of
(non)commutativity in GiNaC). For example, you can use
`set_return_type(return_types::noncommutative, TINFO_matrix)' to make
GiNaC treat your function like a matrix. By default, functions inherit
the commutation properties of their first argument.

     set_symmetry(const symmetry & s)

specifies the symmetry properties of the function with respect to its
arguments. *Note Indexed objects::, for an explanation of symmetry
specifications. GiNaC will automatically rearrange the arguments of
symmetric functions into a canonical order.


File: ginac.info,  Node: Adding classes,  Next: A Comparison With Other CAS,  Prev: Symbolic functions,  Up: Extending GiNaC

Adding classes
==============

If you are doing some very specialized things with GiNaC you may find
that you have to implement your own algebraic classes to fit your
needs. This section will explain how to do this by giving the example
of a simple 'string' class. After reading this section you will know
how to properly declare a GiNaC class and what the minimum required
member functions are that you have to implement. We only cover the
implementation of a 'leaf' class here (i.e. one that doesn't contain
subexpressions). Creating a container class like, for example, a class
representing tensor products is more involved but this section should
give you enough information so you can consult the source to GiNaC's
predefined classes if you want to implement something more complicated.

GiNaC's run-time type information system
----------------------------------------

All algebraic classes (that is, all classes that can appear in
expressions) in GiNaC are direct or indirect subclasses of the class
`basic'. So a `basic *' (which is essentially what an `ex' is)
represents a generic pointer to an algebraic class. Occasionally it is
necessary to find out what the class of an object pointed to by a
`basic *' really is.  Also, for the unarchiving of expressions it must
be possible to find the `unarchive()' function of a class given the
class name (as a string). A system that provides this kind of
information is called a run-time type information (RTTI) system. The
C++ language provides such a thing (see the standard header file
`<typeinfo>') but for efficiency reasons GiNaC implements its own,
simpler RTTI.

The RTTI in GiNaC is based on two mechanisms:

   * The `basic' class declares a member variable `tinfo_key' which
     holds an unsigned integer that identifies the object's class.
     These numbers are defined in the `tinfos.h' header file for the
     built-in GiNaC classes. They all start with `TINFO_'.

   * By means of some clever tricks with static members, GiNaC
     maintains a list of information for all classes derived from
     `basic'. The information available includes the class names, the
     `tinfo_key's, and pointers to the unarchiving functions. This
     class registry is defined in the `registrar.h' header file.


The disadvantage of this proprietary RTTI implementation is that there's
a little more to do when implementing new classes (C++'s RTTI works more
or less automatic) but don't worry, most of the work is simplified by
macros.

A minimalistic example
----------------------

Now we will start implementing a new class `mystring' that allows
placing character strings in algebraic expressions (this is not very
useful, but it's just an example). This class will be a direct subclass
of `basic'. You can use this sample implementation as a starting point
for your own classes.

The code snippets given here assume that you have included some header
files as follows:

     #include <iostream>
     #include <string>
     #include <stdexcept>
     using namespace std;
     
     #include <ginac/ginac.h>
     using namespace GiNaC;

The first thing we have to do is to define a `tinfo_key' for our new
class. This can be any arbitrary unsigned number that is not already
taken by one of the existing classes but it's better to come up with
something that is unlikely to clash with keys that might be added in
the future. The numbers in `tinfos.h' are modeled somewhat after the
class hierarchy which is not a requirement but we are going to stick
with this scheme:

     const unsigned TINFO_mystring = 0x42420001U;

Now we can write down the class declaration. The class stores a C++
`string' and the user shall be able to construct a `mystring' object
from a C or C++ string:

     class mystring : public basic
     {
         GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
     
     public:
         mystring(const string &s);
         mystring(const char *s);
     
     private:
         string str;
     };
     
     GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)

The `GINAC_DECLARE_REGISTERED_CLASS' and
`GINAC_IMPLEMENT_REGISTERED_CLASS' macros are defined in `registrar.h'.
They take the name of the class and its direct superclass as arguments
and insert all required declarations for the RTTI system. The
`GINAC_DECLARE_REGISTERED_CLASS' should be the first line after the
opening brace of the class definition. The
`GINAC_IMPLEMENT_REGISTERED_CLASS' may appear anywhere else in the
source (at global scope, of course, not inside a function).

`GINAC_DECLARE_REGISTERED_CLASS' contains, among other things the
declarations of the default and copy constructor, the destructor, the
assignment operator and a couple of other functions that are required.
It also defines a type `inherited' which refers to the superclass so you
don't have to modify your code every time you shuffle around the class
hierarchy.  `GINAC_IMPLEMENT_REGISTERED_CLASS' implements the copy
constructor, the destructor and the assignment operator.

Now there are nine member functions we have to implement to get a
working class:

   * `mystring()', the default constructor.

   * `void destroy(bool call_parent)', which is used in the destructor
     and the assignment operator to free dynamically allocated members.
     The `call_parent' specifies whether the `destroy()' function of
     the superclass is to be called also.

   * `void copy(const mystring &other)', which is used in the copy
     constructor and assignment operator to copy the member variables
     over from another object of the same class.

   * `void archive(archive_node &n)', the archiving function. This
     stores all information needed to reconstruct an object of this
     class inside an `archive_node'.

   * `mystring(const archive_node &n, const lst &sym_lst)', the
     unarchiving constructor. This constructs an instance of the class
     from the information found in an `archive_node'.

   * `ex unarchive(const archive_node &n, const lst &sym_lst)', the
     static unarchiving function. It constructs a new instance by
     calling the unarchiving constructor.

   * `int compare_same_type(const basic &other)', which is used
     internally by GiNaC to establish a canonical sort order for terms.
     It returns 0, +1 or -1, depending on the relative order of this
     object and the `other' object. If it returns 0, the objects are
     considered equal.  *Note:* This has nothing to do with the
     (numeric) ordering relationship expressed by `<', `>=' etc (which
     cannot be defined for non-numeric classes). For example,
     `numeric(1).compare_same_type(numeric(2))' may return +1 even
     though 1 is clearly smaller than 2. Every GiNaC class must provide
     a `compare_same_type()' function, even those representing objects
     for which no reasonable algebraic ordering relationship can be
     defined.

   * And, of course, `mystring(const string &s)' and `mystring(const
     char *s)' which are the two constructors we declared.


Let's proceed step-by-step. The default constructor looks like this:

     mystring::mystring() : inherited(TINFO_mystring)
     {
         // dynamically allocate resources here if required
     }

The golden rule is that in all constructors you have to set the
`tinfo_key' member to the `TINFO_*' value of your class. Otherwise it
will be set by the constructor of the superclass and all hell will break
loose in the RTTI. For your convenience, the `basic' class provides a
constructor that takes a `tinfo_key' value, which we are using here
(remember that in our case `inherited = basic').  If the superclass
didn't have such a constructor, we would have to set the `tinfo_key' to
the right value manually.

In the default constructor you should set all other member variables to
reasonable default values (we don't need that here since our `str'
member gets set to an empty string automatically). The constructor(s)
are of course also the right place to allocate any dynamic resources
you require.

Next, the `destroy()' function:

     void mystring::destroy(bool call_parent)
     {
         // free dynamically allocated resources here if required
         if (call_parent)
             inherited::destroy(call_parent);
     }

This function is where we free all dynamically allocated resources.  We
don't have any so we're not doing anything here, but if we had, for
example, used a C-style `char *' to store our string, this would be the
place to `delete[]' the string storage. If `call_parent' is true, we
have to call the `destroy()' function of the superclass after we're
done (to mimic C++'s automatic invocation of superclass destructors
where `destroy()' is called from outside a destructor).

The `copy()' function just copies over the member variables from
another object:

     void mystring::copy(const mystring &other)
     {
         inherited::copy(other);
         str = other.str;
     }

We can simply overwrite the member variables here. There's no need to
worry about dynamically allocated storage.  The assignment operator
(which is automatically defined by `GINAC_IMPLEMENT_REGISTERED_CLASS',
as you recall) calls `destroy()' before it calls `copy()'. You have to
explicitly call the `copy()' function of the superclass here so all the
member variables will get copied.

Next are the three functions for archiving. You have to implement them
even if you don't plan to use archives, but the minimum required
implementation is really simple.  First, the archiving function:

     void mystring::archive(archive_node &n) const
     {
         inherited::archive(n);
         n.add_string("string", str);
     }

The only thing that is really required is calling the `archive()'
function of the superclass. Optionally, you can store all information
you deem necessary for representing the object into the passed
`archive_node'.  We are just storing our string here. For more
information on how the archiving works, consult the `archive.h' header
file.

The unarchiving constructor is basically the inverse of the archiving
function:

     mystring::mystring(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
     {
         n.find_string("string", str);
     }

If you don't need archiving, just leave this function empty (but you
must invoke the unarchiving constructor of the superclass). Note that
we don't have to set the `tinfo_key' here because it is done
automatically by the unarchiving constructor of the `basic' class.

Finally, the unarchiving function:

     ex mystring::unarchive(const archive_node &n, const lst &sym_lst)
     {
         return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
     }

You don't have to understand how exactly this works. Just copy these
four lines into your code literally (replacing the class name, of
course).  It calls the unarchiving constructor of the class and unless
you are doing something very special (like matching `archive_node's to
global objects) you don't need a different implementation. For those
who are interested: setting the `dynallocated' flag puts the object
under the control of GiNaC's garbage collection.  It will get deleted
automatically once it is no longer referenced.

Our `compare_same_type()' function uses a provided function to compare
the string members:

     int mystring::compare_same_type(const basic &other) const
     {
         const mystring &o = static_cast<const mystring &>(other);
         int cmpval = str.compare(o.str);
         if (cmpval == 0)
             return 0;
         else if (cmpval < 0)
             return -1;
         else
             return 1;
     }

Although this function takes a `basic &', it will always be a reference
to an object of exactly the same class (objects of different classes
are not comparable), so the cast is safe. If this function returns 0,
the two objects are considered equal (in the sense that A-B=0), so you
should compare all relevant member variables.

Now the only thing missing is our two new constructors:

     mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
     {
         // dynamically allocate resources here if required
     }
     
     mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
     {
         // dynamically allocate resources here if required
     }

No surprises here. We set the `str' member from the argument and
remember to pass the right `tinfo_key' to the `basic' constructor.

That's it! We now have a minimal working GiNaC class that can store
strings in algebraic expressions. Let's confirm that the RTTI works:

     ex e = mystring("Hello, world!");
     cout << is_a<mystring>(e) << endl;
      // -> 1 (true)
     
     cout << e.bp->class_name() << endl;
      // -> mystring

Obviously it does. Let's see what the expression `e' looks like:

     cout << e << endl;
      // -> [mystring object]

Hm, not exactly what we expect, but of course the `mystring' class
doesn't yet know how to print itself. This is done in the `print()'
member function. Let's say that we wanted to print the string surrounded
by double quotes:

     class mystring : public basic
     {
         ...
     public:
         void print(const print_context &c, unsigned level = 0) const;
         ...
     };
     
     void mystring::print(const print_context &c, unsigned level) const
     {
         // print_context::s is a reference to an ostream
         c.s << '\"' << str << '\"';
     }

The `level' argument is only required for container classes to
correctly parenthesize the output. Let's try again to print the
expression:

     cout << e << endl;
      // -> "Hello, world!"

Much better. The `mystring' class can be used in arbitrary expressions:

     e += mystring("GiNaC rulez");
     cout << e << endl;
      // -> "GiNaC rulez"+"Hello, world!"

(GiNaC's automatic term reordering is in effect here), or even

     e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
     cout << e << endl;
      // -> "One string"^(2*sin(-"Another string"+Pi))

Whether this makes sense is debatable but remember that this is only an
example. At least it allows you to implement your own symbolic
algorithms for your objects.

Note that GiNaC's algebraic rules remain unchanged:

     e = mystring("Wow") * mystring("Wow");
     cout << e << endl;
      // -> "Wow"^2
     
     e = pow(mystring("First")-mystring("Second"), 2);
     cout << e.expand() << endl;
      // -> -2*"First"*"Second"+"First"^2+"Second"^2

There's no way to, for example, make GiNaC's `add' class perform string
concatenation. You would have to implement this yourself.

Automatic evaluation
--------------------

When dealing with objects that are just a little more complicated than
the simple string objects we have implemented, chances are that you
will want to have some automatic simplifications or canonicalizations
performed on them.  This is done in the evaluation member function
`eval()'. Let's say that we wanted all strings automatically converted
to lowercase with non-alphabetic characters stripped, and empty strings
removed:

     class mystring : public basic
     {
         ...
     public:
         ex eval(int level = 0) const;
         ...
     };
     
     ex mystring::eval(int level) const
     {
         string new_str;
         for (int i=0; i<str.length(); i++) {
             char c = str[i];
             if (c >= 'A' && c <= 'Z')
                 new_str += tolower(c);
             else if (c >= 'a' && c <= 'z')
                 new_str += c;
         }
     
         if (new_str.length() == 0)
             return 0;
         else
             return mystring(new_str).hold();
     }

The `level' argument is used to limit the recursion depth of the
evaluation.  We don't have any subexpressions in the `mystring' class
so we are not concerned with this.  If we had, we would call the
`eval()' functions of the subexpressions with `level - 1' as the
argument if `level != 1'.  The `hold()' member function sets a flag in
the object that prevents further evaluation.  Otherwise we might end up
in an endless loop.  When you want to return the object unmodified, use
`return this->hold();'.

Let's confirm that it works:

     ex e = mystring("Hello, world!") + mystring("!?#");
     cout << e << endl;
      // -> "helloworld"
     
     e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
     cout << e << endl;
      // -> 3*"wow"

Other member functions
----------------------

We have implemented only a small set of member functions to make the
class work in the GiNaC framework. For a real algebraic class, there
are probably some more functions that you might want to re-implement:

     bool info(unsigned inf) const;
     ex evalf(int level = 0) const;
     ex series(const relational & r, int order, unsigned options = 0) const;
     ex derivative(const symbol & s) const;

If your class stores sub-expressions you will probably want to override

     unsigned nops() cont;
     ex op(int i) const;
     ex & let_op(int i);
     ex map(map_function & f) const;
     ex subs(const lst & ls, const lst & lr, bool no_pattern = false) const;

`let_op()' is a variant of `op()' that allows write access. The default
implementation of `map()' uses it, so you have to implement either
`let_op()' or `map()'.

If your class stores any data that is not accessible via `op()', you
should also implement

     unsigned calchash(void) const;

This function returns an `unsigned' hash value for the object which
will allow GiNaC to compare and canonicalize expressions much more
efficiently. You should consult the implementation of some of the
built-in GiNaC classes for examples of hash functions.

You can, of course, also add your own new member functions. Remember
that the RTTI may be used to get information about what kinds of objects
you are dealing with (the position in the class hierarchy) and that you
can always extract the bare object from an `ex' by stripping the `ex'
off using the `ex_to<mystring>(e)' function when that should become a
need.

That's it. May the source be with you!


File: ginac.info,  Node: A Comparison With Other CAS,  Next: Advantages,  Prev: Adding classes,  Up: Top

A Comparison With Other CAS
***************************

This chapter will give you some information on how GiNaC compares to
other, traditional Computer Algebra Systems, like _Maple_,
_Mathematica_ or _Reduce_, where it has advantages and disadvantages
over these systems.

* Menu:

* Advantages::                       Strengths of the GiNaC approach.
* Disadvantages::                    Weaknesses of the GiNaC approach.
* Why C++?::                         Attractiveness of C++.


File: ginac.info,  Node: Advantages,  Next: Disadvantages,  Prev: A Comparison With Other CAS,  Up: A Comparison With Other CAS

Advantages
==========

GiNaC has several advantages over traditional Computer Algebra Systems,
like

   * familiar language: all common CAS implement their own proprietary
     grammar which you have to learn first (and maybe learn again when
     your vendor decides to `enhance' it).  With GiNaC you can write
     your program in common C++, which is standardized.

   * structured data types: you can build up structured data types using
     `struct's or `class'es together with STL features instead of using
     unnamed lists of lists of lists.

   * strongly typed: in CAS, you usually have only one kind of variables
     which can hold contents of an arbitrary type.  This 4GL like
     feature is nice for novice programmers, but dangerous.

   * development tools: powerful development tools exist for C++, like
     fancy editors (e.g. with automatic indentation and syntax
     highlighting), debuggers, visualization tools, documentation
     generators...

   * modularization: C++ programs can easily be split into modules by
     separating interface and implementation.

   * price: GiNaC is distributed under the GNU Public License which
     means that it is free and available with source code.  And there
     are excellent C++-compilers for free, too.

   * extendable: you can add your own classes to GiNaC, thus extending
     it on a very low level.  Compare this to a traditional CAS that
     you can usually only extend on a high level by writing in the
     language defined by the parser.  In particular, it turns out to be
     almost impossible to fix bugs in a traditional system.

   * multiple interfaces: Though real GiNaC programs have to be written
     in some editor, then be compiled, linked and executed, there are
     more ways to work with the GiNaC engine.  Many people want to play
     with expressions interactively, as in traditional CASs.
     Currently, two such windows into GiNaC have been implemented and
     many more are possible: the tiny `ginsh' that is part of the
     distribution exposes GiNaC's types to a command line and second,
     as a more consistent approach, an interactive interface to the
     Cint C++ interpreter has been put together (called GiNaC-cint)
     that allows an interactive scripting interface consistent with the
     C++ language.  It is available from the usual GiNaC FTP-site.

   * seamless integration: it is somewhere between difficult and
     impossible to call CAS functions from within a program written in
     C++ or any other programming language and vice versa.  With GiNaC,
     your symbolic routines are part of your program.  You can easily
     call third party libraries, e.g. for numerical evaluation or
     graphical interaction.  All other approaches are much more
     cumbersome: they range from simply ignoring the problem (i.e.
     _Maple_) to providing a method for `embedding' the system (i.e.
     _Yacas_).

   * efficiency: often large parts of a program do not need symbolic
     calculations at all.  Why use large integers for loop variables or
     arbitrary precision arithmetics where `int' and `double' are
     sufficient?  For pure symbolic applications, GiNaC is comparable in
     speed with other CAS.



File: ginac.info,  Node: Disadvantages,  Next: Why C++?,  Prev: Advantages,  Up: A Comparison With Other CAS

Disadvantages
=============

Of course it also has some disadvantages:

   * advanced features: GiNaC cannot compete with a program like
     _Reduce_ which exists for more than 30 years now or _Maple_ which
     grows since 1981 by the work of dozens of programmers, with
     respect to mathematical features.  Integration, factorization,
     non-trivial simplifications, limits etc. are missing in GiNaC (and
     are not planned for the near future).

   * portability: While the GiNaC library itself is designed to avoid
     any platform dependent features (it should compile on any ANSI
     compliant C++ compiler), the currently used version of the CLN
     library (fast large integer and arbitrary precision arithmetics)
     can only by compiled without hassle on systems with the C++
     compiler from the GNU Compiler Collection (GCC).(1) GiNaC uses
     recent language features like explicit constructors, mutable
     members, RTTI, `dynamic_cast's and STL, so ANSI compliance is meant
     literally.  Recent GCC versions starting at 2.95.3, although
     itself not yet ANSI compliant, support all needed features.


---------- Footnotes ----------

(1) This is because CLN uses PROVIDE/REQUIRE like macros to let the
compiler gather all static initializations, which works for GNU C++
only.  Feel free to contact the authors in case you really believe that
you need to use a different compiler.  We have occasionally used other
compilers and may be able to give you advice.


File: ginac.info,  Node: Why C++?,  Next: Internal Structures,  Prev: Disadvantages,  Up: A Comparison With Other CAS

Why C++?
========

Why did we choose to implement GiNaC in C++ instead of Java or any other
language?  C++ is not perfect: type checking is not strict (casting is
possible), separation between interface and implementation is not
complete, object oriented design is not enforced.  The main reason is
the often scolded feature of operator overloading in C++.  While it may
be true that operating on classes with a `+' operator is rarely
meaningful, it is perfectly suited for algebraic expressions.  Writing
3x+5y as `3*x+5*y' instead of `x.times(3).plus(y.times(5))' looks much
more natural.  Furthermore, the main developers are more familiar with
C++ than with any other programming language.


File: ginac.info,  Node: Internal Structures,  Next: Expressions are reference counted,  Prev: Why C++?,  Up: Top

Internal Structures
*******************

* Menu:

* Expressions are reference counted::
* Internal representation of products and sums::


File: ginac.info,  Node: Expressions are reference counted,  Next: Internal representation of products and sums,  Prev: Internal Structures,  Up: Internal Structures

Expressions are reference counted
=================================

An expression is extremely light-weight since internally it works like a
handle to the actual representation and really holds nothing more than a
pointer to some other object.  What this means in practice is that
whenever you create two `ex' and set the second equal to the first no
copying process is involved. Instead, the copying takes place as soon
as you try to change the second.  Consider the simple sequence of code:

     #include <iostream>
     #include <ginac/ginac.h>
     using namespace std;
     using namespace GiNaC;
     
     int main()
     {
         symbol x("x"), y("y"), z("z");
         ex e1, e2;
     
         e1 = sin(x + 2*y) + 3*z + 41;
         e2 = e1;                // e2 points to same object as e1
         cout << e2 << endl;     // prints sin(x+2*y)+3*z+41
         e2 += 1;                // e2 is copied into a new object
         cout << e2 << endl;     // prints sin(x+2*y)+3*z+42
     }

The line `e2 = e1;' creates a second expression pointing to the object
held already by `e1'.  The time involved for this operation is
therefore constant, no matter how large `e1' was.  Actual copying,
however, must take place in the line `e2 += 1;' because `e1' and `e2'
are not handles for the same object any more.  This concept is called
"copy-on-write semantics".  It increases performance considerably
whenever one object occurs multiple times and represents a simple
garbage collection scheme because when an `ex' runs out of scope its
destructor checks whether other expressions handle the object it points
to too and deletes the object from memory if that turns out not to be
the case.  A slightly less trivial example of differentiation using the
chain-rule should make clear how powerful this can be:

     {
         symbol x("x"), y("y");
     
         ex e1 = x + 3*y;
         ex e2 = pow(e1, 3);
         ex e3 = diff(sin(e2), x);   // first derivative of sin(e2) by x
         cout << e1 << endl          // prints x+3*y
              << e2 << endl          // prints (x+3*y)^3
              << e3 << endl;         // prints 3*(x+3*y)^2*cos((x+3*y)^3)
     }

Here, `e1' will actually be referenced three times while `e2' will be
referenced two times.  When the power of an expression is built, that
expression needs not be copied.  Likewise, since the derivative of a
power of an expression can be easily expressed in terms of that
expression, no copying of `e1' is involved when `e3' is constructed.
So, when `e3' is constructed it will print as
`3*(x+3*y)^2*cos((x+3*y)^3)' but the argument of `cos()' only holds a
reference to `e2' and the factor in front is just `3*e1^2'.

As a user of GiNaC, you cannot see this mechanism of copy-on-write
semantics.  When you insert an expression into a second expression, the
result behaves exactly as if the contents of the first expression were
inserted.  But it may be useful to remember that this is not what
happens.  Knowing this will enable you to write much more efficient
code.  If you still have an uncertain feeling with copy-on-write
semantics, we recommend you have a look at the C++-FAQ lite
(http://www.cerfnet.com/~mpcline/c++-faq-lite/) by Marshall Cline.
Chapter 16 covers this issue and presents an implementation which is
pretty close to the one in GiNaC.


File: ginac.info,  Node: Internal representation of products and sums,  Next: Package Tools,  Prev: Expressions are reference counted,  Up: Internal Structures

Internal representation of products and sums
============================================

Although it should be completely transparent for the user of GiNaC a
short discussion of this topic helps to understand the sources and also
explain performance to a large degree.  Consider the unexpanded
symbolic expression 2*d^3*(4*a+5*b-3) which could naively be
represented by a tree of linear containers for addition and
multiplication, one container for exponentiation with base and exponent
and some atomic leaves of symbols and numbers in this fashion:

<picture missing>

However, doing so results in a rather deeply nested tree which will
quickly become inefficient to manipulate.  We can improve on this by
representing the sum as a sequence of terms, each one being a pair of a
purely numeric multiplicative coefficient and its rest.  In the same
spirit we can store the multiplication as a sequence of terms, each
having a numeric exponent and a possibly complicated base, the tree
becomes much more flat:

<picture missing>

The number `3' above the symbol `d' shows that `mul' objects are
treated similarly where the coefficients are interpreted as _exponents_
now.  Addition of sums of terms or multiplication of products with
numerical exponents can be coded to be very efficient with such a
pair-wise representation.  Internally, this handling is performed by
most CAS in this way.  It typically speeds up manipulations by an order
of magnitude.  The overall multiplicative factor `2' and the additive
term `-3' look somewhat out of place in this representation, however,
since they are still carrying a trivial exponent and multiplicative
factor `1' respectively.  Within GiNaC, this is avoided by adding a
field that carries an overall numeric coefficient.  This results in the
realistic picture of internal representation for 2*d^3*(4*a+5*b-3):

<picture missing>

This also allows for a better handling of numeric radicals, since
`sqrt(2)' can now be carried along calculations.  Now it should be
clear, why both classes `add' and `mul' are derived from the same
abstract class: the data representation is the same, only the semantics
differs.  In the class hierarchy, methods for polynomial expansion and
the like are reimplemented for `add' and `mul', but the data structure
is inherited from `expairseq'.


File: ginac.info,  Node: Package Tools,  Next: ginac-config,  Prev: Internal representation of products and sums,  Up: Top

Package Tools
*************

If you are creating a software package that uses the GiNaC library,
setting the correct command line options for the compiler and linker
can be difficult. GiNaC includes two tools to make this process easier.

* Menu:

* ginac-config::   A shell script to detect compiler and linker flags.
* AM_PATH_GINAC::  Macro for GNU automake.


File: ginac.info,  Node: ginac-config,  Next: AM_PATH_GINAC,  Prev: Package Tools,  Up: Package Tools

`ginac-config'
==============

`ginac-config' is a shell script that you can use to determine the
compiler and linker command line options required to compile and link a
program with the GiNaC library.

`ginac-config' takes the following flags:

`--version'
     Prints out the version of GiNaC installed.

`--cppflags'
     Prints '-I' flags pointing to the installed header files.

`--libs'
     Prints out the linker flags necessary to link a program against
     GiNaC.

`--prefix[=PREFIX]'
     If PREFIX is specified, overrides the configured value of
     `$prefix'.  (And of exec-prefix, unless `--exec-prefix' is also
     specified) Otherwise, prints out the configured value of `$prefix'.

`--exec-prefix[=PREFIX]'
     If PREFIX is specified, overrides the configured value of
     `$exec_prefix'.  Otherwise, prints out the configured value of
     `$exec_prefix'.

Typically, `ginac-config' will be used within a configure script, as
described below. It, however, can also be used directly from the
command line using backquotes to compile a simple program. For example:

     c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`

This command line might expand to (for example):

     cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
       -lginac -lcln -lstdc++

Not only is the form using `ginac-config' easier to type, it will work
on any system, no matter how GiNaC was configured.


File: ginac.info,  Node: AM_PATH_GINAC,  Next: Configure script options,  Prev: ginac-config,  Up: Package Tools

`AM_PATH_GINAC'
===============

For packages configured using GNU automake, GiNaC also provides a macro
to automate the process of checking for GiNaC.

     AM_PATH_GINAC([MINIMUM-VERSION, [ACTION-IF-FOUND [, ACTION-IF-NOT-FOUND]]])

This macro:

   * Determines the location of GiNaC using `ginac-config', which is
     either found in the user's path, or from the environment variable
     `GINACLIB_CONFIG'.

   * Tests the installed libraries to make sure that their version is
     later than MINIMUM-VERSION. (A default version will be used if not
     specified)

   * If the required version was found, sets the `GINACLIB_CPPFLAGS'
     variable to the output of `ginac-config --cppflags' and the
     `GINACLIB_LIBS' variable to the output of `ginac-config --libs',
     and calls `AC_SUBST()' for these variables so they can be used in
     generated makefiles, and then executes ACTION-IF-FOUND.

   * If the required version was not found, sets `GINACLIB_CPPFLAGS' and
     `GINACLIB_LIBS' to empty strings, and executes ACTION-IF-NOT-FOUND.


This macro is in file `ginac.m4' which is installed in
`$datadir/aclocal'. Note that if automake was installed with a
different `--prefix' than GiNaC, you will either have to manually move
`ginac.m4' to automake's `$datadir/aclocal', or give aclocal the `-I'
option when running it.

* Menu:

* Configure script options::  Configuring a package that uses AM_PATH_GINAC.
* Example package::           Example of a package using AM_PATH_GINAC.


File: ginac.info,  Node: Configure script options,  Next: Example package,  Prev: AM_PATH_GINAC,  Up: AM_PATH_GINAC

Configuring a package that uses `AM_PATH_GINAC'
-----------------------------------------------

Simply make sure that `ginac-config' is in your path, and run the
configure script.

Notes:

   * The directory where the GiNaC libraries are installed needs to be
     found by your system's dynamic linker.

     This is generally done by

          editing `/etc/ld.so.conf' and running `ldconfig'

     or by

          setting the environment variable `LD_LIBRARY_PATH',

     or, as a last resort,

          giving a `-R' or `-rpath' flag (depending on your linker) when
          running configure, for instance:
               LDFLAGS=-R/home/cbauer/lib ./configure

   * You can also specify a `ginac-config' not in your path by setting
     the `GINACLIB_CONFIG' environment variable to the name of the
     executable

   * If you move the GiNaC package from its installed location, you
     will either need to modify `ginac-config' script manually to point
     to the new location or rebuild GiNaC.


Advanced note:

   * configure flags

          --with-ginac-prefix=PREFIX
          --with-ginac-exec-prefix=PREFIX

     are provided to override the prefix and exec-prefix that were
     stored in the `ginac-config' shell script by GiNaC's configure.
     You are generally better off configuring GiNaC with the right path
     to begin with.


File: ginac.info,  Node: Example package,  Next: Bibliography,  Prev: Configure script options,  Up: AM_PATH_GINAC

Example of a package using `AM_PATH_GINAC'
------------------------------------------

The following shows how to build a simple package using automake and
the `AM_PATH_GINAC' macro. The program used here is `simple.cpp':

     #include <ginac/ginac.h>
     
     int main(void)
     {
         GiNaC::symbol x("x");
         GiNaC::ex a = GiNaC::sin(x);
         std::cout << "Derivative of " << a
                   << " is " << a.diff(x) << std::endl;
         return 0;
     }

You should first read the introductory portions of the automake Manual,
if you are not already familiar with it.

Two files are needed, `configure.in', which is used to build the
configure script:

     dnl Process this file with autoconf to produce a configure script.
     AC_INIT(simple.cpp)
     AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
     
     AC_PROG_CXX
     AC_PROG_INSTALL
     AC_LANG_CPLUSPLUS
     
     AM_PATH_GINAC(0.9.0, [
       LIBS="$LIBS $GINACLIB_LIBS"
       CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
     ], AC_MSG_ERROR([need to have GiNaC installed]))
     
     AC_OUTPUT(Makefile)

The only command in this which is not standard for automake is the
`AM_PATH_GINAC' macro.

That command does the following: If a GiNaC version greater or equal
than 0.7.0 is found, then it adds `$GINACLIB_LIBS' to `$LIBS' and
`$GINACLIB_CPPFLAGS' to `$CPPFLAGS'. Otherwise, it dies with the error
message `need to have GiNaC installed'

And the `Makefile.am', which will be used to build the Makefile.

     ## Process this file with automake to produce Makefile.in
     bin_PROGRAMS = simple
     simple_SOURCES = simple.cpp

This `Makefile.am', says that we are building a single executable, from
a single source file `simple.cpp'. Since every program we are building
uses GiNaC we simply added the GiNaC options to `$LIBS' and
`$CPPFLAGS', but in other circumstances, we might want to specify them
on a per-program basis: for instance by adding the lines:

     simple_LDADD = $(GINACLIB_LIBS)
     INCLUDES = $(GINACLIB_CPPFLAGS)

to the `Makefile.am'.

To try this example out, create a new directory and add the three files
above to it.

Now execute the following commands:

     $ automake --add-missing
     $ aclocal
     $ autoconf

You now have a package that can be built in the normal fashion

     $ ./configure
     $ make
     $ make install


File: ginac.info,  Node: Bibliography,  Next: Concept Index,  Prev: Example package,  Up: Top

Bibliography
************

   - `ISO/IEC 14882:1998: Programming Languages: C++'

   - `CLN: A Class Library for Numbers', Bruno Haible <haible@ilog.fr>

   - `The C++ Programming Language', Bjarne Stroustrup, 3rd Edition,
     ISBN 0-201-88954-4, Addison Wesley

   - `C++ FAQs', Marshall Cline, ISBN 0-201-58958-3, 1995, Addison
     Wesley

   - `Algorithms for Computer Algebra', Keith O. Geddes, Stephen R.
     Czapor, and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer
     Academic Publishers, Norwell, Massachusetts

   - `Computer Algebra: Systems and Algorithms for Algebraic
     Computation', James H. Davenport, Yvon Siret, and Evelyne
     Tournier, ISBN 0-12-204230-1, 1988, Academic Press, London

   - `Computer Algebra Systems - A Practical Guide', Michael J. Wester
     (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester

   - `The Art of Computer Programming, Vol 2: Seminumerical Algorithms',
     Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley

   - `The Role of gamma5 in Dimensional Regularization', Dirk Kreimer,
     hep-ph/9401354



File: ginac.info,  Node: Concept Index,  Prev: Bibliography,  Up: Top

Concept Index
*************

* Menu:

* abs():                                 Built-in Functions.
* accuracy:                              Numbers.
* acos():                                Built-in Functions.
* acosh():                               Built-in Functions.
* add <1>:                               Fundamental containers.
* add:                                   Internal representation of products and sums.
* advocacy:                              A Comparison With Other CAS.
* AM_PATH_GINAC:                         AM_PATH_GINAC.
* antisymmetrize():                      Symmetrization.
* append():                              Lists.
* archive (class):                       Input/Output.
* archiving:                             Input/Output.
* asin():                                Built-in Functions.
* asinh():                               Built-in Functions.
* atan():                                Built-in Functions.
* atanh():                               Built-in Functions.
* atom <1>:                              The Class Hierarchy.
* atom:                                  Symbols.
* Autoconf:                              Configuration.
* beta():                                Built-in Functions.
* binomial():                            Built-in Functions.
* branch cut:                            Built-in Functions.
* building GiNaC:                        Building GiNaC.
* calchash():                            Adding classes.
* Catalan:                               Constants.
* chain rule:                            Symbolic Differentiation.
* charpoly():                            Matrices.
* clifford (class):                      Non-commutative objects.
* CLN <1>:                               Numbers.
* CLN:                                   Installation.
* coeff():                               Polynomial Arithmetic.
* collect():                             Polynomial Arithmetic.
* collect_common_factors():              Polynomial Arithmetic.
* color (class):                         Non-commutative objects.
* color_d():                             Non-commutative objects.
* color_f():                             Non-commutative objects.
* color_h():                             Non-commutative objects.
* color_ONE():                           Non-commutative objects.
* color_T():                             Non-commutative objects.
* color_trace():                         Non-commutative objects.
* complex numbers:                       Numbers.
* configuration:                         Configuration.
* constant (class):                      Constants.
* container <1>:                         Information About Expressions.
* container:                             The Class Hierarchy.
* content():                             Polynomial Arithmetic.
* contravariant:                         Indexed objects.
* Converting ex to other classes:        Information About Expressions.
* copy-on-write:                         Expressions are reference counted.
* cos():                                 Built-in Functions.
* cosh():                                Built-in Functions.
* covariant:                             Indexed objects.
* csgn():                                Built-in Functions.
* DECLARE_FUNCTION:                      Symbolic functions.
* degree():                              Polynomial Arithmetic.
* delta_tensor():                        Indexed objects.
* denom():                               Rational Expressions.
* denominator:                           Rational Expressions.
* determinant():                         Matrices.
* diag_matrix():                         Matrices.
* diff():                                Symbolic Differentiation.
* differentiation:                       Symbolic Differentiation.
* Digits:                                Numbers.
* dirac_gamma():                         Non-commutative objects.
* dirac_gamma5():                        Non-commutative objects.
* dirac_gammaL():                        Non-commutative objects.
* dirac_gammaR():                        Non-commutative objects.
* dirac_ONE():                           Non-commutative objects.
* dirac_slash():                         Non-commutative objects.
* dirac_trace():                         Non-commutative objects.
* divide():                              Polynomial Arithmetic.
* Dummy index:                           Indexed objects.
* epsilon_tensor():                      Indexed objects.
* Euler:                                 Constants.
* Euler numbers:                         Symbolic Differentiation.
* eval() <1>:                            Adding classes.
* eval():                                Automatic evaluation.
* evalf():                               Constants.
* evalm():                               Matrices.
* evaluation <1>:                        Automatic evaluation.
* evaluation <2>:                        Adding classes.
* evaluation:                            Symbolic functions.
* ex_to<...>():                          Information About Expressions.
* exceptions:                            Error handling.
* exp():                                 Built-in Functions.
* expand() <1>:                          Polynomial Arithmetic.
* expand():                              Indexed objects.
* expression (class ex):                 Expressions.
* factorial():                           Built-in Functions.
* factorization:                         Polynomial Arithmetic.
* find():                                Pattern Matching and Advanced Substitutions.
* fraction:                              Numbers.
* function (class):                      Mathematical functions.
* Gamma function <1>:                    Mathematical functions.
* Gamma function:                        Built-in Functions.
* garbage collection:                    Expressions are reference counted.
* GCD:                                   Polynomial Arithmetic.
* gcd():                                 Polynomial Arithmetic.
* get_dimension():                       Indexed objects.
* get_free_indices():                    Indexed objects.
* get_value():                           Indexed objects.
* ginac-config:                          ginac-config.
* ginsh <1>:                             Fundamental containers.
* ginsh <2>:                             What it can do for you.
* ginsh:                                 What does not belong into GiNaC.
* GMP:                                   Numbers.
* has() <1>:                             Pattern Matching and Advanced Substitutions.
* has():                                 Expressions.
* Hermite polynomial:                    How to use it from within C++.
* hierarchy of classes <1>:              Symbols.
* hierarchy of classes:                  Adding classes.
* history of GiNaC:                      Introduction.
* hold() <1>:                            Adding classes.
* hold():                                Symbolic functions.
* hyperbolic function:                   Mathematical functions.
* I:                                     Numbers.
* I/O:                                   Input/Output.
* idx (class):                           Indexed objects.
* indexed (class):                       Indexed objects.
* info():                                Information About Expressions.
* input of expressions:                  Input/Output.
* installation:                          Installing GiNaC.
* inverse():                             Matrices.
* is_a<...>():                           Information About Expressions.
* is_equal():                            Information About Expressions.
* is_exactly_a<...>():                   Information About Expressions.
* is_zero():                             Information About Expressions.
* Laurent expansion:                     Series Expansion.
* LCM:                                   Polynomial Arithmetic.
* lcm():                                 Polynomial Arithmetic.
* ldegree():                             Polynomial Arithmetic.
* let_op():                              Adding classes.
* lgamma():                              Built-in Functions.
* Li2():                                 Built-in Functions.
* lists:                                 Lists.
* log():                                 Built-in Functions.
* lorentz_eps():                         Indexed objects.
* lorentz_g():                           Indexed objects.
* lst (class):                           Lists.
* lst_to_matrix():                       Matrices.
* Machin's formula:                      Series Expansion.
* map():                                 Applying a Function on Subexpressions.
* match():                               Pattern Matching and Advanced Substitutions.
* matrix (class):                        Matrices.
* metric_tensor():                       Indexed objects.
* mul <1>:                               Internal representation of products and sums.
* mul:                                   Fundamental containers.
* ncmul (class):                         Non-commutative objects.
* nops() <1>:                            Information About Expressions.
* nops():                                Lists.
* normal():                              Rational Expressions.
* numer():                               Rational Expressions.
* numer_denom():                         Rational Expressions.
* numerator:                             Rational Expressions.
* numeric (class):                       Numbers.
* op() <1>:                              Information About Expressions.
* op():                                  Lists.
* Order() <1>:                           Series Expansion.
* Order():                               Built-in Functions.
* output of expressions:                 Input/Output.
* pair-wise representation:              Internal representation of products and sums.
* Pattern matching:                      Pattern Matching and Advanced Substitutions.
* Pi:                                    Constants.
* pole_error (class):                    Error handling.
* polynomial <1>:                        Methods and Functions.
* polynomial:                            Fundamental containers.
* polynomial division:                   Polynomial Arithmetic.
* pow():                                 Fundamental containers.
* power <1>:                             Internal representation of products and sums.
* power:                                 Fundamental containers.
* prem():                                Polynomial Arithmetic.
* prepend():                             Lists.
* primpart():                            Polynomial Arithmetic.
* print_context (class):                 Input/Output.
* printing:                              Input/Output.
* product rule <1>:                      Symbolic functions.
* product rule:                          Symbolic Differentiation.
* pseries (class):                       Series Expansion.
* pseudo-remainder:                      Polynomial Arithmetic.
* psi():                                 Built-in Functions.
* quo():                                 Polynomial Arithmetic.
* quotient:                              Polynomial Arithmetic.
* radical:                               Internal representation of products and sums.
* rational:                              Numbers.
* reference counting:                    Expressions are reference counted.
* REGISTER_FUNCTION:                     Symbolic functions.
* relational (class) <1>:                Relations.
* relational (class):                    Information About Expressions.
* rem():                                 Polynomial Arithmetic.
* remainder:                             Polynomial Arithmetic.
* remove_first():                        Lists.
* remove_last():                         Lists.
* representation:                        Internal representation of products and sums.
* return_type() <1>:                     Information About Expressions.
* return_type():                         Non-commutative objects.
* return_type_tinfo() <1>:               Non-commutative objects.
* return_type_tinfo():                   Information About Expressions.
* rounding:                              Numbers.
* RTTI:                                  Adding classes.
* series():                              Series Expansion.
* simplification:                        Rational Expressions.
* simplify_indexed():                    Indexed objects.
* sin():                                 Built-in Functions.
* sinh():                                Built-in Functions.
* spinidx (class):                       Indexed objects.
* spinor_metric():                       Indexed objects.
* sqrfree():                             Polynomial Arithmetic.
* sqrt():                                Built-in Functions.
* square-free decomposition:             Polynomial Arithmetic.
* STL:                                   Advantages.
* subs() <1>:                            Pattern Matching and Advanced Substitutions.
* subs() <2>:                            Mathematical functions.
* subs() <3>:                            Symbols.
* subs() <4>:                            Indexed objects.
* subs() <5>:                            Methods and Functions.
* subs():                                Substituting Expressions.
* sy_anti():                             Indexed objects.
* sy_cycl():                             Indexed objects.
* sy_none():                             Indexed objects.
* sy_symm():                             Indexed objects.
* symbol (class):                        Symbols.
* symbolic_matrix():                     Matrices.
* symmetrize():                          Symmetrization.
* symmetrize_cyclic():                   Symmetrization.
* symmetry (class):                      Indexed objects.
* tan():                                 Built-in Functions.
* tanh():                                Built-in Functions.
* Taylor expansion:                      Series Expansion.
* temporary replacement:                 Rational Expressions.
* tensor (class):                        Indexed objects.
* tgamma():                              Built-in Functions.
* to_rational():                         Rational Expressions.
* trace():                               Matrices.
* transpose():                           Matrices.
* Tree traversal <1>:                    Applying a Function on Subexpressions.
* Tree traversal:                        Input/Output.
* trigonometric function:                Mathematical functions.
* unit():                                Polynomial Arithmetic.
* unit_matrix():                         Matrices.
* variance:                              Indexed objects.
* varidx (class):                        Indexed objects.
* viewgar:                               Input/Output.
* wildcard (class):                      Pattern Matching and Advanced Substitutions.
* Zeta function:                         What it can do for you.
* zeta():                                Built-in Functions.



Tag Table:
Node: Top795
Node: Introduction1651
Node: A Tour of GiNaC4833
Node: How to use it from within C++5264
Node: What it can do for you7614
Node: Installation12783
Node: Prerequisites13327
Node: Configuration14435
Ref: Configuration-Footnote-117915
Node: Building GiNaC18273
Node: Installing GiNaC20013
Ref: Installing GiNaC-Footnote-121427
Node: Basic Concepts21892
Node: Expressions23131
Node: Automatic evaluation24559
Node: Error handling26997
Node: The Class Hierarchy28663
Node: Symbols30973
Node: Numbers32959
Node: Constants40519
Node: Fundamental containers41163
Node: Lists43989
Node: Mathematical functions46203
Node: Relations48085
Node: Matrices49435
Node: Indexed objects55123
Node: Non-commutative objects80457
Node: Methods and Functions94545
Node: Information About Expressions96961
Node: Substituting Expressions103358
Node: Pattern Matching and Advanced Substitutions105253
Node: Applying a Function on Subexpressions114973
Node: Polynomial Arithmetic119508
Node: Rational Expressions129309
Node: Symbolic Differentiation132446
Node: Series Expansion134364
Node: Symmetrization137818
Node: Built-in Functions139096
Node: Input/Output141680
Node: Extending GiNaC152687
Node: What does not belong into GiNaC153460
Node: Symbolic functions154638
Node: Adding classes163750
Node: A Comparison With Other CAS181993
Node: Advantages182587
Node: Disadvantages185961
Ref: Disadvantages-Footnote-1187244
Node: Why C++?187570
Node: Internal Structures188386
Node: Expressions are reference counted188641
Node: Internal representation of products and sums192148
Node: Package Tools194633
Node: ginac-config195122
Node: AM_PATH_GINAC196665
Node: Configure script options198281
Node: Example package199759
Node: Bibliography202223
Node: Concept Index203397

End Tag Table
