# Yacas user's function reference

- Running Yacas

- Interactive session commands
- Command-line options

- Startup configuration
- Arithmetic and other operations on numbers
- Number theory
- Calculus and elementary functions
- Special functions
- Differential Equations
- Transforms
- Linear Algebra
- Polynomials
- List operations
- Functional operators
- Control flow functions
- Predicates
- Propositional logic theorem prover
- Constants
- Variables
- Input/output and plotting
- String manipulation
- Probability and Statistics

- Probability
- Statistics

- Platform-dependent packages
- Yacas plugins

- The file scanner plugin
- The regular expressions plugin
- The GSL library plugin
- The example plugin
- The Forth plugin

- GNU Free Documentation License
- Programming
- Arbitrary-precision numerical programming
- Error reporting
- Built-in (core) functions

- Full listing of core functions

- The Yacas plugin structure
- Generic objects
- The Yacas test suite
- Glossary
- GNU General Public License

- Preamble
- NO WARRANTY

- GNU Free Documentation License

?, ?? , quit, restart, Ctrl-C .

# Running Yacas

This section describes the special commands for the Yacas interactive sessions (for example, to restart or to exit the interpreter). These commands are not functions but special directives that only apply while running Yacas interactively. They should not be used in scripts.

##

Interactive session commands

###

?, ?? -- get online help

##### Internal function

##### Calling format:

    ?function
    ??

##### Parameters:

`function` -- name of function or keyword to request help on

##### Description:

The directive `?` is used to request online help while running an interactive session. A name of a function or a help keyword can be specified. The corresponding reference manual section will be shown. (There is a special glossary section that contains keywords and explains some of the Yacas terminology.)

Function names should be given without quotes. Note that function names are case-sensitive in Yacas but the `?` facility will also respond to all lowercase names.

The directive `??` displays the index to all documentation.

Documentation is displayed in a Web browser of your choice. The default choice is `lynx`. An alternative browser can be set using the `SetHelpBrowser` function.

##### Examples:

Get help on a function:

    In> ?Random

The function name can be given in all lowercase:

    In> ?makefunctionplugin

Get help on a keyword:

    In> ?arity

Browse all documentation:

    In> ??

##### See also:

Help , SetHelpBrowser .

###

quit, restart, Ctrl-C -- basic session control

##### Internal function

##### Calling format:

    quit
    restart

##### Description:

Type `quit` or `restart` at the Yacas prompt to exit or to restart the interpreter.

The directives `quit` and `restart` are not reserved words or variable names. They take effect only when typed as first characters at a prompt.

Pressing `Ctrl-C` will stop the currently running calculation. If there is no currently running calculation, `Ctrl-C` will quit the interpreter.

When the interpreter quits, it saves the command history (so quitting by `Ctrl-C` does not mean a "crash").

##### Examples:

To be effective, the directive must be typed immediately after the prompt:

    In> quit
    Quitting...

We can use variables named `quit`:

    In> 1+quit
    Out> quit+1;

There is no effect if we type some spaces first:

    In>       restart
    Out> restart;

##### See also:

Exit .

##

Command-line options

The default operation of `Yacas` is to run in the interactive console mode. `Yacas` accepts several options that modify its operation. Here is a summary of options:

- filename ... (read and execute a file or several files)
- `-c` (omit line prompts)
- `-d` (print default directory)
- `-v` (print version information)
- `-f` (execute standard input as one statement)
- `-p` (do not use terminal capabilities)
- `-t` (enable extra history features)
- `--archive` filename (use a given library archive file)
- `--dlldir` directory (specify default directory for plugins)
- `--init` filename (use a given initial file)
- `--patchload` (use `PatchLoad` to load files)
- `--read-eval-print` expression (call this expression for the read-eval-print loop)
- `--rootdir` directory (specify default directory for scripts)
- `--server` port (start `Yacas` as a network server on given port)
- `--verbose-debug` (turn on showing some additional debugging information on screen)
- `--disable-compiled-plugins` (disable loading of compiled plugins, loading the script versions instead)
- `--stacksize` size (change size of stack arguments are stored on)
- `--execute` expression (run expression from the command line)

Options can be combined, for example

    yacas -pc filename

will read and execute the file `filename` non-interactively without using terminal capabilities and without printing prompts.

Here is a more detailed description of the command-line options.

    yacas -c

Inhibit printing of prompts `In>` and `Out>`. Useful for non-interactive sessions.

    yacas -f

Reads standard input as one file, but executes only the first statement in it. (You may want to use a statement block to have several statements executed.)

    yacas -p

Does not use terminal capabilities, no fancy editing on the command line and no escape sequences printed. Useful for non-interactive sessions.

    yacas -t

Enable some extra history recall functionality in console mode: after executing a command from the history list, the next unmodified command from the history list will be automatically entered on the command line.

    yacas [options] {filename}

Reads and executes commands in the filename and exits. Equivalent to `Load()`.

    yacas -v

Prints version information and exits. (This is the same information as returned by `Version()`.)

    yacas -d

Prints the path to the `Yacas` default library directory (this information is compiled into the `Yacas` executable) and exits.

    yacas --patchload

Will load every file on the command line with the `PatchLoad` command instead of the normal `Load` command, This is useful for generating HTML pages for a web site using the `Yacas` scripting language, much like you can do with the PHP scripting language.

    yacas --init [file]

Tells the system to load `file` as the initialization file. By default it loads the file `yacasinit.ys` from the scripts directory. Thus for customization one has two options: write a `~/.yacasrc` file with initialization code (as it is loaded after the initialization script is loaded), or write a custom initialization script that first uses `yacasinit.ys` and adds some extra custom code.

    yacas --read-eval-print [expression]

Call `expression` for the read-eval-print loop. The default read-eval-print loop is implemented in the initialization script `yacasinit.ys` as the function `REP`. The default behavior is therefore equivalent to `--read-eval-print` `REP()`.

There is also a fallback read-eval-print loop in the kernel; it can be invoked by passing an empty string to this command line option, as `--read-eval-print ""`.

An alternative way to replace the default read-eval-print loop is to write a custom initialization script that implements the read-eval-print loop function `REP()` instead of `yacasinit.ys`.

Care has to be taken with this option because a Yacas session may become unusable if the read-eval-print expression doesn't function correctly.

    yacas --server <port>

On some platforms server mode can be enabled at build time by passing the flag `--enable-server` to the `./configure` script. `Yacas` then allows you to pass the flag `--server` with a port number behind it, and the `Yacas` executable will listen to the socket behind that port instead of waiting for user input on the console.

Commands can be sent to the server by sending a text line as one block of data, and the server will respond back with another text block.

One can test this function by using `telnet`. First, set up the server by calling

    yacas --server 9734

and then invoke `telnet` in another window, for example:

    telnet 127.0.0.1 9734

Then type a line of Yacas input and hit Enter. The result will be one line that you will get back from the Yacas server.

Some security measures and resource management measures have been taken. No more than 10 connections can be alive at any time, a calculation cannot take more than 30 seconds, and `Yacas` operates in the secure mode, much like calling an expression by passing it as an argument to the `Secure` function. This means that no system calls are allowed, and no writing to local files, amongst other things. Something that has not been taken care of yet is memory use. A calculation could take up all memory, but not for longer than 30 seconds.

The server is single-threaded, but has persistent sessions for at most 10 users at a time, from which it can service requests in a sequenial order. To make the service multi-threaded, a solution might be to have a proxy in front of the service listening to the port, redirecting it to different processes which get started up for users (this has not been tried yet).

    yacas --rootdir [directory]

Tells the system where to find the library scripts. Here, `directory` is a path that is passed to `DefaultDirectory`. It is also possible to give a list of directories, separated by a colon, e.g. `yacas --rootdir scripts/:morescripts/`. Note that it is not necessary to append a trailing slash to the directory names.

    yacas --dlldir [directory]

Tells the system where to find the plugins. Here `directory` is a path that is passed to `DllDirectory`. It is also possible to give a list of directories separated by a colon. The default value is specified at compile time, usually `/usr/local/lib/yacas`.

    yacas --archive [file]

Use a compressed archive instead of the script library.

`Yacas` has an experimental system where files can be compressed into one file, and accessed through this command line option. The advantages are:

- Smaller disk/memory use (useful if `Yacas` is used on small hand-held computers).
- No problems with directory path separators: "`path`/`file`" will always resolve to the right file, no matter what platform (read: Windows) it runs on.
- The start-up time of the program might improve a little, since a smaller file is loaded from disk (disk access being slow), and then decompressed in memory, which might be a lot faster than loading from disk.

An additional savings is due to the fact that the script files are stripped from white spaces and comments, making them smaller and faster loading.

To prepare the compressed library archive, run `./configure` with the command line option `--enable-archive`.

The result should be the archive file `scripts.dat`. Then launch `Yacas` with the command line option `--archive scripts.dat`, with the file `scripts.dat` in the current directory.

The reason that the `scripts.dat` file is not built automatically is that it is not tested, at this time, that the build process works on all platforms. (Right now it works on Unix, MacOSX, and Win32.)

Alternatively, configure Yacas with

    ./configure --enable-archive

and the archive file `scripts.dat` will be created in the `ramscripts/` subdirectory.

When an archive is present, Yacas will try to load it before it looks for scripts from the library directories. Typing

    make archivetest -f makefile.compressor

in the `ramscripts/` directory runs all the test scripts using the archived files.

The currently supported compression schemes are uncompressed and compressed with `minilzo`. Script file stripping (removing whitespace and comments) may be disabled by editing `compressor.cpp` (variable `strip_script`).

    yacas --disable-compiled-plugins

Disable loading of compiled scripts, in favor of scripts themselves. This is useful when developing the scripts that need to be compiled in the end, or when the scripts have not been compiled yet.

    yacas --stacksize <size>

Yacas maintains an internal stack for arguments. For nested function calls, all arguments currently used are on this stack. The size of this stack is 50000 be default.

For a function that would take 4 arguments and has one return value, there would be 5 places reserved on this stack, and the function could call itself recursively 10000 steps deep.

This differs from the MaxEvalDepth mechanism. The MaxEvalDepth mechanism allows one to specify the number of separate stack frames (number of calls, nested), instead of the number of arguments pushed on the stack. MaxEvalDepth was introduced to protect the normal `C++` stack.

    yacas --execute <expression>

This instructs Yacas to run a certain expression, passed in over the command line, before dropping to the read-eval-print loop. This can be used to load a file before dropping to the command line without exiting (if there are files to run specified on the command line, Yacas will exit after running these scripts). Alternatively, the expression can exit the interpreter immediately by calling `Exit();`. When used in combination with `-pc`, the Yacas interpreter can be used to calculate something and print the result to standard output. Example:

    user% ./yacas -pc --execute '[Echo("answer ",D(x)Sin(x));Exit();]'
    answer Cos(x)
    user%

DefaultDirectory , DllDirectory , PrettyPrinter, GetPrettyPrinter , PrettyReader, GetPrettyReader , MaxEvalDepth , Help , HistorySize .

# Startup configuration

Yacas allows you to configure a few things at startup. The file `~/.yacasrc` is written in the Yacas language and will be executed when Yacas is run. The following functions can be useful in the `~/.yacasrc` file.

###

DefaultDirectory -- add directory to path for Yacas scripts

##### Internal function

##### Calling format:

    DefaultDirectory(path)

##### Parameters:

`path` -- a string containing a full path where yacas script files reside

##### Description:

When loading files, yacas is also allowed to look in the folder "path". `path` will be prepended to the file name before trying to load the file. This means that "path" should end with a forward slash (under Unix-like operating systems).

Yacas first tries to load a file from the current directory, and otherwise it tries to load from directories defined with this function, in the order they are defined. Note there will be at least one directory specified at start-up time, defined during compilation. This is the directory Yacas searches for the initialization scripts and standard scripts.

##### Examples:

    In> DefaultDirectory("/home/user/myscripts/");
    Out> True;

##### See also:

Load , Use , DefLoad , FindFile , DllDirectory .

###

DllDirectory -- add directory to path for Yacas plugins

##### Internal function

##### Calling format:

    DllDirectory(path)

##### Parameters:

`path` -- a string containing a full path where Yacas plugins reside

##### Description:

This command adds "path" to the list of directories in which Yacas looks for plugins.

Note there will be at least one directory specified at start-up time. If the command line option `--dlldir` is given, the default directory for plugins will be the argument to the option. Otherwise, it is a directory specified at compile time.

This command always evaluates to `True`.

##### Examples:

    In> DllDirectory("/home/user/yacas/plugins/");
    Out> True;

##### See also:

DefaultDirectory , DllLoad .

###

PrettyPrinter, GetPrettyPrinter -- set/get routine to use as pretty-printer

##### Standard library

##### Calling format:

    PrettyPrinter(printer)
    PrettyPrinter()
    GetPrettyPrinter()

##### Parameters:

`printer` -- a string containing the name of a function that can "pretty-print" an expression.

##### Description:

This function sets up the function printer to print out the results on the command line. This can be reset to the internal printer with `PrettyPrinter()` (when no argument is given, the system returns to the default).

Currently implemented prettyprinters are: `PrettyForm`, `TeXForm`, `Print`, `OMForm`, `CForm` and `DefaultPrint`.

`GetPrettyPrinter()` returns the current pretty printer, or it returns an empty string if the default pretty printer is used.

##### Examples:

    In> Taylor(x,0,5)Sin(x)
    Out> x-x^3/6+x^5/120;
    In> PrettyPrinter("PrettyForm");

    True

    In> Taylor(x,0,5)Sin(x)

         3    5
        x    x
    x - -- + ---
        6    120

    In> PrettyPrinter();
    Out> True;
    In> Taylor(x,0,5)Sin(x)
    Out> x-x^3/6+x^5/120;

##### See also:

PrettyForm , Write , TeXForm , CForm , OMForm , PrettyReader , GetPrettyReader .

###

PrettyReader, GetPrettyReader -- set/get routine to use as pretty-printer

##### Standard library

##### Calling format:

    PrettyReader(reader)
    PrettyReader()
    GetPrettyReader()

##### Parameters:

`reader` -- a string containing the name of a function that can read an expression from current input.

##### Description:

This function sets up the function reader to read in the input on the command line. This can be reset to the internal reader with `PrettyReader()` (when no argument is given, the system returns to the default).

Currently implemented PrettyReaders are: `LispRead`, `OMRead`.

`GetPrettyReader()` returns the current reader, or it returns an empty string if the default pretty printer is used.

##### Examples:

    In> Taylor(x,0,5)Sin(x)
    Out> x-x^3/6+x^5/120
    In> PrettyReader("LispRead")
    Out> True
    In> (Taylor x 0 5 (Sin x))
    Out> x-x^3/6+x^5/120

##### See also:

Read , LispRead , OMRead , PrettyPrinter , GetPrettyPrinter .

###

MaxEvalDepth -- set depth of recursion stack

##### Internal function

##### Calling format:

    MaxEvalDepth(n)

##### Parameters:

`n` -- integer

##### Description:

Sets the maximum depth of recursive function call. An error message is printed when too many recursive calls are executed, and this function can be used to increase or decrease the limit as necessary.

###

Help -- get documentation for some command

##### Standard library

##### Calling format:

    Help()
    Help(function)

##### Parameters:

`function` -- a string containing the name of a function to show help for

##### Description:

When help is requested by the user, by typing `?`function or `??`, the functions `Help()` (for `??`) and `Help(`function`)` (for `?`function) are called. By default, `lynx` is used as a browser (another recommended text-only browser is `links`). The help resides in the subdirectory `documentation/` under the directory the math scripts were installed in. So the help files can be found using `FindFile`.

The function `Help` is auxiliary and should not be used to actually get online help.

##### Examples:

To use `netscape` for browsing help, enter the following commands:

    Help(_f) <-- SystemCall("netscape " :
      FindFile("documentation/ref.html"):"#":f);
    Help() := SystemCall("netscape " :
      FindFile("documentation/books.html"));

##### See also:

SystemCall , FindFile , ? .

###

HistorySize -- set size of history file

##### Internal function

##### Calling format:

    HistorySize(n)

##### Parameters:

`n` -- number of lines to store in history file

##### Description:

When exiting, yacas saves the command line history to a file `~/.yacas_history`. By default it will save the last 1024 lines. The default can be overridden with this function. Passing -1 tells the system to save all lines.

##### Examples:

    In> HistorySize(200)
    Out> True;
    In> quit

##### See also:

quit .

+, -, *, /, ^ , Div, Mod , Gcd , Lcm , <<, >> , FromBase, ToBase , Precision , GetPrecision , N , Rationalize , IntLog , IntNthRoot , NthRoot , ContFrac , ContFracList, ContFracEval , GuessRational, NearRational, BracketRational , Decimal , TruncRadian , Floor , Ceil , Round , Pslq .

# Arithmetic and other operations on numbers

Besides the usual arithmetical operations, Yacas defines some more advanced operations on numbers. Many of them also work on polynomials.

###

+, -, *, /, ^ -- arithmetic operations

##### Standard library

##### Calling format:

    x+y
    +x

Precedence: 70

    x-y

Precedence: left-side: 70 , right-side: 40

    -x
    x*y

Precedence: 40

    x/y

Precedence: 30

    x^y

Precedence: 20

##### Parameters:

`x` and `y` -- objects for which arithmetic operations are defined

##### Description:

These are the basic arithmetic operations. They can work on integers, rational numbers, complex numbers, vectors, matrices and lists.

These operators are implemented in the standard math library (as opposed to being built-in). This means that they can be extended by the user.

##### Examples:

    In> 2+3
    Out> 5;
    In> 2*3
    Out> 6;

###

Div, Mod -- division with remainder

##### Standard library

##### Calling format:

    Div(x,y)
    Mod(x,y)

##### Parameters:

`x`, `y` -- integers or univariate polynomials

##### Description:

`Div` performs integer division and `Mod` returns the remainder after division. `Div` and `Mod` are also defined for polynomials.

If `Div(x,y)` returns "a" and `Mod(x,y)` equals "b", then these numbers satisfy x=a*y+b and 0<=b<y.

##### Examples:

    In> Div(5,3)
    Out> 1;
    In> Mod(5,3)
    Out> 2;

##### See also:

Gcd , Lcm .

###

Gcd -- greatest common divisor

##### Standard library

##### Calling format:

    Gcd(n,m)
    Gcd(list)

##### Parameters:

`n`, `m` -- integers or Gaussian integers or univariate polynomials

`list` -- a list of all integers or all univariate polynomials

##### Description:

This function returns the greatest common divisor of "n" and "m". The gcd is the largest number that divides "n" and "m". It is also known as the highest common factor (hcf). The library code calls `MathGcd`, which is an internal function. This function implements the "binary Euclidean algorithm" for determining the greatest common divisor:

##### Routine for calculating `Gcd(n,m)`

- if n=m then return n
- if both n and m are even then return 2*Gcd(n/2,m/2)
- if exactly one of n or m (say n) is even then return Gcd(n/2,m)
- if both n and m are odd and, say, n>m then return Gcd((n-m)/2,m)

This is a rather fast algorithm on computers that can efficiently shift integers. When factoring Gaussian integers, a slower recursive algorithm is used.

If the second calling form is used, `Gcd` will return the greatest common divisor of all the integers or polynomials in "list". It uses the identity

Gcd(a,b,c)=Gcd(Gcd(a,b),c).

##### Examples:

    In> Gcd(55,10)
    Out> 5;
    In> Gcd({60,24,120})
    Out> 12;
    In> Gcd( 7300 + 12*I, 2700 + 100*I)
    Out> Complex(-4,4);

##### See also:

Lcm .

###

Lcm -- least common multiple

##### Standard library

##### Calling format:

    Lcm(n,m)
    Lcm(list)

##### Parameters:

`n`, `m` -- integers or univariate polynomials `list` -- list of integers

##### Description:

This command returns the least common multiple of "n" and "m" or all of the integers in the list `list`. The least common multiple of two numbers "n" and "m" is the lowest number which is an integer multiple of both "n" and "m". It is calculated with the formula

Lcm(n,m)=Div(n*m,Gcd(n,m)).

This means it also works on polynomials, since `Div`, `Gcd` and multiplication are also defined for them.

##### Examples:

    In> Lcm(60,24)
    Out> 120;
    In> Lcm({3,5,7,9})
    Out> 315;

##### See also:

Gcd .

###

<<, >> -- shift operators

##### Standard library

##### Calling format:

    n<<m
    n>>m

##### Parameters:

`n`, `m` -- integers

##### Description:

These operators shift integers to the left or to the right. They are similar to the C shift operators. These are sign-extended shifts, so they act as multiplication or division by powers of 2.

##### Examples:

    In> 1 << 10
    Out> 1024;
    In> -1024 >> 10
    Out> -1;

###

FromBase, ToBase -- conversion from/to non-decimal base

##### Internal function

##### Calling format:

    FromBase(base,"string")
    ToBase(base, number)

##### Parameters:

`base` -- integer, base to convert to/from

`number` -- integer, number to write out in a different base

`"string"` -- string representing a number in a different base

##### Description:

In Yacas, all numbers are written in decimal notation (base 10). The two functions `FromBase`, `ToBase` convert numbers between base 10 and a different base. Numbers in non-decimal notation are represented by strings.

`FromBase` converts an integer, written as a string in base `base`, to base 10. `ToBase` converts `number`, written in base 10, to base `base`.

Non-integer arguments are not supported.

##### Examples:

Write the binary number `111111` as a decimal number:

    In> FromBase(2,"111111")
    Out> 63;

Write the (decimal) number `255` in hexadecimal notation:

    In> ToBase(16,255)
    Out> "ff";

##### See also:

PAdicExpand .

###

Precision -- set the precision

##### Internal function

##### Calling format:

    Precision(n)

##### Parameters:

`n` -- integer, new value of precision

##### Description:

This command sets the number of decimal digits to be used in calculations. All subsequent floating point operations will allow for at least `n` digits of mantissa.

This is not the number of digits after the decimal point. For example, `123.456` has 3 digits after the decimal point and 6 digits of mantissa. The number `123.456` is adequately computed by specifying `Precision(6)`.

The call `Precision(n)` will not guarantee that all results are precise to `n` digits.

When the precision is changed, all variables containing previously calculated values remain unchanged. The `Precision` function only makes all further calculations proceed with a different precision.

Also, when typing floating-point numbers, the current value of `Precision` is used to implicitly determine the number of precise digits in the number.

##### Examples:

    In> Precision(10)
    Out> True;
    In> N(Sin(1))
    Out> 0.8414709848;
    In> Precision(20)
    Out> True;
    In> x:=N(Sin(1))
    Out> 0.84147098480789650665;

The value `x` is not changed by a `Precision()` call:

    In> [ Precision(10); x; ]
    Out> 0.84147098480789650665;

The value `x` is rounded off to 10 digits after an arithmetic operation:

    In> x+0.
    Out> 0.8414709848;

In the above operation, `0.` was interpreted as a number which is precise to 10 digits (the user does not need to type `0.0000000000` for this to happen). So the result of `x+0.` is precise only to 10 digits.

##### See also:

GetPrecision , N .

###

GetPrecision -- get the current precision

##### Internal function

##### Calling format:

    GetPrecision()

##### Description:

This command returns the current precision, as set by `Precision`.

##### Examples:

    In> GetPrecision();
    Out> 10;
    In> Precision(20);
    Out> True;
    In> GetPrecision();
    Out> 20;

##### See also:

Precision , N .

###

N -- compute numerical approximation

##### Standard library

##### Calling format:

    N(expr)
    N(expr, prec)

##### Parameters:

`expr` -- expression to evaluate

`prec` -- integer, precision to use

##### Description:

This function forces Yacas to give a numerical approximation to the expression "expr", using "prec" digits if the second calling sequence is used, and the precision as set by `SetPrecision` otherwise. This overrides the normal behaviour, in which expressions are kept in symbolic form (eg. `Sqrt(2)` instead of `1.41421`).

Application of the `N` operator will make Yacas calculate floating point representations of functions whenever possible. In addition, the variable `Pi` is bound to the value of Pi calculated at the current precision. (This value is a "cached constant", so it is not recalculated each time `N` is used, unless the precision is increased.)

##### Examples:

    In> 1/2
    Out> 1/2;
    In> N(1/2)
    Out> 0.5;
    In> Sin(1)
    Out> Sin(1);
    In> N(Sin(1),10)
    Out> 0.8414709848;
    In> Pi
    Out> Pi;
    In> N(Pi,20)
    Out> 3.14159265358979323846;

##### See also:

Precision , GetPrecision , Pi , CachedConstant .

###

Rationalize -- convert floating point numbers to fractions

##### Standard library

##### Calling format:

    Rationalize(expr)

##### Parameters:

`expr` -- an expression containing real numbers

##### Description:

This command converts every real number in the expression "expr" into a rational number. This is useful when a calculation needs to be done on floating point numbers and the algorithm is unstable. Converting the floating point numbers to rational numbers will force calculations to be done with infinite precision (by using rational numbers as representations).

It does this by finding the smallest integer n such that multiplying the number with 10^n is an integer. Then it divides by 10^n again, depending on the internal gcd calculation to reduce the resulting division of integers.

##### Examples:

    In> {1.2,3.123,4.5}
    Out> {1.2,3.123,4.5};
    In> Rationalize(%)
    Out> {6/5,3123/1000,9/2};

##### See also:

IsRational .

###

IntLog -- integer part of logarithm

##### Standard library

##### Calling format:

    IntLog(n, base)

##### Parameters:

`n`, `base` -- positive integers

##### Description:

`IntLog` calculates the integer part of the logarithm of `n` in base `base`. The algorithm uses only integer math and may be faster than computing

Ln(n)/Ln(base)
with multiple precision floating-point math and rounding off to get the integer part.

This function can also be used to quickly count the digits in a given number.

##### Examples:

Count the number of bits:

    In> IntLog(257^8, 2)
    Out> 64;

Count the number of decimal digits:

    In> IntLog(321^321, 10)
    Out> 804;

##### See also:

IntNthRoot , Div , Mod , Ln .

###

IntNthRoot -- integer part of n-th root

##### Standard library

##### Calling format:

    IntNthRoot(x, n)

##### Parameters:

`x`, `n` -- positive integers

##### Description:

`IntNthRoot` calculates the integer part of the n-th root of x. The algorithm uses only integer math and may be faster than computing x^(1/n) with floating-point and rounding.

This function is used to test numbers for prime powers.

##### Example:

    In> IntNthRoot(65537^111, 37)
    Out> 281487861809153;

##### See also:

IntLog , MathPower , IsPrimePower .

###

NthRoot -- calculate/simplify nth root of an integer

##### Standard library

##### Calling format:

    NthRoot(m,n)

##### Parameters:

`m` -- a non-negative integer (m>0)

`n` -- a positive integer greater than 1 ( n>1)

##### Description:

`NthRoot(m,n)` calculates the integer part of the n-th root m^(1/n) and returns a list `{f,r}`. `f` and `r` are both positive integers that satisfy f^n*r= m. In other words, f is the largest integer such that m divides f^n and r is the remaining factor.

For large `m` and small `n` `NthRoot` may work quite slowly. Every result `{f,r}` for given `m`, `n` is saved in a lookup table, thus subsequent calls to `NthRoot` with the same values `m`, `n` will be executed quite fast.

##### Example:

    In> NthRoot(12,2)
    Out> {2,3};
    In> NthRoot(81,3)
    Out> {3,3};
    In> NthRoot(3255552,2)
    Out> {144,157};
    In> NthRoot(3255552,3)
    Out> {12,1884};

##### See also:

IntNthRoot , Factors , MathPower .

###

ContFrac -- continued fraction expansion

##### Standard library

##### Calling format:

    ContFrac(x)
    ContFrac(x, depth)

##### Parameters:

`x` -- number or polynomial to expand in continued fractions

`depth` -- integer, maximum required depth of result

##### Description:

This command returns the continued fraction expansion of `x`, which should be either a floating point number or a polynomial. If `depth` is not specified, it defaults to 6. The remainder is denoted by `rest`.

This is especially useful for polynomials, since series expansions that converge slowly will typically converge a lot faster if calculated using a continued fraction expansion.

##### Examples:

    In> PrettyForm(ContFrac(N(Pi)))

                 1
    --------------------------- + 3
               1
    ----------------------- + 7
            1
    ------------------ + 15
          1
    -------------- + 1
       1
    -------- + 292
    rest + 1

    Out> True;
    In> PrettyForm(ContFrac(x^2+x+1, 3))

           x
    ---------------- + 1
             x
    1 - ------------
           x
        -------- + 1
        rest + 1

    Out> True;

##### See also:

ContFracList , NearRational , GuessRational , PAdicExpand , N .

###

ContFracList, ContFracEval -- manipulate continued fractions

##### Standard library

##### Calling format:

    ContFracList(frac)
    ContFracList(frac, depth)
    ContFracEval(list)
    ContFracEval(list, rest)

##### Parameters:

`frac` -- a number to be expanded

`depth` -- desired number of terms

`list` -- a list of coefficients

`rest` -- expression to put at the end of the continued fraction

##### Description:

The function `ContFracList` computes terms of the continued fraction representation of a rational number `frac`. It returns a list of terms of length `depth`. If `depth` is not specified, it returns all terms.

The function `ContFracEval` converts a list of coefficients into a continued fraction expression. The optional parameter `rest` specifies the symbol to put at the end of the expansion. If it is not given, the result is the same as if `rest=0`.

##### Examples:

    In> A:=ContFracList(33/7 + 0.000001)
    Out> {4,1,2,1,1,20409,2,1,13,2,1,4,1,1,3,3,2};
    In> ContFracEval(Take(A, 5))
    Out> 33/7;
    In> ContFracEval(Take(A,3), remainder)
    Out> 1/(1/(remainder+2)+1)+4;

##### See also:

ContFrac , GuessRational .

###

GuessRational, NearRational, BracketRational -- find optimal rational approximations

##### Standard library

##### Calling format:

    GuessRational(x)
    GuessRational(x, digits)
    NearRational(x)
    NearRational(x, digits)
    BracketRational(x, eps)

##### Parameters:

`x` -- a number to be approximated (must be already evaluated to floating-point)

`digits` -- desired number of decimal digits (integer)

`eps` -- desired precision

##### Description:

The functions `GuessRational(x)` and `NearRational(x)` attempt to find "optimal" rational approximations to a given value `x`. The approximations are "optimal" in the sense of having smallest numerators and denominators among all rational numbers close to `x`. This is done by computing a continued fraction representation of `x` and truncating it at a suitably chosen term. Both functions return a rational number which is an approximation of `x`.

Unlike the function `Rationalize()` which converts floating-point numbers to rationals without loss of precision, the functions `GuessRational()` and `NearRational()` are intended to find the best rational that is approximately equal to a given value.

The function `GuessRational()` is useful if you have obtained a floating-point representation of a rational number and you know approximately how many digits its exact representation should contain. This function takes an optional second parameter `digits` which limits the number of decimal digits in the denominator of the resulting rational number. If this parameter is not given, it defaults to half the current precision. This function truncates the continuous fraction expansion when it encounters an unusually large value (see example). This procedure does not always give the "correct" rational number; a rule of thumb is that the floating-point number should have at least as many digits as the combined number of digits in the numerator and the denominator of the correct rational number.

The function `NearRational(x)` is useful if one needs to approximate a given value, i.e. to find an "optimal" rational number that lies in a certain small interval around a certain value `x`. This function takes an optional second parameter `digits` which has slightly different meaning: it specifies the number of digits of precision of the approximation; in other words, the difference between `x` and the resulting rational number should be at most one digit of that precision. The parameter `digits` also defaults to half of the current precision.

The function `BracketRational(x,eps)` can be used to find approximations with a given relative precision from above and from below. This function returns a list of two rational numbers `{r1,r2}` such that r1<x<r2 and Abs(r2-r1)<Abs(x*eps). The argument `x` must be already evaluated to enough precision so that this approximation can be meaningfully found. If the approximation with the desired precision cannot be found, the function returns an empty list.

##### Examples:

Start with a rational number and obtain a floating-point approximation:

    In> x:=N(956/1013)
    Out> 0.9437314906
    In> Rationalize(x)
    Out> 4718657453/5000000000;
    In> V(GuessRational(x))

    GuessRational: using 10 terms of the
      continued fraction
    Out> 956/1013;
    In> ContFracList(x)
    Out> {0,1,16,1,3,2,1,1,1,1,508848,3,1,2,1,2,2};

The first 10 terms of this continued fraction correspond to the correct continued fraction for the original rational number.

    In> NearRational(x)
    Out> 218/231;

This function found a different rational number closeby because the precision was not high enough.

    In> NearRational(x, 10)
    Out> 956/1013;

Find an approximation to Ln(10) good to 8 digits:

    In> BracketRational(N(Ln(10)), 10^(-8))
    Out> {12381/5377,41062/17833};

##### See also:

ContFrac , ContFracList , Rationalize .

###

Decimal -- decimal representation of a rational

##### Standard library

##### Calling format:

    Decimal(frac)

##### Parameters:

`frac` -- a rational number

##### Description:

This function returns the infinite decimal representation of a rational number `frac`. It returns a list, with the first element being the number before the decimal point and the last element the sequence of digits that will repeat forever. All the intermediate list elements are the initial digits before the period sets in.

##### Examples:

    In> Decimal(1/22)
    Out> {0,0,{4,5}};
    In> N(1/22,30)
    Out> 0.045454545454545454545454545454;

##### See also:

N .

###

TruncRadian -- remainder modulo 2*Pi

##### Standard library

##### Calling format:

    TruncRadian(r)

##### Parameters:

`r` -- a number

##### Description:

`TruncRadian` calculates Mod(r,2*Pi), returning a value between 0 and 2*Pi. This function is used in the trigonometry functions, just before doing a numerical calculation using a Taylor series. It greatly speeds up the calculation if the value passed is a large number.

The library uses the formula

TruncRadian(r)=r-Floor(r/(2*Pi))*2*Pi,
where r and 2*Pi are calculated with twice the precision used in the environment to make sure there is no rounding error in the significant digits.

##### Examples:

    In> 2*Pi()
    Out> 6.283185307;
    In> TruncRadian(6.28)
    Out> 6.28;
    In> TruncRadian(6.29)
    Out> 0.0068146929;

##### See also:

Sin , Cos , Tan .

###

Floor -- round a number downwards

##### Standard library

##### Calling format:

    Floor(x)

##### Parameters:

`x` -- a number

##### Description:

This function returns Floor(x), the largest integer smaller than or equal to x.

##### Examples:

    In> Floor(1.1)
    Out> 1;
    In> Floor(-1.1)
    Out> -2;

##### See also:

Ceil , Round .

###

Ceil -- round a number upwards

##### Standard library

##### Calling format:

    Ceil(x)

##### Parameters:

`x` -- a number

##### Description:

This function returns Ceil(x), the smallest integer larger than or equal to x.

##### Examples:

    In> Ceil(1.1)
    Out> 2;
    In> Ceil(-1.1)
    Out> -1;

##### See also:

Floor , Round .

###

Round -- round a number to the nearest integer

##### Standard library

##### Calling format:

    Round(x)

##### Parameters:

`x` -- a number

##### Description:

This function returns the integer closest to x. Half-integers (i.e. numbers of the form n+0.5, with n an integer) are rounded upwards.

##### Examples:

    In> Round(1.49)
    Out> 1;
    In> Round(1.51)
    Out> 2;
    In> Round(-1.49)
    Out> -1;
    In> Round(-1.51)
    Out> -2;

##### See also:

Floor , Ceil .

###

Pslq -- search for integer relations between reals

##### Standard library

##### Calling format:

    Pslq(xlist,precision)

##### Parameters:

`xlist` -- list of numbers

`precision` -- required number of digits precision of calculation

##### Description:

This function is an integer relation detection algorithm. This means that, given the numbers x[i] in the list "xlist", it tries to find integer coefficients a[i] such that a[1]*x[1] + ... + a[n]*x[n]=0. The list of integer coefficients is returned.

The numbers in "xlist" must evaluate to floating point numbers if the `N` operator is applied on them.

##### Example:

    In> Pslq({ 2*Pi+3*Exp(1), Pi, Exp(1) },20)
    Out> {1,-2,-3};

Note: in this example the system detects correctly that 1*(2*Pi+3*e)+(-2)*Pi+(-3)*e=0.

##### See also:

N .

IsPrime, IsSmallPrime , IsComposite , IsCoprime , IsSquareFree , IsPrimePower , NextPrime , IsTwinPrime , IsIrregularPrime , IsCarmichaelNumber , Factors , IsAmicablePair , Factor , Divisors , DivisorsSum , ProperDivisors , ProperDivisorsSum , Moebius , CatalanNumber , FermatNumber , HarmonicNumber , StirlingNumber1 , StirlingNumber2 , DivisorsList , SquareFreeDivisorsList , MoebiusDivisorsList , SumForDivisors , RamanujanSum , Cyclotomic , PAdicExpand , IsQuadraticResidue, LegendreSymbol, JacobiSymbol , GaussianFactors , GaussianNorm , IsGaussianUnit , IsGaussianPrime , GaussianGcd .

# Number theory

This chapter describes functions that are of interest in number theory. These functions typically operate on integers. Some of these functions work quite slowly.

###

IsPrime, IsSmallPrime -- test for a prime number

##### Standard library

##### Calling format:

    IsPrime(n)
    IsSmallPrime(n)

##### Parameters:

`n` -- integer to test

##### Description:

The commands checks whether n, which should be a positive integer, is a prime number. A number n is a prime number if it is only divisible by 1 and itself. As a special case, 1 is not considered a prime number. The first prime numbers are 2, 3, 5, ...

The function `IsShortPrime` only works for numbers n<=65537 but it is very fast.

The function `IsPrime` operates on all numbers and uses different algorithms depending on the magnitude of the number n. For small numbers n<=65537, a constant-time table lookup is performed. (The function `IsShortPrime` is used for that.) For numbers n between 65537 and 34155071728321, the function uses the Rabin-Miller test together with table lookups to guarantee correct results.

For even larger numbers a version of the probabilistic Rabin-Miller test is executed. The test can sometimes mistakenly mark a number as prime while it is in fact composite, but a prime number will never be mistakenly declared composite. The parameters of the test are such that the probability for a false result is less than 10^(-24).

##### Examples:

    In> IsPrime(1)
    Out> False;
    In> IsPrime(2)
    Out> True;
    In> IsPrime(10)
    Out> False;
    In> IsPrime(23)
    Out> True;
    In> Select("IsPrime", 1 .. 100)
    Out> {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,
      53,59,61,67,71,73,79,83,89,97};

##### See also:

IsPrimePower , Factors .

###

IsComposite -- test for a composite number

##### Standard library

##### Calling format:

    IsComposite(n)

##### Parameters:

`n` -- positive integer

##### Description:

This function is the logical negation of `IsPrime`, except for the number 1, which is neither prime nor composite.

##### Examples:

    In> IsComposite(1)
    Out> False;
    In> IsComposite(7)
    Out> False;
    In> IsComposite(8)
    Out> True;
    In> Select(IsComposite,1 .. 20)
    Out> {4,6,8,9,10,12,14,15,16,18,20};

##### See also:

IsPrime .

###

IsCoprime -- test if integers are coprime

##### Standard library

##### Calling format:

    IsCoprime(m,n)
    IsCoprime(list)

##### Parameters:

`m`,`n` -- positive integers

`list` -- list of positive integers

##### Description:

This function returns `True` if the given pair or list of integers are coprime, also called relatively prime. A pair or list of numbers are coprime if they share no common factors.

##### Examples:

    In> IsCoprime({3,4,5,8})
    Out> False;
    In> IsCoprime(15,17)
    Out> True;

##### See also:

Prime .

###

IsSquareFree -- test for a square-free number

##### Standard library

##### Calling format:

    IsSquareFree(n)

##### Parameters:

`n` -- positive integer

##### Description:

This function uses the `Moebius` function to tell if the given number is square-free, which means it has distinct prime factors. If Moebius(n)!=0, then `n` is square free. All prime numbers are trivially square-free.

##### Examples:

    In> IsSquareFree(37)
    Out> True;
    In> IsSquareFree(4)
    Out> False;
    In> IsSquareFree(16)
    Out> False;
    In> IsSquareFree(18)
    Out> False;

##### See also:

Moebius , SquareFreeDivisorsList .

###

IsPrimePower -- test for a power of a prime number

##### Standard library

##### Calling format:

    IsPrimePower(n)

##### Parameters:

`n` -- integer to test

##### Description:

This command tests whether "n", which should be a positive integer, is a prime power, that is whether it is of the form p^m, with "p" prime and "m" an integer.

This function does not try to decompose the number n into factors. Instead we check for all prime numbers r=2, 3, ... that the r-th root of n is an integer, and we find such r and m that n=m^r, we check that m is a prime. If it is not a prime, we execute the same function call on m.

##### Examples:

    In> IsPrimePower(9)
    Out> True;
    In> IsPrimePower(10)
    Out> False;
    In> Select("IsPrimePower", 1 .. 50)
    Out> {2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,
      29,31,32,37,41,43,47,49};

##### See also:

IsPrime , Factors .

###

NextPrime -- generate a prime following a number

##### Standard library

##### Calling format:

    NextPrime(i)

##### Parameters:

`i` -- integer value

##### Description:

The function finds the smallest prime number that is greater than the given integer value.

The routine generates "candidate numbers" using the formula n+2*Mod(-n,3) where n is an odd number (this generates the sequence 5, 7, 11, 13, 17, 19, ...) and `IsPrime()` to test whether the next candidate number is in fact prime.

##### Example:

    In> NextPrime(5)
    Out> 7;

##### See also:

IsPrime .

###

IsTwinPrime -- test for a twin prime

##### Standard library

##### Calling format:

    IsTwinPrime(n)

##### Parameters:

`n` -- positive integer

##### Description:

This function returns `True` if `n` is a twin prime. By definition, a twin prime is a prime number n such that n+2 is also a prime number.

##### Examples:

    In> IsTwinPrime(101)
    Out> True;
    In> IsTwinPrime(7)
    Out> False;
    In> Select(IsTwinPrime, 1 .. 100)
    Out> {3,5,11,17,29,41,59,71};

##### See also:

IsPrime .

###

IsIrregularPrime -- test for an irregular prime

##### Standard library

##### Calling format:

    IsIrregularPrime(n)

##### Parameters:

`n` -- positive integer

##### Description:

This function returns `True` if `n` is an irregular prime. A prime number n is irregular if and only if `n` divides the numerator of a Bernoulli number B[2*i], where 2*i+1<n. Small irregular primes are quite rare; the only irregular primes under 100 are 37, 59 and 67. Asymptotically, roughly 40`%` of primes are irregular.

##### Examples:

    In> IsIrregularPrime(5)
    Out> False;
    In> Select(IsIrregularPrime,1 .. 100)
    Out> {37,59,67};

##### See also:

IsPrime .

###

IsCarmichaelNumber -- test for a Carmichael number

##### Standard library

##### Calling format:

    IsCarmichaelNumber(n)

##### Parameters:

`n` -- positive integer

##### Description:

This function returns `True` if `n` is a Carmichael number, also called an absolute pseudoprime. They have the property that b^(n-1)%n==1 for all b satisfying Gcd(b,n)==1. These numbers cannot be proved composite by Fermat's little theorem. Because the previous property is extremely slow to test, the following equivalent property is tested by Yacas: for all prime factors p[i] of n, (n-1)%(p[i]-1)==0 and n must be square free. Also, Carmichael numbers must be odd and have at least three prime factors. Although these numbers are rare (there are only 43 such numbers between 1 and 10^6), it has recently been proven that there are infinitely many of them.

##### Examples:

    In> IsCarmichaelNumber(561)
    Out> True;
    In> Time(Select(IsCarmichaelNumber,1 .. 10000))
    504.19 seconds taken
    Out> {561,1105,1729,2465,2821,6601,8911};

##### See also:

IsSquareFree , IsComposite .

###

Factors -- factorization

##### Standard library

##### Calling format:

    Factors(x)

##### Parameters:

`x` -- integer or univariate polynomial

##### Description:

This function decomposes the integer number `x` into a product of numbers. Alternatively, if `x` is a univariate polynomial, it is decomposed in irreducible polynomials.

The factorization is returned as a list of pairs. The first member of each pair is the factor, while the second member denotes the power to which this factor should be raised. So the factorization x=p1^n1*...*p9^n9 is returned as `{{p1,n1}, ..., {p9,n9}}`.

##### Examples:

    In> Factors(24);
    Out> {{2,3},{3,1}};
    In> Factors(2*x^3 + 3*x^2 - 1);
    Out> {{2,1},{x+1,2},{x-1/2,1}};

##### See also:

Factor , IsPrime , GaussianFactors .

###

IsAmicablePair -- test for a pair of amicable numbers

##### Standard library

##### Calling format:

    IsAmicablePair(m,n)

##### Parameters:

`m`, `n` -- positive integers

##### Description:

This function tests if a pair of numbers are amicable. A pair of numbers m, n has this property if the sum of the proper divisors of m is n and the sum of the proper divisors of n is m.

##### Examples:

    In> IsAmicablePair(200958394875, 209194708485 )
    Out> True;
    In> IsAmicablePair(220, 284)
    Out> True;

##### See also:

ProperDivisorsSum .

###

Factor -- factorization, in pretty form

##### Standard library

##### Calling format:

    Factor(x)

##### Parameters:

`x` -- integer or univariate polynomial

##### Description:

This function factorizes "x", similarly to `Factors`, but it shows the result in a nicer human readable format.

##### Examples:

    In> PrettyForm(Factor(24));

     3
    2  * 3

    Out> True;
    In> PrettyForm(Factor(2*x^3 + 3*x^2 - 1));

                 2   /     1 \
    2 * ( x + 1 )  * | x - - |
                     \     2 /

    Out> True;

##### See also:

Factors , IsPrime , PrettyForm .

###

Divisors -- number of divisors

##### Standard library

##### Calling format:

    Divisors(n)

##### Parameters:

`n` -- positive integer

##### Description:

`Divisors` returns the number of positive divisors of a number. A number is prime if and only if it has two divisors, 1 and itself.

##### Examples:

    In> Divisors(180)
    Out> 18;
    In> Divisors(37)
    Out> 2;

##### See also:

DivisorsSum , ProperDivisors , ProperDivisorsSum , Moebius .

###

DivisorsSum -- the sum of divisors

##### Standard library

##### Calling format:

    DivisorsSum(n)

##### Parameters:

`n` -- positive integer

##### Description:

`DivisorsSum` returns the sum all numbers that divide it. A number `n` is prime if and only if the sum of its divisors are `n+1`.

##### Examples:

    In> DivisorsSum(180)
    Out> 546;
    In> DivisorsSum(37)
    Out> 38;

##### See also:

DivisorsSum , ProperDivisors , ProperDivisorsSum , Moebius .

###

ProperDivisors -- the number of proper divisors

##### Standard library

##### Calling format:

    ProperDivisors(n)

##### Parameters:

`n` -- positive integer

##### Description:

`ProperDivisors` returns the number of proper divisors, i.e `Divisors(n)-1`, since `n` is not counted. An integer n is prime if and only if it has 1 proper divisor.

##### Examples:

    In> ProperDivisors(180)
    Out> 17;
    In> ProperDivisors(37)
    Out> 1;

##### See also:

DivisorsSum , ProperDivisors , ProperDivisorsSum , Moebius .

###

ProperDivisorsSum -- the sum of proper divisors

##### Standard library

##### Calling format:

    ProperDivisorsSum(n)

##### Parameters:

`n` -- positive integer

##### Description:

`ProperDivisorsSum` returns the sum of proper divisors, i.e. `ProperDivisors(n)-n`, since `n` is not counted. `n` is prime if and only if `ProperDivisorsSum(n)==1`.

##### Examples:

    In> ProperDivisorsSum(180)
    Out> 366;
    In> ProperDivisorsSum(37)
    Out> 1;

##### See also:

DivisorsSum , ProperDivisors , ProperDivisorsSum , Moebius .

###

Moebius -- the Moebius function

##### Standard library

##### Calling format:

    Moebius(n)

##### Parameters:

`n` -- positive integer

##### Description:

The Moebius function is 0 when a prime factor is repeated (which means it is not square-free) and is (-1)^r if n has r distinct factors. Also, Moebius(1)==1.

##### Examples:

    In> Moebius(10)
    Out> 1;
    In> Moebius(11)
    Out> -1;
    In> Moebius(12)
    Out> 0;
    In> Moebius(13)
    Out> -1;

##### See also:

DivisorsSum , ProperDivisors , ProperDivisorsSum , MoebiusDivisorsList .

###

CatalanNumber -- return the `n`th Catalan Number

##### Standard library

##### Calling format:

    CatalanNumber(n)

##### Parameters:

`n` -- positive integer

##### Description:

This function returns the `n`-th Catalan number, defined as Bin(2*n,n)/(n+1).

##### Examples:

    In> CatalanNumber(10)
    Out> 16796;
    In> CatalanNumber(5)
    Out> 42;

##### See also:

Bin .

###

FermatNumber -- return the `n`th Fermat Number

##### Standard library

##### Calling format:

    FermatNumber(n)

##### Parameters:

`n` -- positive integer

##### Description:

This function returns the `n`-th Fermat number, which is defined as 2^2^n+1.

##### Examples:

    In> FermatNumber(7)
    Out> 340282366920938463463374607431768211457;

##### See also:

Factor .

###

HarmonicNumber -- return the `n`th Harmonic Number

##### Standard library

##### Calling format:

    HarmonicNumber(n)
    HarmonicNumber(n,r)

##### Parameters:

`n`, `r` -- positive integers

##### Description:

This function returns the `n`-th Harmonic number, which is defined as Sum(k,1,n,1/k). If given a second argument, the Harmonic number of order r is returned, which is defined as Sum(k,1,n,k^(-r)).

##### Examples:

    In> HarmonicNumber(10)
    Out> 7381/2520;
    In> HarmonicNumber(15)
    Out> 1195757/360360;
    In> HarmonicNumber(1)
    Out> 1;
    In> HarmonicNumber(4,3)
    Out> 2035/1728;

##### See also:

Sum .

###

StirlingNumber1 -- return the `n,m`th Stirling Number of the first kind

##### Standard library

##### Calling format:

    StirlingNumber1(n,m)

##### Parameters:

`n`, `m` -- positive integers

##### Description:

This function returns the signed Stirling Number of the first kind. All Stirling Numbers are integers. If m>n, then `StirlingNumber1` returns 0.

##### Examples:

    In> StirlingNumber1(10,5)
    Out> -269325;
    In> StirlingNumber1(3,6)
    Out> 0;

##### See also:

StirlingNumber2 .

###

StirlingNumber2 -- return the `n,m`th Stirling Number of the second kind

##### Standard library

##### Calling format:

StirlingNumber1(n,m)

##### Parameters:

`n`, `m` -- positive integers

##### Description:

This function returns the Stirling Number of the second kind. All Stirling Numbers are positive integers. If m>n, then `StirlingNumber2` returns 0.

##### Examples:

    In> StirlingNumber2(3,6)
    Out> 0;
    In> StirlingNumber2(10,4)
    Out> 34105;

##### See also:

StirlingNumber1 .

###

DivisorsList -- the list of divisors

##### Standard library

##### Calling format:

    DivisorsList(n)

##### Parameters:

`n` -- positive integer

##### Description:

`DivisorsList` creates a list of the divisors of n. This is useful for loops like

    ForEach(d,DivisorsList(n))

##### Examples:

    In> DivisorsList(18)
    Out> {1,2,3,6,9,18};

##### See also:

DivisorsSum .

###

SquareFreeDivisorsList -- the list of square-free divisors

##### Standard library

##### Calling format:

    SquareFreeDivisorsList(n)

##### Parameters:

`n` -- positive integer

##### Description:

`SquareFreeDivisorsList` creates a list of the square-free divisors of n. Square-free numbers are numbers that have only simple prime factors (no prime powers). For example, 18=2*3*3 is not square-free because it contains a square of 3 as a factor.

##### Examples:

    In> SquareFreeDivisorsList(18)
    Out> {1,2,3,6};

##### See also:

DivisorsList .

###

MoebiusDivisorsList -- the list of divisors and Moebius values

##### Standard library

##### Calling format:

    MoebiusDivisorsList(n)

##### Parameters:

`n` -- positive integer

##### Description:

Returns a list of pairs of the form `{d,m}`, where `d` runs through the squarefree divisors of n and m=Moebius(d). This is more efficient than making a list of all square-free divisors of n and then computing `Moebius` on each of them. It is useful for computing the cyclotomic polynomials. It can be useful in other computations based on the Moebius inversion formula.

##### Examples:

    In> MoebiusDivisorsList(18)
    Out> {{1,1},{2,-1},{3,-1},{6,1}};

##### See also:

DivisorsList , Moebius .

###

SumForDivisors -- loop over divisors

##### Standard library

##### Calling format:

    SumForDivisors(var,n,expr)

##### Parameters:

`var` -- atom, variable name

`n` -- positive integer

`expr` -- expression depending on `var`

##### Description:

This function performs the sum of the values of the expression `expr` while the variable `var` runs through the divisors of `n`. For example, `SumForDivisors(d, 10, d^2)` sums d^2 where d runs through the divisors of 10. This kind of computation is frequently used in number theory.

##### See also:

DivisorsList .

###

RamanujanSum -- compute the "Ramanujan sum"

##### Standard library

##### Calling format:

    RamanujanSum(k,n)

##### Parameters:

`k`, `n` -- positive integers

##### Description:

This function computes the Ramanujan sum, i.e. the sum of the n-th powers of the k-th primitive roots of the unit:

Sum(l,1,k,Exp(2*Pi*I*(l*n)/k))

where l runs thought the integers between 1 and k-1 that are coprime to l.

The computation is done by using the formula in T. M. Apostol, Introduction to Analytic Theory (Springer-Verlag), Theorem 8.6.

###

Cyclotomic -- construct the cyclotomic polynomial

##### Standard library

##### Calling format:

    Cyclotomic(n,x)

##### Parameters:

`n` -- positive integer

`x` -- variable

##### Description:

Returns the cyclotomic polynomial in the variable `x` (which is the minimal polynomial of the n-th primitive roots of the unit, over the field of rational numbers).

For n even, we write n=m*k, where k is a power of 2 and m is odd, and reduce it to the case of even m since

Cyclotomic(n,x)=Cyclotomic(m,-x^(k/2)).

If m=1, n is a power of 2, and Cyclotomic(n,x)=x^k+1.

For n odd, the algorithm is based on the formula

Cyclotomic(n,x):=Prod((x^(n/d)-1)^mu(d)),
where d runs through the divisors of n. In order to compute this in a efficient way, we use the function `MoebiusDivisorsList`. Then we compute in `poly1` the product of x^(n/d)-1 with mu(d)=1 , and in `poly2` the product of these polynomials with mu(d)= -1. Finally we compute the quotient `poly1`/`poly2`.

##### See also:

RamanujanSum .

###

PAdicExpand -- p-adic expansion

##### Standard library

##### Calling format:

    PAdicExpand(n, p)

##### Parameters:

`n` -- number or polynomial to expand

`p` -- base to expand in

##### Description:

This command computes the p-adic expansion of n. In other words, n is expanded in powers of p. The argument n can be either an integer or a univariate polynomial. The base p should be of the same type.

##### Examples:

    In> PrettyForm(PAdicExpand(1234, 10));

                   2     3
    3 * 10 + 2 * 10  + 10  + 4

    Out> True;
    In> PrettyForm(PAdicExpand(x^3, x-1));

                                 2            3
    3 * ( x - 1 ) + 3 * ( x - 1 )  + ( x - 1 )  + 1

    Out> True;

##### See also:

Mod , ContFrac , FromBase , ToBase .

###

IsQuadraticResidue, LegendreSymbol, JacobiSymbol -- functions related to finite groups

##### Standard library

##### Calling format:

    IsQuadraticResidue(m,n)
    LegendreSymbol(m,n)
    JacobiSymbol(m,n)

##### Parameters:

`m`, `n` -- integers, n must be odd and positive

##### Description:

A number m is a "quadratic residue modulo n" if there exists a number k such that k^2:=Mod(m,n).

The Legendre symbol (m/ n) is defined as +1 if m is a quadratic residue modulo n and -1 if it is a non-residue. The Legendre symbol is equal to 0 if m/n is an integer.

The Jacobi symbol [m/n;] is defined as the product of the Legendre symbols of the prime factors f[i] of n=f[1]^p[1]*...*f[s]^p[s],

[m/n;]:=[m/f[1];]^p[1]*...*[m/f[s];]^p[s].
(Here we used the same notation [a/b;] for the Legendre and the Jacobi symbols; this is confusing but seems to be the current practice.) The Jacobi symbol is equal to 0 if m, n are not mutually prime (have a common factor). The Jacobi symbol and the Legendre symbol have values +1, -1 or 0. If n is prime, then the Jacobi symbol is the same as the Legendre symbol.

The Jacobi symbol can be efficiently computed without knowing the full factorization of the number n.

##### Examples:

    In> IsQuadraticResidue(9,13)
    Out> True;
    In> LegendreSymbol(15,23)
    Out> -1;
    In> JacobiSymbol(7,15)
    Out> -1;

##### See also:

Gcd .

###

GaussianFactors -- factorization in Gaussian integers

##### Standard library

##### Calling format:

    GaussianFactors(z)

##### Parameters:

`z` -- Gaussian integer

##### Description:

This function decomposes a Gaussian integer number `z` into a product of Gaussian prime factors. A Gaussian integer is a complex number with integer real and imaginary parts. A Gaussian integer z can be decomposed into Gaussian primes essentially in a unique way (up to Gaussian units and associated prime factors), i.e. one can write z as

z=u*p[1]^n[1]*...*p[s]^n[s],
where u is a Gaussian unit and p[1], p[2], ..., p[s] are Gaussian primes.

The factorization is returned as a list of pairs. The first member of each pair is the factor (a Gaussian integer) and the second member denotes the power to which this factor should be raised. So the factorization is returned as a list, e.g. `{{p1,n1}, {p2,n2}, ...}`.

##### Examples:

    In> GaussianFactors(5)
    Out> {{Complex(2,1),1},{Complex(2,-1),1}};
    In> GaussianFactors(3+I)
    Out> {{Complex(1,1),1},{Complex(2,-1),1}};

##### See also:

Factors , IsGaussianPrime , IsGaussianUnit .

###

GaussianNorm -- norm of a Gaussian integer

##### Standard library

##### Calling format:

    GaussianNorm(z)

##### Parameters:

`z` -- Gaussian integer

##### Description:

This function returns the norm of a Gaussian integer z=a+b*I, defined as a^2+b^2.

##### Examples:

    In> GaussianNorm(2+I)
    Out> 5;

##### See also:

IsGaussianInteger .

###

IsGaussianUnit -- test for a Gaussian unit

##### Standard library

##### Calling format:

    IsGaussianUnit(z)

##### Parameters:

`z` -- a Gaussian integer

##### Description:

This function returns `True` if the argument is a unit in the Gaussian integers and `False` otherwise. A unit in a ring is an element that divides any other element.

There are four "units" in the ring of Gaussian integers, which are 1, -1, I, and -I.

##### Examples:

    In> IsGaussianInteger(I)
    Out> True;
    In> IsGaussianUnit(5+6*I)
    Out> False;

##### See also:

IsGaussianInteger , IsGaussianPrime , GaussianNorm .

###

IsGaussianPrime -- test for a Gaussian prime

##### Standard library

##### Calling format:

    IsGaussianPrime(z)

##### Parameters:

`z` -- a complex or real number

##### Description:

This function returns `True` if the argument is a Gaussian prime and `False` otherwise.

A prime element x of a ring is divisible only by the units of the ring and by associates of x. ("Associates" of x are elements of the form x*u where u is a unit of the ring).

Gaussian primes are Gaussian integers z=a+b*I that satisfy one of the following properties:

- If Re(z) and Im(z) are nonzero then z is a Gaussian prime if and only if Re(z)^2+Im(z)^2 is an ordinary prime.
- If Re(z)==0 then z is a Gaussian prime if and only if Im(z) is an ordinary prime and Im(z):=Mod(3,4).
- If Im(z)==0 then z is a Gaussian prime if and only if Re(z) is an ordinary prime and Re(z):=Mod(3,4).

##### Examples:

    In> IsGaussianPrime(13)
    Out> False;
    In> IsGaussianPrime(2+2*I)
    Out> False;
    In> IsGaussianPrime(2+3*I)
    Out> True;
    In> IsGaussianPrime(3)
    Out> True;

##### See also:

IsGaussianInteger , GaussianFactors .

###

GaussianGcd -- greatest common divisor in Gaussian integers

##### Standard library

##### Calling format:

    GaussianGcd(z,w)

##### Parameters:

`z`, `w` -- Gaussian integers

##### Description:

This function returns the greatest common divisor, in the ring of Gaussian integers, computed using Euclid's algorithm. Note that in the Gaussian integers, the greatest common divisor is only defined up to a Gaussian unit factor.

##### Examples:

    In> GaussianGcd(2+I,5)
    Out> Complex(2,1);

The GCD of two mutually prime Gaussian integers might come out to be equal to some Gaussian unit instead of 1:

    In> GaussianGcd(2+I,3+I)
    Out> -1;

##### See also:

Gcd , Lcm , IsGaussianUnit .

Sin, Cos, Tan , ArcSin, ArcCos, ArcTan , Exp , Ln , Sqrt , Abs , Sign , Complex , Re , Im , I , Conjugate , Arg , RootsOfUnity , !, !!, ***, Subfactorial , Bin , Eulerian , Add , Sum , Average , Factorize , Min , Max , IsZero , IsRational , Numer , Denom , Commutator , Taylor , InverseTaylor , ReversePoly , BigOh , Newton , D , Curl , Diverge , Integrate , Simplify , RadSimp , FactorialSimplify , LnExpand , LnCombine , Rationalize , Solve , OldSolve , SuchThat , Eliminate , PSolve , Pi , Random, RandomSeed , RngCreate, RngSeed, Rng , Limit , TrigSimpCombine , LagrangeInterpolant , Fibonacci .

# Calculus and elementary functions

In this chapter, some facilities for doing calculus are described. These include functions implementing differentiation, integration, standard mathematical functions, and solving of equations.

###

Sin, Cos, Tan -- trigonometric functions

##### Standard library

##### Calling format:

    Sin(x)
    Cos(x)
    Tan(x)

##### Parameters:

`x` -- argument to the function, in radians

##### Description:

These functions represent the trigonometric functions sine, cosine, and tangent respectively. Yacas leaves them alone even if x is a number, trying to keep the result as exact as possible. The floating point approximations of these functions can be forced by using the `N` function.

Yacas knows some trigonometric identities, so it can simplify to exact results even if `N` is not used. This is the case, for instance, when the argument is a multiple of Pi/6 or Pi/4.

These functions are threaded, meaning that if the argument `x` is a list, the function is applied to all entries in the list.

##### Examples:

    In> Sin(1)
    Out> Sin(1);
    In> N(Sin(1),20)
    Out> 0.84147098480789650665;
    In> Sin(Pi/4)
    Out> Sqrt(2)/2;

##### See also:

ArcSin , ArcCos , ArcTan , N , Pi .

###

ArcSin, ArcCos, ArcTan -- inverse trigonometric functions

##### Standard library

##### Calling format:

    ArcSin(x)
    ArcCos(x)
    ArcTan(x)

##### Parameters:

`x` -- argument to the function

##### Description:

These functions represent the inverse trigonometric functions. For instance, the value of ArcSin(x) is the number y such that Sin(y) equals x.

Note that the number y is not unique. For instance, Sin(0) and Sin(Pi) both equal 0, so what should ArcSin(0) be? In Yacas, it is agreed that the value of ArcSin(x) should be in the interval [-Pi/2, Pi/2]. The same goes for ArcTan(x). However, ArcCos(x) is in the interval [0,Pi].

Usually, Yacas leaves these functions alone unless it is forced to do a numerical evaluation by the `N` function. If the argument is -1, 0, or 1 however, Yacas will simplify the expression. If the argument is complex, the expression will be rewritten using the `Ln` function.

These functions are threaded, meaning that if the argument `x` is a list, the function is applied to all entries in the list.

##### Examples:

    In> ArcSin(1)
    Out> Pi/2;

    In> ArcSin(1/3)
    Out> ArcSin(1/3);
    In> Sin(ArcSin(1/3))
    Out> 1/3;

    In> x:=N(ArcSin(0.75))
    Out> 0.848062;
    In> N(Sin(x))
    Out> 0.7499999477;

##### See also:

Sin , Cos , Tan , N , Pi , Ln .

###

Exp -- exponential function

##### Standard library

##### Calling format:

    Exp(x)

##### Parameters:

`x` -- argument to the function

##### Description:

This function calculates e raised to the power x, where e is the mathematic constant 2.71828... One can use `Exp(1)` to represent e.

This function is threaded, meaning that if the argument `x` is a list, the function is applied to all entries in the list.

##### Examples:

    In> Exp(0)
    Out> 1;
    In> Exp(I*Pi)
    Out> -1;
    In> N(Exp(1))
    Out> 2.7182818284;

##### See also:

Ln , Sin , Cos , Tan , N .

###

Ln -- natural logarithm

##### Standard library

##### Calling format:

    Ln(x)

##### Parameters:

`x` -- argument to the function

##### Description:

This function calculates the natural logarithm of "x". This is the inverse function of the exponential function, `Exp`, i.e. Ln(x)=y implies that Exp(y)=x. For complex arguments, the imaginary part of the logarithm is in the interval (- Pi, Pi]. This is compatible with the branch cut of `Arg`.

This function is threaded, meaning that if the argument `x` is a list, the function is applied to all entries in the list.

##### Examples:

    In> Ln(1)
    Out> 0;
    In> Ln(Exp(x))
    Out> x;
    In> D(x) Ln(x)
    Out> 1/x;

##### See also:

Exp , Arg .

###

Sqrt -- square root

##### Standard library

##### Calling format:

    Sqrt(x)

##### Parameters:

`x` -- argument to the function

##### Description:

This function calculates the square root of "x". If the result is not rational, the call is returned unevaluated unless a numerical approximation is forced with the `N` function. This function can also handle negative and complex arguments.

This function is threaded, meaning that if the argument `x` is a list, the function is applied to all entries in the list.

##### Examples:

    In> Sqrt(16)
    Out> 4;
    In> Sqrt(15)
    Out> Sqrt(15);
    In> N(Sqrt(15))
    Out> 3.8729833462;
    In> Sqrt(4/9)
    Out> 2/3;
    In> Sqrt(-1)
    Out> Complex(0,1);

##### See also:

Exp , ^ , N .

###

Abs -- absolute value or modulus

##### Standard library

##### Calling format:

    Abs(x)

##### Parameters:

`x` -- argument to the function

##### Description:

This function returns the absolute value (also called the modulus) of "x". If "x" is positive, the absolute value is "x" itself; if "x" is negative, the absolute value is "-x". For complex "x", the modulus is the "r" in the polar decomposition x=r*Exp(I*phi).

This function is connected to the `Sign` function by the identity "Abs(x) * Sign(x) = x" for real "x".

This function is threaded, meaning that if the argument `x` is a list, the function is applied to all entries in the list.

##### Examples:

    In> Abs(2);
    Out> 2;
    In> Abs(-1/2);
    Out> -1/2;
    In> Abs(3+4*I);
    Out> 5;

##### See also:

Sign , Arg .

###

Sign -- sign of a number

##### Standard library

##### Calling format:

    Sign(x)

##### Parameters:

`x` -- argument to the function

##### Description:

This function returns the sign of the real number x. It is "1" for positive numbers and "-1" for negative numbers. Somewhat arbitrarily, `Sign(0)` is defined to be 1.

This function is connected to the `Abs` function by the identity Abs(x)*Sign(x)=x for real x.

This function is threaded, meaning that if the argument `x` is a list, the function is applied to all entries in the list.

##### Examples:

    In> Sign(2)
    Out> 1;
    In> Sign(-3)
    Out> -1;
    In> Sign(0)
    Out> 1;
    In> Sign(-3) * Abs(-3)
    Out> -3;

##### See also:

Arg , Abs .

###

Complex -- construct a complex number

##### Standard library

##### Calling format:

    Complex(r, c)

##### Parameters:

`r` -- real part

`c` -- imaginary part

##### Description:

This function represents the complex number "r + I*c", where "I" is the imaginary unit. It is the standard representation used in Yacas to represent complex numbers. Both "r" and "c" are supposed to be real.

Note that, at the moment, many functions in Yacas assume that all numbers are real unless it is obvious that it is a complex number. Hence `Im(Sqrt(x))` evaluates to `0` which is only true for nonnegative "x".

##### Examples:

    In> I
    Out> Complex(0,1);
    In> 3+4*I
    Out> Complex(3,4);
    In> Complex(-2,0)
    Out> -2;

##### See also:

Re , Im , I , Abs , Arg .

###

Re -- real part of a complex number

##### Standard library

##### Calling format:

    Re(x)

##### Parameters:

`x` -- argument to the function

##### Description:

This function returns the real part of the complex number "x".

##### Examples:

    In> Re(5)
    Out> 5;
    In> Re(I)
    Out> 0;
    In> Re(Complex(3,4))
    Out> 3;

##### See also:

Complex , Im .

###

Im -- imaginary part of a complex number

##### Standard library

##### Calling format:

    Im(x)

##### Parameters:

`x` -- argument to the function

##### Description:

This function returns the imaginary part of the complex number "x".

##### Examples:

    In> Im(5)
    Out> 0;
    In> Im(I)
    Out> 1;
    In> Im(Complex(3,4))
    Out> 4;

##### See also:

Complex , Re .

###

I -- imaginary unit

##### Standard library

##### Calling format:

    I

##### Description:

This symbol represents the imaginary unit, which equals the square root of -1. It evaluates to `Complex(0,1)`.

##### Examples:

    In> I
    Out> Complex(0,1);
    In> I = Sqrt(-1)
    Out> True;

##### See also:

Complex .

###

Conjugate -- complex conjugate

##### Standard library

##### Calling format:

    Conjugate(x)

##### Parameters:

`x` -- argument to the function

##### Description:

This function returns the complex conjugate of "x". The complex conjugate of a+I*b is a-I*b. This function assumes that all unbound variables are real.

##### Examples:

    In> Conjugate(2)
    Out> 2;
    In> Conjugate(Complex(a,b))
    Out> Complex(a,-b);

##### See also:

Complex , Re , Im .

###

Arg -- argument of a complex number

##### Standard library

##### Calling format:

    Arg(x)

##### Parameters:

`x` -- argument to the function

##### Description:

This function returns the argument of "x". The argument is the angle with the positive real axis in the Argand diagram, or the angle "phi" in the polar representation r*Exp(I*phi) of "x". The result is in the range (-Pi, Pi], that is, excluding -Pi but including Pi. The argument of 0 is `Undefined`.

##### Examples:

    In> Arg(2)
    Out> 0;
    In> Arg(-1)
    Out> Pi;
    In> Arg(1+I)
    Out> Pi/4;

##### See also:

Abs , Sign .

###

RootsOfUnity -- find the n complex roots of unity

##### Standard library

##### Calling format:

    RootsOfUnity(n)

##### Parameters:

`n` -- positive integer

##### Description:

This function returns the list of n roots of unity. The first element of this list is the primitive n-th root of unity, and the last element is simply `1`.

##### Examples:

    In> RootsOfUnity(3)
    Out> {Complex(-1/2,Sqrt(3/4)),Complex(-1/2,-Sqrt(3/4)),1};
    In> RootsOfUnity(1)
    Out> {1};
    In> RootsOfUnity(2)
    Out> {-1,1};

##### See also:

I , Complex .

###

!, !!, ***, Subfactorial -- factorial and related functions

##### Standard library

##### Calling format:

    n!
    n!!
    a *** b
    Subfactorial(m)

##### Parameters:

`m` -- integer `n` -- integer, half-integer, or list `a`, `b` -- numbers

##### Description:

The factorial function `n!` calculates the factorial of integer or half-integer numbers. For nonnegative integers, n! :=n*(n-1)*(n-2)*...*1. The factorial of half-integers is defined via Euler's Gamma function, z! :=Gamma(z+1). If n=0 the function returns 1.

The "double factorial" function `n!!` calculates n*(n-2)*(n-4)*.... This product terminates either with 1 or with 2 depending on whether n is odd or even. If n=0 the function returns 1.

The "partial factorial" function `a *** b` calculates the product a*(a+1)*... which is terminated at the least integer not greater than b. The arguments a and b do not have to be integers; for integer arguments, `a *** b` = b! /(a-1)!. This function is sometimes a lot faster than evaluating the two factorials, especially if a and b are close together. If a>b the function returns 1.

The `Subfactorial` function can be interpreted as the number of permutations of `m` objects in which no object appears in its natural place, also called "derangements."

The factorial functions are threaded, meaning that if the argument `n` is a list, the function will be applied to each element of the list.

Note: For reasons of Yacas syntax, the factorial sign `!` cannot precede other non-letter symbols such as `+` or `*`. Therefore, you should enter a space after `!` in expressions such as `x! +1`.

The factorial functions terminate and print an error message if the arguments are too large (currently the limit is n<65535) because exact factorials of such large numbers are computationally expensive and most probably not useful. One can call `LnGammaNum()` to evaluate logarithms of such factorials to desired precision.

##### Examples:

    In> 5!
    Out> 120;
    In> 1 * 2 * 3 * 4 * 5
    Out> 120;
    In> (1/2)!
    Out> Sqrt(Pi)/2;
    In> 7!!;
    Out> 105;
    In> 1/3 *** 10;
    Out> 17041024000/59049;
    In> Subfactorial(10)
    Out> 1334961;

##### See also:

Bin , Factorize , Gamma , !! , *** , Subfactorial .

###

Bin -- binomial coefficients

##### Standard library

##### Calling format:

    Bin(n, m)

##### Parameters:

`n`, `m` -- integers

##### Description:

This function calculates the binomial coefficient "n" above "m", which equals

n! /(m! *(n-m)!)

This is equal to the number of ways to choose "m" objects out of a total of "n" objects if order is not taken into account. The binomial coefficient is defined to be zero if "m" is negative or greater than "n"; `Bin(0,0)`=1.

##### Examples:

    In> Bin(10, 4)
    Out> 210;
    In> 10! / (4! * 6!)
    Out> 210;

##### See also:

! , Eulerian .

###

Eulerian -- Eulerian numbers

##### Standard library

##### Calling format:

    Eulerian(n,m)

##### Parameters:

`n`, `m` --- integers

##### Description:

The Eulerian numbers can be viewed as a generalization of the binomial coefficients, and are given explicitly by

Sum(j,0,k+1,(-1)^j*Bin(n+1,j)*(k-j+1)^n)
.

##### Examples:

    In> Eulerian(6,2)
    Out> 302;
    In> Eulerian(10,9)
    Out> 1;

##### See also:

Bin .

###

Add -- find sum of a list of values

##### Standard library

##### Calling format:

    Add(val1, val2, ...)
    Add({list})

##### Parameters:

`val1`, `val2` -- expressions

`{list}` -- list of expressions to add

##### Description:

This function adds all its arguments and returns their sum. It accepts any number of arguments. The arguments can be also passed as a list.

##### Examples:

    In> Add(1,4,9);
    Out> 14;
    In> Add(1 .. 10);
    Out> 55;

##### See also:

Average .

###

Sum -- find sum of a sequence

##### Standard library

##### Calling format:

    Sum(var, from, to, body)

##### Parameters:

`var` -- variable to iterate over

`from` -- integer value to iterate from

`to` -- integer value to iterate up to

`body` -- expression to evaluate for each iteration

##### Description:

The command finds the sum of the sequence generated by an iterative formula. The expression "body" is evaluated while the variable "var" ranges over all integers from "from" up to "to", and the sum of all the results is returned. Obviously, "to" should be greater than or equal to "from".

Warning: `Sum` does not evaluate its arguments `var` and `body` until the actual loop is run.

##### Examples:

    In> Sum(i, 1, 3, i^2);
    Out> 14;

##### See also:

Factorize .

###

Average -- average of a list of values

##### Standard library

##### Calling format:

    Average(list)

##### Parameters:

`list` -- list of values to average

##### Description:

This command calculates the (arithmetical) average of all the entries in "list", which is the sum of all entries divided by the number of entries.

##### Examples:

    In> Average({1,2,3,4,5});
    Out> 3;
    In> Average({2,6,7});
    Out> 5;

##### See also:

Add .

###

Factorize -- product of a list of values

##### Standard library

##### Calling format:

    Factorize(list)
    Factorize(var, from, to, body)

##### Parameters:

`list` -- list of values to multiply

`var` -- variable to iterate over

`from` -- integer value to iterate from

`to` -- integer value to iterate up to

`body` -- expression to evaluate for each iteration

##### Description:

The first form of the `Factorize` command simply multiplies all the entries in "list" and returns their product.

If the second calling sequence is used, the expression "body" is evaluated while the variable "var" ranges over all integers from "from" up to "to", and the product of all the results is returned. Obviously, "to" should be greater than or equal to "from".

##### Examples:

    In> Factorize({1,2,3,4});
    Out> 24;
    In> Factorize(i, 1, 4, i);
    Out> 24;

##### See also:

Sum , Apply .

###

Min -- minimum of a number of values

##### Standard library

##### Calling format:

    Min(x,y)
    Min(list)

##### Parameters:

`x`, `y` -- pair of values to determine the minimum of

`list` -- list of values from which the minimum is sought

##### Description:

This function returns the minimum value of its argument(s). If the first calling sequence is used, the smaller of "x" and "y" is returned. If one uses the second form, the smallest of the entries in "list" is returned. In both cases, this function can only be used with numerical values and not with symbolic arguments.

##### Examples:

    In> Min(2,3);
    Out> 2;
    In> Min({5,8,4});
    Out> 4;

##### See also:

Max , Sum , Average .

###

Max -- maximum of a number of values

##### Standard library

##### Calling format:

    Max(x,y)
    Max(list)

##### Parameters:

`x`, `y` -- pair of values to determine the maximum of

`list` -- list of values from which the maximum is sought

##### Description:

This function returns the maximum value of its argument(s). If the first calling sequence is used, the larger of "x" and "y" is returned. If one uses the second form, the largest of the entries in "list" is returned. In both cases, this function can only be used with numerical values and not with symbolic arguments.

##### Examples:

    In> Max(2,3);
    Out> 3;
    In> Max({5,8,4});
    Out> 8;

##### See also:

Min , Sum , Average .

###

IsZero -- test whether argument is zero

##### Standard library

##### Calling format:

    IsZero(n)

##### Parameters:

`n` -- number to test

##### Description:

`IsZero(n)` evaluates to `True` if "n" is zero. In case "n" is not a number, the function returns `False`.

##### Examples:

    In> IsZero(3.25)
    Out> False;
    In> IsZero(0)
    Out> True;
    In> IsZero(x)
    Out> False;

##### See also:

IsNumber , IsNotZero .

###

IsRational -- test whether argument is a rational

##### Standard library

##### Calling format:

    IsRational(expr)

##### Parameters:

`expr` -- expression to test

##### Description:

This commands tests whether the expression "expr" is a rational number, i.e. an integer or a fraction of integers.

##### Examples:

    In> IsRational(5)
    Out> False;
    In> IsRational(2/7)
    Out> True;
    In> IsRational(0.5)
    Out> False;
    In> IsRational(a/b)
    Out> False;
    In> IsRational(x + 1/x)
    Out> False;

##### See also:

Numer , Denom .

###

Numer -- numerator of an expression

##### Standard library

##### Calling format:

    Numer(expr)

##### Parameters:

`expr` -- expression to determine numerator of

##### Description:

This function determines the numerator of the rational expression "expr" and returns it. As a special case, if its argument is numeric but not rational, it returns this number. If "expr" is neither rational nor numeric, the function returns unevaluated.

##### Examples:

    In> Numer(2/7)
    Out> 2;
    In> Numer(a / x^2)
    Out> a;
    In> Numer(5)
    Out> 5;

##### See also:

Denom , IsRational , IsNumber .

###

Denom -- denominator of an expression

##### Standard library

##### Calling format:

    Denom(expr)

##### Parameters:

`expr` -- expression to determine denominator of

##### Description:

This function determines the denominator of the rational expression "expr" and returns it. As a special case, if its argument is numeric but not rational, it returns `1`. If "expr" is neither rational nor numeric, the function returns unevaluated.

##### Examples:

    In> Denom(2/7)
    Out> 7;
    In> Denom(a / x^2)
    Out> x^2;
    In> Denom(5)
    Out> 1;

##### See also:

Numer , IsRational , IsNumber .

###

Commutator -- commutator of two objects

##### Standard library

##### Calling format:

    Commutator(a, b)

##### Parameters:

`a`, `b` -- two objects whose commutator should be computed

##### Description:

This command computes the commutator of 'a" and "b", i.e. the expression "a b - b a". For numbers and other objects for which multiplication is commutative, the commutator is zero. But this is not necessarily the case for matrices.

##### Examples:

    In> Commutator(2,3)
    Out> 0;
    In> PrettyPrinter("PrettyForm");

    True

    In> A := { {0,x}, {0,0} }

    /              \
    | ( 0 ) ( x )  |
    |              |
    | ( 0 ) ( 0 )  |
    \              /

    In> B := { {y,0}, {0,z} }

    /              \
    | ( y ) ( 0 )  |
    |              |
    | ( 0 ) ( z )  |
    \              /

    In> Commutator(A,B)

    /                          \
    | ( 0 ) ( x * z - y * x )  |
    |                          |
    | ( 0 ) ( 0 )              |
    \                          /

###

Taylor -- univariate Taylor series expansion

##### Standard library

##### Calling format:

    Taylor(var, at, order) expr

##### Parameters:

`var` -- variable

`at` -- point to get Taylor series around

`order` -- order of approximation

`expr` -- expression to get Taylor series for

##### Description:

This function returns the Taylor series expansion of the expression "expr" with respect to the variable "var" around "at" up to order "order". This is a polynomial which agrees with "expr" at the point "var = at", and furthermore the first "order" derivatives of the polynomial at this point agree with "expr". Taylor expansions around removable singularities are correctly handled by taking the limit as "var" approaches "at".

##### Examples:

    In> PrettyForm(Taylor(x,0,9) Sin(x))

         3    5      7       9
        x    x      x       x
    x - -- + --- - ---- + ------
        6    120   5040   362880

    Out> True;

##### See also:

D , InverseTaylor , ReversePoly , BigOh .

###

InverseTaylor -- Taylor expansion of inverse

##### Standard library

##### Calling format:

    InverseTaylor(var, at, order) expr

##### Parameters:

`var` -- variable

`at` -- point to get inverse Taylor series around

`order` -- order of approximation

`expr` -- expression to get inverse Taylor series for

##### Description:

This function builds the Taylor series expansion of the inverse of the expression "expr" with respect to the variable "var" around "at" up to order "order". It uses the function `ReversePoly` to perform the task.

##### Examples:

    In> PrettyPrinter("PrettyForm")

    True

    In> exp1 := Taylor(x,0,7) Sin(x)

         3    5      7
        x    x      x
    x - -- + --- - ----
        6    120   5040

    In> exp2 := InverseTaylor(x,0,7) ArcSin(x)

     5      7     3
    x      x     x
    --- - ---- - -- + x
    120   5040   6

    In> Simplify(exp1-exp2)

    0

##### See also:

ReversePoly , Taylor , BigOh .

###

ReversePoly -- solve h(f(x))=g(x)+O(x^n) for h

##### Standard library

##### Calling format:

    ReversePoly(f, g, var, newvar, degree)

##### Parameters:

`f`, `g` -- functions of "var"

`var` -- a variable

`newvar` -- a new variable to express the result in

`degree` -- the degree of the required solution

##### Description:

This function returns a polynomial in "newvar", say "h(newvar)", with the property that "h(f(var))" equals "g(var)" up to order "degree". The degree of the result will be at most "degree-1". The only requirement is that the first derivative of "f" should not be zero.

This function is used to determine the Taylor series expansion of the inverse of a function "f": if we take "g(var)=var", then "h(f(var))=var" (up to order "degree"), so "h" will be the inverse of "f".

##### Examples:

    In> f(x):=Eval(Expand((1+x)^4))
    Out> True;
    In> g(x) := x^2
    Out> True;
    In> h(y):=Eval(ReversePoly(f(x),g(x),x,y,8))
    Out> True;
    In> BigOh(h(f(x)),x,8)
    Out> x^2;
    In> h(x)
    Out> (-2695*(x-1)^7)/131072+(791*(x-1)^6)
    /32768 +(-119*(x-1)^5)/4096+(37*(x-1)^4)
    /1024+(-3*(x-1)^3)/64+(x-1)^2/16;

##### See also:

InverseTaylor , Taylor , BigOh .

###

BigOh -- drop all terms of a certain order in a polynomial

##### Standard library

##### Calling format:

    BigOh(poly, var, degree)

##### Parameters:

`poly` -- a univariate polynomial

`var` -- a free variable

`degree` -- positive integer

##### Description:

This function drops all terms of order "degree" or higher in "poly", which is a polynomial in the variable "var".

##### Examples:

    In> BigOh(1+x+x^2+x^3,x,2)
    Out> x+1;

##### See also:

Taylor , InverseTaylor .

###

Newton -- solve an equation numerically with Newton's method

##### Standard library

##### Calling format:

    Newton(expr, var, initial, accuracy)
    Newton(expr, var, initial, accuracy,min,max)

##### Parameters:

`expr` -- an expression to find a zero for

`var` -- free variable to adjust to find a zero

`initial` -- initial value for "var" to use in the search

`accuracy` -- minimum required accuracy of the result

`min` -- minimum value for "var" to use in the search

`max` -- maximum value for "var" to use in the search

##### Description:

This function tries to numerically find a zero of the expression `expr`, which should depend only on the variable `var`. It uses the value `initial` as an initial guess.

The function will iterate using Newton's method until it estimates that it has come within a distance `accuracy` of the correct solution, and then it will return its best guess. In particular, it may loop forever if the algorithm does not converge.

When `min` and `max` are supplied, the Newton iteration takes them into account by returning `Fail` if it failed to find a root in the given range. Note this doesn't mean there isn't a root, just that this algorithm failed to find it due to the trial values going outside of the bounds.

##### Examples:

    In> Newton(Sin(x),x,3,0.0001)
    Out> 3.1415926535;
    In> Newton(x^2-1,x,2,0.0001,-5,5)
    Out> 1;
    In> Newton(x^2+1,x,2,0.0001,-5,5)
    Out> Fail;

##### See also:

Solve , NewtonNum .

###

D -- differentiation

##### Standard library

##### Calling format:

    D(var) expr
    D(list) expr
    D(var,n) expr

##### Parameters:

`var` -- variable

`list` -- a list of variables

`expr` -- expression to take derivatives of

`n` -- order of derivative

##### Description:

This function calculates the derivative of the expression `expr` with respect to the variable `var` and returns it. If the third calling format is used, the `n`-th derivative is determined. Yacas knows how the differentiate standard functions such as `Ln` and `Sin`.

The `D` operator is threaded in both `var` and `expr`. This means that if either of them is a list, the function is applied to each entry in the list. The results are collected in another list which is returned. If both `var` and `expr` are a list, their lengths should be equal. In this case, the first entry in the list `expr` is differentiated with respect to the first entry in the list `var`, the second entry in `expr` is differentiated with respect to the second entry in `var`, and so on.

The `D` operator returns the original function if n=0, a common mathematical idiom that simplifies many formulae.

##### Examples:

    In> D(x)Sin(x*y)
    Out> y*Cos(x*y);
    In> D({x,y,z})Sin(x*y)
    Out> {y*Cos(x*y),x*Cos(x*y),0};
    In> D(x,2)Sin(x*y)
    Out> -Sin(x*y)*y^2;
    In> D(x){Sin(x),Cos(x)}
    Out> {Cos(x),-Sin(x)};

##### See also:

Integrate , Taylor , Diverge , Curl .

###

Curl -- curl of a vector field

##### Standard library

##### Calling format:

    Curl(vector, basis)

##### Parameters:

`vector` -- vector field to take the curl of

`basis` -- list of variables forming the basis

##### Description:

This function takes the curl of the vector field "vector" with respect to the variables "basis". The curl is defined in the usual way,

    Curl(f,x) = {
        D(x[2]) f[3] - D(x[3]) f[2],
        D(x[3]) f[1] - D(x[1]) f[3],
        D(x[1]) f[2] - D(x[2]) f[1]
    }

Both "vector" and "basis" should be lists of length 3.

##### Example:

    In> Curl({x*y,x*y,x*y},{x,y,z})
    Out> {x,-y,y-x};

##### See also:

D , Diverge .

###

Diverge -- divergence of a vector field

##### Standard library

##### Calling format:

    Diverge(vector, basis)

##### Parameters:

`vector` -- vector field to calculate the divergence of

`basis` -- list of variables forming the basis

##### Description:

This function calculates the divergence of the vector field "vector" with respect to the variables "basis". The divergence is defined as

    Diverge(f,x) = D(x[1]) f[1] + ...
        + D(x[n]) f[n],

where `n` is the length of the lists "vector" and "basis". These lists should have equal length.

##### Example:

    In> Diverge({x*y,x*y,x*y},{x,y,z})
    Out> y+x;

##### See also:

D , Curl .

###

Integrate -- integration

##### Standard library

##### Calling format:

    Integrate(var, x1, x2) expr
    Integrate(var) expr

##### Parameters:

`var` -- atom, variable to integrate over

`x1` -- first point of definite integration

`x2` -- second point of definite integration

`expr` -- expression to integrate

##### Description:

This function integrates the expression `expr` with respect to the variable `var`. The first calling format is used to perform definite integration: the integration is carried out from var=x1 to var=x2". The second form is for indefinite integration.

Some simple integration rules have currently been implemented. Polynomials, some quotients of polynomials, trigonometric functions and their inverses, hyperbolic functions and their inverses, `Exp`, and `Ln`, and products of these functions with polynomials can be integrated.

##### Examples:

    In> Integrate(x,a,b) Cos(x)
    Out> Sin(b)-Sin(a);
    In> Integrate(x) Cos(x)
    Out> Sin(x);

##### See also:

D , UniqueConstant .

###

Simplify -- try to simplify an expression

##### Standard library

##### Calling format:

    Simplify(expr)

##### Parameters:

`expr` -- expression to simplify

##### Description:

This function tries to simplify the expression `expr` as much as possible. It does this by grouping powers within terms, and then grouping similar terms.

##### Examples:

    In> a*b*a^2/b-a^3
    Out> (b*a^3)/b-a^3;
    In> Simplify(a*b*a^2/b-a^3)
    Out> 0;

##### See also:

TrigSimpCombine , RadSimp .

###

RadSimp -- simplify expression with nested radicals

##### Standard library

##### Calling format:

    RadSimp(expr)

##### Parameters:

`expr` -- an expression containing nested radicals

##### Description:

This function tries to write the expression "expr" as a sum of roots of integers: Sqrt(e1)+Sqrt(e2)+..., where e1, e2 and so on are natural numbers. The expression "expr" may not contain free variables.

It does this by trying all possible combinations for e1, e2, ... Every possibility is numerically evaluated using `N` and compared with the numerical evaluation of "expr". If the approximations are equal (up to a certain margin), this possibility is returned. Otherwise, the expression is returned unevaluated.

Note that due to the use of numerical approximations, there is a small chance that the expression returned by `RadSimp` is close but not equal to `expr`. The last example underneath illustrates this problem. Furthermore, if the numerical value of `expr` is large, the number of possibilities becomes exorbitantly big so the evaluation may take very long.

##### Examples:

    In> RadSimp(Sqrt(9+4*Sqrt(2)))
    Out> Sqrt(8)+1;
    In> RadSimp(Sqrt(5+2*Sqrt(6)) \
      +Sqrt(5-2*Sqrt(6)))
    Out> Sqrt(12);
    In> RadSimp(Sqrt(14+3*Sqrt(3+2
    *Sqrt(5-12*Sqrt(3-2*Sqrt(2))))))
    Out> Sqrt(2)+3;

But this command may yield incorrect results:

    In> RadSimp(Sqrt(1+10^(-6)))
    Out> 1;

##### See also:

Simplify , N .

###

FactorialSimplify -- Simplify hypergeometric expressions containing factorials

##### Standard library

##### Calling format:

    FactorialSimplify(expression)

##### Parameters:

`expression` -- expression to simplify

##### Description:

`FactorialSimplify` takes an expression that may contain factorials, and tries to simplify it. An expression like (n+1)! /n! would simplify to n+1.

The following steps are taken to simplify:

- binomials are expanded into factorials
- the expression is flattened as much as possible, to reduce it to a sum of simple rational terms
- expressions like p^n/p^m are reduced to p^(n-m) if n-m is an integer
- expressions like n! /m! are simplified if n-m is an integer

The function `Simplify` is used to determine if the relevant expressions n-m are integers.

##### Example:

    In> FactorialSimplify( (n-k+1)! / (n-k)! )
    Out> n+1-k
    In> FactorialSimplify(n! / Bin(n,k))
    Out> k! *(n-k)!
    In> FactorialSimplify(2^(n+1)/2^n)
    Out> 2

##### See also:

Simplify , ! , Bin .

###

LnExpand -- expand a logarithmic expression using standard logarithm rules

##### Standard library

##### Calling format:

    LnExpand(expr)

##### Parameters:

`expr` -- the logarithm of an expression

##### Description:

`LnExpand` takes an expression of the form Ln(expr), and applies logarithm rules to expand this into multiple `Ln` expressions where possible. An expression like Ln(a*b^n) would be expanded to Ln(a)+n*Ln(b).

If the logarithm of an integer is discovered, it is factorised using `Factors` and expanded as though `LnExpand` had been given the factorised form. So Ln(18) goes to Ln(x)+2*Ln(3).

##### Example:

    In> LnExpand(Ln(a*b^n))
    Out> Ln(a)+Ln(b)*n
    In> LnExpand(Ln(a^m/b^n))
    Out> Ln(a)*m-Ln(b)*n
    In> LnExpand(Ln(60))
    Out> 2*Ln(2)+Ln(3)+Ln(5)
    In> LnExpand(Ln(60/25))
    Out> 2*Ln(2)+Ln(3)-Ln(5)

##### See also:

Ln , LnCombine , Factors .

###

LnCombine -- combine logarithmic expressions using standard logarithm rules

##### Standard library

##### Calling format:

    LnCombine(expr)

##### Parameters:

`expr` -- an expression possibly containing multiple `Ln` terms to be combined

##### Description:

`LnCombine` finds `Ln` terms in the expression it is given, and combines them using logarithm rules. It is intended to be the exact converse of `LnExpand`.

##### Example:

    In> LnCombine(Ln(a)+Ln(b)*n)
    Out> Ln(a*b^n)
    In> LnCombine(2*Ln(2)+Ln(3)-Ln(5))
    Out> Ln(12/5)

##### See also:

Ln , LnExpand .

###

Rationalize -- convert floating point numbers to fractions

##### Standard library

##### Calling format:

    Rationalize(expr)

##### Parameters:

`expr` -- an expression containing real numbers

##### Description:

This command converts every real number in the expression "expr" into a rational number. This is useful when a calculation needs to be done on floating point numbers and the algorithm is unstable. Converting the floating point numbers to rational numbers will force calculations to be done with infinite precision (by using rational numbers as representations).

It does this by finding the smallest integer n such that multiplying the number with 10^n is an integer. Then it divides by 10^n again, depending on the internal gcd calculation to reduce the resulting division of integers.

##### Examples:

    In> x:={1.2, 3.123, 4.5}
    Out> {1.2,3.123,4.5};
    In> Rationalize(x)
    Out> {6/5,3123/1000,9/2};

##### See also:

IsRational .

###

Solve -- solve an equation

##### Standard library

##### Calling format:

    Solve(eq, var)

##### Parameters:

`eq` -- equation to solve

`var` -- variable to solve for

##### Description:

This command tries to solve an equation. If `eq` does not contain the `==` operator, it is assumed that the user wants to solve $eq == 0$. The result is a list of equations of the form `var == value`, each representing a solution of the given equation. The `Where` operator can be used to substitute this solution in another expression. If the given equation `eq` does not have any solutions, or if `Solve` is unable to find any, then an empty list is returned.

The current implementation is far from perfect. In particular, the user should keep the following points in mind:

- `Solve` cannot solve all equations. If it is given a equation it can not solve, it raises an error via `Check`. Unfortunately, this is not displayed by the inline pretty-printer; call `PrettyPrinter` to change this. If an equation cannot be solved analytically, you may want to call `Newton` to get a numerical solution.
- Systems of equations are not handled yet. For linear systems, `MatrixSolve` can be used. The old version of `Solve`, with the name `OldSolve` might be able to solve nonlinear systems of equations.
- The periodicity of the trigonometric functions `Sin`, `Cos`, and `Tan` is not taken into account. The same goes for the (imaginary) periodicity of `Exp`. This causes `Solve` to miss solutions.
- It is assumed that all denominators are nonzero. Hence, a solution reported by `Solve` may in fact fail to be a solution because a denominator vanishes.
- In general, it is wise not to have blind trust in the results returned by `Solve`. A good strategy is to substitute the solutions back in the equation.

##### Examples:

First a simple example, where everything works as it should. The quadratic equation x^2+x==0 is solved. Then the result is checked by substituting it back in the quadratic.

    In> quadratic := x^2+x;
    Out> x^2+x;
    In> Solve(quadratic, x);
    Out> {x==0,x==(-1)};
    In> quadratic Where %;
    Out> {0,0};

If one tries to solve the equation Exp(x)==Sin(x), one finds that `Solve` can not do this.

    In> PrettyPrinter("DefaultPrint");
    Out> True;
    In> Solve(Exp(x) == Sin(x), x);
    Error: Solve'Fails: cannot solve equation Exp(x)-Sin(x) for x
    Out> {};

The equation Cos(x)==1/2 has an infinite number of solutions, namely x==(2*k+1/3)*Pi and x==(2*k-1/3)*Pi for any integer k. However, `Solve` only reports the solutions with k==0.

    In> Solve(Cos(x) == 1/2, x);
    Out> {x==Pi/3,x== -Pi/3};

For the equation x/Sin(x)==0, a spurious solution at x==0 is returned. However, the fraction is undefined at that point.

    In> Solve(x / Sin(x) == 0, x);
    Out> {x==0};

At first sight, the equation Sqrt(x)==a seems to have the solution x==a^2. However, this is not true for eg. a== -1.

    In> PrettyPrinter("DefaultPrint");
    Out> True;
    In> Solve(Sqrt(x) == a, x);
    Error: Solve'Fails: cannot solve equation Sqrt(x)-a for  x
    Out> {};
    In> Solve(Sqrt(x) == 2, x);
    Out> {x==4};
    In> Solve(Sqrt(x) == -1, x);
    Out> {};

##### See also:

Check , MatrixSolve , Newton , OldSolve , PrettyPrinter , PSolve , Where , == .

###

OldSolve -- old version of `Solve`

##### Standard library

##### Calling format:

    OldSolve(eq, var)
    OldSolve(eqlist, varlist)

##### Parameters:

`eq` -- single identity equation

`var` -- single variable

`eqlist` -- list of identity equations

`varlist` -- list of variables

##### Description:

This is an older version of `Solve`. It is retained for two reasons. The first one is philosophical: it is good to have multiple algorithms available. The second reason is more practical: the newer version cannot handle systems of equations, but `OldSolve` can.

This command tries to solve one or more equations. Use the first form to solve a single equation and the second one for systems of equations.

The first calling sequence solves the equation "eq" for the variable "var". Use the `==` operator to form the equation. The value of "var" which satisfies the equation, is returned. Note that only one solution is found and returned.

To solve a system of equations, the second form should be used. It solves the system of equations contained in the list "eqlist" for the variables appearing in the list "varlist". A list of results is returned, and each result is a list containing the values of the variables in "varlist". Again, at most a single solution is returned.

The task of solving a single equation is simply delegated to `SuchThat`. Multiple equations are solved recursively: firstly, an equation is sought in which one of the variables occurs exactly once; then this equation is solved with `SuchThat`; and finally the solution is substituted in the other equations by `Eliminate` decreasing the number of equations by one. This suffices for all linear equations and a large group of simple nonlinear equations.

##### Examples:

    In> OldSolve(a+x*y==z,x)
    Out> (z-a)/y;
    In> OldSolve({a*x+y==0,x+z==0},{x,y})
    Out> {{-z,z*a}};

This means that "x = (z-a)/y" is a solution of the first equation and that "x = -z", "y = z*a" is a solution of the systems of equations in the second command.

An example which `OldSolve` cannot solve:

    In> OldSolve({x^2-x == y^2-y,x^2-x == y^3+y},{x,y});
    Out> {};

##### See also:

Solve , SuchThat , Eliminate , PSolve , == .

###

SuchThat -- special purpose solver

##### Standard library

##### Calling format:

    SuchThat(expr, var)

##### Parameters:

`expr` -- expression to make zero

`var` -- variable (or subexpression) to solve for

##### Description:

This functions tries to find a value of the variable "var" which makes the expression "expr" zero. It is also possible to pass a subexpression as "var", in which case `SuchThat` will try to solve for that subexpression.

Basically, only expressions in which "var" occurs only once are handled; in fact, `SuchThat` may even give wrong results if the variables occurs more than once. This is a consequence of the implementation, which repeatedly applies the inverse of the top function until the variable "var" is reached.

##### Examples:

    In> SuchThat(a+b*x, x)
    Out> (-a)/b;
    In> SuchThat(Cos(a)+Cos(b)^2, Cos(b))
    Out> Cos(a)^(1/2);
    In> A:=Expand(a*x+b*x+c, x)
    Out> (a+b)*x+c;
    In> SuchThat(A, x)
    Out> (-c)/(a+b);

##### See also:

Solve , OldSolve , Subst , Simplify .

###

Eliminate -- substitute and simplify

##### Standard library

##### Calling format:

    Eliminate(var, value, expr)

##### Parameters:

`var` -- variable (or subexpression) to substitute

`value` -- new value of "var"

`expr` -- expression in which the substitution should take place

##### Description:

This function uses `Subst` to replace all instances of the variable (or subexpression) "var" in the expression "expr" with "value", calls `Simplify` to simplify the resulting expression, and returns the result.

##### Examples:

    In> Subst(Cos(b), c) (Sin(a)+Cos(b)^2/c)
    Out> Sin(a)+c^2/c;
    In> Eliminate(Cos(b), c, Sin(a)+Cos(b)^2/c)
    Out> Sin(a)+c;

##### See also:

SuchThat , Subst , Simplify .

###

PSolve -- solve a polynomial equation

##### Standard library

##### Calling format:

    PSolve(poly, var)

##### Parameters:

`poly` -- a polynomial in "var"

`var` -- a variable

##### Description:

This commands returns a list containing the roots of "poly", considered as a polynomial in the variable "var". If there is only one root, it is not returned as a one-entry list but just by itself. A double root occurs twice in the result, and similarly for roots of higher multiplicity. All polynomials of degree up to 4 are handled.

##### Examples:

    In> PSolve(b*x+a,x)
    Out> -a/b;
    In> PSolve(c*x^2+b*x+a,x)
    Out> {(Sqrt(b^2-4*c*a)-b)/(2*c),(-(b+
    Sqrt(b^2-4*c*a)))/(2*c)};

##### See also:

Solve , Factor .

###

Pi -- numerical approximation of Pi

##### Internal function

##### Calling format:

    Pi()

##### Description:

This commands returns the value of the mathematical constant pi at the current precision, as set by `Precision`. Usually this function should not be called directly. The constant `Pi` can (and should) be used to represent the exact value of pi, as it is recognized by the simplification rules. When the function `N` is invoked on an expression, `Pi` will be replaced with the value returned by `Pi()`.

##### Examples:

    In> Pi()
    Out> 3.14159265358979323846;
    In> Sin(3*Pi/2)
    Out> -1;
    In> Sin(3*Pi()/2)
    Out> Sin(4.7123889804);
    In> Precision(35)
    Out> True;
    In> Pi()
    Out> 3.14159265358979323846264338327950288;

##### See also:

N , Pi , Precision .

###

Random, RandomSeed -- (pseudo-) random number generator

##### Standard library

##### Calling format:

    Random()
    RandomSeed(init)

*PARAMS `init` -- positive integer, initial random seed

##### Description:

The function `Random` returns a random number, uniformly distributed in the interval between 0 and 1. The same sequence of random numbers is generated in every Yacas session.

The random number generator can be initialized by calling `RandomSeed` with an integer value. Each seed value will result in the same sequence of pseudo-random numbers.

##### See also:

RandomInteger , RandomPoly , Rng .

###

RngCreate, RngSeed, Rng -- manipulate random number generators as objects

##### Standard library

##### Calling format:

    RngCreate()
    RngCreate(init)
    RngCreate(option=value,...)
    RngSeed(r, init)
    Rng(r)

##### Parameters:

`init` -- integer, initial seed value

`option` -- atom, option name

`value` -- atom, option value

`r` -- a list, RNG object

##### Description:

These commands are an object-oriented interface to (pseudo-)random number generators (RNGs).

`RngCreate` returns a list which is a well-formed RNG object. Its value should be saved in a variable and used to call `Rng` and `RngSeed`.

`Rng(r)` returns a floating-point random number between 0 and 1 and updates the RNG object `r`. (Currently, the Gaussian option makes a RNG return a complex random number instead of a real random number.)

`RngSeed(r,init)` re-initializes the RNG object `r` with the seed value `init`. The seed value should be a positive integer.

The `RngCreate` function accepts several options as arguments. Currently the following options are available:

- `seed` -- specify initial seed value, must be a positive integer
- `dist` -- specify the distribution of the random number; currently `flat` and `gauss` are implemented, and the default is the flat (uniform) distribution
- `engine` -- specify the RNG engine; currently `default` and `advanced` are available ("advanced" is slower but has much longer period)

If the initial seed is not specified, the value of 76544321 will be used.

The `gauss` option will create a RNG object that generates pairs of Gaussian distributed random numbers as a complex random number. The real and the imaginary parts of this number are independent random numbers taken from a Gaussian (i.e. "normal") distribution with unit variance.

Note that unlike the global `Random` function, the RNG objects created with `RngCreate` are independent RNGs and do not affect each other. They generate independent streams of pseudo-random numbers. However, the `Random` function is slightly faster.

##### Examples:

    In> r1:=RngCreate(seed=1,dist=gauss)
    Out> {"GaussianRNGDist","RNGEngine'LCG'2",{1}}
    In> Rng(r1)
    Out> Complex(-1.6668466417,0.228904004);
    In> Rng(r1);
    Out> Complex(0.0279296109,-0.5382405341);

The second RNG gives a uniform distribution (default option) but uses a more complicated algorithm:

    In> [r2:=RngCreate(engine=advanced);Rng(r2);]
    Out> 0.3653615377;

The generator `r1` can be re-initialized with seed 1 again to obtain the same sequence:

    In> RngSeed(r1, 1)
    Out> True;
    In> Rng(r1)
    Out> Complex(-1.6668466417,0.228904004);

##### See also:

Random .

###

Limit -- limit of an expression

##### Standard library

##### Calling format:

    Limit(var, val) expr
    Limit(var, val, dir) expr

##### Parameters:

`var` -- a variable

`val` -- a number

`dir` -- a direction (`Left` or `Right`)

`expr` -- an expression

##### Description:

This command tries to determine the value that the expression "expr" converges to when the variable "var" approaches "val". One may use `Infinity` or `-Infinity` for "val". The result of `Limit` may be one of the symbols `Undefined` (meaning that the limit does not exist), `Infinity`, or `-Infinity`.

The second calling sequence is used for unidirectional limits. If one gives "dir" the value `Left`, the limit is taken as "var" approaches "val" from the positive infinity; and `Right` will take the limit from the negative infinity.

##### Examples:

    In> Limit(x,0) Sin(x)/x
    Out> 1;
    In> Limit(x,0) (Sin(x)-Tan(x))/(x^3)
    Out> -1/2;
    In> Limit(x,0) 1/x
    Out> Undefined;
    In> Limit(x,0,Left) 1/x
    Out> -Infinity;
    In> Limit(x,0,Right) 1/x
    Out> Infinity;

###

TrigSimpCombine -- combine products of trigonometric functions

##### Standard library

##### Calling format:

    TrigSimpCombine(expr)

##### Parameters:

`expr` -- expression to simplify

##### Description:

This function applies the product rules of trigonometry, e.g. Cos(u)*Sin(v)=1/2*(Sin(v-u)+Sin(v+u)). As a result, all products of the trigonometric functions `Cos` and `Sin` disappear. The function also tries to simplify the resulting expression as much as possible by combining all similar terms.

This function is used in for instance `Integrate`, to bring down the expression into a simpler form that hopefully can be integrated easily.

##### Examples:

    In> PrettyPrinter("PrettyForm");

    True

    In> TrigSimpCombine(Cos(a)^2+Sin(a)^2)

    1

    In> TrigSimpCombine(Cos(a)^2-Sin(a)^2)

    Cos( -2 * a )

    Out>
    In> TrigSimpCombine(Cos(a)^2*Sin(b))

    Sin( b )   Sin( -2 * a + b )
    -------- + -----------------
       2               4

        Sin( -2 * a - b )
      - -----------------
                4

##### See also:

Simplify , Integrate , Expand , Sin , Cos , Tan .

###

LagrangeInterpolant -- polynomial interpolation

##### Standard library

##### Calling format:

    LagrangeInterpolant(xlist, ylist, var)

##### Parameters:

`xlist` -- list of argument values

`ylist` -- list of function values

`var` -- free variable for resulting polynomial

##### Description:

This function returns a polynomial in the variable "var" which interpolates the points "(xlist, ylist)". Specifically, the value of the resulting polynomial at "xlist[1]" is "ylist[1]", the value at "xlist[2]" is "ylist[2]", etc. The degree of the polynomial is not greater than the length of "xlist".

The lists "xlist" and "ylist" should be of equal length. Furthermore, the entries of "xlist" should be all distinct to ensure that there is one and only one solution.

This routine uses the Lagrange interpolant formula to build up the polynomial.

##### Examples:

    In> f := LagrangeInterpolant({0,1,2}, \
      {0,1,1}, x);
    Out> (x*(x-1))/2-x*(x-2);
    In> Eval(Subst(x,0) f);
    Out> 0;
    In> Eval(Subst(x,1) f);
    Out> 1;
    In> Eval(Subst(x,2) f);
    Out> 1;

    In> PrettyPrinter("PrettyForm");

    True

    In> LagrangeInterpolant({x1,x2,x3}, {y1,y2,y3}, x)

    y1 * ( x - x2 ) * ( x - x3 )
    ----------------------------
     ( x1 - x2 ) * ( x1 - x3 )

      y2 * ( x - x1 ) * ( x - x3 )
    + ----------------------------
       ( x2 - x1 ) * ( x2 - x3 )

      y3 * ( x - x1 ) * ( x - x2 )
    + ----------------------------
       ( x3 - x1 ) * ( x3 - x2 )

##### See also:

Subst .

###

Fibonacci -- Fibonacci sequence

##### Standard library

##### Calling format:

    Fibonacci(n)

##### Parameters:

`n` -- an integer

##### Description:

This command calculates and returns the "n"-th Fibonacci number.

The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, ... where every number is the sum of the two preceding numbers. Formally, it is defined by F(1)=1, F(2)=1, and F(n+1)=F(n)+F(n-1), where F(n) denotes the n-th Fibonacci number.

##### Examples:

    In> Fibonacci(4)
    Out> 3;
    In> Fibonacci(8)
    Out> 21;
    In> Table(Fibonacci(i), i, 1, 10, 1)
    Out> {1,1,2,3,5,8,13,21,34,55};

Gamma, GammaNum , Zeta, ZetaNum , Bernoulli, BernoulliArray , Euler , LambertW, LambertWNum , gamma , GoldenRatio , Catalan .

# Special functions

In this chapter, special and transcendental mathematical functions are described.

###

Gamma, GammaNum -- Euler's Gamma function

##### Standard library

##### Calling format:

    Gamma(x)
    GammaNum(number)

##### Parameters:

`x` -- expression

`number` -- expression that can be evaluated to a number

##### Description:

`Gamma(x)` is an interface to Euler's Gamma function Gamma(x). It returns exact values on integer and half-integer arguments. `GammaNum(x)` or equivalently `N(Gamma(x)` takes a numeric parameter and always returns a floating-point number in the current precision.

Note that Euler's constant gamma<=>0.57722 is the lowercase `gamma` in Yacas.

##### Examples:

    In> Precision(30)
    Out> True;
    In> Gamma(1.3)
    Out> Gamma(1.3);
    In> N(Gamma(1.3))
    Out> 0.897470696306277188493754954771;
    In> Gamma(1.5)
    Out> Sqrt(Pi)/2;
    In> GammaNum(1.5);
    Out> 0.88622692545275801364908374167;

##### See also:

! , N , gamma .

###

Zeta, ZetaNum -- Riemann's Zeta function

##### Standard library

##### Calling format:

    Zeta(x)
    ZetaNum(number)

##### Parameters:

`x` -- expression

`number` -- expression that can be evaluated to a number

##### Description:

`Zeta(x)` is an interface to Riemann's Zeta function zeta(s). It returns exact values on integer and half-integer arguments. `ZetaNum(x)` or equivalently `N(Zeta(x)` takes a numeric parameter and always returns a floating-point number in the current precision.

##### Examples:

    In> Precision(30)
    Out> True;
    In> Zeta(1)
    Out> Infinity;
    In> Zeta(1.3)
    Out> Zeta(1.3);
    In> N(Zeta(1.3))
    Out> 3.93194921180954422697490751058798;
    In> Zeta(2)
    Out> Pi^2/6;
    In> ZetaNum(2);
    Out> 1.64493406684822643647241516664602;

##### See also:

! , N .

###

Bernoulli, BernoulliArray -- Bernoulli numbers and polynomials

##### Standard library

##### Calling format:

    Bernoulli(index)
    BernoulliArray(index)
    Bernoulli(index, x)

##### Parameters:

`x` -- expression that will be the variable in the polynomial

`index` -- expression that can be evaluated to an integer

##### Description:

`Bernoulli(n)` evaluates the n-th Bernoulli number. `Bernoulli(n, x)` returns the n-th Bernoulli polynomial in the variable x. The polynomial is returned in the Horner form.

An auxiliary function `BernoulliArray(n)` might be useful too: it returns an array (of type `GenericArray`) of Bernoulli numbers up to n. The array is 1-based, so that the n-th Bernoulli number is `BernoulliArray(n)[n+1]`.

##### Example:

    In> Bernoulli(20);
    Out> -174611/330;
    In> Bernoulli(4, x);
    Out> ((x-2)*x+1)*x^2-1/30;

##### See also:

Gamma , Zeta .

###

Euler -- Euler numbers and polynomials

##### Standard library

##### Calling format:

    Euler(index)
    Euler(index,x)

##### Parameters:

`x` -- expression that will be the variable in the polynomial

`index` -- expression that can be evaluated to an integer

##### Description:

`Euler(n)` evaluates the n-th Euler number. `Euler(n,x)` returns the n-th Euler polynomial in the variable x.

##### Examples:

    In> Euler(6)
    Out> -61;
    In> A:=Euler(5,x)
    Out> (x-1/2)^5+(-10*(x-1/2)^3)/4+(25*(x-1/2))/16;
    In> Simplify(A)
    Out> (2*x^5-5*x^4+5*x^2-1)/2;

##### See also:

Bin .

###

LambertW, LambertWNum -- Lambert's W function

##### Standard library

##### Calling format:

    LambertW(x)
    LambertWNum(x)

##### Parameters:

`x` -- expression, argument of the function

##### Description:

Lambert's W function is (a multiple-valued, complex function) defined for any (complex) z by

W(z)*Exp(W(z))=z.
This function is sometimes useful to represent solutions of transcendental equations. For example, the equation Ln(x)=3*x can be "solved" by writing x= -3*W(-1/3). It is also possible to take a derivative or integrate this function "explicitly".

For real arguments x, W(x) is real if x>= -Exp(-1).

`LambertWNum` is an auxiliary function that computes the numerical value of the principal branch of Lambert's W function for real arguments x>= -Exp(-1) to current precision.

##### Examples:

    In> LambertW(0)
    Out> 0;
    In> N(LambertW(-0.24/Sqrt(3*Pi)))
    Out> -0.0851224014;

##### See also:

Exp .

###

gamma -- Euler's constant gamma

##### Standard library

##### Calling format:

    N(gamma)
    gamma()

##### Description:

These functions compute Euler's constant gamma<=>0.57722...

The constant is available symbolically as `gamma` or numerically as a function `gamma()`. This is a "cached constant" which is recalculated only when precision is increased. The numerical value of the constant can also be obtained as `N(gamma)`. The low-level numerical computations are performed by the routine `GammaConstNum`.

Note that Euler's Gamma function Gamma(x) is the capitalized `Gamma` in Yacas.

##### Examples:

    In> gamma+Pi
    Out> gamma+Pi;
    In> N(gamma+Pi)
    Out> 3.7188083184;
    In> [Precision(20);V(gamma());]

    CachedConstant: Info: constant gamma is being
      recalculated at precision 20
    GammaConstNum: Info: used 56 iterations at
      working precision 24
    Out> 0.57721566490153286061;

##### See also:

Gamma , N , CachedConstant .

###

GoldenRatio -- the Golden Ratio

##### Standard library

##### Calling format:

    N(GoldenRatio)
    GoldenRatio()

##### Description:

These functions compute the "golden ratio"

phi<=>1.6180339887<=>(1+Sqrt(5))/2.

The ancient Greeks defined the "golden ratio" as follows: If one divides a length 1 into two pieces x and 1-x, such that the ratio of 1 to x is the same as the ratio of x to 1-x, then 1/x<=>1.618... is the "golden ratio".

The constant is available symbolically as `GoldenRatio` or numerically as a function `GoldenRatio()`. This is a "cached constant" which is recalculated only when precision is increased. The numerical value of the constant can also be obtained as `N(GoldenRatio)`.

##### Examples:

    In> x:=GoldenRatio - 1
    Out> GoldenRatio-1;
    In> N(x)
    Out> 0.6180339887;
    In> N(1/GoldenRatio)
    Out> 0.6180339887;
    In> [Precision(20);V(GoldenRatio());]

    CachedConstant: Info: constant GoldenRatio is
    being recalculated at precision 20
    Out> 1.6180339887498948482;

##### See also:

N , CachedConstant .

###

Catalan -- Catalan's Constant

##### Standard library

##### Calling format:

    N(Catalan)
    Catalan()

##### Description:

These functions compute Catalan's Constant Catalan<=>0.9159655941.

The constant is available symbolically as `Catalan` or numerically as a function `Catalan()`. This is a "cached constant" which is recalculated only when precision is increased. The numerical value of the constant can also be obtained as `N(Catalan)`. The low-level numerical computations are performed by the routine `CatalanConstNum`.

##### Examples:

    In> N(Catalan)
    Out> 0.9159655941;
    In> DirichletBeta(2)
    Out> Catalan;
    In> [Precision(20);V(Catalan());]

    CachedConstant: Info: constant Catalan is
    being recalculated at precision 20
    Out> 0.91596559417721901505;

##### See also:

N , CachedConstant .

OdeSolve , OdeTest , OdeOrder .

# Differential Equations

In this chapter, some facilities for solving differential equations are described. Currently only simple equations without auxiliary conditions are supported.

###

OdeSolve -- general ODE solver

##### Standard library

##### Calling format:

    OdeSolve(expr1==expr2)

##### Parameters:

`expr1,expr2` -- expressions containing a function to solve for

##### Description:

This function currently can solve second order homogeneous linear real constant coefficient equations. The solution is returned with unique constants generated by `UniqueConstant`. The roots of the auxiliary equation are used as the arguments of exponentials. If the roots are complex conjugate pairs, then the solution returned is in the form of exponentials, sines and cosines.

First and second derivatives are entered as `y',y''`. Higher order derivatives may be entered as `y(n)`, where `n` is any integer.

##### Examples:

    In> OdeSolve( y'' + y == 0 )
    Out> C42*Sin(x)+C43*Cos(x);
    In> OdeSolve( 2*y'' + 3*y' + 5*y == 0 )
    Out> Exp(((-3)*x)/4)*(C78*Sin(Sqrt(31/16)*x)+C79*Cos(Sqrt(31/16)*x));
    In> OdeSolve( y'' - 4*y == 0 )
    Out> C132*Exp((-2)*x)+C136*Exp(2*x);
    In> OdeSolve( y'' +2*y' + y == 0 )
    Out> (C183+C184*x)*Exp(-x);

##### See also:

Solve , RootsWithMultiples .

###

OdeTest -- test the solution of an ODE

##### Standard library

##### Calling format:

    OdeTest(eqn,testsol)

##### Parameters:

`eqn` -- equation to test

`testsol` -- test solution

##### Description:

This function automates the verification of the solution of an ODE. It can also be used to quickly see how a particular equation operates on a function.

##### Examples:

    In> OdeTest(y''+y,Sin(x)+Cos(x))
    Out> 0;
    In> OdeTest(y''+2*y,Sin(x)+Cos(x))
    Out> Sin(x)+Cos(x);

##### See also:

OdeSolve .

###

OdeOrder -- return order of an ODE

##### Standard library

##### Calling format:

    OdeOrder(eqn)

##### Parameters:

`eqn` -- equation

##### Description:

This function returns the order of the differential equation, which is order of the highest derivative. If no derivatives appear, zero is returned.

##### Examples:

    In> OdeOrder(y'' + 2*y' == 0)
    Out> 2;
    In> OdeOrder(Sin(x)*y(5) + 2*y' == 0)
    Out> 5;
    In> OdeOrder(2*y + Sin(y) == 0)
    Out> 0;

##### See also:

OdeSolve .

LaplaceTransform .

# Transforms

In this chapter, some facilities for various transforms are described.

###

LaplaceTransform -- Laplace Transform

##### Standard library

##### Calling format:

    LaplaceTransform(t,s,func)

##### Parameters:

`t` -- independent variable that is being transformed

`s` -- independent variable that is being transformed into

`f` -- function

##### Description:

This function attempts to take the function `f(t)` and find the Laplace transform of it,`F(s)`, which is defined as `Integrate(t,0,Infinity) Exp(-s*t)*f`. This is also sometimes referred to the "unilateral" Laplace tranform. `LaplaceTransform` can transform most elementary functions that do not require a convolution integral, as well as any polynomial times an elementary function. If a transform cannot be found then `LaplaceTransform` will return unevaluated. This can happen for function which are not of "exponential order", which means that they grow faster than exponential functions.

##### Examples:

    In> LaplaceTransform(t,s,2*t^5+ t^2/2 )
    Out> 240/s^6+2/(2*s^3);
    In> LaplaceTransform(t,s,t*Sin(2*t)*Exp(-3*t) )
    Out> (2*(s+3))/(2*(2*(((s+3)/2)^2+1))^2);
    In> LaplaceTransform(t,s, BesselJ(3,2*t) )
    Out> (Sqrt((s/2)^2+1)-s/2)^3/(2*Sqrt((s/2)^2+1));
    In> LaplaceTransform(t,s,Exp(t^2)); // not of exponential order
    Out> LaplaceTransform(t,s,Exp(t^2));
    In> LaplaceTransform(p,q,Ln(p))
    Out> -(gamma+Ln(q))/q;

LeviCivita , Permutations , Dot, . , InProduct , CrossProduct , Outer, o , ZeroVector , BaseVector , Identity , ZeroMatrix , Diagonal , DiagonalMatrix , OrthogonalBasis , OrthonormalBasis , IsScalar , IsVector , IsMatrix , IsSquareMatrix , Normalize , Transpose , Determinant , Trace , Inverse , Minor , CoFactor , MatrixPower , SolveMatrix , CharacteristicEquation , EigenValues , EigenVectors , IsHermitian , IsOrthogonal , IsDiagonal , IsLowerTriangular, IsUpperTriangular , IsSymmetric , IsSkewSymmetric , IsUnitary , IsIdempotent , JacobianMatrix , VandermondeMatrix , HessianMatrix , Sparsity , HilbertMatrix , HilbertInverseMatrix , ToeplitzMatrix , WronskianMatrix , SylvesterMatrix , MatrixSolve , Cholesky , RandomIntegerMatrix .

# Linear Algebra

This chapter describes the commands for doing linear algebra. They can be used to manipulate vectors, represented as lists, and matrices, represented as lists of lists.

###

LeviCivita -- totally anti-symmetric Levi-Civita symbol

##### Standard library

##### Calling format:

    LeviCivita(list)

##### Parameters:

`list` -- a list of integers 1 .. n in some order

##### Description:

`LeviCivita` implements the Levi-Civita symbol. This is generally useful for tensor calculus. `list` should be a list of integers, and this function returns 1 if the integers are in successive order, eg. `LeviCivita( {1,2,3,...} )` would return 1. Swapping two elements of this list would return -1. So, `LeviCivita( {2,1,3} )` would evaluate to -1.

##### Examples:

    In> LeviCivita({1,2,3})
    Out> 1;
    In> LeviCivita({2,1,3})
    Out> -1;
    In> LeviCivita({2,2,3})
    Out> 0;

##### See also:

Permutations .

###

Permutations -- get all permutations of a list

##### Standard library

##### Calling format:

    Permutations(list)

##### Parameters:

`list` -- a list of elements

##### Description:

Permutations returns a list with all the permutations of the original list.

##### Examples:

    In> Permutations({a,b,c})
    Out> {{a,b,c},{a,c,b},{c,a,b},{b,a,c},
    {b,c,a},{c,b,a}};

##### See also:

LeviCivita .

###

Dot, . -- get dot product of tensors

##### Standard library

##### Calling format:

    Dot(t1,t2)
    t1 . t2

Precedence: 30

##### Parameters:

`t1,t2` -- tensor lists (currently only vectors and matrices are supported)

##### Description:

`Dot` returns the dot (aka inner) product of two tensors t1 and t2. The last index of t1 and the first index of t2 are contracted. Currently `Dot` works only for vectors and matrices. `Dot`-multiplication of two vectors, a matrix with a vector (and vice versa) or two matrices yields either a scalar, a vector or a matrix.

##### Examples:

    In> Dot({1,2},{3,4})
    Out> 11;
    In> Dot({{1,2},{3,4}},{5,6})
    Out> {17,39};
    In> Dot({5,6},{{1,2},{3,4}})
    Out> {23,34};
    In> Dot({{1,2},{3,4}},{{5,6},{7,8}})
    Out> {{19,22},{43,50}};

    Or, using the "."-Operator:

    In> {1,2} . {3,4}
    Out> 11;
    In> {{1,2},{3,4}} . {5,6}
    Out> {17,39};
    In> {5,6} . {{1,2},{3,4}}
    Out> {23,34};
    In> {{1,2},{3,4}} . {{5,6},{7,8}}
    Out> {{19,22},{43,50}};

##### See also:

Outer , Cross , IsScalar , IsVector , IsMatrix .

###

InProduct -- inner product of vectors (deprecated)

##### Standard library

##### Calling format:

    InProduct(a,b)

##### Parameters:

`a`, `b` -- vectors of equal length

##### Description:

The inner product of the two vectors "a" and "b" is returned. The vectors need to have the same size.

This function is superceded by the `.` operator.

##### Examples:

    In> {a,b,c} . {d,e,f};
    Out> a*d+b*e+c*f;

##### See also:

Dot , CrossProduct .

###

CrossProduct -- outer product of vectors

##### Standard library

##### Calling format:

    CrossProduct(a,b)
    a X b

Precedence: 30

##### Parameters:

`a`, `b` -- three-dimensional vectors

##### Description:

The cross product of the vectors "a" and "b" is returned. The result is perpendicular to both "a" and "b" and its length is the product of the lengths of the vectors. Both "a" and "b" have to be three-dimensional.

##### Examples:

    In> {a,b,c} X {d,e,f};
    Out> {b*f-c*e,c*d-a*f,a*e-b*d};

##### See also:

InProduct .

###

Outer, o -- get outer tensor product

##### Standard library

##### Calling format:

    Outer(t1,t2)
    t1 o t2

Precedence: 30

##### Parameters:

`t1,t2` -- tensor lists (currently only vectors are supported)

##### Description:

`Outer` returns the outer product of two tensors t1 and t2. Currently `Outer` work works only for vectors, i.e. tensors of rank 1. The outer product of two vectors yields a matrix.

##### Examples:

    In> Outer({1,2},{3,4,5})
    Out> {{3,4,5},{6,8,10}};
    In> Outer({a,b},{c,d})
    Out> {{a*c,a*d},{b*c,b*d}};

    Or, using the "o"-Operator:

    In> {1,2} o {3,4,5}
    Out> {{3,4,5},{6,8,10}};
    In> {a,b} o {c,d}
    Out> {{a*c,a*d},{b*c,b*d}};

##### See also:

Dot , Cross .

###

ZeroVector -- create a vector with all zeroes

##### Standard library

##### Calling format:

    ZeroVector(n)

##### Parameters:

`n` -- length of the vector to return

##### Description:

This command returns a vector of length "n", filled with zeroes.

##### Examples:

    In> ZeroVector(4)
    Out> {0,0,0,0};

##### See also:

BaseVector , ZeroMatrix , IsZeroVector .

###

BaseVector -- base vector

##### Standard library

##### Calling format:

    BaseVector(k, n)

##### Parameters:

`k` -- index of the base vector to construct

`n` -- dimension of the vector

##### Description:

This command returns the "k"-th base vector of dimension "n". This is a vector of length "n" with all zeroes except for the "k"-th entry, which contains a 1.

##### Examples:

    In> BaseVector(2,4)
    Out> {0,1,0,0};

##### See also:

ZeroVector , Identity .

###

Identity -- make identity matrix

##### Standard library

##### Calling format:

    Identity(n)

##### Parameters:

`n` -- size of the matrix

##### Description:

This commands returns the identity matrix of size "n" by "n". This matrix has ones on the diagonal while the other entries are zero.

##### Examples:

    In> Identity(3)
    Out> {{1,0,0},{0,1,0},{0,0,1}};

##### See also:

BaseVector , ZeroMatrix , DiagonalMatrix .

###

ZeroMatrix -- make a zero matrix

##### Standard library

##### Calling format:

    ZeroMatrix(n)
    ZeroMatrix(n, m)

##### Parameters:

`n` -- number of rows

`m` -- number of columns

##### Description:

This command returns a matrix with `n` rows and `m` columns, completely filled with zeroes. If only given one parameter, it returns the square `n` by `n` zero matrix.

##### Examples:

    In> ZeroMatrix(3,4)
    Out> {{0,0,0,0},{0,0,0,0},{0,0,0,0}};
    In> ZeroMatrix(3)
    Out> {{0,0,0},{0,0,0},{0,0,0}};

##### See also:

ZeroVector , Identity .

###

Diagonal -- extract the diagonal from a matrix

##### Standard library

##### Calling format:

    Diagonal(A)

##### Parameters:

`A` -- matrix

##### Description:

This command returns a vector of the diagonal components of the matrix `A`.

##### Examples:

    In> Diagonal(5*Identity(4))
    Out> {5,5,5,5};
    In> Diagonal(HilbertMatrix(3))
    Out> {1,1/3,1/5};

##### See also:

DiagonalMatrix , IsDiagonal .

###

DiagonalMatrix -- construct a diagonal matrix

##### Standard library

##### Calling format:

    DiagonalMatrix(d)

##### Parameters:

`d` -- list of values to put on the diagonal

##### Description:

This command constructs a diagonal matrix, that is a square matrix whose off-diagonal entries are all zero. The elements of the vector "d" are put on the diagonal.

##### Examples:

    In> DiagonalMatrix(1 .. 4)
    Out> {{1,0,0,0},{0,2,0,0},{0,0,3,0},{0,0,0,4}};

##### See also:

Identity , ZeroMatrix .

###

OrthogonalBasis -- create an orthogonal basis

##### Standard library

##### Calling format:

    OrthogonalBasis(W)

##### Parameters:

`W` - A linearly independent set of row vectors (aka a matrix)

##### Description:

Given a linearly independent set `W` (constructed of rows vectors), this command returns an orthogonal basis `V` for `W`, which means that span(V) = span(W) and `InProduct(V[i],V[j]) = 0` when `i != j`. This function uses the Gram-Schmidt orthogonalization process.

##### Examples:

    In> OrthogonalBasis({{1,1,0},{2,0,1},{2,2,1}})
    Out> {{1,1,0},{1,-1,1},{-1/3,1/3,2/3}};

##### See also:

OrthonormalBasis , InProduct .

###

OrthonormalBasis -- create an orthonormal basis

##### Standard library

##### Calling format:

    OrthonormalBasis(W)

##### Parameters:

`W` - A linearly independent set of row vectors (aka a matrix)

##### Description:

Given a linearly independent set `W` (constructed of rows vectors), this command returns an orthonormal basis `V` for `W`. This is done by first using `OrthogonalBasis(W)`, then dividing each vector by its magnitude, so as the give them unit length.

##### Examples:

    In> OrthonormalBasis({{1,1,0},{2,0,1},{2,2,1}})
    Out> {{Sqrt(1/2),Sqrt(1/2),0},{Sqrt(1/3),-Sqrt(1/3),Sqrt(1/3)},
    	{-Sqrt(1/6),Sqrt(1/6),Sqrt(2/3)}};

##### See also:

OrthogonalBasis , InProduct , Normalize .

###

IsScalar -- test for a scalar

##### Standard library

##### Calling format:

    IsScalar(expr)

##### Parameters:

`expr` -- a mathematical object

##### Description:

`IsScalar` returns `True` if `expr` is a scalar, `False` otherwise. Something is considered to be a scalar if it's not a list.

##### Examples:

    In> IsScalar(7)
    Out> True;
    In> IsScalar(Sin(x)+x)
    Out> True;
    In> IsScalar({x,y})
    Out> False;

##### See also:

IsList , IsVector , IsMatrix .

###

IsVector -- test for a vector

##### Standard library

##### Calling format:

    IsVector(expr)

    IsVector(pred,expr)

##### Parameters:

`expr` -- expression to test

`pred` -- predicate test (e.g. IsNumber, IsInteger, ...)

##### Description:

`IsVector(expr)` returns `True` if `expr` is a vector, `False` otherwise. Something is considered to be a vector if it's a list of scalars. `IsVector(pred,expr)` returns `True` if `expr` is a vector and if the predicate test `pred` returns `True` when applied to every element of the vector `expr`, `False` otherwise.

##### Examples:

    In> IsVector({a,b,c})
    Out> True;
    In> IsVector({a,{b},c})
    Out> False;
    In> IsVector(IsInteger,{1,2,3})
    Out> True;
    In> IsVector(IsInteger,{1,2.5,3})
    Out> False;

##### See also:

IsList , IsScalar , IsMatrix .

###

IsMatrix -- test for a matrix

##### Standard library

##### Calling format:

    IsMatrix(expr)

    IsMatrix(pred,expr)

##### Parameters:

`expr` -- expression to test

`pred` -- predicate test (e.g. IsNumber, IsInteger, ...)

##### Description:

`IsMatrix(expr)` returns `True` if `expr` is a matrix, `False` otherwise. Something is considered to be a matrix if it's a list of vectors of equal length. `IsMatrix(pred,expr)` returns `True` if `expr` is a matrix and if the predicate test `pred` returns `True` when applied to every element of the matrix `expr`, `False` otherwise.

##### Examples:

    In> IsMatrix(1)
    Out> False;
    In> IsMatrix({1,2})
    Out> False;
    In> IsMatrix({{1,2},{3,4}})
    Out> True;
    In> IsMatrix(IsRational,{{1,2},{3,4}})
    Out> False;
    In> IsMatrix(IsRational,{{1/2,2/3},{3/4,4/5}})
    Out> True;

##### See also:

IsList , IsVector .

###

IsSquareMatrix -- test for a square matrix

##### Standard library

##### Calling format:

    IsSquareMatrix(expr)

    IsSquareMatrix(pred,expr)

##### Parameters:

`expr` -- expression to test

`pred` -- predicate test (e.g. IsNumber, IsInteger, ...)

##### Description:

`IsSquareMatrix(expr)` returns `True` if `expr` is a square matrix, `False` otherwise. Something is considered to be a square matrix if it's a matrix having the same number of rows and columns. `IsMatrix(pred,expr)` returns `True` if `expr` is a square matrix and if the predicate test `pred` returns `True` when applied to every element of the matrix `expr`, `False` otherwise.

##### Examples:

    In> IsSquareMatrix({{1,2},{3,4}});
    Out> True;
    In> IsSquareMatrix({{1,2,3},{4,5,6}});
    Out> False;
    In> IsSquareMatrix(IsBoolean,{{1,2},{3,4}});
    Out> False;
    In> IsSquareMatrix(IsBoolean,{{True,False},{False,True}});
    Out> True;

##### See also:

IsMatrix .

###

Normalize -- normalize a vector

##### Standard library

##### Calling format:

    Normalize(v)

##### Parameters:

`v` -- a vector

##### Description:

Return the normalized (unit) vector parallel to `v`: a vector having the same direction but with length 1.

##### Examples:

    In> v:=Normalize({3,4})
    Out> {3/5,4/5};
    In> v . v
    Out> 1;

##### See also:

InProduct , CrossProduct .

###

Transpose -- get transpose of a matrix

##### Standard library

##### Calling format:

    Transpose(M)

##### Parameters:

`M` -- a matrix

##### Description:

`Transpose` returns the transpose of a matrix M. Because matrices are just lists of lists, this is a useful operation too for lists.

##### Examples:

    In> Transpose({{a,b}})
    Out> {{a},{b}};

###

Determinant -- determinant of a matrix

##### Standard library

##### Calling format:

    Determinant(M)

##### Parameters:

`M` -- a matrix

##### Description:

Returns the determinant of a matrix M.

##### Examples:

    In> A:=DiagonalMatrix(1 .. 4)
    Out> {{1,0,0,0},{0,2,0,0},{0,0,3,0},{0,0,0,4}};
    In> Determinant(A)
    Out> 24;

###

Trace -- trace of a matrix

##### Standard library

##### Calling format:

    Trace(M)

##### Parameters:

`M` -- a matrix

##### Description:

`Trace` returns the trace of a matrix M (defined as the sum of the elements on the diagonal of the matrix).

##### Examples:

    In> A:=DiagonalMatrix(1 .. 4)
    Out> {{1,0,0,0},{0,2,0,0},{0,0,3,0},{0,0,0,4}};
    In> Trace(A)
    Out> 10;

###

Inverse -- get inverse of a matrix

##### Standard library

##### Calling format:

    Inverse(M)

##### Parameters:

`M` -- a matrix

##### Description:

Inverse returns the inverse of matrix M. The determinant of M should be non-zero. Because this function uses `Determinant` for calculating the inverse of a matrix, you can supply matrices with non-numeric (symbolic) matrix elements.

##### Examples:

    In> A:=DiagonalMatrix({a,b,c})
    Out> {{a,0,0},{0,b,0},{0,0,c}};
    In> B:=Inverse(A)
    Out> {{(b*c)/(a*b*c),0,0},{0,(a*c)/(a*b*c),0},
    {0,0,(a*b)/(a*b*c)}};
    In> Simplify(B)
    Out> {{1/a,0,0},{0,1/b,0},{0,0,1/c}};

##### See also:

Determinant .

###

Minor -- get principal minor of a matrix

##### Standard library

##### Calling format:

    Minor(M,i,j)

##### Parameters:

`M` -- a matrix

`i`, `j` - positive integers

##### Description:

Minor returns the minor of a matrix around the element ( i, j). The minor is the determinant of the matrix obtained from M by deleting the i-th row and the j-th column.

##### Examples:

    In> A := {{1,2,3}, {4,5,6}, {7,8,9}};
    Out> {{1,2,3},{4,5,6},{7,8,9}};
    In> PrettyForm(A);

    /                    \
    | ( 1 ) ( 2 ) ( 3 )  |
    |                    |
    | ( 4 ) ( 5 ) ( 6 )  |
    |                    |
    | ( 7 ) ( 8 ) ( 9 )  |
    \                    /
    Out> True;
    In> Minor(A,1,2);
    Out> -6;
    In> Determinant({{2,3}, {8,9}});
    Out> -6;

##### See also:

CoFactor , Determinant , Inverse .

###

CoFactor -- cofactor of a matrix

##### Standard library

##### Calling format:

    CoFactor(M,i,j)

##### Parameters:

`M` -- a matrix

`i`, `j` - positive integers

##### Description:

`CoFactor` returns the cofactor of a matrix around the element ( i, j). The cofactor is the minor times (-1)^(i+j).

##### Examples:

    In> A := {{1,2,3}, {4,5,6}, {7,8,9}};
    Out> {{1,2,3},{4,5,6},{7,8,9}};
    In> PrettyForm(A);

    /                    \
    | ( 1 ) ( 2 ) ( 3 )  |
    |                    |
    | ( 4 ) ( 5 ) ( 6 )  |
    |                    |
    | ( 7 ) ( 8 ) ( 9 )  |
    \                    /
    Out> True;
    In> CoFactor(A,1,2);
    Out> 6;
    In> Minor(A,1,2);
    Out> -6;
    In> Minor(A,1,2) * (-1)^(1+2);
    Out> 6;

##### See also:

Minor , Determinant , Inverse .

###

MatrixPower -- get nth power of a square matrix

##### Standard library

##### Calling format:

    MatrixPower(mat,n)

##### Parameters:

`mat` -- a square matrix

`n` -- an integer

##### Description:

`MatrixPower(mat,n)` returns the `n`th power of a square matrix `mat`. For positive `n` it evaluates dot products of `mat` with itself. For negative `n` the nth power of the inverse of `mat` is returned. For `n`=0 the identity matrix is returned.

##### Example:

    In> A:={{1,2},{3,4}}
    Out> {{1,2},{3,4}};
    In> MatrixPower(A,0)
    Out> {{1,0},{0,1}};
    In> MatrixPower(A,1)
    Out> {{1,2},{3,4}};
    In> MatrixPower(A,3)
    Out> {{37,54},{81,118}};
    In> MatrixPower(A,-3)
    Out> {{-59/4,27/4},{81/8,-37/8}};

##### See also:

IsSquareMatrix , Inverse , Dot .

###

SolveMatrix -- solve a linear system

##### Standard library

##### Calling format:

    SolveMatrix(M,v)

##### Parameters:

`M` -- a matrix

`v` -- a vector

##### Description:

`SolveMatrix` returns the vector x that satisfies the equation M*x=v. The determinant of M should be non-zero.

##### Examples:

    In> A := {{1,2}, {3,4}};
    Out> {{1,2},{3,4}};
    In> v := {5,6};
    Out> {5,6};
    In> x := SolveMatrix(A, v);
    Out> {-4,9/2};
    In> A * x;
    Out> {5,6};

##### See also:

Inverse , Solve , PSolve , Determinant .

###

CharacteristicEquation -- get characteristic polynomial of a matrix

##### Standard library

##### Calling format:

    CharacteristicEquation(matrix,var)

##### Parameters:

`matrix` -- a matrix

`var` -- a free variable

##### Description:

CharacteristicEquation returns the characteristic equation of "matrix", using "var". The zeros of this equation are the eigenvalues of the matrix, Det(matrix-I*var);

##### Examples:

    In> A:=DiagonalMatrix({a,b,c})
    Out> {{a,0,0},{0,b,0},{0,0,c}};
    In> B:=CharacteristicEquation(A,x)
    Out> (a-x)*(b-x)*(c-x);
    In> Expand(B,x)
    Out> (b+a+c)*x^2-x^3-((b+a)*c+a*b)*x+a*b*c;

##### See also:

EigenValues , EigenVectors .

###

EigenValues -- get eigenvalues of a matrix

##### Standard library

##### Calling format:

    EigenValues(matrix)

##### Parameters:

`matrix` -- a square matrix

##### Description:

EigenValues returns the eigenvalues of a matrix. The eigenvalues x of a matrix M are the numbers such that M*v=x*v for some vector.

It first determines the characteristic equation, and then factorizes this equation, returning the roots of the characteristic equation Det(matrix-x*identity).

##### Examples:

    In> M:={{1,2},{2,1}}
    Out> {{1,2},{2,1}};
    In> EigenValues(M)
    Out> {3,-1};

##### See also:

EigenVectors , CharacteristicEquation .

###

EigenVectors -- get eigenvectors of a matrix

##### Standard library

##### Calling format:

    EigenVectors(A,eigenvalues)

##### Parameters:

`matrix` -- a square matrix

`eigenvalues` -- list of eigenvalues as returned by `EigenValues`

##### Description:

`EigenVectors` returns a list of the eigenvectors of a matrix. It uses the eigenvalues and the matrix to set up n equations with n unknowns for each eigenvalue, and then calls `Solve` to determine the values of each vector.

##### Examples:

    In> M:={{1,2},{2,1}}
    Out> {{1,2},{2,1}};
    In> e:=EigenValues(M)
    Out> {3,-1};
    In> EigenVectors(M,e)
    Out> {{-ki2/ -1,ki2},{-ki2,ki2}};

##### See also:

EigenValues , CharacteristicEquation .

###

IsHermitian -- test for a Hermitian matrix

##### Standard library

##### Calling format:

    IsHermitian(A)

##### Parameters:

`A` -- a square matrix

##### Description:

IsHermitian(A) returns `True` if `A` is Hermitian and `False` otherwise. A is a Hermitian matrix iff Conjugate( Transpose A )= A. If A is a real matrix, it must be symmetric to be Hermitian.

##### Examples:

    In> IsHermitian({{0,I},{-I,0}})
    Out> True;
    In> IsHermitian({{0,I},{2,0}})
    Out> False;

##### See also:

IsUnitary .

###

IsOrthogonal -- test for an orthogonal matrix

##### Standard library

##### Calling format:

    IsOrthogonal(A)

##### Parameters:

`A` -- square matrix

##### Description:

`IsOrthogonal(A)` returns `True` if `A` is orthogonal and `False` otherwise. A is orthogonal iff A*Transpose( A) = Identity, or equivalently Inverse( A) = Transpose( A).

##### Examples:

    In> A := {{1,2,2},{2,1,-2},{-2,2,-1}};
    Out> {{1,2,2},{2,1,-2},{-2,2,-1}};
    In> PrettyForm(A/3)

    /                      \
    | / 1 \  / 2 \ / 2 \   |
    | | - |  | - | | - |   |
    | \ 3 /  \ 3 / \ 3 /   |
    |                      |
    | / 2 \  / 1 \ / -2 \  |
    | | - |  | - | | -- |  |
    | \ 3 /  \ 3 / \ 3  /  |
    |                      |
    | / -2 \ / 2 \ / -1 \  |
    | | -- | | - | | -- |  |
    | \ 3  / \ 3 / \ 3  /  |
    \                      /
    Out> True;
    In> IsOrthogonal(A/3)
    Out> True;

###

IsDiagonal -- test for a diagonal matrix

##### Standard library

##### Calling format:

    IsDiagonal(A)

##### Parameters:

`A` -- a matrix

##### Description:

`IsDiagonal(A)` returns `True` if `A` is a diagonal square matrix and `False` otherwise.

##### Examples:

    In> IsDiagonal(Identity(5))
    Out> True;
    In> IsDiagonal(HilbertMatrix(5))
    Out> False;

###

IsLowerTriangular, IsUpperTriangular -- test for a lower or an upper triangular matrix

##### Standard library

##### Calling format:

    IsLowerTriangular(A)
    IsUpperTriangular(A)

##### Parameters:

`A` -- a matrix

##### Description:

A lower/upper triangular matrix is a square matrix which has all zero entries above/below the diagonal.

`IsLowerTriangular(A)` returns `True` if `A` is a lower triangular matrix and `False` otherwise. `IsUpperTriangular(A)` returns `True` if `A` is an upper triangular matrix and `False` otherwise.

##### Examples:

    In> IsUpperTriangular(Identity(5))
    Out> True;
    In> IsLowerTriangular(Identity(5))
    Out> True;
    In> IsLowerTriangular({{1,2},{0,1}})
    Out> False;
    In> IsUpperTriangular({{1,2},{0,1}})
    Out> True;

A non-square matrix cannot be triangular:

    In> IsUpperTriangular({{1,2,3},{0,1,2}})
    Out> False;

##### See also:

IsDiagonal .

###

IsSymmetric -- test for a symmetric matrix

##### Standard library

##### Calling format:

    IsSymmetric(A)

##### Parameters:

`A` -- a matrix

##### Description:

`IsSymmetric(A)` returns `True` if `A` is symmetric and `False` otherwise. A is symmetric iff Transpose ( A) = A.

##### Examples:

    In> A := {{1,0,0,0,1},{0,2,0,0,0},{0,0,3,0,0},
      {0,0,0,4,0},{1,0,0,0,5}};
    In> PrettyForm(A)

    /                                \
    | ( 1 ) ( 0 ) ( 0 ) ( 0 ) ( 1 )  |
    |                                |
    | ( 0 ) ( 2 ) ( 0 ) ( 0 ) ( 0 )  |
    |                                |
    | ( 0 ) ( 0 ) ( 3 ) ( 0 ) ( 0 )  |
    |                                |
    | ( 0 ) ( 0 ) ( 0 ) ( 4 ) ( 0 )  |
    |                                |
    | ( 1 ) ( 0 ) ( 0 ) ( 0 ) ( 5 )  |
    \                                /
    Out> True;
    In> IsSymmetric(A)
    Out> True;

##### See also:

IsHermitian , IsSkewSymmetric .

###

IsSkewSymmetric -- test for a skew-symmetric matrix

##### Standard library

##### Calling format:

    IsSkewSymmetric(A)

##### Parameters:

`A` -- a square matrix

##### Description:

`IsSkewSymmetric(A)` returns `True` if `A` is skew symmetric and `False` otherwise. A is skew symmetric iff Transpose(A) =-A.

##### Examples:

    In> A := {{0,-1},{1,0}}
    Out> {{0,-1},{1,0}};
    In> PrettyForm(%)

    /               \
    | ( 0 ) ( -1 )  |
    |               |
    | ( 1 ) ( 0 )   |
    \               /
    Out> True;
    In> IsSkewSymmetric(A);
    Out> True;

##### See also:

IsSymmetric , IsHermitian .

###

IsUnitary -- test for a unitary matrix

##### Standard library

##### Calling format:

    IsUnitary(A)

##### Parameters:

`A` -- a square matrix

##### Description:

This function tries to find out if A is unitary.

A matrix A is orthogonal iff A^(-1) = Transpose( Conjugate(A) ). This is equivalent to the fact that the columns of A build an orthonormal system (with respect to the scalar product defined by `InProduct`).

##### Examples:

    In> IsUnitary({{0,I},{-I,0}})
    Out> True;
    In> IsUnitary({{0,I},{2,0}})
    Out> False;

##### See also:

IsHermitian , IsSymmetric .

###

IsIdempotent -- test for an idempotent matrix

##### Standard library

##### Calling format:

    IsIdempotent(A)

##### Parameters:

`A` -- a square matrix

##### Description:

`IsIdempotent(A)` returns `True` if `A` is idempotent and `False` otherwise. A is idempotent iff A^2=A. Note that this also implies that A raised to any power is also equal to A.

##### Examples:

    In> IsIdempotent(ZeroMatrix(10,10));
    Out> True;
    In> IsIdempotent(Identity(20))
    Out> True;

###

JacobianMatrix -- calculate the Jacobian matrix of n functions in n variables

##### Standard library

##### Calling format:

    JacobianMatrix(functions,variables)

##### Parameters:

`functions` -- an n-dimensional vector of functions

`variables` -- an n-dimensional vector of variables

##### Description:

The function `JacobianMatrix` calculates the Jacobian matrix of n functions in n variables.

The ( i, j)-th element of the Jacobian matrix is defined as the derivative of i-th function with respect to the j-th variable.

##### Examples:

    In> JacobianMatrix( {Sin(x),Cos(y)}, {x,y} );
    Out> {{Cos(x),0},{0,-Sin(y)}};
    In> PrettyForm(%)

    /                                 \
    | ( Cos( x ) ) ( 0 )              |
    |                                 |
    | ( 0 )        ( -( Sin( y ) ) )  |
    \                                 /

###

VandermondeMatrix -- create the Vandermonde matrix

##### Standard library

##### Calling format:

    VandermondeMatrix(vector)

##### Parameters:

`vector` -- an n-dimensional vector

##### Description:

The function `VandermondeMatrix` calculates the Vandermonde matrix of a vector.

The ( i, j)-th element of the Vandermonde matrix is defined as i^(j-1).

##### Examples:

    In> VandermondeMatrix({1,2,3,4})
    Out> {{1,1,1,1},{1,2,3,4},{1,4,9,16},{1,8,27,64}};
    In>PrettyForm(%)

    /                            \
    | ( 1 ) ( 1 ) ( 1 )  ( 1 )   |
    |                            |
    | ( 1 ) ( 2 ) ( 3 )  ( 4 )   |
    |                            |
    | ( 1 ) ( 4 ) ( 9 )  ( 16 )  |
    |                            |
    | ( 1 ) ( 8 ) ( 27 ) ( 64 )  |
    \                            /

###

HessianMatrix -- create the Hessian matrix

##### Standard library

##### Calling format:

    HessianMatrix(function,var)

##### Parameters:

`function` -- a function in n variables

`var` -- an n-dimensional vector of variables

##### Description:

The function `HessianMatrix` calculates the Hessian matrix of a vector.

If f(x) is a function of an n-dimensional vector x, then the ( i, j)-th element of the Hessian matrix of the function f(x) is defined as Deriv(x[i])Deriv(x[j])f(x). If the third order mixed partials are continuous, then the Hessian matrix is symmetric (a standard theorem of calculus).

The Hessian matrix is used in the second derivative test to discern if a critical point is a local maximum, a local minimum or a saddle point.

##### Examples:

    In> HessianMatrix(3*x^2-2*x*y+y^2-8*y, {x,y} )
    Out> {{6,-2},{-2,2}};
    In> PrettyForm(%)

    /                \
    | ( 6 )  ( -2 )  |
    |                |
    | ( -2 ) ( 2 )   |
    \                /

###

Sparsity -- get the sparsity of a matrix

##### Standard library

##### Calling format:

    Sparsity(matrix)

##### Parameters:

`matrix` -- a matrix

##### Description:

The function `Sparsity` returns a number between `0` and `1` which represents the percentage of zero entries in the matrix. Although there is no definite critical value, a sparsity of `0.75` or more is almost universally considered a "sparse" matrix. These type of matrices can be handled in a different manner than "full" matrices which speedup many calculations by orders of magnitude.

##### Examples:

    In> Sparsity(Identity(2))
    Out> 0.5;
    In> Sparsity(Identity(10))
    Out> 0.9;
    In> Sparsity(HankelMatrix(10))
    Out> 0.45;
    In> Sparsity(HankelMatrix(100))
    Out> 0.495;
    In> Sparsity(HilbertMatrix(10))
    Out> 0;
    In> Sparsity(ZeroMatrix(10,10))
    Out> 1;

###

HilbertMatrix -- create a Hilbert matrix

##### Standard library

##### Calling format:

    HilbertMatrix(n)
    HilbertMatrix(n,m)

##### Parameters:

`n,m` -- positive integers

##### Description:

The function `HilbertMatrix` returns the `n` by `m` Hilbert matrix if given two arguments, and the square `n` by `n` Hilbert matrix if given only one. The Hilbert matrix is defined as `A(i,j) = 1/(i+j-1)`. The Hilbert matrix is extremely sensitive to manipulate and invert numerically.

##### Examples:

    In> PrettyForm(HilbertMatrix(4))

    /                          \
    | ( 1 ) / 1 \ / 1 \ / 1 \  |
    |       | - | | - | | - |  |
    |       \ 2 / \ 3 / \ 4 /  |
    |                          |
    | / 1 \ / 1 \ / 1 \ / 1 \  |
    | | - | | - | | - | | - |  |
    | \ 2 / \ 3 / \ 4 / \ 5 /  |
    |                          |
    | / 1 \ / 1 \ / 1 \ / 1 \  |
    | | - | | - | | - | | - |  |
    | \ 3 / \ 4 / \ 5 / \ 6 /  |
    |                          |
    | / 1 \ / 1 \ / 1 \ / 1 \  |
    | | - | | - | | - | | - |  |
    | \ 4 / \ 5 / \ 6 / \ 7 /  |
    \                          /

##### See also:

HilbertInverseMatrix .

###

HilbertInverseMatrix -- create a Hilbert inverse matrix

##### Standard library

##### Calling format:

    HilbertInverseMatrix(n)

##### Parameters:

`n` -- positive integer

##### Description:

The function `HilbertInverseMatrix` returns the `n` by `n` inverse of the corresponding Hilbert matrix. All Hilbert inverse matrices have integer entries that grow in magnitude rapidly.

##### Examples:

    In> PrettyForm(HilbertInverseMatrix(4))

    /                                         \
    | ( 16 )   ( -120 )  ( 240 )   ( -140 )   |
    |                                         |
    | ( -120 ) ( 1200 )  ( -2700 ) ( 1680 )   |
    |                                         |
    | ( 240 )  ( -2700 ) ( 6480 )  ( -4200 )  |
    |                                         |
    | ( -140 ) ( 1680 )  ( -4200 ) ( 2800 )   |
    \                                         /

##### See also:

HilbertMatrix .

###

ToeplitzMatrix -- create a Toeplitz matrix

##### Standard library

##### Calling format:

    ToeplitzMatrix(N)

##### Parameters:

`N` -- an n-dimensional row vector

##### Description:

The function `ToeplitzMatrix` calculates the Toeplitz matrix given an n-dimensional row vector. This matrix has the same entries in all diagonal columns, from upper left to lower right.

##### Examples:

    In> PrettyForm(ToeplitzMatrix({1,2,3,4,5}))

    /                                \
    | ( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 )  |
    |                                |
    | ( 2 ) ( 1 ) ( 2 ) ( 3 ) ( 4 )  |
    |                                |
    | ( 3 ) ( 2 ) ( 1 ) ( 2 ) ( 3 )  |
    |                                |
    | ( 4 ) ( 3 ) ( 2 ) ( 1 ) ( 2 )  |
    |                                |
    | ( 5 ) ( 4 ) ( 3 ) ( 2 ) ( 1 )  |
    \                                /

###

WronskianMatrix -- create the Wronskian matrix

##### Standard library

##### Calling format:

    WronskianMatrix(func,var)

##### Parameters:

`func` -- an n-dimensional vector of functions

`var` -- a variable to differentiate with respect to

##### Description:

The function `WronskianMatrix` calculates the Wronskian matrix of n functions.

The Wronskian matrix is created by putting each function as the first element of each column, and filling in the rest of each column by the ( i-1)-th derivative, where i is the current row.

The Wronskian matrix is used to verify that the n functions are linearly independent, usually solutions to a differential equation. If the determinant of the Wronskian matrix is zero, then the functions are dependent, otherwise they are independent.

##### Examples:

    In> WronskianMatrix({Sin(x),Cos(x),x^4},x);
    Out> {{Sin(x),Cos(x),x^4},{Cos(x),-Sin(x),4*x^3},
      {-Sin(x),-Cos(x),12*x^2}};
    In> PrettyForm(%)

    /                                                 \
    | ( Sin( x ) )      ( Cos( x ) )      /  4 \      |
    |                                     \ x  /      |
    |                                                 |
    | ( Cos( x ) )      ( -( Sin( x ) ) ) /      3 \  |
    |                                     \ 4 * x  /  |
    |                                                 |
    | ( -( Sin( x ) ) ) ( -( Cos( x ) ) ) /       2 \ |
    |                                     \ 12 * x  / |
    \                                                 /

The last element is a linear combination of the first two, so the determinant is zero:

    In> A:=Determinant( WronskianMatrix( {x^4,x^3,2*x^4
      + 3*x^3},x ) )
    Out> x^4*3*x^2*(24*x^2+18*x)-x^4*(8*x^3+9*x^2)*6*x
      +(2*x^4+3*x^3)*4*x^3*6*x-4*x^6*(24*x^2+18*x)+x^3
      *(8*x^3+9*x^2)*12*x^2-(2*x^4+3*x^3)*3*x^2*12*x^2;
    In> Simplify(A)
    Out> 0;

###

SylvesterMatrix -- calculate the Sylvester matrix of two polynomials

##### Standard library

##### Calling format:

    SylvesterMatrix(poly1,poly2,variable)

##### Parameters:

`poly1` -- polynomial

`poly2` -- polynomial

`variable` -- variable to express the matrix for

##### Description:

The function `SylvesterMatrix` calculates the Sylvester matrix for a pair of polynomials.

The Sylvester matrix is closely related to the resultant, which is defined as the determinant of the Sylvester matrix. Two polynomials share common roots only if the resultant is zero.

##### Examples:

    In> ex1:= x^2+2*x-a
    Out> x^2+2*x-a;
    In> ex2:= x^2+a*x-4
    Out> x^2+a*x-4;
    In> A:=SylvesterMatrix(ex1,ex2,x)
    Out> {{1,2,-a,0},{0,1,2,-a},
      {1,a,-4,0},{0,1,a,-4}};
    In> B:=Determinant(A)
    Out> 16-a^2*a- -8*a-4*a+a^2- -2*a^2-16-4*a;
    In> Simplify(B)
    Out> 3*a^2-a^3;

The above example shows that the two polynomials have common zeros if a=3.

##### See also:

Determinant , Simplify , Solve , PSolve .

###

MatrixSolve -- solve a system of equations

##### Standard library

##### Calling format:

    MatrixSolve(A,b)

##### Parameters:

`A` -- coefficient matrix

`b` -- row vector

##### Description:

`MatrixSolve` solves the matrix equations `A*x = b` using Gaussian Elimination with Backward substitution. If your matrix is triangular or diagonal, it will be recognized as such and a faster algorithm will be used.

##### Examples:

    In> A:={{2,4,-2,-2},{1,2,4,-3},{-3,-3,8,-2},{-1,1,6,-3}};
    Out> {{2,4,-2,-2},{1,2,4,-3},{-3,-3,8,-2},{-1,1,6,-3}};
    In> b:={-4,5,7,7};
    Out> {-4,5,7,7};
    In> MatrixSolve(A,b);
    Out> {1,2,3,4};

###

Cholesky -- find the Cholesky Decomposition

##### Standard library

##### Calling format:

    Cholesky(A)

##### Parameters:

`A` -- a square positive definite matrix

##### Description:

`Cholesky` returns a upper triangular matrix `R` such that `Transpose(R)*R = A`. The matrix `A` must be positive definite, `Cholesky` will notify the user if the matrix is not. Some families of positive definite matrices are all symmetric matrices, diagonal matrices with positive elements and Hilbert matrices.

##### Examples:

    In> A:={{4,-2,4,2},{-2,10,-2,-7},{4,-2,8,4},{2,-7,4,7}}
    Out> {{4,-2,4,2},{-2,10,-2,-7},{4,-2,8,4},{2,-7,4,7}};
    In> R:=Cholesky(A);
    Out> {{2,-1,2,1},{0,3,0,-2},{0,0,2,1},{0,0,0,1}};
    In> Transpose(R)*R = A
    Out> True;
    In> Cholesky(4*Identity(5))
    Out> {{2,0,0,0,0},{0,2,0,0,0},{0,0,2,0,0},{0,0,0,2,0},{0,0,0,0,2}};
    In> Cholesky(HilbertMatrix(3))
    Out> {{1,1/2,1/3},{0,Sqrt(1/12),Sqrt(1/12)},{0,0,Sqrt(1/180)}};
    In> Cholesky(ToeplitzMatrix({1,2,3}))
    In function "Check" :
    CommandLine(1) : "Cholesky: Matrix is not positive definite"

##### See also:

IsSymmetric , IsDiagonal , Diagonal .

###

RandomIntegerMatrix -- generate a matrix of random integers

##### Standard library

##### Calling format:

    RandomIntegerMatrix(rows,cols,from,to)

##### Parameters:

`rows` -- number of rows in matrix

`cols` -- number of cols in matrix

`from` -- lower bound

`to` -- upper bound

##### Description:

This function generates a `rows x cols` matrix of random integers. All entries lie between "from" and "to", including the boundaries, and are uniformly distributed in this interval.

##### Examples:

    In> PrettyForm( RandomIntegerMatrix(5,5,-2^10,2^10) )

    /                                               \
    | ( -506 ) ( 749 )  ( -574 ) ( -674 ) ( -106 )  |
    |                                               |
    | ( 301 )  ( 151 )  ( -326 ) ( -56 )  ( -277 )  |
    |                                               |
    | ( 777 )  ( -761 ) ( -161 ) ( -918 ) ( -417 )  |
    |                                               |
    | ( -518 ) ( 127 )  ( 136 )  ( 797 )  ( -406 )  |
    |                                               |
    | ( 679 )  ( 854 )  ( -78 )  ( 503 )  ( 772 )   |
    \                                               /

##### See also:

RandomIntegerVector , RandomPoly .

Expand , Degree , Coef , Content , PrimitivePart , LeadingCoef , Monic , RandomPoly , Horner , ExpandBrackets , EvaluateHornerScheme , OrthoP , OrthoH , OrthoG , OrthoL , OrthoT, OrthoU , OrthoPSum, OrthoHSum, OrthoLSum, OrthoGSum, OrthoTSum, OrthoUSum , OrthoPoly , OrthoPolySum , SquareFree , FindRealRoots , NumRealRoots , MinimumBound, MaximumBound .

# Polynomials

This chapter contains commands to manipulate polynomials. This includes functions for constructing and evaluating orthogonal polynomials.

###

Expand -- transform a polynomial to an expanded form

##### Standard library

##### Calling format:

    Expand(expr)
    Expand(expr, var)
    Expand(expr, varlist)

##### Parameters:

`expr` -- a polynomial expression

`var` -- a variable

`varlist` -- a list of variables

##### Description:

This command brings a polynomial in expanded form, in which polynomials are represented in the form c0+c1*x+c2*x^2+...+c[n]*x^n. In this form, it is easier to test whether a polynomial is zero, namely by testing whether all coefficients are zero.

If the polynomial "expr" contains only one variable, the first calling sequence can be used. Otherwise, the second form should be used which explicitly mentions that "expr" should be considered as a polynomial in the variable "var". The third calling form can be used for multivariate polynomials. Firstly, the polynomial "expr" is expanded with respect to the first variable in "varlist". Then the coefficients are all expanded with respect to the second variable, and so on.

##### Examples:

    In> PrettyPrinter("PrettyForm");

    True

    In> Expand((1+x)^5);

     5        4         3         2
    x  + 5 * x  + 10 * x  + 10 * x  + 5 * x + 1

    In> Expand((1+x-y)^2, x);

     2                                2
    x  + 2 * ( 1 - y ) * x + ( 1 - y )

    In> Expand((1+x-y)^2, {x,y});

     2                         2
    x  + ( -2 * y + 2 ) * x + y  - 2 * y + 1

##### See also:

ExpandBrackets .

###

Degree -- degree of a polynomial

##### Standard library

##### Calling format:

    Degree(expr)
    Degree(expr, var)

##### Parameters:

`expr` -- a polynomial

`var` -- a variable occurring in "expr"

##### Description:

This command returns the degree of the polynomial "expr" with respect to the variable "var". The degree is the highest power of "var" occurring in the polynomial. If only one variable occurs in "expr", the first calling sequence can be used. Otherwise the user should use the second form in which the variable is explicitly mentioned.

##### Examples:

    In> Degree(x^5+x-1);
    Out> 5;
    In> Degree(a+b*x^3, a);
    Out> 1;
    In> Degree(a+b*x^3, x);
    Out> 3;

##### See also:

Expand , Coef .

###

Coef -- coefficient of a polynomial

##### Standard library

##### Calling format:

    Coef(expr, var, order)

##### Parameters:

`expr` -- a polynomial

`var` -- a variable occurring in "expr"

`order` -- integer or list of integers

##### Description:

This command returns the coefficient of "var" to the power "order" in the polynomial "expr". The parameter "order" can also be a list of integers, in which case this function returns a list of coefficients.

##### Examples:

    In> e := Expand((a+x)^4,x)
    Out> x^4+4*a*x^3+(a^2+(2*a)^2+a^2)*x^2+
    (a^2*2*a+2*a^3)*x+a^4;
    In> Coef(e,a,2)
    Out> 6*x^2;
    In> Coef(e,a,0 .. 4)
    Out> {x^4,4*x^3,6*x^2,4*x,1};

##### See also:

Expand , Degree , LeadingCoef .

###

Content -- content of a univariate polynomial

##### Standard library

##### Calling format:

    Content(expr)

##### Parameters:

`expr` -- univariate polynomial

##### Description:

This command determines the content of a univariate polynomial. The content is the greatest common divisor of all the terms in the polynomial. Every polynomial can be written as the product of the content with the primitive part.

##### Examples:

    In> poly := 2*x^2 + 4*x;
    Out> 2*x^2+4*x;
    In> c := Content(poly);
    Out> 2*x;
    In> pp := PrimitivePart(poly);
    Out> x+2;
    In> Expand(pp*c);
    Out> 2*x^2+4*x;

##### See also:

PrimitivePart , Gcd .

###

PrimitivePart -- primitive part of a univariate polynomial

##### Standard library

##### Calling format:

    PrimitivePart(expr)

##### Parameters:

`expr` -- univariate polynomial

##### Description:

This command determines the primitive part of a univariate polynomial. The primitive part is what remains after the content (the greatest common divisor of all the terms) is divided out. So the product of the content and the primitive part equals the original polynomial.

##### Examples:

    In> poly := 2*x^2 + 4*x;
    Out> 2*x^2+4*x;
    In> c := Content(poly);
    Out> 2*x;
    In> pp := PrimitivePart(poly);
    Out> x+2;
    In> Expand(pp*c);
    Out> 2*x^2+4*x;

##### See also:

Content .

###

LeadingCoef -- leading coefficient of a polynomial

##### Standard library

##### Calling format:

    LeadingCoef(poly)
    LeadingCoef(poly, var)

##### Parameters:

`poly` -- a polynomial

`var` -- a variable

##### Description:

This function returns the leading coefficient of "poly", regarded as a polynomial in the variable "var". The leading coefficient is the coefficient of the term of highest degree. If only one variable appears in the expression "poly", it is obvious that it should be regarded as a polynomial in this variable and the first calling sequence may be used.

##### Examples:

    In> poly := 2*x^2 + 4*x;
    Out> 2*x^2+4*x;
    In> lc := LeadingCoef(poly);
    Out> 2;
    In> m := Monic(poly);
    Out> x^2+2*x;
    In> Expand(lc*m);
    Out> 2*x^2+4*x;

    In> LeadingCoef(2*a^2 + 3*a*b^2 + 5, a);
    Out> 2;
    In> LeadingCoef(2*a^2 + 3*a*b^2 + 5, b);
    Out> 3*a;

##### See also:

Coef , Monic .

###

Monic -- monic part of a polynomial

##### Standard library

##### Calling format:

    Monic(poly)
    Monic(poly, var)

##### Parameters:

`poly` -- a polynomial

`var` -- a variable

##### Description:

This function returns the monic part of "poly", regarded as a polynomial in the variable "var". The monic part of a polynomial is the quotient of this polynomial by its leading coefficient. So the leading coefficient of the monic part is always one. If only one variable appears in the expression "poly", it is obvious that it should be regarded as a polynomial in this variable and the first calling sequence may be used.

##### Examples:

    In> poly := 2*x^2 + 4*x;
    Out> 2*x^2+4*x;
    In> lc := LeadingCoef(poly);
    Out> 2;
    In> m := Monic(poly);
    Out> x^2+2*x;
    In> Expand(lc*m);
    Out> 2*x^2+4*x;

    In> Monic(2*a^2 + 3*a*b^2 + 5, a);
    Out> a^2+(a*3*b^2)/2+5/2;
    In> Monic(2*a^2 + 3*a*b^2 + 5, b);
    Out> b^2+(2*a^2+5)/(3*a);

##### See also:

LeadingCoef .

###

RandomPoly -- construct a random polynomial

##### Standard library

##### Calling format:

    RandomPoly(var,deg,coefmin,coefmax)

##### Parameters:

`var` -- free variable for resulting univariate polynomial

`deg` -- degree of resulting univariate polynomial

`coefmin` -- minimum value for coefficients

`coefmax` -- maximum value for coefficients

##### Description:

RandomPoly generates a random polynomial in variable "var", of degree "deg", with integer coefficients ranging from "coefmin" to "coefmax" (inclusive). The coefficients are uniformly distributed in this interval, and are independent of each other.

##### Examples:

    In> RandomPoly(x,3,-10,10)
    Out> 3*x^3+10*x^2-4*x-6;
    In> RandomPoly(x,3,-10,10)
    Out> -2*x^3-8*x^2+8;

##### See also:

Random , RandomIntegerVector .

###

`Div` and `Mod` for polynomials

##### Standard library

`Div` and `Mod` are also defined for polynomials.

##### See also:

Div , Mod .

###

Horner -- convert a polynomial into the Horner form

##### Standard library

##### Calling format:

    Horner(expr, var)

##### Parameters:

`expr` -- a polynomial in "var"

`var` -- a variable

##### Description:

This command turns the polynomial "expr", considered as a univariate polynomial in "var", into Horner form. A polynomial in normal form is an expression such as

c[0]+c[1]*x+...+c[n]*x^n.

If one converts this polynomial into Horner form, one gets the equivalent expression

(...(c[n]*x+c[n-1])*x+...+c[1])*x+c[0].

Both expression are equal, but the latter form gives a more efficient way to evaluate the polynomial as the powers have disappeared.

##### Examples:

    In> expr1:=Expand((1+x)^4)
    Out> x^4+4*x^3+6*x^2+4*x+1;
    In> Horner(expr1,x)
    Out> (((x+4)*x+6)*x+4)*x+1;

##### See also:

Expand , ExpandBrackets , EvaluateHornerScheme .

###

ExpandBrackets -- expand all brackets

##### Standard library

##### Calling format:

    ExpandBrackets(expr)

##### Parameters:

`expr` -- an expression

##### Description:

This command tries to expand all the brackets by repeatedly using the distributive laws a*(b+c)=a*b+a*c and (a+b)*c=a*c+b*c. It goes further than `Expand`, in that it expands all brackets.

##### Examples:

    In> Expand((a-x)*(b-x),x)
    Out> x^2-(b+a)*x+a*b;
    In> Expand((a-x)*(b-x),{x,a,b})
    Out> x^2-(b+a)*x+b*a;
    In> ExpandBrackets((a-x)*(b-x))
    Out> a*b-x*b+x^2-a*x;

##### See also:

Expand .

###

EvaluateHornerScheme -- fast evaluation of polynomials

##### Standard library

##### Calling format:

    EvaluateHornerScheme(coeffs,x)

##### Parameters:

`coeffs` -- a list of coefficients

`x` -- expression

##### Description:

This function evaluates a polynomial given as a list of its coefficients, using the Horner scheme. The list of coefficients starts with the 0-th power.

##### Example:

    In> EvaluateHornerScheme({a,b,c,d},x)
    Out> a+x*(b+x*(c+x*d));

##### See also:

Horner .

###

OrthoP -- Legendre and Jacobi orthogonal polynomials

##### Standard library

##### Calling format:

    OrthoP(n, x);
    OrthoP(n, a, b, x);

##### Parameters:

`n` -- degree of polynomial

`x` -- point to evaluate polynomial at

`a`, `b` -- parameters for Jacobi polynomial

##### Description:

The first calling format with two arguments evaluates the Legendre polynomial of degree `n` at the point `x`. The second form does the same for the Jacobi polynomial with parameters `a` and `b`, which should be both greater than -1.

The Jacobi polynomials are orthogonal with respect to the weight function (1-x)^a*(1+x)^b on the interval [-1,1]. They satisfy the recurrence relation

P(n,a,b,x)=(2*n+a+b-1)/(2*n+a+b-2)*

(a^2-b^2+x*(2*n+a+b-2)*(n+a+b))/(2*n*(n+a+b))*P(n-1,a,b,x)

-((n+a-1)*(n+b-1)*(2*n+a+b))/(n*(n+a+b)*(2*n+a+b-2))*P(n-2,a,b,x)
for n>1, with P(0,a,b,x)=1,

P(1,a,b,x)=(a-b)/2+x*(1+(a+b)/2).

Legendre polynomials are a special case of Jacobi polynomials with the specific parameter values a=b=0. So they form an orthogonal system with respect to the weight function identically equal to 1 on the interval [-1,1], and they satisfy the recurrence relation

P(n,x)=(2*n-1)*x/(2*n)*P(n-1,x)-(n-1)/n*P(n-2,x)
for n>1, with P(0,x)=1, P(1,x)=x.

Most of the work is performed by the internal function `OrthoPoly`.

##### Examples:

    In> PrettyPrinter("PrettyForm");

    True

    In> OrthoP(3, x);

        /      2     \
        | 5 * x    3 |
    x * | ------ - - |
        \   2      2 /

    In> OrthoP(3, 1, 2, x);

    1       /     / 21 * x   7 \   7 \
    - + x * | x * | ------ - - | - - |
    2       \     \   2      2 /   2 /

    In> Expand(%)

          3        2
    21 * x  - 7 * x  - 7 * x + 1
    ----------------------------
                 2

    In> OrthoP(3, 1, 2, 0.5);

    -0.8124999999

##### See also:

OrthoPSum , OrthoG , OrthoPoly .

###

OrthoH -- Hermite orthogonal polynomials

##### Standard library

##### Calling format:

    OrthoH(n, x);

##### Parameters:

`n` -- degree of polynomial

`x` -- point to evaluate polynomial at

##### Description:

This function evaluates the Hermite polynomial of degree `n` at the point `x`.

The Hermite polynomials are orthogonal with respect to the weight function Exp(-x^2/2) on the entire real axis. They satisfy the recurrence relation

H(n,x)=2*x*H(n-1,x)-2*(n-1)*H(n-2,x)
for n>1, with H(0,x)=1, H(1,x)=2*x.

Most of the work is performed by the internal function `OrthoPoly`.

##### Examples:

    In> OrthoH(3, x);
    Out> x*(8*x^2-12);
    In> OrthoH(6, 0.5);
    Out> 31;

##### See also:

OrthoHSum , OrthoPoly .

###

OrthoG -- Gegenbauer orthogonal polynomials

##### Standard library

##### Calling format:

    OrthoG(n, a, x);

##### Parameters:

`n` -- degree of polynomial

`a` -- parameter

`x` -- point to evaluate polynomial at

##### Description:

This function evaluates the Gegenbauer (or ultraspherical) polynomial with parameter `a` and degree `n` at the point `x`. The parameter `a` should be greater than -1/2.

The Gegenbauer polynomials are orthogonal with respect to the weight function (1-x^2)^(a-1/2) on the interval [-1,1]. Hence they are connected to the Jacobi polynomials via

G(n,a,x)=P(n,a-1/2,a-1/2,x).
They satisfy the recurrence relation

G(n,a,x)=2*(1+(a-1)/n)*x*G(n-1,a,x)

-(1+2*(a-2)/n)*G(n-2,a,x)
for n>1, with G(0,a,x)=1, G(1,a,x)=2*x.

Most of the work is performed by the internal function `OrthoPoly`.

##### Examples:

    In> OrthoG(5, 1, x);
    Out> x*((32*x^2-32)*x^2+6);
    In> OrthoG(5, 2, -0.5);
    Out> 2;

##### See also:

OrthoP , OrthoT , OrthoU , OrthoGSum , OrthoPoly .

###

OrthoL -- Laguerre orthogonal polynomials

##### Standard library

##### Calling format:

    OrthoL(n, a, x);

##### Parameters:

`n` -- degree of polynomial

`a` -- parameter

`x` -- point to evaluate polynomial at

##### Description:

This function evaluates the Laguerre polynomial with parameter `a` and degree `n` at the point `x`. The parameter `a` should be greater than -1.

The Laguerre polynomials are orthogonal with respect to the weight function x^a*Exp(-x) on the positive real axis. They satisfy the recurrence relation

L(n,a,x)=(2+(a-1-x)/n)*L(n-1,a,x)

-(1-(a-1)/n)*L(n-2,a,x)
for n>1, with L(0,a,x)=1, L(1,a,x)=a+1-x.

Most of the work is performed by the internal function `OrthoPoly`.

##### Examples:

    In> OrthoL(3, 1, x);
    Out> x*(x*(2-x/6)-6)+4;
    In> OrthoL(3, 1/2, 0.25);
    Out> 1.2005208334;

##### See also:

OrthoLSum , OrthoPoly .

###

OrthoT, OrthoU -- Chebyshev polynomials

##### Standard library

##### Calling format:

    OrthoT(n, x);
    OrthoU(n, x);

##### Parameters:

`n` -- degree of polynomial

`x` -- point to evaluate polynomial at

##### Description:

These functions evaluate the Chebyshev polynomials of the first kind T(n,x) and of the second kind U(n,x), of degree "n" at the point "x". (The name of this Russian mathematician is also sometimes spelled "Tschebyscheff".)

The Chebyshev polynomials are orthogonal with respect to the weight function (1-x^2)^(-1/2). Hence they are a special case of the Gegenbauer polynomials G(n,a,x), with a=0. They satisfy the recurrence relations

T(n,x)=2*x*T(n-1,x)-T(n-2,x),

U(n,x)=2*x*U(n-1,x)-U(n-2,x)
for n>1, with T(0,x)=1, T(1,x)=x, U(0,x)=1, U(1,x)=2*x.

##### Examples:

    In> OrthoT(3, x);
    Out> 2*x*(2*x^2-1)-x;
    In> OrthoT(10, 0.9);
    Out> -0.2007474688;
    In> OrthoU(3, x);
    Out> 4*x*(2*x^2-1);
    In> OrthoU(10, 0.9);
    Out> -2.2234571776;

##### See also:

OrthoG , OrthoTSum , OrthoUSum , OrthoPoly .

###

OrthoPSum, OrthoHSum, OrthoLSum, OrthoGSum, OrthoTSum, OrthoUSum -- sums of series of orthogonal polynomials

##### Standard library

##### Calling format:

    OrthoPSum(c, x);
    OrthoPSum(c, a, b, x);
    OrthoHSum(c, x);
    OrthoLSum(c, a, x);
    OrthoGSum(c, a, x);
    OrthoTSum(c, x);
    OrthoUSum(c, x);

##### Parameters:

`c` -- list of coefficients

`a`, `b` -- parameters of specific polynomials

`x` -- point to evaluate polynomial at

##### Description:

These functions evaluate the sum of series of orthogonal polynomials at the point `x`, with given list of coefficients `c` of the series and fixed polynomial parameters `a`, `b` (if applicable).

The list of coefficients starts with the lowest order, so that for example OrthoLSum(c, a, x) = c[1] L[0](a,x) + c[2] L[1](a,x) + ... + c[N] L[N-1](a,x).

See pages for specific orthogonal polynomials for more details on the parameters of the polynomials.

Most of the work is performed by the internal function `OrthoPolySum`. The individual polynomials entering the series are not computed, only the sum of the series.

##### Examples:

    In> Expand(OrthoPSum({1,0,0,1/7,1/8}, 3/2, \
      2/3, x));
    Out> (7068985*x^4)/3981312+(1648577*x^3)/995328+
    (-3502049*x^2)/4644864+(-4372969*x)/6967296
    +28292143/27869184;

##### See also:

OrthoP , OrthoG , OrthoH , OrthoL , OrthoT , OrthoU , OrthoPolySum .

###

OrthoPoly -- internal function for constructing orthogonal polynomials

##### Standard library

##### Calling format:

    OrthoPoly(name, n, par, x)

##### Parameters:

`name` -- string containing name of orthogonal family

`n` -- degree of the polynomial

`par` -- list of values for the parameters

`x` -- point to evaluate at

##### Description:

This function is used internally to construct orthogonal polynomials. It returns the `n`-th polynomial from the family `name` with parameters `par` at the point `x`.

All known families are stored in the association list `KnownOrthoPoly`. The name serves as key. At the moment the following names are known to Yacas: `"Jacobi"`, `"Gegenbauer"`, `"Laguerre"`, `"Hermite"`, `"Tscheb1"`, and `"Tscheb2"`. The value associated to the key is a pure function that takes two arguments: the order `n` and the extra parameters `p`, and returns a list of two lists: the first list contains the coefficients `A,B` of the n=1 polynomial, i.e. A+B*x; the second list contains the coefficients `A,B,C` in the recurrence relation, i.e. P[n]=(A+B*x)*P[n-1]+C*P[n-2]. (There are only 3 coefficients in the second list, because none of the polynomials use C+D*x instead of C in the recurrence relation. This is assumed in the implementation!)

If the argument `x` is numerical, the function `OrthoPolyNumeric` is called. Otherwise, the function `OrthoPolyCoeffs` computes a list of coefficients, and `EvaluateHornerScheme` converts this list into a polynomial expression.

##### See also:

OrthoP , OrthoG , OrthoH , OrthoL , OrthoT , OrthoU , OrthoPolySum .

###

OrthoPolySum -- internal function for computing series of orthogonal polynomials

##### Standard library

##### Calling format:

    OrthoPolySum(name, c, par, x)

##### Parameters:

`name` -- string containing name of orthogonal family

`c` -- list of coefficients

`par` -- list of values for the parameters

`x` -- point to evaluate at

##### Description:

This function is used internally to compute series of orthogonal polynomials. It is similar to the function `OrthoPoly` and returns the result of the summation of series of polynomials from the family `name` with parameters `par` at the point `x`, where `c` is the list of coefficients of the series.

The algorithm used to compute the series without first computing the individual polynomials is the Clenshaw-Smith recurrence scheme. (See the algorithms book for explanations.)

If the argument `x` is numerical, the function `OrthoPolySumNumeric` is called. Otherwise, the function `OrthoPolySumCoeffs` computes the list of coefficients of the resulting polynomial, and `EvaluateHornerScheme` converts this list into a polynomial expression.

##### See also:

OrthoPSum , OrthoGSum , OrthoHSum , OrthoLSum , OrthoTSum , OrthoUSum , OrthoPoly .

###

SquareFree -- return the square-free part of polynomial

##### Standard library

##### Calling format:

    SquareFree(p)

##### Parameters:

`p` - a polynomial in `x`

##### Description:

Given a polynomial

p=p[1]^n[1]*...*p[m]^n[m]
with irreducible polynomials p[i], return the square-free version part (with all the factors having multiplicity 1):

p[1]*...*p[m]

##### Examples:

    In> Expand((x+1)^5)
    Out> x^5+5*x^4+10*x^3+10*x^2+5*x+1;
    In> SquareFree(%)
    Out> (x+1)/5;
    In> Monic(%)
    Out> x+1;

##### See also:

FindRealRoots , NumRealRoots , MinimumBound , MaximumBound , Factor .

###

FindRealRoots -- find the real roots of a polynomial

##### Standard library

##### Calling format:

    FindRealRoots(p)

##### Parameters:

`p` - a polynomial in `x`

##### Description:

Return a list with the real roots of p. It tries to find the real-valued roots, and thus requires numeric floating point calculations. The precision of the result can be improved by increasing the calculation precision.

##### Examples:

    In> p:=Expand((x+3.1)^5*(x-6.23))
    Out> x^6+9.27*x^5-0.465*x^4-300.793*x^3-
    1394.2188*x^2-2590.476405*x-1783.5961073;
    In> FindRealRoots(p)
    Out> {-3.1,6.23};

##### See also:

SquareFree , NumRealRoots , MinimumBound , MaximumBound , Factor .

###

NumRealRoots -- return the number of real roots of a polynomial

##### Standard library

##### Calling format:

    NumRealRoots(p)

##### Parameters:

`p` - a polynomial in `x`

##### Description:

Returns the number of real roots of a polynomial p. The polynomial must use the variable `x` and no other variables.

##### Examples:

    In> NumRealRoots(x^2-1)
    Out> 2;
    In> NumRealRoots(x^2+1)
    Out> 0;

##### See also:

FindRealRoots , SquareFree , MinimumBound , MaximumBound , Factor .

###

MinimumBound, MaximumBound -- return upper/lower bounds on the absolute values of real roots of a polynomial

##### Standard library

##### Calling format:

    MinimumBound(p)
    MaximumBound(p)

##### Parameters:

`p` - a polynomial in x

##### Description:

Return minimum and maximum bounds for the absolute values of the real roots of a polynomial `p`. The polynomial has to be converted to one with rational coefficients first, and be made square-free. The polynomial must use the variable `x`.

##### Examples:

    In> p:=SquareFree(Rationalize((x-3.1)*(x+6.23)))
    Out> (-40000*x^2-125200*x+772520)/870489;
    In> MinimumBound(p)
    Out> 5000000000/2275491039;
    In> N(%)
    Out> 2.1973279236;
    In> MaximumBound(p)
    Out> 10986639613/1250000000;
    In> N(%)
    Out> 8.7893116904;

##### See also:

SquareFree , NumRealRoots , FindRealRoots , Factor .

Head , Tail , Length , Map , MapSingle , RandomIntegerVector , MakeVector , Select , Nth , DestructiveReverse , Reverse , List , UnList , Listify , Concat , Delete , Insert , DestructiveDelete , DestructiveInsert , Replace , DestructiveReplace , FlatCopy , Contains , Find , Append , DestructiveAppend , RemoveDuplicates , Push , Pop , PopFront , PopBack , Swap , Count , Intersection , Union , Difference , FillList , Drop , Take , Partition , Assoc , AssocIndices , AssocDelete , Flatten , UnFlatten , Type , NrArgs , VarList, VarListArith, VarListSome , FuncList, FuncListArith, FuncListSome , BubbleSort, HeapSort , PrintList , Table , TableForm , GlobalPop, GlobalPush .

# List operations

Most objects that can be of variable size are represented as lists (linked lists internally). Yacas does implement arrays, which are faster when the number of elements in a collection of objects doesn't change. Operations on lists have better support in the current system.

###

Head -- the first element of a list

##### Internal function

##### Calling format:

    Head(list)

##### Parameters:

`list` -- a list

##### Description:

This function returns the first element of a list. If it is applied to a general expression, it returns the first operand. An error is returned if "list" is an atom.

##### Examples:

    In> Head({a,b,c})
    Out> a;
    In> Head(f(a,b,c));
    Out> a;

##### See also:

Tail , Length .

###

Tail -- returns a list without its first element

##### Internal function

##### Calling format:

    Tail(list)

##### Parameters:

`list` -- a list

##### Description:

This function returns "list" without its first element.

##### Examples:

    In> Tail({a,b,c})
    Out> {b,c};

##### See also:

Head , Length .

###

Length -- the length of a list or string

##### Internal function

##### Calling format:

    Length(object)

##### Parameters:

`object` -- a list, array or string

##### Description:

Length returns the length of a list. This function also works on strings and arrays.

##### Examples:

    In> Length({a,b,c})
    Out> 3;
    In> Length("abcdef");
    Out> 6;

##### See also:

Head , Tail , Nth , Count .

###

Map -- apply an n-ary function to all entries in a list

##### Standard library

##### Calling format:

    Map(fn, list)

##### Parameters:

`fn` -- function to apply

`list` -- list of lists of arguments

##### Description:

This function applies "fn" to every list of arguments to be found in "list". So the first entry of "list" should be a list containing the first, second, third, ... argument to "fn", and the same goes for the other entries of "list". The function can either be given as a string or as a pure function.

##### Examples:

    In> MapSingle("Sin",{a,b,c});
    Out> {Sin(a),Sin(b),Sin(c)};
    In> Map("+",{{a,b},{c,d}});
    Out> {a+c,b+d};

##### See also:

MapSingle , MapArgs .

###

MapSingle -- apply a unary function to all entries in a list

##### Standard library

##### Calling format:

    MapSingle(fn, list)

##### Parameters:

`fn` -- function to apply

`list` -- list of arguments

##### Description:

The function "fn" is successively applied to all entries in "list", and a list containing the respective results is returned. The function can be given either as a string or as a pure function.

The `/@` operator provides a shorthand for `MapSingle`.

##### Examples:

    In> MapSingle("Sin",{a,b,c});
    Out> {Sin(a),Sin(b),Sin(c)};
    In> MapSingle({{x},x^2}, {a,2,c});
    Out> {a^2,4,c^2};

##### See also:

Map , MapArgs , /@ .

###

RandomIntegerVector -- generate a vector of random integers

##### Standard library

##### Calling format:

    RandomIntegerVector(nr, from, to)

##### Parameters:

`nr` -- number of integers to generate

`from` -- lower bound

`to` -- upper bound

##### Description:

This function generates a list with "nr" random integers. All entries lie between "from" and "to", including the boundaries, and are uniformly distributed in this interval.

##### Examples:

    In> RandomIntegerVector(4,-3,3)
    Out> {0,3,2,-2};

##### See also:

Random , RandomPoly .

###

MakeVector -- vector of uniquely numbered variable names

##### Standard library

##### Calling format:

    MakeVector(var,n)

##### Parameters:

`var` -- free variable

`n` -- length of the vector

##### Description:

A list of length "n" is generated. The first entry contains the identifier "var" with the number 1 appended to it, the second entry contains "var" with the suffix 2, and so on until the last entry which contains "var" with the number "n" appended to it.

##### Examples:

    In> MakeVector(a,3)
    Out> {a1,a2,a3};

##### See also:

RandomIntegerVector , ZeroVector .

###

Select -- select entries satisfying some predicate

##### Standard library

##### Calling format:

    Select(pred, list)

##### Parameters:

`pred` -- a predicate

`list` -- a list of elements to select from

##### Description:

`Select` returns a sublist of "list" which contains all the entries for which the predicate "pred" returns `True` when applied to this entry.

##### Examples:

    In> Select("IsInteger",{a,b,2,c,3,d,4,e,f})
    Out> {2,3,4};

##### See also:

Length , Find , Count .

###

Nth -- return the n-th element of a list

##### Internal function

##### Calling format:

    Nth(list, n)

##### Parameters:

`list` -- list to choose from

`n` -- index of entry to pick

##### Description:

The entry with index "n" from "list" is returned. The first entry has index 1. It is possible to pick several entries of the list by taking "n" to be a list of indices.

More generally, `Nth` returns the n-th operand of the expression passed as first argument.

An alternative but equivalent form of `Nth(list, n)` is `list[n]`.

##### Examples:

    In> lst := {a,b,c,13,19};
    Out> {a,b,c,13,19};
    In> Nth(lst, 3);
    Out> c;
    In> lst[3];
    Out> c;
    In> Nth(lst, {3,4,1});
    Out> {c,13,a};
    In> Nth(b*(a+c), 2);
    Out> a+c;

##### See also:

Select , Nth .

###

DestructiveReverse -- reverse a list destructively

##### Internal function

##### Calling format:

    DestructiveReverse(list)

##### Parameters:

`list` -- list to reverse

##### Description:

This command reverses "list" in place, so that the original is destroyed. This means that any variable bound to "list" will now have an undefined content, and should not be used any more. The reversed list is returned.

Destructive commands are faster than their nondestructive counterparts. `Reverse` is the non-destructive version of this function.

##### Examples:

    In> lst := {a,b,c,13,19};
    Out> {a,b,c,13,19};
    In> revlst := DestructiveReverse(lst);
    Out> {19,13,c,b,a};
    In> lst;
    Out> {a};

##### See also:

FlatCopy , Reverse .

###

Reverse -- return the reversed list (without touching the original)

##### Standard library

##### Calling format:

    Reverse(list)

##### Parameters:

`list` -- list to reverse

##### Description:

This function returns a list reversed, without changing the original list. It is similar to `DestructiveReverse`, but safer and slower.

##### Example:

    In> lst:={a,b,c,13,19}
    Out> {a,b,c,13,19};
    In> revlst:=Reverse(lst)
    Out> {19,13,c,b,a};
    In> lst
    Out> {a,b,c,13,19};

##### See also:

FlatCopy , DestructiveReverse .

###

List -- construct a list

##### Internal function

##### Calling format:

    List(expr1, expr2, ...)

##### Parameters:

`expr1`, `expr2` -- expressions making up the list

##### Description:

A list is constructed whose first entry is "expr1", the second entry is "expr2", and so on. This command is equivalent to the expression "`expr1, expr2, ...`".

##### Examples:

    In> List();
    Out> {};
    In> List(a,b);
    Out> {a,b};
    In> List(a,{1,2},d);
    Out> {a,{1,2},d};

##### See also:

UnList , Listify .

###

UnList -- convert a list to a function application

##### Internal function

##### Calling format:

    UnList(list)

##### Parameters:

`list` -- list to be converted

##### Description:

This command converts a list to a function application. The first entry of "list" is treated as a function atom, and the following entries are the arguments to this function. So the function referred to in the first element of "list" is applied to the other elements.

Note that "list" is evaluated before the function application is formed, but the resulting expression is left unevaluated. The functions `UnList()` and `Hold()` both stop the process of evaluation.

##### Examples:

    In> UnList({Cos, x});
    Out> Cos(x);
    In> UnList({f});
    Out> f();
    In> UnList({Taylor,x,0,5,Cos(x)});
    Out> Taylor(x,0,5)Cos(x);
    In> Eval(%);
    Out> 1-x^2/2+x^4/24;

##### See also:

List , Listify , Hold .

###

Listify -- convert a function application to a list

##### Internal function

##### Calling format:

    Listify(expr)

##### Parameters:

`expr` -- expression to be converted

##### Description:

The parameter "expr" is expected to be a compound object, i.e. not an atom. It is evaluated and then converted to a list. The first entry in the list is the top-level operator in the evaluated expression and the other entries are the arguments to this operator. Finally, the list is returned.

##### Examples:

    In> Listify(Cos(x));
    Out> {Cos,x};
    In> Listify(3*a);
    Out> {*,3,a};

##### See also:

List , UnList , IsAtom .

###

Concat -- concatenate lists

##### Internal function

##### Calling format:

    Concat(list1, list2, ...)

##### Parameters:

`list1`, `list2`, ... -- lists to concatenate

##### Description:

The lists "list1", "list2", ... are evaluated and concatenated. The resulting big list is returned.

##### Examples:

    In> Concat({a,b}, {c,d});
    Out> {a,b,c,d};
    In> Concat({5}, {a,b,c}, {{f(x)}});
    Out> {5,a,b,c,{f(x)}};

##### See also:

ConcatStrings , : , Insert .

###

Delete -- delete an element from a list

##### Internal function

##### Calling format:

    Delete(list, n)

##### Parameters:

`list` -- list from which an element should be removed

`n` -- index of the element to remove

##### Description:

This command deletes the n-th element from "list". The first parameter should be a list, while "n" should be a positive integer less than or equal to the length of "list". The entry with index "n" is removed (the first entry has index 1), and the resulting list is returned.

##### Examples:

    In> Delete({a,b,c,d,e,f}, 4);
    Out> {a,b,c,e,f};

##### See also:

DestructiveDelete , Insert , Replace .

###

Insert -- insert an element into a list

##### Internal function

##### Calling format:

    Insert(list, n, expr)

##### Parameters:

`list` -- list in which "expr" should be inserted

`n` -- index at which to insert

`expr` -- expression to insert in "list"

##### Description:

The expression "expr" is inserted just before the n-th entry in "list". The first parameter "list" should be a list, while "n" should be a positive integer less than or equal to the length of "list" plus one. The expression "expr" is placed between the entries in "list" with entries "n-1" and "n". There are two border line cases: if "n" is 1, the expression "expr" is placed in front of the list (just as by the `:` operator); if "n" equals the length of "list" plus one, the expression "expr" is placed at the end of the list (just as by `Append`). In any case, the resulting list is returned.

##### Examples:

    In> Insert({a,b,c,d}, 4, x);
    Out> {a,b,c,x,d};
    In> Insert({a,b,c,d}, 5, x);
    Out> {a,b,c,d,x};
    In> Insert({a,b,c,d}, 1, x);
    Out> {x,a,b,c,d};

##### See also:

DestructiveInsert , : , Append , Delete , Remove .

###

DestructiveDelete -- delete an element destructively from a list

##### Internal function

##### Calling format:

    DestructiveDelete(list, n)

##### Parameters:

`list` -- list from which an element should be removed

`n` -- index of the element to remove

##### Description:

This is the destructive counterpart of `Delete`. This command yields the same result as the corresponding call to `Delete`, but the original list is modified. So if a variable is bound to "list", it will now be bound to the list with the n-th entry removed.

Destructive commands run faster than their nondestructive counterparts because the latter copy the list before they alter it.

##### Examples:

    In> lst := {a,b,c,d,e,f};
    Out> {a,b,c,d,e,f};
    In> Delete(lst, 4);
    Out> {a,b,c,e,f};
    In> lst;
    Out> {a,b,c,d,e,f};
    In> DestructiveDelete(lst, 4);
    Out> {a,b,c,e,f};
    In> lst;
    Out> {a,b,c,e,f};

##### See also:

Delete , DestructiveInsert , DestructiveReplace .

###

DestructiveInsert -- insert an element destructively into a list

##### Internal function

##### Calling format:

    DestructiveInsert(list, n, expr)

##### Parameters:

`list` -- list in which "expr" should be inserted

`n` -- index at which to insert

`expr` -- expression to insert in "list"

##### Description:

This is the destructive counterpart of `Insert`. This command yields the same result as the corresponding call to `Insert`, but the original list is modified. So if a variable is bound to "list", it will now be bound to the list with the expression "expr" inserted.

Destructive commands run faster than their nondestructive counterparts because the latter copy the list before they alter it.

##### Examples:

    In> lst := {a,b,c,d};
    Out> {a,b,c,d};
    In> Insert(lst, 2, x);
    Out> {a,x,b,c,d};
    In> lst;
    Out> {a,b,c,d};
    In> DestructiveInsert(lst, 2, x);
    Out> {a,x,b,c,d};
    In> lst;
    Out> {a,x,b,c,d};

##### See also:

Insert , DestructiveDelete , DestructiveReplace .

###

Replace -- replace an entry in a list

##### Internal function

##### Calling format:

    Replace(list, n, expr)

##### Parameters:

`list` -- list of which an entry should be replaced

`n` -- index of entry to replace

`expr` -- expression to replace the n-th entry with

##### Description:

The n-th entry of "list" is replaced by the expression "expr". This is equivalent to calling `Delete` and `Insert` in sequence. To be precise, the expression `Replace(list, n, expr)` has the same result as the expression `Insert(Delete(list, n), n, expr)`.

##### Examples:

    In> Replace({a,b,c,d,e,f}, 4, x);
    Out> {a,b,c,x,e,f};

##### See also:

Delete , Insert , DestructiveReplace .

###

DestructiveReplace -- replace an entry destructively in a list

##### Internal function

##### Calling format:

    DestructiveReplace(list, n, expr)

##### Parameters:

`list` -- list of which an entry should be replaced

`n` -- index of entry to replace

`expr` -- expression to replace the n-th entry with

##### Description:

This is the destructive counterpart of `Replace`. This command yields the same result as the corresponding call to `Replace`, but the original list is modified. So if a variable is bound to "list", it will now be bound to the list with the expression "expr" inserted.

Destructive commands run faster than their nondestructive counterparts because the latter copy the list before they alter it.

##### Examples:

    In> lst := {a,b,c,d,e,f};
    Out> {a,b,c,d,e,f};
    In> Replace(lst, 4, x);
    Out> {a,b,c,x,e,f};
    In> lst;
    Out> {a,b,c,d,e,f};
    In> DestructiveReplace(lst, 4, x);
    Out> {a,b,c,x,e,f};
    In> lst;
    Out> {a,b,c,x,e,f};

##### See also:

Replace , DestructiveDelete , DestructiveInsert .

###

FlatCopy -- copy the top level of a list

##### Internal function

##### Calling format:

    FlatCopy(list)

##### Parameters:

`list` -- list to be copied

##### Description:

A copy of "list" is made and returned. The list is not recursed into, only the first level is copied. This is useful in combination with the destructive commands that actually modify lists in place (for efficiency).

##### Examples:

The following shows a possible way to define a command that reverses a list nondestructively.

    In> reverse(l_IsList) <-- DestructiveReverse \
      (FlatCopy(l));
    Out> True;
    In> lst := {a,b,c,d,e};
    Out> {a,b,c,d,e};
    In> reverse(lst);
    Out> {e,d,c,b,a};
    In> lst;
    Out> {a,b,c,d,e};

###

Contains -- test whether a list contains a certain element

##### Standard library

##### Calling format:

    Contains(list, expr)

##### Parameters:

`list` -- list to examine

`expr` -- expression to look for in "list"

##### Description:

This command tests whether "list" contains the expression "expr" as an entry. It returns `True` if it does and `False` otherwise. Only the top level of "list" is examined. The parameter "list" may also be a general expression, in that case the top-level operands are tested for the occurrence of "expr".

##### Examples:

    In> Contains({a,b,c,d}, b);
    Out> True;
    In> Contains({a,b,c,d}, x);
    Out> False;
    In> Contains({a,{1,2,3},z}, 1);
    Out> False;
    In> Contains(a*b, b);
    Out> True;

##### See also:

Find , Count .

###

Find -- get the index at which a certain element occurs

##### Standard library

##### Calling format:

    Find(list, expr)

##### Parameters:

`list` -- the list to examine

`expr` -- expression to look for in "list"

##### Description:

This commands returns the index at which the expression "expr" occurs in "list". If "expr" occurs more than once, the lowest index is returned. If "expr" does not occur at all, `-1` is returned.

##### Examples:

    In> Find({a,b,c,d,e,f}, d);
    Out> 4;
    In> Find({1,2,3,2,1}, 2);
    Out> 2;
    In> Find({1,2,3,2,1}, 4);
    Out> -1;

##### See also:

Contains .

###

Append -- append an entry at the end of a list

##### Standard library

##### Calling format:

    Append(list, expr)

##### Parameters:

`list` -- list to append "expr" to

`expr` -- expression to append to the list

##### Description:

The expression "expr" is appended at the end of "list" and the resulting list is returned.

Note that due to the underlying data structure, the time it takes to append an entry at the end of a list grows linearly with the length of the list, while the time for prepending an entry at the beginning is constant.

##### Examples:

    In> Append({a,b,c,d}, 1);
    Out> {a,b,c,d,1};

##### See also:

Concat , : , DestructiveAppend .

###

DestructiveAppend -- destructively append an entry to a list

##### Internal function

##### Calling format:

    DestructiveAppend(list, expr)

##### Parameters:

`list` -- list to append "expr" to

`expr` -- expression to append to the list

##### Description:

This is the destructive counterpart of `Append`. This command yields the same result as the corresponding call to `Append`, but the original list is modified. So if a variable is bound to "list", it will now be bound to the list with the expression "expr" inserted.

Destructive commands run faster than their nondestructive counterparts because the latter copy the list before they alter it.

##### Examples:

    In> lst := {a,b,c,d};
    Out> {a,b,c,d};
    In> Append(lst, 1);
    Out> {a,b,c,d,1};
    In> lst
    Out> {a,b,c,d};
    In> DestructiveAppend(lst, 1);
    Out> {a,b,c,d,1};
    In> lst;
    Out> {a,b,c,d,1};

##### See also:

Concat , : , Append .

###

RemoveDuplicates -- remove any duplicates from a list

##### Standard library

##### Calling format:

    RemoveDuplicates(list)

##### Parameters:

`list` -- list to act on

##### Description:

This command removes all duplicate elements from a given list and returns the resulting list. To be precise, the second occurrence of any entry is deleted, as are the third, the fourth, etc.

##### Examples:

    In> RemoveDuplicates({1,2,3,2,1});
    Out> {1,2,3};
    In> RemoveDuplicates({a,1,b,1,c,1});
    Out> {a,1,b,c};

###

Push -- add an element on top of a stack

##### Standard library

##### Calling format:

    Push(stack, expr)

##### Parameters:

`stack` -- a list (which serves as the stack container)

`expr` -- expression to push on "stack"

##### Description:

This is part of a simple implementation of a stack, internally represented as a list. This command pushes the expression "expr" on top of the stack, and returns the stack afterwards.

##### Examples:

    In> stack := {};
    Out> {};
    In> Push(stack, x);
    Out> {x};
    In> Push(stack, x2);
    Out> {x2,x};
    In> PopFront(stack);
    Out> x2;

##### See also:

Pop , PopFront , PopBack .

###

Pop -- remove an element from a stack

##### Standard library

##### Calling format:

    Pop(stack, n)

##### Parameters:

`stack` -- a list (which serves as the stack container)

`n` -- index of the element to remove

##### Description:

This is part of a simple implementation of a stack, internally represented as a list. This command removes the element with index "n" from the stack and returns this element. The top of the stack is represented by the index 1. Invalid indices, for example indices greater than the number of element on the stack, lead to an error.

##### Examples:

    In> stack := {};
    Out> {};
    In> Push(stack, x);
    Out> {x};
    In> Push(stack, x2);
    Out> {x2,x};
    In> Push(stack, x3);
    Out> {x3,x2,x};
    In> Pop(stack, 2);
    Out> x2;
    In> stack;
    Out> {x3,x};

##### See also:

Push , PopFront , PopBack .

###

PopFront -- remove an element from the top of a stack

##### Standard library

##### Calling format:

    PopFront(stack)

##### Parameters:

`stack` -- a list (which serves as the stack container)

##### Description:

This is part of a simple implementation of a stack, internally represented as a list. This command removes the element on the top of the stack and returns it. This is the last element that is pushed onto the stack.

##### Examples:

    In> stack := {};
    Out> {};
    In> Push(stack, x);
    Out> {x};
    In> Push(stack, x2);
    Out> {x2,x};
    In> Push(stack, x3);
    Out> {x3,x2,x};
    In> PopFront(stack);
    Out> x3;
    In> stack;
    Out> {x2,x};

##### See also:

Push , Pop , PopBack .

###

PopBack -- remove an element from the bottom of a stack

##### Standard library

##### Calling format:

    PopBack(stack)

##### Parameters:

`stack` -- a list (which serves as the stack container)

##### Description:

This is part of a simple implementation of a stack, internally represented as a list. This command removes the element at the bottom of the stack and returns this element. Of course, the stack should not be empty.

##### Examples:

    In> stack := {};
    Out> {};
    In> Push(stack, x);
    Out> {x};
    In> Push(stack, x2);
    Out> {x2,x};
    In> Push(stack, x3);
    Out> {x3,x2,x};
    In> PopBack(stack);
    Out> x;
    In> stack;
    Out> {x3,x2};

##### See also:

Push , Pop , PopFront .

###

Swap -- swap two elements in a list

##### Standard library

##### Calling format:

    Swap(list, i1, i2)

##### Parameters:

`list` -- the list in which a pair of entries should be swapped

`i1, i2` -- indices of the entries in "list" to swap

##### Description:

This command swaps the pair of entries with entries "i1" and "i2" in "list". So the element at index "i1" ends up at index "i2" and the entry at "i2" is put at index "i1". Both indices should be valid to address elements in the list. Then the updated list is returned.

`Swap()` works also on generic arrays.

##### Examples:

    In> lst := {a,b,c,d,e,f};
    Out> {a,b,c,d,e,f};
    In> Swap(lst, 2, 4);
    Out> {a,d,c,b,e,f};

##### See also:

Replace , DestructiveReplace , ArrayCreate .

###

Count -- count the number of occurrences of an expression

##### Standard library

##### Calling format:

    Count(list, expr)

##### Parameters:

`list` -- the list to examine

`expr` -- expression to look for in "list"

##### Description:

This command counts the number of times that the expression "expr" occurs in "list" and returns this number.

##### Examples:

    In> lst := {a,b,c,b,a};
    Out> {a,b,c,b,a};
    In> Count(lst, a);
    Out> 2;
    In> Count(lst, c);
    Out> 1;
    In> Count(lst, x);
    Out> 0;

##### See also:

Length , Select , Contains .

###

Intersection -- return the intersection of two lists

##### Standard library

##### Calling format:

    Intersection(l1, l2)

##### Parameters:

`l1`, `l2` -- two lists

##### Description:

The intersection of the lists "l1" and "l2" is determined and returned. The intersection contains all elements that occur in both lists. The entries in the result are listed in the same order as in "l1". If an expression occurs multiple times in both "l1" and "l2", then it will occur the same number of times in the result.

##### Examples:

    In> Intersection({a,b,c}, {b,c,d});
    Out> {b,c};
    In> Intersection({a,e,i,o,u}, {f,o,u,r,t,e,e,n});
    Out> {e,o,u};
    In> Intersection({1,2,2,3,3,3}, {1,1,2,2,3,3});
    Out> {1,2,2,3,3};

##### See also:

Union , Difference .

###

Union -- return the union of two lists

##### Standard library

##### Calling format:

    Union(l1, l2)

##### Parameters:

`l1`, `l2` -- two lists

##### Description:

The union of the lists "l1" and "l2" is determined and returned. The union contains all elements that occur in one or both of the lists. In the resulting list, any element will occur only once.

##### Examples:

    In> Union({a,b,c}, {b,c,d});
    Out> {a,b,c,d};
    In> Union({a,e,i,o,u}, {f,o,u,r,t,e,e,n});
    Out> {a,e,i,o,u,f,r,t,n};
    In> Union({1,2,2,3,3,3}, {2,2,3,3,4,4});
    Out> {1,2,3,4};

##### See also:

Intersection , Difference .

###

Difference -- return the difference of two lists

##### Standard library

##### Calling format:

    Difference(l1, l2)

##### Parameters:

`l1`, `l2` -- two lists

##### Description:

The difference of the lists "l1" and "l2" is determined and returned. The difference contains all elements that occur in "l1" but not in "l2". The order of elements in "l1" is preserved. If a certain expression occurs "n1" times in the first list and "n2" times in the second list, it will occur "n1-n2" times in the result if "n1" is greater than "n2" and not at all otherwise.

##### Examples:

    In> Difference({a,b,c}, {b,c,d});
    Out> {a};
    In> Difference({a,e,i,o,u}, {f,o,u,r,t,e,e,n});
    Out> {a,i};
    In> Difference({1,2,2,3,3,3}, {2,2,3,4,4});
    Out> {1,3,3};

##### See also:

Intersection , Union .

###

FillList -- fill a list with a certain expression

##### Standard library

##### Calling format:

    FillList(expr, n)

##### Parameters:

`expr` -- expression to fill the list with

`n` -- the length of the list to construct

##### Description:

This command creates a list of length "n" in which all slots contain the expression "expr" and returns this list.

##### Examples:

    In> FillList(x, 5);
    Out> {x,x,x,x,x};

##### See also:

MakeVector , ZeroVector , RandomIntegerVector .

###

Drop -- drop a range of elements from a list

##### Standard library

##### Calling format:

    Drop(list, n)
    Drop(list, -n)
    Drop(list, {m,n})

##### Parameters:

`list` -- list to act on

`n`, `m` -- positive integers describing the entries to drop

##### Description:

This command removes a sublist of "list" and returns a list containing the remaining entries. The first calling sequence drops the first "n" entries in "list". The second form drops the last "n" entries. The last invocation drops the elements with indices "m" through "n".

##### Examples:

    In> lst := {a,b,c,d,e,f,g};
    Out> {a,b,c,d,e,f,g};
    In> Drop(lst, 2);
    Out> {c,d,e,f,g};
    In> Drop(lst, -3);
    Out> {a,b,c,d};
    In> Drop(lst, {2,4});
    Out> {a,e,f,g};

##### See also:

Take , Select , Remove .

###

Take -- take a sublist from a list, dropping the rest

##### Standard library

##### Calling format:

    Take(list, n)
    Take(list, -n)
    Take(list, {m,n})

##### Parameters:

`list` -- list to act on

`n`, `m` -- positive integers describing the entries to take

##### Description:

This command takes a sublist of "list", drops the rest, and returns the selected sublist. The first calling sequence selects the first "n" entries in "list". The second form takes the last "n" entries. The last invocation selects the sublist beginning with entry number "m" and ending with the "n"-th entry.

##### Examples:

    In> lst := {a,b,c,d,e,f,g};
    Out> {a,b,c,d,e,f,g};
    In> Take(lst, 2);
    Out> {a,b};
    In> Take(lst, -3);
    Out> {e,f,g};
    In> Take(lst, {2,4});
    Out> {b,c,d};

##### See also:

Drop , Select , Remove .

###

Partition -- partition a list in sublists of equal length

##### Standard library

##### Calling format:

    Partition(list, n)

##### Parameters:

`list` -- list to partition

`n` -- length of partitions

##### Description:

This command partitions "list" into non-overlapping sublists of length "n" and returns a list of these sublists. The first "n" entries in "list" form the first partition, the entries from position "n+1" up to "2n" form the second partition, and so on. If "n" does not divide the length of "list", the remaining entries will be thrown away. If "n" equals zero, an empty list is returned.

##### Examples:

    In> Partition({a,b,c,d,e,f,}, 2);
    Out> {{a,b},{c,d},{e,f}};
    In> Partition(1 .. 11, 3);
    Out> {{1,2,3},{4,5,6},{7,8,9}};

##### See also:

Take , Permutations .

###

Assoc -- return element stored in association list

##### Standard library

##### Calling format:

    Assoc(key, alist)

##### Parameters:

`key` -- string, key under which element is stored

`alist` -- association list to examine

##### Description:

The association list "alist" is searched for an entry stored with index "key". If such an entry is found, it is returned. Otherwise the atom `Empty` is returned.

Association lists are represented as a list of two-entry lists. The first element in the two-entry list is the key, the second element is the value stored under this key.

The call `Assoc(key, alist)` can (probably more intuitively) be accessed as `alist[key]`.

##### Examples:

    In> writer := {};
    Out> {};
    In> writer["Iliad"] := "Homer";
    Out> True;
    In> writer["Henry IV"] := "Shakespeare";
    Out> True;
    In> writer["Ulysses"] := "James Joyce";
    Out> True;
    In> Assoc("Henry IV", writer);
    Out> {"Henry IV","Shakespeare"};
    In> Assoc("War and Peace", writer);
    Out> Empty;

##### See also:

AssocIndices , [] , := , AssocDelete .

###

AssocIndices -- return the keys in an association list

##### Standard library

##### Calling format:

    AssocIndices(alist)

##### Parameters:

`alist` -- association list to examine

##### Description:

All the keys in the association list "alist" are assembled in a list and this list is returned.

##### Examples:

    In> writer := {};
    Out> {};
    In> writer["Iliad"] := "Homer";
    Out> True;
    In> writer["Henry IV"] := "Shakespeare";
    Out> True;
    In> writer["Ulysses"] := "James Joyce";
    Out> True;
    In> AssocIndices(writer);
    Out> {"Iliad","Henry IV","Ulysses"};

##### See also:

Assoc , AssocDelete .

###

AssocDelete -- delete an entry in an association list

##### Standard library

##### Calling format:

    AssocDelete(alist, "key")
    AssocDelete(alist, {key, value})

##### Parameters:

`alist` -- association list

`"key"` -- string, association key

`value` -- value of the key to be deleted

##### Description:

The key `"key"` in the association list `alist` is deleted. (The list itself is modified.) If the key was found and successfully deleted, returns `True`, otherwise if the given key was not found, the function returns `False`.

The second, longer form of the function deletes the entry that has both the specified key and the specified value. It can be used for two purposes:

- to make sure that we are deleting the right value;
- if several values are stored on the same key, to delete the specified entry (see the last example).

At most one entry is deleted.

##### Examples:

    In> writer := {};
    Out> {};
    In> writer["Iliad"] := "Homer";
    Out> True;
    In> writer["Henry IV"] := "Shakespeare";
    Out> True;
    In> writer["Ulysses"] := "James Joyce";
    Out> True;
    In> AssocDelete(writer, "Henry IV")
    Out> True;
    In> AssocDelete(writer, "Henry XII")
    Out> False;
    In> writer
    Out> {{"Ulysses","James Joyce"},
      {"Iliad","Homer"}};
    In> DestructiveAppend(writer,
      {"Ulysses", "Dublin"});
    Out> {{"Iliad","Homer"},{"Ulysses","James Joyce"},
      {"Ulysses","Dublin"}};
    In> writer["Ulysses"];
    Out> "James Joyce";
    In> AssocDelete(writer,{"Ulysses","James Joyce"});
    Out> True;
    In> writer
    Out> {{"Iliad","Homer"},{"Ulysses","Dublin"}};

##### See also:

Assoc , AssocIndices .

###

Flatten -- flatten expression w.r.t. some operator

##### Standard library

##### Calling format:

    Flatten(expression,operator)

##### Parameters:

`expression` -- an expression

`operator` -- string with the contents of an infix operator.

##### Description:

Flatten flattens an expression with respect to a specific operator, converting the result into a list. This is useful for unnesting an expression. Flatten is typically used in simple simplification schemes.

##### Examples:

    In> Flatten(a+b*c+d,"+");
    Out> {a,b*c,d};
    In> Flatten({a,{b,c},d},"List");
    Out> {a,b,c,d};

##### See also:

UnFlatten .

###

UnFlatten -- inverse operation of Flatten

##### Standard library

##### Calling format:

    UnFlatten(list,operator,identity)

##### Parameters:

`list` -- list of objects the operator is to work on

`operator` -- infix operator

`identity` -- identity of the operator

##### Description:

UnFlatten is the inverse operation of Flatten. Given a list, it can be turned into an expression representing for instance the addition of these elements by calling UnFlatten with "+" as argument to operator, and 0 as argument to identity (0 is the identity for addition, since a+0=a). For multiplication the identity element would be 1.

##### Examples:

    In> UnFlatten({a,b,c},"+",0)
    Out> a+b+c;
    In> UnFlatten({a,b,c},"*",1)
    Out> a*b*c;

##### See also:

Flatten .

###

Type -- return the type of an expression

##### Internal function

##### Calling format:

    Type(expr)

##### Parameters:

`expr` -- expression to examine

##### Description:

The type of the expression "expr" is represented as a string and returned. So, if "expr" is a list, the string `"List"` is returned. In general, the top-level operator of "expr" is returned. If the argument "expr" is an atom, the result is the empty string `""`.

##### Examples:

    In> Type({a,b,c});
    Out> "List";
    In> Type(a*(b+c));
    Out> "*";
    In> Type(123);
    Out> "";

##### See also:

IsAtom , NrArgs .

###

NrArgs -- return number of top-level arguments

##### Standard library

##### Calling format:

    NrArgs(expr)

##### Parameters:

`expr` -- expression to examine

##### Description:

This function evaluates to the number of top-level arguments of the expression "expr". The argument "expr" may not be an atom, since that would lead to an error.

##### Examples:

    In> NrArgs(f(a,b,c))
    Out> 3;
    In> NrArgs(Sin(x));
    Out> 1;
    In> NrArgs(a*(b+c));
    Out> 2;

##### See also:

Type , Length .

###

VarList, VarListArith, VarListSome -- list of variables appearing in an expression

##### Standard library

##### Calling format:

    VarList(expr)
    VarListArith(expr)
    VarListSome(expr, list)

##### Parameters:

`expr` -- an expression

`list` -- a list of function atoms

##### Description:

The command `VarList(expr)` returns a list of all variables that appear in the expression `expr`. The expression is traversed recursively.

The command `VarListSome` looks only at arguments of functions in the `list`. All other functions are considered "opaque" (as if they do not contain any variables) and their arguments are not checked. For example, `VarListSome(a + Sin(b-c))` will return `{a, b, c}`, but `VarListSome(a*Sin(b-c), {*})` will not look at arguments of `Sin()` and will return `{a,Sin(b-c)}`. Here `Sin(b-c)` is considered a "variable" because the function `Sin` does not belong to `list`.

The command `VarListArith` returns a list of all variables that appear arithmetically in the expression `expr`. This is implemented through `VarListSome` by restricting to the arithmetic functions `+`, `-`, `*`, `/`. Arguments of other functions are not checked.

Note that since the operators "`+`" and "`-`" are prefix as well as infix operators, it is currently required to use `Atom("+")` to obtain the unevaluated atom "`+`".

##### Examples:

    In> VarList(Sin(x))
    Out> {x};
    In> VarList(x+a*y)
    Out> {x,a,y};
    In> VarListSome(x+a*y, {Atom("+")})
    Out> {x,a*y};
    In> VarListArith(x+y*Cos(Ln(x)/x))
    Out> {x,y,Cos(Ln(x)/x)}
    In> VarListArith(x+a*y^2-1)
    Out> {x,a,y^2};

##### See also:

IsFreeOf , IsVariable , FuncList , HasExpr , HasFunc .

###

FuncList, FuncListArith, FuncListSome -- list of functions used in an expression

##### Standard library

##### Calling format:

    FuncList(expr)
    FuncListArith(expr)
    FuncListSome(expr, list)

##### Parameters:

`expr` -- an expression

`list` -- list of function atoms to be considered "transparent"

##### Description:

The command `FuncList(expr)` returns a list of all function atoms that appear in the expression `expr`. The expression is recursively traversed.

The command `FuncListSome(expr, list)` does the same, except it only looks at arguments of a given `list` of functions. All other functions become "opaque" (as if they do not contain any other functions). For example, `FuncListSome(a + Sin(b-c))` will see that the expression has a "`-`" operation and return `{+,Sin,-}`, but `FuncListSome(a + Sin(b-c), {+})` will not look at arguments of `Sin()` and will return `{+,Sin}`.

`FuncListArith` is defined through `FuncListSome` to look only at arithmetic operations `+`, `-`, `*`, `/`.

Note that since the operators "`+`" and "`-`" are prefix as well as infix operators, it is currently required to use `Atom("+")` to obtain the unevaluated atom "`+`".

##### Examples:

    In> FuncList(x+y*Cos(Ln(x)/x))
    Out> {+,*,Cos,/,Ln};
    In> FuncListArith(x+y*Cos(Ln(x)/x))
    Out> {+,*,Cos};
    In> FuncListSome({a+b*2,c/d},{List})
    Out> {List,+,/};

##### See also:

VarList , HasExpr , HasFunc .

###

BubbleSort, HeapSort -- sort a list

##### Standard library

##### Calling format:

    BubbleSort(list, compare)
    HeapSort(list, compare)

##### Parameters:

`list` -- list to sort

`compare` -- function used to compare elements of `list`

##### Description:

This command returns `list` after it is sorted using `compare` to compare elements. The function `compare` should accept two arguments, which will be elements of `list`, and compare them. It should return `True` if in the sorted list the second argument should come after the first one, and `False` otherwise.

The function `BubbleSort` uses the so-called "bubble sort" algorithm to do the sorting by swapping elements that are out of order. This algorithm is easy to implement, though it is not particularly fast. The sorting time is proportional to n^2 where n is the length of the list.

The function `HeapSort` uses a recursive algorithm "heapsort" and is much faster for large lists. The sorting time is proportional to n*Ln(n) where n is the length of the list.

##### Examples:

    In> BubbleSort({4,7,23,53,-2,1}, "<");
    Out> {-2,1,4,7,23,53};
    In> HeapSort({4,7,23,53,-2,1}, ">");
    Out> {53,23,7,4,1,-2};

###

PrintList -- print list with padding

##### Standard library

##### Calling format:

    PrintList(list)
    PrintList(list, padding);

##### Parameters:

`list` -- a list to be printed

`padding` -- (optional) a string

##### Description:

Prints `list` and inserts the `padding` string between each pair of items of the list. Items of the list which are strings are printed without quotes, unlike `Write()`. Items of the list which are themselves lists are printed inside braces `{}`. If padding is not specified, a standard one is used (comma, space).

##### Examples:

    In> PrintList({a,b,{c, d}}, " .. ")
    Out> " a ..  b .. { c ..  d}";

##### See also:

Write , WriteString .

###

Table -- evaluate while some variable ranges over interval

##### Standard library

##### Calling format:

    Table(body, var, from, to, step)

##### Parameters:

`body` -- expression to evaluate multiple times

`var` -- variable to use as loop variable

`from` -- initial value for "var"

`to` -- final value for "var"

`step` -- step size with which "var" is incremented

##### Description:

This command generates a list of values from "body", by assigning variable "var" values from "from" up to "to", incrementing "step" each time. So, the variable "var" first gets the value "from", and the expression "body" is evaluated. Then the value "from"+"step" is assigned to "var" and the expression "body" is again evaluated. This continues, incrementing "var" with "step" on every iteration, until "var" exceeds "to". At that moment, all the results are assembled in a list and this list is returned.

##### Examples:

    In> Table(i!, i, 1, 9, 1);
    Out> {1,2,6,24,120,720,5040,40320,362880};
    In> Table(i, i, 3, 16, 4);
    Out> {3,7,11,15};
    In> Table(i^2, i, 10, 1, -1);
    Out> {100,81,64,49,36,25,16,9,4,1};

##### See also:

For , MapSingle , .. , TableForm .

###

TableForm -- print each entry in a list on a line

##### Standard library

##### Calling format:

    TableForm(list)

##### Parameters:

`list` -- list to print

##### Description:

This functions writes out the list `list` in a better readable form, by printing every element in the list on a separate line.

##### Examples:

    In> TableForm(Table(i!, i, 1, 10, 1));

    1
     2
     6
     24
     120
     720
     5040
     40320
     362880
     3628800
    Out> True;

##### See also:

PrettyForm , Echo , Table .

###

GlobalPop, GlobalPush -- save/restore variables using a global stack

##### Standard library

##### Calling format:

    GlobalPop(var)
    GlobalPop()
    GlobalPush(expr)

##### Parameters:

`var` -- atom, name of variable to restore from the stack

`expr` -- expression, value to save on the stack

##### Description:

These functions operate with a global stack, currently implemented as a list `GlobalStack`.

`GlobalPush` stores a value on the stack. `GlobalPop` removes the last pushed value from the stack. If a variable name is given, the variable is assigned, otherwise the popped value is returned.

If the global stack is empty, an error message is printed.

##### Examples:

    In> GlobalPush(3)
    Out> 3;
    In> GlobalPush(Sin(x))
    Out> Sin(x);
    In> GlobalPop(x)
    Out> Sin(x);
    In> GlobalPop(x)
    Out> 3;
    In> x
    Out> 3;

##### See also:

Push , PopFront .

: , @ , /@ , .. , NFunction , Where , AddTo .

# Functional operators

These operators can help the user to program in the style of functional programming languages such as Miranda or Haskell.

###

: -- prepend item to list, or concatenate strings

##### Standard library

##### Calling format:

    item : list
    string1 : string2

Precedence: 70

##### Parameters:

`item` -- an item to be prepended to a list

`list` -- a list

`string1` -- a string

`string2` -- a string

##### Description:

The first form prepends "item" as the first entry to the list "list". The second form concatenates the strings "string1" and "string2".

##### Examples:

    In> a:b:c:{}
    Out> {a,b,c};
    In> "This":"Is":"A":"String"
    Out> "ThisIsAString";

##### See also:

Concat , ConcatStrings .

###

@ -- apply a function

##### Standard library

##### Calling format:

    fn @ arglist

Precedence: 600

##### Parameters:

`fn` -- function to apply

`arglist` -- single argument, or a list of arguments

##### Description:

This function is a shorthand for `Apply`. It applies the function "fn" to the argument(s) in "arglist" and returns the result. The first parameter "fn" can either be a string containing the name of a function or a pure function.

##### Examples:

    In> "Sin" @ a
    Out> Sin(a);
    In> {{a},Sin(a)} @ a
    Out> Sin(a);
    In> "f" @ {a,b}
    Out> f(a,b);

##### See also:

Apply .

###

/@ -- apply a function to all entries in a list

##### Standard library

##### Calling format:

    fn /@ list

Precedence: 600

##### Parameters:

`fn` -- function to apply

`list` -- list of arguments

##### Description:

This function is a shorthand for `MapSingle`. It successively applies the function "fn" to all the entries in "list" and returns a list contains the results. The parameter "fn" can either be a string containing the name of a function or a pure function.

##### Examples:

    In> "Sin" /@ {a,b}
    Out> {Sin(a),Sin(b)};
    In> {{a},Sin(a)*a} /@ {a,b}
    Out> {Sin(a)*a,Sin(b)*b};

##### See also:

MapSingle , Map , MapArgs .

###

.. -- construct a list of consecutive integers

##### Standard library

##### Calling format:

    n .. m

Precedence: 600

##### Parameters:

`n` -- integer. the first entry in the list

`m` -- integer, the last entry in the list

##### Description:

This command returns the list `{n, n+1, n+2, ..., m}`. If `m` is smaller than `n`, the empty list is returned. Note that the `..` operator should be surrounded by spaces to keep the parser happy, if "n" is a number. So one should write "`1 .. 4`" instead of "`1..4`".

##### Example:

    In> 1 .. 4
    Out> {1,2,3,4};

##### See also:

Table .

###

NFunction -- make wrapper for numeric functions

##### Standard library

##### Calling format:

    NFunction("newname","funcname", {arglist})

##### Parameters:

`"newname"` -- name of new function

`"funcname"` -- name of an existing function

`arglist` -- symbolic list of arguments

##### Description:

This function will define a function named "newname" with the same arguments as an existing function named "funcname". The new function will evaluate and return the expression "funcname(arglist)" only when all items in the argument list `arglist` are numbers, and return unevaluated otherwise.

This can be useful when plotting functions defined through other Yacas routines that cannot return unevaluated.

If the numerical calculation does not return a number (for example, it might return the atom `nan`, "not a number", for some arguments), then the new function will return `Undefined`.

##### Examples:

    In> f(x) := N(Sin(x));
    Out> True;
    In> NFunction("f1", "f", {x});
    Out> True;
    In> f1(a);
    Out> f1(a);
    In> f1(0);
    Out> 0;

Suppose we need to define a complicated function `t(x)` which cannot be evaluated unless `x` is a number:

    In> t(x) := If(x<=0.5, 2*x, 2*(1-x));
    Out> True;
    In> t(0.2);
    Out> 0.4;
    In> t(x);
    In function "If" :
    bad argument number 1 (counting from 1)
    CommandLine(1) : Invalid argument

Then, we can use `NFunction()` to define a wrapper `t1(x)` around `t(x)` which will not try to evaluate `t(x)` unless `x` is a number.

    In> NFunction("t1", "t", {x})
    Out> True;
    In> t1(x);
    Out> t1(x);
    In> t1(0.2);
    Out> 0.4;

Now we can plot the function.

    In> Plot2D(t1(x), -0.1: 1.1)
    Out> True;

##### See also:

MacroRule .

###

Where -- substitute result into expression

##### Standard library

##### Calling format:

    expr Where x==v
    expr Where x1==v1 And x2==v2 And ...
    expr Where {x1==v1 And x2==v2,x1==v3
      And x2==v4,...}

##### Parameters:

`expr` - expression to evaluate

`x` - variable to set

`v` - value to substitute for variable

##### Description:

The operator `Where` fills in values for variables, in its simplest form. It accepts sets of variable/value pairs defined as

    var1==val1 And var2==val2 And ...

and fills in the corresponding values. Lists of value pairs are also possible, as:

    {var1==val1 And var2==val2, var1==val3
      And var2==val4}

These values might be obtained through `Solve`.

##### Examples:

    In> x^2+y^2 Where x==2
    Out> y^2+4;
    In> x^2+y^2 Where x==2 And y==3
    Out> 13;
    In> x^2+y^2 Where {x==2 And y==3}
    Out> {13};
    In> x^2+y^2 Where {x==2 And y==3,x==4 And y==5}
    Out> {13,41};

##### See also:

Solve , AddTo .

###

AddTo -- add an equation to a set of equations or set of set of equations

##### Standard library

##### Calling format:

    eq1 AddTo eq2

##### Parameters:

`eq` - (set of) set of equations

##### Description:

Given two (sets of) sets of equations, the command AddTo combines multiple sets of equations into one.

A list `a,b` means that a is a solution, OR b is a solution. AddTo then acts as a AND operation:

    (a or b) and (c or d) =>
    (a or b) Addto (c or d) =>
    (a and c) or (a and d) or (b and c)
      or (b and d)

This function is useful for adding an identity to an already existing set of equations. Suppose a solve command returned `a>=0 And x==a,a<0 And x== -a` from an expression x==Abs(a), then a new identity a==2 could be added as follows:

    In> a==2 AddTo {a>=0 And x==a,a<0 And x== -a}
    Out> {a==2 And a>=0 And x==a,a==2 And a<0
      And x== -a};

Passing this set of set of identities back to solve, solve should recognize that the second one is not a possibility any more, since a==2 And a<0 can never be true at the same time.

##### Examples:

    In> {A==2,c==d} AddTo {b==3 And d==2}
    Out> {A==2 And b==3 And d==2,c==d
      And b==3 And d==2};
    In> {A==2,c==d} AddTo {b==3, d==2}
    Out> {A==2 And b==3,A==2 And d==2,c==d
      And b==3,c==d And d==2};

##### See also:

Where , Solve .

MaxEvalDepth , Hold , Eval , While , Until , If , SystemCall , Function , Macro , Use , For , ForEach , Apply , MapArgs , Subst , WithValue , /:, /:: , SetHelpBrowser , TraceStack , TraceExp , TraceRule , Time .

# Control flow functions

###

MaxEvalDepth -- set the maximum evaluation depth

##### Internal function

##### Calling format:

    MaxEvalDepth(n)

##### Parameters:

`n` -- new maximum evaluation depth

##### Description:

Use this command to set the maximum evaluation depth to the integer "n". The default value is 1000. The function `MaxEvalDepth` returns `True`.

The point of having a maximum evaluation depth is to catch any infinite recursion. For example, after the definition `f(x) := f(x)`, evaluating the expression `f(x)` would call `f(x)`, which would call `f(x)`, etc. The interpreter will halt if the maximum evaluation depth is reached. Also indirect recursion, e.g. the pair of definitions `f(x) := g(x)` and `g(x) := f(x)`, will be caught.

##### Examples:

An example of an infinite recursion, caught because the maximum evaluation depth is reached.

    In> f(x) := f(x)
    Out> True;
    In> f(x)
    Error on line 1 in file [CommandLine]
    Max evaluation stack depth reached.
    Please use MaxEvalDepth to increase the stack
    size as needed.

However, a long calculation may cause the maximum evaluation depth to be reached without the presence of infinite recursion. The function `MaxEvalDepth` is meant for these cases.

    In> 10 # g(0) <-- 1;
    Out> True;
    In> 20 # g(n_IsPositiveInteger) <-- \
      2 * g(n-1);
    Out> True;
    In> g(1001);
    Error on line 1 in file [CommandLine]
    Max evaluation stack depth reached.
    Please use MaxEvalDepth to increase the stack
    size as needed.

    In> MaxEvalDepth(10000);
    Out> True;
    In> g(1001);
    Out> 21430172143725346418968500981200036211228096234
    1106721488750077674070210224987224498639675763139171
    6255189345835106293650374290571384628087196915514939
    7149607869135549648461970842149210124742283755908364
    3060929499671638825347975351183310878921541258291423
    92955373084335320859663305248773674411336138752;

###

Hold -- keep expression unevaluated

##### Internal function

##### Calling format:

    Hold(expr)

##### Parameters:

`expr` -- expression to keep unevaluated

##### Description:

The expression "expr" is returned unevaluated. This is useful to prevent the evaluation of a certain expression in a context in which evaluation normally takes place.

The function `UnList()` also leaves its result unevaluated. Both functions stop the process of evaluation (no more rules will be applied).

##### Examples:

    In> Echo({ Hold(1+1), "=", 1+1 });
     1+1 = 2
    Out> True;

##### See also:

Eval , HoldArg , UnList .

###

Eval -- force evaluation of expression

##### Internal function

##### Calling format:

    Eval(expr)

##### Parameters:

`expr` -- expression to evaluate

##### Description:

This function explicitly requests an evaluation of the expression "expr", and returns the result of this evaluation.

##### Examples:

    In> a := x;
    Out> x;
    In> x := 5;
    Out> 5;
    In> a;
    Out> x;
    In> Eval(a);
    Out> 5;

The variable `a` is bound to `x`, and `x` is bound to 5. Hence evaluating `a` will give `x`. Only when an extra evaluation of `a` is requested, the value 5 is returned.

Note that the behavior would be different if we had exchanged the assignments. If the assignment `a := x` were given while `x` had the value 5, the variable `a` would also get the value 5 because the assignment operator `:=` evaluates the right-hand side.

##### See also:

Hold , HoldArg , := .

###

While -- loop while a condition is met

##### Internal function

##### Calling format:

    While(pred) body

##### Parameters:

`pred` -- predicate deciding whether to keep on looping

`body` -- expression to loop over

##### Description:

Keep on evaluating "body" while "pred" evaluates to `True`. More precisely, `While` evaluates the predicate "pred", which should evaluate to either `True` or `False`. If the result is `True`, the expression "body" is evaluated and then the predicate "pred" is again evaluated. If it is still `True`, the expressions "body" and "pred" are again evaluated and so on until "pred" evaluates to `False`. At that point, the loop terminates and `While` returns `True`.

In particular, if "pred" immediately evaluates to `False`, the body is never executed. `While` is the fundamental looping construct on which all other loop commands are based. It is equivalent to the `while` command in the programming language C.

##### Examples:

    In> x := 0;
    Out> 0;
    In> While (x! < 10^6) \
      [ Echo({x, x!}); x++; ];
     0  1
     1  1
     2  2
     3  6
     4  24
     5  120
     6  720
     7  5040
     8  40320
     9  362880
    Out> True;

##### See also:

Until , For .

###

Until -- loop until a condition is met

##### Standard library

##### Calling format:

    Until(pred) body

##### Parameters:

`pred` -- predicate deciding whether to stop

`body` -- expression to loop over

##### Description:

Keep on evaluating "body" until "pred" becomes `True`. More precisely, `Until` first evaluates the expression "body". Then the predicate "pred" is evaluated, which should yield either `True` or `False`. In the latter case, the expressions "body" and "pred" are again evaluated and this continues as long as "pred" is `False`. As soon as "pred" yields `True`, the loop terminates and `Until` returns `True`.

The main difference with `While` is that `Until` always evaluates the body at least once, but `While` may not evaluate the body at all. Besides, the meaning of the predicate is reversed: `While` stops if "pred" is `False` while `Until` stops if "pred" is `True`. The command `Until(pred) body;` is equivalent to `pred; While(Not pred) body;`. In fact, the implementation of `Until` is based on the internal command `While`. The `Until` command can be compared to the `do ... while` construct in the programming language C.

##### Examples:

    In> x := 0;
    Out> 0;
    In> Until (x! > 10^6) \
      [ Echo({x, x!}); x++; ];
     0  1
     1  1
     2  2
     3  6
     4  24
     5  120
     6  720
     7  5040
     8  40320
     9  362880
    Out> True;

##### See also:

While , For .

###

If -- branch point

##### Internal function

##### Calling format:

    If(pred, then)
    If(pred, then, else)

##### Parameters:

`pred` -- predicate to test

`then` -- expression to evaluate if "pred" is `True`

`else` -- expression to evaluate if "pred" is `False`

##### Description:

This command implements a branch point. The predicate "pred" is evaluated, which should result in either `True` or `False`. In the first case, the expression "then" is evaluated and returned. If the predicate yields `False`, the expression "else" (if present) is evaluated and returned. If there is no "else" branch (i.e. if the first calling sequence is used), the `If` expression returns `False`.

##### Examples:

The sign function is defined to be 1 if its argument is positive and -1 if its argument is negative. A possible implementation is

    In> mysign(x) := If (IsPositiveReal(x), 1, -1);
    Out> True;
    In> mysign(Pi);
    Out> 1;
    In> mysign(-2.5);
    Out> -1;

Note that this will give incorrect results, if "x" cannot be numerically approximated.

    In> mysign(a);
    Out> -1;

Hence a better implementation would be

    In> mysign(_x)_IsNumber(N(x)) <-- If \
      (IsPositiveReal(x), 1, -1);
    Out> True;

###

SystemCall -- pass a command to the shell

##### Internal function

##### Calling format:

    SystemCall(str)

##### Parameters:

`str` -- string containing the command to call

##### Description:

The command contained in the string "str" is executed by the underlying operating system (OS). The return value of `SystemCall` is `True` or `False` according to the exit code of the command.

The `SystemCall` function is not allowed in the body of the `Secure` command and will lead to an error.

##### Examples:

In a UNIX environment, the command `SystemCall("ls")` would print the contents of the current directory.

    In> SystemCall("ls")
    AUTHORS
    COPYING
    ChangeLog

... (truncated to save space)

    Out> True;

The standard UNIX command `test` returns success or failure depending on conditions. For example, the following command will check if a directory exists:

    In> SystemCall("test -d scripts/")
    Out> True;

Check that a file exists:

    In> SystemCall("test -f COPYING")
    Out> True;
    In> SystemCall("test -f nosuchfile.txt")
    Out> False;

##### See also:

Secure .

###

Function -- declare or define a function

##### Standard library

##### Calling format:

    Function() func(arglist)
    Function() func(arglist, ...)
    Function("op", {arglist}) body
    Function("op", {arglist, ...}) body

##### Parameters:

`func(args)` -- function declaration, e.g. `f(x,y)`

`"op"` -- string, name of the function

`{arglist}` -- list of atoms, formal arguments to the function

`...` -- literal ellipsis symbol "`...`" used to denote a variable number of arguments

`body` -- expression comprising the body of the function

##### Description:

This command can be used to define a new function with named arguments.

The number of arguments of the new function and their names are determined by the list `arglist`. If the ellipsis "`...`" follows the last atom in `arglist`, a function with a variable number of arguments is declared (using `RuleBaseListed`). Note that the ellipsis cannot be the only element of `arglist` and must be preceded by an atom.

A function with variable number of arguments can take more arguments than elements in `arglist`; in this case, it obtains its last argument as a list containing all extra arguments.

The short form of the `Function` call merely declares a `RuleBase` for the new function but does not define any function body. This is a convenient shorthand for `RuleBase` and `RuleBaseListed`, when definitions of the function are to be supplied by rules. If the new function has been already declared with the same number of arguments (with or without variable arguments), `Function` returns false and does nothing.

The second, longer form of the `Function` call declares a function and also defines a function body. It is equivalent to a single rule such as `op(_arg1, _arg2) <-- body`. The rule will be declared at precedence 1025. Any previous rules associated with `"op"` (with the same arity) will be discarded. More complicated functions (with more than one body) can be defined by adding more rules.

##### Examples:

This will declare a new function with two or more arguments, but define no rules for it. This is equivalent to `RuleBase ("f1", {x, y, ...})`.

    In> Function() f1(x,y,...);
    Out> True;
    In> Function() f1(x,y);
    Out> False;

This defines a function `FirstOf` which returns the first element of a list. Equivalent definitions would be `FirstOf(_list) <-- list[1]` or `FirstOf(list) := list[1]`.

    In> Function("FirstOf", {list})  list[1];
    Out> True;
    In> FirstOf({a,b,c});
    Out> a;

The following function will print all arguments to a string:

    In> Function("PrintAll",{x, ...}) If(IsList(x),
      PrintList(x), ToString()Write(x));
    Out> True;
    In> PrintAll(1):
    Out> " 1";
    In> PrintAll(1,2,3);
    Out> " 1 2 3";

##### See also:

TemplateFunction , Rule , RuleBase , RuleBaseListed , := , Retract .

###

Macro -- declare or define a macro

##### Standard library

##### Calling format:

    Macro() func(arglist)
    Macro() func(arglist, ...)
    Macro("op", {arglist}) body
    Macro("op", {arglist, ...}) body

##### Parameters:

`func(args)` -- function declaration, e.g. `f(x,y)`

`"op"` -- string, name of the function

`{arglist}` -- list of atoms, formal arguments to the function

`...` -- literal ellipsis symbol "`...`" used to denote a variable number of arguments

`body` -- expression comprising the body of the function

##### Description:

This does the same as `Function`, but for macros. One can define a macro easily with this function, in stead of having to use `DefMacroRuleBase`.

##### Examples:

the following example defines a looping function.

    In> Macro("myfor",{init,pred,inc,body}) [@init;While(@pred)[@body;@inc;];True;];
    Out> True;
    In> a:=10
    Out> 10;

Here this new macro `myfor` is used to loop, using a variable `a` from the calling environment.

    In> myfor(i:=1,i<10,i++,Echo(a*i))
    10
    20
    30
    40
    50
    60
    70
    80
    90
    Out> True;
    In> i
    Out> 10;

##### See also:

Function , DefMacroRuleBase .

###

Use -- load a file, but not twice

##### Internal function

##### Calling format:

    Use(name)

##### Parameters:

`name` -- name of the file to load

##### Description:

If the file "name" has been loaded before, either by an earlier call to `Use` or via the `DefLoad` mechanism, nothing happens. Otherwise all expressions in the file are read and evaluated. `Use` always returns `True`.

The purpose of this function is to make sure that the file will at least have been loaded, but is not loaded twice.

##### See also:

Load , DefLoad , DefaultDirectory .

###

For -- C-style `for` loop

##### Standard library

##### Calling format:

    For(init, pred, incr) body

##### Parameters:

`init` -- expression for performing the initialization

`pred` -- predicate deciding whether to continue the loop

`incr` -- expression to increment the counter

`body` -- expression to loop over

##### Description:

This commands implements a C style `for` loop. First of all, the expression "init" is evaluated. Then the predicate "pred" is evaluated, which should return `True` or `False`. Next the loop is executed as long as the predicate yields `True`. One traversal of the loop consists of the subsequent evaluations of "body", "incr", and "pred". Finally, the value `True` is returned.

This command is most often used in a form such as `For(i=1, i<=10, i++) body`, which evaluates `body` with `i` subsequently set to 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.

The expression `For(init, pred, incr) body` is equivalent to `init; While(pred) [body; incr;]`.

##### Examples:

    In> For (i:=1, i<=10, i++) Echo({i, i!});
     1  1
     2  2
     3  6
     4  24
     5  120
     6  720
     7  5040
     8  40320
     9  362880
     10  3628800
    Out> True;

##### See also:

While , Until , ForEach .

###

ForEach -- loop over all entries in list

##### Standard library

##### Calling format:

    ForEach(var, list) body

##### Parameters:

`var` -- looping variable

`list` -- list of values to assign to "var"

`body` -- expression to evaluate with different values of "var"

##### Description:

The expression "body" is evaluated multiple times. The first time, "var" has the value of the first element of "list", then it gets the value of the second element and so on. `ForEach` returns `True`.

##### Examples:

     In> ForEach(i,{2,3,5,7,11}) Echo({i, i!});
     2  2
     3  6
     5  120
     7  5040
     11  39916800
    Out> True;

##### See also:

For .

###

Apply -- apply a function to arguments

##### Standard library

##### Calling format:

    Apply(fn, arglist)

##### Parameters:

`fn` -- function to apply

`arglist` -- list of arguments

##### Description:

This function applies the function "fn" to the arguments in "arglist" and returns the result. The first parameter "fn" can either be a string containing the name of a function or a pure function. Pure functions, modeled after lambda-expressions, have the form "`varlist,body`", where "varlist" is the list of formal parameters. Upon application, the formal parameters are assigned the values in "arglist" (the second parameter of `Apply`) and the "body" is evaluated

An shorthand for `Apply` is provided by the `@` operator.

##### Examples:

    In> Apply("+", {5,9});
    Out> 14;
    In> Apply({{x,y}, x-y^2}, {Cos(a), Sin(a)});
    Out> Cos(a)-Sin(a)^2;

##### See also:

Map , MapSingle , @ .

###

MapArgs -- apply a function to all top-level arguments

##### Standard library

##### Calling format:

    MapArgs(expr, fn)

##### Parameters:

`expr` -- an expression to work on

`fn` -- an operation to perform on each argument

##### Description:

Every top-level argument in "expr" is substituted by the result of applying "fn" to this argument. Here "fn" can be either the name of a function or a pure function.

##### Examples:

    In> MapArgs(f(x,y,z),"Sin");
    Out> f(Sin(x),Sin(y),Sin(z));
    In> MapArgs({3,4,5,6}, {{x},x^2});
    Out> {9,16,25,36};

##### See also:

MapSingle , Map , Apply .

###

Subst -- perform a substitution

##### Standard library

##### Calling format:

    Subst(from, to) expr

##### Parameters:

`from` -- expression to be substituted

`to` -- expression to substitute for "from"

`expr` -- expression in which the substitution takes place

##### Description:

This function substitutes every occurrence of "from" in "expr" by "to". This is a syntactical substitution: only places where "from" occurs as a subexpression are affected.

##### Examples:

    In> Subst(x, Sin(y)) x^2+x+1;
    Out> Sin(y)^2+Sin(y)+1;
    In> Subst(a+b, x) a+b+c;
    Out> x+c;
    In> Subst(b+c, x) a+b+c;
    Out> a+b+c;

The explanation for the last result is that the expression `a+b+c` is internally stored as `(a+b)+c`. Hence `a+b` is a subexpression, but `b+c` is not.

##### See also:

WithValue , /: .

###

WithValue -- temporary assignment during an evaluation

##### Standard library

##### Calling format:

    WithValue(var, val, expr)
    WithValue({var,...}, {val,...}, expr)

##### Parameters:

`var` -- variable to assign to

`val` -- value to be assigned to "var"

`expr` -- expression to evaluate with "var" equal to "val"

##### Description:

First, the expression "val" is assigned to the variable "var". Then, the expression "expr" is evaluated and returned. Finally, the assignment is reversed so that the variable "var" has the same value as it had before `WithValue` was evaluated.

The second calling sequence assigns the first element in the list of values to the first element in the list of variables, the second value to the second variable, etc.

##### Examples:

    In> WithValue(x, 3, x^2+y^2+1);
    Out> y^2+10;
    In> WithValue({x,y}, {3,2}, x^2+y^2+1);
    Out> 14;

##### See also:

Subst , /: .

###

/:, /:: -- local simplification rules

##### Standard library

##### Calling format:

    expression /: patterns
    expressions /:: patterns

Precedence: 20000

##### Parameters:

`expression` -- an expression

`patterns` -- a list of patterns

##### Description:

Sometimes you have an expression, and you want to use specific simplification rules on it that are not done by default. This can be done with the `/:` and the `/::` operators. Suppose we have the expression containing things such as `Ln(a*b)`, and we want to change these into `Ln(a)+Ln(b)`, the easiest way to do this is using the `/:` operator, as follows:

    In> Sin(x)*Ln(a*b)
    Out> Sin(x)*Ln(a*b);
    In> % /: { Ln(_x*_y) <- Ln(x)+Ln(y) }
    Out> Sin(x)*(Ln(a)+Ln(b));

A whole list of simplification rules can be built up in the list, and they will be applied to the expression on the left hand side of `/:` .

The forms the patterns can have are one of:

    pattern <- replacement
    {pattern,replacement}
    {pattern,postpredicate,replacement}

Note that for these local rules, `<-` should be used instead of `<--` which would be used in a global rule.

The `/:` operator traverses an expression much as `Subst` does, that is, top down, trying to apply the rules from the beginning of the list of rules to the end of the list of rules. If the rules cannot be applied to an expression, it will try subexpressions of that expression and so on.

It might be necessary sometimes to use the `/::` operator, which repeatedly applies the `/:` operator until the result doesn't change any more. Caution is required, since rules can contradict each other, which could result in an infinite loop. To detect this situation, just use /: repeatedly on the expression. The repetitive nature should become apparent.

##### Examples:

    In> Sin(u)*Ln(a*b) /: {Ln(_x*_y) <- Ln(x)+Ln(y)}
    Out> Sin(u)*(Ln(a)+Ln(b));
    In> Sin(u)*Ln(a*b) /:: { a <- 2, b <- 3 }
    Out> Sin(u)*Ln(6);

##### See also:

Subst .

###

SetHelpBrowser -- set the HTML browser to use for help

##### Standard library

##### Calling format:

    SetHelpBrowser(helpbrowser)

##### Parameters:

`helpbrowser` -- string containing a html browser to use for help

##### Description:

This function sets the help browser you want to use to browse the help online. It calls the help browser with the html page as first argument. The default value is `lynx`. If you want to use a different browser by default it suffices to create a file `~/.yacasrc` and add a line to set the browser in there.

##### Examples:

    In> SetHelpBrowser("netscape")
    Out> "netscape";
    In> ??

##### See also:

Help .

###

TraceStack -- show calling stack after an error occurs

##### Internal function

##### Calling format:

    TraceStack(expression)

##### Parameters:

`expression` -- an expression to evaluate

##### Description:

TraceStack shows the calling stack after an error occurred. It shows the last few items on the stack, not to flood the screen. These are usually the only items of interest on the stack. This is probably by far the most useful debugging function in Yacas. It shows the last few things it did just after an error was generated somewhere.

For each stack frame, it shows if the function evaluated was a built-in function or a user-defined function, and for the user-defined function, the number of the rule it is trying whether it was evaluating the pattern matcher of the rule, or the body code of the rule.

This functionality is not offered by default because it slows down the evaluation code.

##### Examples:

Here is an example of a function calling itself recursively, causing Yacas to flood its stack:

    In> f(x):=f(Sin(x))
    Out> True;
    In> TraceStack(f(2))
    Debug> 982 :  f (Rule # 0 in body)
    Debug> 983 :  f (Rule # 0 in body)
    Debug> 984 :  f (Rule # 0 in body)
    Debug> 985 :  f (Rule # 0 in body)
    Debug> 986 :  f (Rule # 0 in body)
    Debug> 987 :  f (Rule # 0 in body)
    Debug> 988 :  f (Rule # 0 in body)
    Debug> 989 :  f (Rule # 0 in body)
    Debug> 990 :  f (Rule # 0 in body)
    Debug> 991 :  f (Rule # 0 in body)
    Debug> 992 :  f (Rule # 0 in body)
    Debug> 993 :  f (Rule # 0 in body)
    Debug> 994 :  f (Rule # 0 in body)
    Debug> 995 :  f (User function)
    Debug> 996 :  Sin (Rule # 0 in pattern)
    Debug> 997 :  IsList (Internal function)
    Error on line 1 in file [CommandLine]
    Max evaluation stack depth reached.
    Please use MaxEvalDepth to increase the stack
    size as needed.

##### See also:

TraceExp , TraceRule .

###

TraceExp -- evaluate with tracing enabled

##### Internal function

##### Calling format:

    TraceExp(expr)

##### Parameters:

`expr` -- expression to trace

##### Description:

The expression "expr" is evaluated with the tracing facility turned on. This means that every subexpression, which is evaluated, is shown before and after evaluation. Before evaluation, it is shown in the form `TrEnter(x)`, where `x` denotes the subexpression being evaluated. After the evaluation the line `TrLeave(x,y)` is printed, where `y` is the result of the evaluation. The indentation shows the nesting level.

Note that this command usually generates huge amounts of output. A more specific form of tracing (eg. `TraceRule`) is probably more useful for all but very simple expressions.

##### Examples:

    In> TraceExp(2+3);
      TrEnter(2+3);
        TrEnter(2);
        TrLeave(2, 2);
        TrEnter(3);
        TrLeave(3, 3);
        TrEnter(IsNumber(x));
          TrEnter(x);
          TrLeave(x, 2);
        TrLeave(IsNumber(x),True);
        TrEnter(IsNumber(y));
          TrEnter(y);
          TrLeave(y, 3);
        TrLeave(IsNumber(y),True);
        TrEnter(True);
        TrLeave(True, True);
        TrEnter(MathAdd(x,y));
          TrEnter(x);
          TrLeave(x, 2);
          TrEnter(y);
          TrLeave(y, 3);
        TrLeave(MathAdd(x,y),5);
      TrLeave(2+3, 5);
    Out> 5;

##### See also:

TraceStack , TraceRule .

###

TraceRule -- turn on tracing for a particular function

##### Internal function

##### Calling format:

    TraceRule(template) expr

##### Parameters:

`template` -- template showing the operator to trace

`expr` -- expression to evaluate with tracing on

##### Description:

The tracing facility is turned on for subexpressions of the form "template", and the expression "expr" is evaluated. The template "template" is an example of the function to trace on. Specifically, all subexpressions with the same top-level operator and arity as "template" are shown. The subexpressions are displayed before (indicated with `TrEnter`) and after (`TrLeave`) evaluation. In between, the arguments are shown before and after evaluation (`TrArg`). Only functions defined in scripts can be traced.

This is useful for tracing a function that is called from within another function. This way you can see how your function behaves in the environment it is used in.

##### Examples:

    In> TraceRule(x+y) 2+3*5+4;
        TrEnter(2+3*5+4);
          TrEnter(2+3*5);
              TrArg(2, 2);
              TrArg(3*5, 15);
          TrLeave(2+3*5, 17);
            TrArg(2+3*5, 17);
            TrArg(4, 4);
        TrLeave(2+3*5+4, 21);
    Out> 21;

##### See also:

TraceStack , TraceExp .

###

Time -- measure the time taken by a function

##### Standard library

##### Calling format:

    Time(expr)

##### Parameters:

`expr` -- any expression

##### Description:

The function `Time(expr)` evaluates the expression `expr` and prints the time in seconds needed for the evaluation. The time is printed to the current output stream. The built-in function `GetTime` is used for timing.

The result is the "user time" as reported by the OS, not the real ("wall clock") time. Therefore, any CPU-intensive processes running alongside Yacas will not significantly affect the result of `Time`.

##### Example:

    In> Time([Precision(40);MathLog(1000);])
    0.34 seconds taken
    Out> 6.9077552789821370520539743640530926228033;

##### See also:

GetTime .

< , > , <= , >= , != , = , Not , And , Or , IsFreeOf , IsZeroVector , IsNonObject , IsEven , IsOdd , IsEvenFunction , IsOddFunction , IsFunction , IsAtom , IsString , IsNumber , IsList , IsNumericList , IsBound , IsBoolean , IsNegativeNumber , IsNegativeInteger , IsPositiveNumber , IsPositiveInteger , IsNotZero , IsNonZeroInteger , IsInfinity , IsPositiveReal , IsNegativeReal , IsConstant , IsGaussianInteger , MatchLinear , HasExpr, HasExprArith, HasExprSome , HasFunc, HasFuncArith, HasFuncSome .

# Predicates

A predicate is a function that returns a boolean value, i.e. `True` or `False`. Predicates are often used in patterns, For instance, a rule that only holds for a positive integer would use a pattern such as `n_IsPositiveInteger`.

###

< -- test for "less than"

##### Standard library

##### Calling format:

    e1 < e2

Precedence: 90

##### Parameters:

`e1`, `e2` -- expressions to be compared

##### Description:

The two expression are evaluated. If both results are numeric, they are compared. If the first expression is smaller than the second one, the result is `True` and it is `False` otherwise. If either of the expression is not numeric, after evaluation, the expression is returned with evaluated arguments.

The word "numeric" in the previous paragraph has the following meaning. An expression is numeric if it is either a number (i.e. `IsNumber` returns `True`), or the quotient of two numbers, or an infinity (i.e. `IsInfinity` returns `True`).

##### Examples:

    In> 2 < 5;
    Out> True;
    In> Cos(1) < 5;
    Out> Cos(1)<5;
    In> N(Cos(1)) < 5;
    Out> True

##### See also:

IsNumber , IsInfinity , N .

###

> -- test for "greater than"

##### Standard library

##### Calling format:

    e1 > e2

Precedence: 90

##### Parameters:

`e1`, `e2` -- expressions to be compared

##### Description:

The two expression are evaluated. If both results are numeric, they are compared. If the first expression is larger than the second one, the result is `True` and it is `False` otherwise. If either of the expression is not numeric, after evaluation, the expression is returned with evaluated arguments.

The word "numeric" in the previous paragraph has the following meaning. An expression is numeric if it is either a number (i.e. `IsNumber` returns `True`), or the quotient of two numbers, or an infinity (i.e. `IsInfinity` returns `True`).

##### Examples:

    In> 2 > 5;
    Out> False;
    In> Cos(1) > 5;
    Out> Cos(1)>5;
    In> N(Cos(1)) > 5;
    Out> False

##### See also:

IsNumber , IsInfinity , N .

###

<= -- test for "less or equal"

##### Standard library

##### Calling format:

    e1 <= e2

Precedence: 90

##### Parameters:

`e1`, `e2` -- expressions to be compared

##### Description:

The two expression are evaluated. If both results are numeric, they are compared. If the first expression is smaller than or equals the second one, the result is `True` and it is `False` otherwise. If either of the expression is not numeric, after evaluation, the expression is returned with evaluated arguments.

The word "numeric" in the previous paragraph has the following meaning. An expression is numeric if it is either a number (i.e. `IsNumber` returns `True`), or the quotient of two numbers, or an infinity (i.e. `IsInfinity` returns `True`).

##### Examples:

    In> 2 <= 5;
    Out> True;
    In> Cos(1) <= 5;
    Out> Cos(1)<=5;
    In> N(Cos(1)) <= 5;
    Out> True

##### See also:

IsNumber , IsInfinity , N .

###

>= -- test for "greater or equal"

##### Standard library

##### Calling format:

    e1 >= e2

Precedence: 90

##### Parameters:

`e1`, `e2` -- expressions to be compared

##### Description:

The two expression are evaluated. If both results are numeric, they are compared. If the first expression is larger than or equals the second one, the result is `True` and it is `False` otherwise. If either of the expression is not numeric, after evaluation, the expression is returned with evaluated arguments.

The word "numeric" in the previous paragraph has the following meaning. An expression is numeric if it is either a number (i.e. `IsNumber` returns `True`), or the quotient of two numbers, or an infinity (i.e. `IsInfinity` returns `True`).

##### Examples:

    In> 2 >= 5;
    Out> False;
    In> Cos(1) >= 5;
    Out> Cos(1)>=5;
    In> N(Cos(1)) >= 5;
    Out> False

##### See also:

IsNumber , IsInfinity , N .

###

!= -- test for "not equal"

##### Standard library

##### Calling format:

    e1 != e2

Precedence: 90

##### Parameters:

`e1`, `e2` -- expressions to be compared

##### Description:

Both expressions are evaluated and compared. If they turn out to be equal, the result is `False`. Otherwise, the result is `True`.

The expression `e1 != e2` is equivalent to `Not(e1 = e2)`.

##### Examples:

    In> 1 != 2;
    Out> True;
    In> 1 != 1;
    Out> False;

##### See also:

= .

###

= -- test for equality of expressions

##### Standard library

##### Calling format:

    e1 = e2

Precedence: 90

##### Parameters:

`e1`, `e2` -- expressions to be compared

##### Description:

Both expressions are evaluated and compared. If they turn out to be equal, the result is `True`. Otherwise, the result is `False`. The function `Equals` does the same.

Note that the test is on syntactic equality, not mathematical equality. Hence even if the result is `False`, the expressions can still be mathematically equal; see the examples below. Put otherwise, this function tests whether the two expressions would be displayed in the same way if they were printed.

##### Examples:

    In> e1 := (x+1) * (x-1);
    Out> (x+1)*(x-1);
    In> e2 := x^2 - 1;
    Out> x^2-1;

    In> e1 = e2;
    Out> False;
    In> Expand(e1) = e2;
    Out> True;

##### See also:

!= , Equals .

###

Not -- logical negation

##### Internal function

##### Calling format:

    Not expr

##### Parameters:

`expr` -- a boolean expression

##### Description:

Not returns the logical negation of the argument expr. If `expr` is `False` it returns `True`, and if `expr` is `True`, `Not expr` returns `False`. If the argument is neither `True` nor `False`, it returns the entire expression with evaluated arguments.

##### Examples:

    In> Not True
    Out> False;
    In> Not False
    Out> True;
    In> Not(a)
    Out> Not a;

##### See also:

And , Or .

###

And -- logical conjunction

##### Internal function

##### Calling format:

    a1 And a2

Precedence: 1000

    And(a1, a2, a3, ..., aN)

##### Parameters:

`a`1, ..., `a`N -- boolean values (may evaluate to `True` or `False`)

##### Description:

This function returns `True` if all arguments are true. The `And` operation is "lazy", i.e. it returns `False` as soon as a `False` argument is found (from left to right). If an argument other than `True` or `False` is encountered a new `And` expression is returned with all arguments that didn't evaluate to `True` or `False` yet.

##### Examples:

    In> True And False
    Out> False;
    In> And(True,True)
    Out> True;
    In> False And a
    Out> False;
    In> True And a
    Out> And(a);
    In> And(True,a,True,b)
    Out> b And a;

##### See also:

Or , Not .

###

Or -- logical disjunction

##### Internal function

##### Calling format:

    a1 Or a2

Precedence: 1010

    Or(a1, a2, a3, ..., aN)

##### Parameters:

`a`1, ..., `a`N -- boolean expressions (may evaluate to `True` or `False`)

##### Description:

This function returns `True` if an argument is encountered that is true (scanning from left to right). The `Or` operation is "lazy", i.e. it returns `True` as soon as a `True` argument is found (from left to right). If an argument other than `True` or `False` is encountered, an unevaluated `Or` expression is returned with all arguments that didn't evaluate to `True` or `False` yet.

##### Examples:

    In> True Or False
    Out> True;
    In> False Or a
    Out> Or(a);
    In> Or(False,a,b,True)
    Out> True;

##### See also:

And , Not .

###

IsFreeOf -- test whether expression depends on variable

##### Standard library

##### Calling format:

    IsFreeOf(var, expr)
    IsFreeOf({var, ...}, expr)

##### Parameters:

`expr` -- expression to test

`var` -- variable to look for in "expr"

##### Description:

This function checks whether the expression "expr" (after being evaluated) depends on the variable "var". It returns `False` if this is the case and `True` otherwise.

The second form test whether the expression depends on any of the variables named in the list. The result is `True` if none of the variables appear in the expression and `False` otherwise.

##### Examples:

    In> IsFreeOf(x, Sin(x));
    Out> False;
    In> IsFreeOf(y, Sin(x));
    Out> True;
    In> IsFreeOf(x, D(x) a*x+b);
    Out> True;
    In> IsFreeOf({x,y}, Sin(x));
    Out> False;

The third command returns `True` because the expression `D(x) a*x+b` evaluates to `a`, which does not depend on `x`.

##### See also:

Contains .

###

IsZeroVector -- test whether list contains only zeroes

##### Standard library

##### Calling format:

    IsZeroVector(list)

##### Parameters:

`list` -- list to compare against the zero vector

##### Description:

The only argument given to `IsZeroVector` should be a list. The result is `True` if the list contains only zeroes and `False` otherwise.

##### Examples:

    In> IsZeroVector({0, x, 0});
    Out> False;
    In> IsZeroVector({x-x, 1 - D(x) x});
    Out> True;

##### See also:

IsList , ZeroVector .

###

IsNonObject -- test whether argument is not an `Object()`

##### Standard library

##### Calling format:

    IsNonObject(expr)

##### Parameters:

`expr` -- the expression to examine

##### Description:

This function returns `True` if "expr" is not of the form `Object(...)` and `False` otherwise.

##### Bugs

In fact, the result is always `True`.

##### See also:

Object .

###

IsEven -- test for an even integer

##### Standard library

##### Calling format:

    IsEven(n)

##### Parameters:

`n` -- integer to test

##### Description:

This function tests whether the integer "n" is even. An integer is even if it is divisible by two. Hence the even numbers are 0, 2, 4, 6, 8, 10, etc., and -2, -4, -6, -8, -10, etc.

##### Examples:

    In> IsEven(4);
    Out> True;
    In> IsEven(-1);
    Out> False;

##### See also:

IsOdd , IsInteger .

###

IsOdd -- test for an odd integer

##### Standard library

##### Calling format:

    IsOdd(n)

##### Parameters:

`n` -- integer to test

##### Description:

This function tests whether the integer "n" is odd. An integer is odd if it is not divisible by two. Hence the odd numbers are 1, 3, 5, 7, 9, etc., and -1, -3, -5, -7, -9, etc.

##### Examples:

    In> IsOdd(4);
    Out> False;
    In> IsOdd(-1);
    Out> True;

##### See also:

IsEven , IsInteger .

###

IsEvenFunction -- Return true if function is an even function, False otherwise

###

IsOddFunction -- Return true if function is an odd function, False otherwise

##### Standard library

##### Calling format:

    IsEvenFunction(expression,variable)
    IsOddFunction(expression,variable)

##### Parameters:

`expression` -- mathematical expression `variable` -- variable

##### Description:

These functions return True if Yacas can determine that the function is even or odd respectively. Even functions are defined to be functions that have the property:

f(x)=f(-x)

And odd functions have the property:

f(x)= -f(-x)

`Sin(x)` is an example of an odd function, and `Cos(x)` is an example of an even function.

As a side note, one can decompose a function into an even and an odd part:

f(x)=f _even(x)+f _odd(x)

Where

f _even(x)=(f(x)+f(-x))/2

and

f _odd(x)=(f(x)-f(-x))/2

##### Example:

    In> IsEvenFunction(Cos(b*x),x)
    Out> True
    In> IsOddFunction(Cos(b*x),x)
    Out> False
    In> IsOddFunction(Sin(b*x),x)
    Out> True
    In> IsEvenFunction(Sin(b*x),x)
    Out> False
    In> IsEvenFunction(1/x^2,x)
    Out> True
    In> IsEvenFunction(1/x,x)
    Out> False
    In> IsOddFunction(1/x,x)
    Out> True
    In> IsOddFunction(1/x^2,x)
    Out> False

##### See also:

Sin , Cos .

###

IsFunction -- test for a composite object

##### Internal function

##### Calling format:

    IsFunction(expr)

##### Parameters:

`expr` -- expression to test

##### Description:

This function tests whether "expr" is a composite object, i.e. not an atom. This includes not only obvious functions such as `f(x)`, but also expressions such as `x+5` and lists.

##### Examples:

    In> IsFunction(x+5);
    Out> True;
    In> IsFunction(x);
    Out> False;

##### See also:

IsAtom , IsList , Type .

###

IsAtom -- test for an atom

##### Internal function

##### Calling format:

    IsAtom(expr)

##### Parameters:

`expr` -- expression to test

##### Description:

This function tests whether "expr" is an atom. Numbers, strings, and variables are all atoms.

##### Examples:

    In> IsAtom(x+5);
    Out> Falso;
    In> IsAtom(5);
    Out> True;

##### See also:

IsFunction , IsNumber , IsString .

###

IsString -- test for an string

##### Internal function

##### Calling format:

    IsString(expr)

##### Parameters:

`expr` -- expression to test

##### Description:

This function tests whether "expr" is a string. A string is a text within quotes, e.g. `"duh"`.

##### Examples:

    In> IsString("duh");
    Out> True;
    In> IsString(duh);
    Out> False;

##### See also:

IsAtom , IsNumber .

###

IsNumber -- test for a number

##### Internal function

##### Calling format:

    IsNumber(expr)

##### Parameters:

`expr` -- expression to test

##### Description:

This function tests whether "expr" is a number. There are two kinds of numbers, integers (e.g. 6) and reals (e.g. -2.75 or 6.0). Note that a complex number is represented by the `Complex` function, so `IsNumber` will return `False`.

##### Examples:

    In> IsNumber(6);
    Out> True;
    In> IsNumber(3.25);
    Out> True;
    In> IsNumber(I);
    Out> False;
    In> IsNumber("duh");
    Out> False;

##### See also:

IsAtom , IsString , IsInteger , IsPositiveNumber , IsNegativeNumber , Complex .

###

IsList -- test for a list

##### Internal function

##### Calling format:

    IsList(expr)

##### Parameters:

`expr` -- expression to test

##### Description:

This function tests whether "expr" is a list. A list is a sequence between curly braces, e.g. `{2, 3, 5}`.

##### Examples:

    In> IsList({2,3,5});
    Out> True;
    In> IsList(2+3+5);
    Out> False;

##### See also:

IsFunction .

###

IsNumericList -- test for a list of numbers

##### Standard library

##### Calling format:

    IsNumericList({list})

##### Parameters:

`{list}` -- a list

##### Description:

Returns `True` when called on a list of numbers or expressions that evaluate to numbers using `N()`. Returns `False` otherwise.

##### See also:

N , IsNumber .

###

IsBound -- test for a bound variable

##### Internal function

##### Calling format:

    IsBound(var)

##### Parameters:

`var` -- variable to test

##### Description:

This function tests whether the variable "var" is bound, i.e. whether it has been assigned a value. The argument "var" is not evaluated.

##### Examples:

    In> IsBound(x);
    Out> False;
    In> x := 5;
    Out> 5;
    In> IsBound(x);
    Out> True;

##### See also:

IsAtom .

###

IsBoolean -- test for a Boolean value

##### Standard library

##### Calling format:

    IsBoolean(expression)

##### Parameters:

`expression` -- an expression

##### Description:

IsBoolean returns True if the argument is of a boolean type. This means it has to be either True, False, or an expression involving functions that return a boolean result, e.g. `=`, `>`, `<`, `>=`, `<=`, `!=`, `And`, `Not`, `Or`.

##### Examples:

    In> IsBoolean(a)
    Out> False;
    In> IsBoolean(True)
    Out> True;
    In> IsBoolean(a And b)
    Out> True;

##### See also:

True , False .

###

IsNegativeNumber -- test for a negative number

##### Standard library

##### Calling format:

    IsNegativeNumber(n)

##### Parameters:

`n` -- number to test

##### Description:

`IsNegativeNumber(n)` evaluates to `True` if n is (strictly) negative, i.e. if n<0. If `n` is not a number, the functions return `False`.

##### Examples:

    In> IsNegativeNumber(6);
    Out> False;
    In> IsNegativeNumber(-2.5);
    Out> True;

##### See also:

IsNumber , IsPositiveNumber , IsNotZero , IsNegativeInteger , IsNegativeReal .

###

IsNegativeInteger -- test for a negative integer

##### Standard library

##### Calling format:

    IsNegativeInteger(n)

##### Parameters:

`n` -- integer to test

##### Description:

This function tests whether the integer `n` is (strictly) negative. The negative integers are -1, -2, -3, -4, -5, etc. If `n` is not a integer, the function returns `False`.

##### Examples:

    In> IsNegativeInteger(31);
    Out> False;
    In> IsNegativeInteger(-2);
    Out> True;

##### See also:

IsPositiveInteger , IsNonZeroInteger , IsNegativeNumber .

###

IsPositiveNumber -- test for a positive number

##### Standard library

##### Calling format:

    IsPositiveNumber(n)

##### Parameters:

`n` -- number to test

##### Description:

`IsPositiveNumber(n)` evaluates to `True` if n is (strictly) positive, i.e. if n>0. If `n` is not a number the function returns `False`.

##### Examples:

    In> IsPositiveNumber(6);
    Out> True;
    In> IsPositiveNumber(-2.5);
    Out> False;

##### See also:

IsNumber , IsNegativeNumber , IsNotZero , IsPositiveInteger , IsPositiveReal .

###

IsPositiveInteger -- test for a positive integer

##### Standard library

##### Calling format:

    IsPositiveInteger(n)

##### Parameters:

`n` -- integer to test

##### Description:

This function tests whether the integer `n` is (strictly) positive. The positive integers are 1, 2, 3, 4, 5, etc. If `n` is not a integer, the function returns `False`.

##### Examples:

    In> IsPositiveInteger(31);
    Out> True;
    In> IsPositiveInteger(-2);
    Out> False;

##### See also:

IsNegativeInteger , IsNonZeroInteger , IsPositiveNumber .

###

IsNotZero -- test for a nonzero number

##### Standard library

##### Calling format:

    IsNotZero(n)

##### Parameters:

`n` -- number to test

##### Description:

`IsNotZero(n)` evaluates to `True` if `n` is not zero. In case `n` is not a number, the function returns `False`.

##### Examples:

    In> IsNotZero(3.25);
    Out> True;
    In> IsNotZero(0);
    Out> False;

##### See also:

IsNumber , IsPositiveNumber , IsNegativeNumber , IsNonZeroInteger .

###

IsNonZeroInteger -- test for a nonzero integer

##### Standard library

##### Calling format:

    IsNonZeroInteger(n)

##### Parameters:

`n` -- integer to test

##### Description:

This function tests whether the integer `n` is not zero. If `n` is not an integer, the result is `False`.

##### Examples:

    In> IsNonZeroInteger(0)
    Out> False;
    In> IsNonZeroInteger(-2)
    Out> True;

##### See also:

IsPositiveInteger , IsNegativeInteger , IsNotZero .

###

IsInfinity -- test for an infinity

##### Standard library

##### Calling format:

    IsInfinity(expr)

##### Parameters:

`expr` -- expression to test

##### Description:

This function tests whether `expr` is an infinity. This is only the case if `expr` is either `Infinity` or `-Infinity`.

##### Examples:

    In> IsInfinity(10^1000);
    Out> False;
    In> IsInfinity(-Infinity);
    Out> True;

##### See also:

Integer .

###

IsPositiveReal -- test for a numerically positive value

##### Standard library

##### Calling format:

    IsPositiveReal(expr)

##### Parameters:

`expr` -- expression to test

##### Description:

This function tries to approximate "expr" numerically. It returns `True` if this approximation is positive. In case no approximation can be found, the function returns `False`. Note that round-off errors may cause incorrect results.

##### Examples:

    In> IsPositiveReal(Sin(1)-3/4);
    Out> True;
    In> IsPositiveReal(Sin(1)-6/7);
    Out> False;
    In> IsPositiveReal(Exp(x));
    Out> False;

The last result is because `Exp(x)` cannot be numerically approximated if `x` is not known. Hence Yacas can not determine the sign of this expression.

##### See also:

IsNegativeReal , IsPositiveNumber , N .

###

IsNegativeReal -- test for a numerically negative value

##### Standard library

##### Calling format:

    IsNegativeReal(expr)

##### Parameters:

`expr` -- expression to test

##### Description:

This function tries to approximate `expr` numerically. It returns `True` if this approximation is negative. In case no approximation can be found, the function returns `False`. Note that round-off errors may cause incorrect results.

##### Examples:

    In> IsNegativeReal(Sin(1)-3/4);
    Out> False;
    In> IsNegativeReal(Sin(1)-6/7);
    Out> True;
    In> IsNegativeReal(Exp(x));
    Out> False;

The last result is because `Exp(x)` cannot be numerically approximated if `x` is not known. Hence Yacas can not determine the sign of this expression.

##### See also:

IsPositiveReal , IsNegativeNumber , N .

###

IsConstant -- test for a constant

##### Standard library

##### Calling format:

    IsConstant(expr)

##### Parameters:

`expr` -- some expression

##### Description:

`IsConstant` returns `True` if the expression is some constant or a function with constant arguments. It does this by checking that no variables are referenced in the expression. `Pi` is considered a constant.

##### Examples:

    In> IsConstant(Cos(x))
    Out> False;
    In> IsConstant(Cos(2))
    Out> True;
    In> IsConstant(Cos(2+x))
    Out> False;

##### See also:

IsNumber , IsInteger , VarList .

###

IsGaussianInteger -- test for a Gaussian integer

##### Standard library

##### Calling format:

    IsGaussianInteger(z)

##### Parameters:

`z` -- a complex or real number

##### Description:

This function returns `True` if the argument is a Gaussian integer and `False` otherwise. A Gaussian integer is a generalization of integers into the complex plane. A complex number a+b*I is a Gaussian integer if and only if a and b are integers.

##### Examples:

    In> IsGaussianInteger(5)
    Out> True;
    In> IsGaussianInteger(5+6*I)
    Out> True;
    In> IsGaussianInteger(1+2.5*I)
    Out> False;

##### See also:

IsGaussianUnit , IsGaussianPrime .

###

MatchLinear -- match an expression to a polynomial of degree one in a variable

##### Standard library

##### Calling format:

    MatchLinear(x,expr)

##### Parameters:

`x` -- variable to express the univariate polynomial in

`expr` -- expression to match

##### Description:

`MatchLinear` tries to match an expression to a linear (degree less than two) polynomial. The function returns `True` if it could match, and it stores the resulting coefficients in the variables "`a`" and "`b`" as a side effect. The function calling this predicate should declare local variables "`a`" and "`b`" for this purpose. `MatchLinear` tries to match to constant coefficients which don't depend on the variable passed in, trying to find a form "`a*x+b`" with "`a`" and "`b`" not depending on `x` if `x` is given as the variable.

##### Examples:

    In> MatchLinear(x,(R+1)*x+(T-1))
    Out> True;
    In> {a,b};
    Out> {R+1,T-1};
    In> MatchLinear(x,Sin(x)*x+(T-1))
    Out> False;

##### See also:

Integrate .

###

HasExpr, HasExprArith, HasExprSome -- check for expression containing a subexpression

##### Standard library

##### Calling format:

    HasExpr(expr, x)
    HasExprArith(expr, x)
    HasExprSome(expr, x, list)

##### Parameters:

`expr` -- an expression

`x` -- a subexpression to be found

`list` -- list of function atoms to be considered "transparent"

##### Description:

The command `HasExpr` returns `True` if the expression `expr` contains a literal subexpression `x`. The expression is recursively traversed.

The command `HasExprSome` does the same, except it only looks at arguments of a given `list` of functions. All other functions become "opaque" (as if they do not contain anything).

`HasExprArith` is defined through `HasExprSome` to look only at arithmetic operations `+`, `-`, `*`, `/`.

Note that since the operators "`+`" and "`-`" are prefix as well as infix operators, it is currently required to use `Atom("+")` to obtain the unevaluated atom "`+`".

##### Examples:

    In> HasExpr(x+y*Cos(Ln(z)/z), z)
    Out> True;
    In> HasExpr(x+y*Cos(Ln(z)/z), Ln(z))
    Out> True;
    In> HasExpr(x+y*Cos(Ln(z)/z), z/Ln(z))
    Out> False;
    In> HasExprArith(x+y*Cos(Ln(x)/x), z)
    Out> False;
    In> HasExprSome({a+b*2,c/d},c/d,{List})
    Out> True;
    In> HasExprSome({a+b*2,c/d},c,{List})
    Out> False;

##### See also:

FuncList , VarList , HasFunc .

###

HasFunc, HasFuncArith, HasFuncSome -- check for expression containing a function

##### Standard library

##### Calling format:

    HasFunc(expr, func)
    HasFuncArith(expr, func)
    HasFuncSome(expr, func, list)

##### Parameters:

`expr` -- an expression

`func` -- a function atom to be found

`list` -- list of function atoms to be considered "transparent"

##### Description:

The command `HasFunc` returns `True` if the expression `expr` contains a function `func`. The expression is recursively traversed.

The command `HasFuncSome` does the same, except it only looks at arguments of a given `list` of functions. Arguments of all other functions become "opaque" (as if they do not contain anything).

`HasFuncArith` is defined through `HasFuncSome` to look only at arithmetic operations `+`, `-`, `*`, `/`.

Note that since the operators "`+`" and "`-`" are prefix as well as infix operators, it is currently required to use `Atom("+")` to obtain the unevaluated atom "`+`".

##### Examples:

    In> HasFunc(x+y*Cos(Ln(z)/z), Ln)
    Out> True;
    In> HasFunc(x+y*Cos(Ln(z)/z), Sin)
    Out> False;
    In> HasFuncArith(x+y*Cos(Ln(x)/x), Cos)
    Out> True;
    In> HasFuncArith(x+y*Cos(Ln(x)/x), Ln)
    Out> False;
    In> HasFuncSome({a+b*2,c/d},/,{List})
    Out> True;
    In> HasFuncSome({a+b*2,c/d},*,{List})
    Out> False;

##### See also:

FuncList , VarList , HasExpr .

CanProve .

# Propositional logic theorem prover

###

CanProve -- try to prove statement

##### Standard library

##### Calling format:

    CanProve(proposition)

##### Parameters:

`proposition` -- an expression with logical operations

##### Description:

Yacas has a small built-in propositional logic theorem prover. It can be invoked with a call to `CanProve`.

An example of a proposition is: "if a implies b and b implies c then a implies c". Yacas supports the following logical operations:

`Not` : negation, read as "not"

`And` : conjunction, read as "and"

`Or` : disjunction, read as "or"

`=>` : implication, read as "implies"

The abovementioned proposition would be represented by the following expression,

    ( (a=>b) And (b=>c) ) => (a=>c)

Yacas can prove that is correct by applying `CanProve` to it:

    In> CanProve(( (a=>b) And (b=>c) ) => (a=>c))
    Out> True;

It does this in the following way: in order to prove a proposition p, it suffices to prove that Not p is false. It continues to simplify Not p using the rules:

    Not  ( Not x)      --> x

(eliminate double negation),

    x=>y  -->  Not x  Or  y

(eliminate implication),

    Not (x And y)  -->  Not x  Or  Not y

(De Morgan's law),

    Not (x Or y) -->  Not x  And  Not y

(De Morgan's law),

    (x And y) Or z --> (x Or z) And (y Or z)

(distribution),

    x Or (y And z) --> (x Or y) And (x Or z)

(distribution), and the obvious other rules, such as,

    True Or x --> True

etc. The above rules will translate a proposition into a form

    (p1  Or  p2  Or  ...)  And  (q1  Or  q2
       Or  ...)  And ...

If any of the clauses is false, the entire expression will be false. In the next step, clauses are scanned for situations of the form:

    (p Or Y)  And  ( Not p Or Z) --> (Y Or Z)

If this combination `(Y Or Z)` is empty, it is false, and thus the entire proposition is false.

As a last step, the algorithm negates the result again. This has the added advantage of simplifying the expression further.

##### Examples:

    In> CanProve(a  Or   Not a)
    Out> True;
    In> CanProve(True  Or  a)
    Out> True;
    In> CanProve(False  Or  a)
    Out> a;
    In> CanProve(a  And   Not a)
    Out> False;
    In> CanProve(a  Or b Or (a And b))
    Out> a Or b;

##### See also:

True , False , And , Or , Not .

% , True, False , EndOfFile , Infinity , Pi , Undefined .

# Constants

###

% -- previous result

##### Internal function

##### Calling format:

    %

##### Description:

`%` evaluates to the previous result on the command line. `%` is a global variable that is bound to the previous result from the command line. Using `%` will evaluate the previous result. (This uses the functionality offered by the `LazyGlobal` command).

Typical examples are `Simplify(%)` and `PrettyForm(%)` to simplify and show the result in a nice form respectively.

##### Examples:

    In> Taylor(x,0,5)Sin(x)
    Out> x-x^3/6+x^5/120;
    In> PrettyForm(%)

         3    5
        x    x
    x - -- + ---
        6    120

##### See also:

LazyGlobal .

###

True, False -- boolean constants

##### Internal function

##### Calling format:

    True
    False

##### Description:

`True` and `False` are typically a result of boolean expressions such as `2 < 3` or `True And False`.

##### See also:

And , Or , Not .

###

EndOfFile -- end-of-file marker

##### Internal function

##### Calling format:

    EndOfFile

##### Description:

End of file marker when reading from file. If a file contains the expression `EndOfFile;` the operation will stop reading the file at that point.

###

Infinity -- constant representing mathematical infinity

##### Standard library

##### Calling format:

    Infinity

##### Description:

Infinity represents infinitely large values. It can be the result of certain calculations.

Note that for most analytic functions Yacas understands `Infinity` as a positive number. Thus `Infinity*2` will return `Infinity`, and `a < Infinity` will evaluate to `True`.

##### Examples:

    In> 2*Infinity
    Out> Infinity;
    In> 2<Infinity
    Out> True;

###

Pi -- mathematical constant, pi

##### Standard library

##### Calling format:

    Pi

##### Description:

Pi symbolically represents the exact value of pi. When the `N()` function is used, `Pi` evaluates to a numerical value according to the current precision. This is performed by the function `Pi()` which always returns the numerical value. It is better to use `Pi` than `Pi()` except in numerical calculations, because exact simplification will be possible.

This is a "cached constant" which is recalculated only when precision is increased.

##### Examples:

    In> Sin(3*Pi/2)
    Out> -1;
    In> Sin(3*Pi()/2)
    Out> Sin(4.7123889804);
    In> Pi+1
    Out> Pi+1;
    In> N(Pi)
    Out> 3.14159265358979323846;

##### See also:

Sin , Cos , Precision , N , Pi() , CachedConstant .

###

Undefined -- constant signifying an undefined result

##### Standard library

##### Calling format:

    Undefined

##### Description:

`Undefined` is a token that can be returned by a function when it considers its input to be invalid or when no meaningful answer can be given. The result is then "undefined".

Most functions also return `Undefined` when evaluated on it.

##### Examples:

    In> 2*Infinity
    Out> Infinity;
    In> 0*Infinity
    Out> Undefined;
    In> Sin(Infinity);
    Out> Undefined;
    In> Undefined+2*Exp(Undefined);
    Out> Undefined;

##### See also:

Infinity .

:= , Set , Clear , Local , ++ , -- , Object , LazyGlobal , UniqueConstant .

# Variables

###

:= -- assign a variable or a list; define a function

##### Standard library

##### Calling format:

    var := expr
    {var1, var2, ...} := {expr1, expr2, ...}
    var[i] := expr
    fn(arg1, arg2, ...) := expr

Precedence: 10000

##### Parameters:

`var` -- atom, variable which should be assigned

`expr` -- expression to assign to the variable or body of function

`i` -- index (can be integer or string)

`fn` -- atom, name of a new function to define

`arg1`, `arg2` -- atoms, names of arguments of the new function `fn`

##### Description:

The `:=` operator can be used in a number of ways. In all cases, some sort of assignment or definition takes place.

The first form is the most basic one. It evaluates the expression on the right-hand side and assigns it to the variable named on the left-hand side. The left-hand side is not evaluated. The evaluated expression is also returned.

The second form is a small extension, which allows one to do multiple assignments. The first entry in the list on the right-hand side is assigned to the first variable mentioned in the left-hand side, the second entry on the right-hand side to the second variable on the left-hand side, etc. The list on the right-hand side must have at least as many entries as the list on the left-hand side. Any excess entries are silently ignored. The result of the expression is the list of values that have been assigned.

The third form allows one to change an entry in the list. If the index "i" is an integer, the "i"-th entry in the list is changed to the expression on the right-hand side. It is assumed that the length of the list is at least "i". If the index "i" is a string, then "var" is considered to be an associative list (sometimes called hash table), and the key "i" is paired with the value "exp". In both cases, the right-hand side is evaluated before the assignment and the result of the assignment is `True`.

The last form defines a function. For example, the assignment `fn(x) := x^2` removes any rules previously associated with `fn(x)` and defines the rule `fn(_x) <-- x^2`. Note that the left-hand side may take a different form if `fn` is defined to be a prefix, infix or bodied function. This case is special since the right-hand side is not evaluated immediately, but only when the function `fn` is used. If this takes time, it may be better to force an immediate evaluation with `Eval` (see the last example). If the expression on the right hand side begins with `Eval()`, then it will be evaluated before defining the new function.

A variant of the function definition can be used to make a function accepting a variable number of arguments. The last argument

##### Examples:

A simple assignment:

    In> a := Sin(x) + 3;
    Out> Sin(x)+3;
    In> a;
    Out> Sin(x)+3;

Multiple assignments:

    In> {a,b,c} := {1,2,3};
    Out> {1,2,3};
    In> a;
    Out> 1;
    In> b+c;
    Out> 5;

Assignment to a list:

    In> xs := { 1,2,3,4,5 };
    Out> {1,2,3,4,5};
    In> xs[3] := 15;
    Out> True;
    In> xs;
    Out> {1,2,15,4,5};

Building an associative list:

    In> alist := {};
    Out> {};
    In> alist["cherry"] := "red";
    Out> True;
    In> alist["banana"] := "yellow";
    Out> True;
    In> alist["cherry"];
    Out> "red";
    In> alist;
    Out> {{"banana","yellow"},{"cherry","red"}};

Defining a function:

    In> f(x) := x^2;
    Out> True;
    In> f(3);
    Out> 9;
    In> f(Sin(a));
    Out> Sin(a)^2;

Defining a function with variable number of arguments:

    In> f(x, ...) := If(IsList(x),Sum(x),x);
    Out> True;
    In> f(2);
    Out> 2;
    In> f(1,2,3);
    Out> 6;

Defining a new infix operator:

    In> Infix("*&*",10);
    Out> True;
    In> x1 *&* x2 := x1/x2 + x2/x1;
    Out> True;
    In> Sin(a) *&* Cos(a);
    Out> Tan(1)+Cos(1)/Sin(1);
    In> Clear(a);
    Out> True;
    In> Sin(a) *&* Exp(a);
    Out> Sin(a)/Exp(a)+Exp(a)/Sin(a);

In the following example, it may take some time to compute the Taylor expansion. This has to be done every time the function `f` is called.

    In> f(a) := Taylor(x,0,25) Sin(x);
    Out> True;
    In> f(1);
    Out> x-x^3/6+x^5/120-x^7/5040+x^9/362880-
    x^11/39916800+x^13/6227020800-x^15/
    1307674368000+x^17/355687428096000-x^19/
    121645100408832000+x^21/51090942171709440000
    -x^23/25852016738884976640000+x^25
    /15511210043330985984000000;
    In> f(2);
    Out> x-x^3/6+x^5/120-x^7/5040+x^9/362880-
    x^11/39916800+x^13/6227020800-x^15
    /1307674368000+x^17/355687428096000-x^19/
    121645100408832000+x^21/51090942171709440000
    -x^23/25852016738884976640000+x^25/
    15511210043330985984000000;

The remedy is to evaluate the Taylor expansion immediately. Now the expansion is computed only once.

    In> f(a) := Eval(Taylor(x,0,25) Sin(x));
    Out> True;
    In> f(1);
    Out> x-x^3/6+x^5/120-x^7/5040+x^9/362880-
    x^11/39916800+x^13/6227020800-x^15/
    1307674368000+x^17/355687428096000-x^19/
    121645100408832000+x^21/51090942171709440000
    -x^23/25852016738884976640000+x^25
    /15511210043330985984000000;
    In> f(2);
    Out> x-x^3/6+x^5/120-x^7/5040+x^9/362880-
    x^11/39916800+x^13/6227020800-x^15
    /1307674368000+x^17/355687428096000-x^19/
    121645100408832000+x^21/51090942171709440000
    -x^23/25852016738884976640000+x^25/
    15511210043330985984000000;

##### See also:

Set , Clear , [] , Rule , Infix , Eval , Function .

###

Set -- assignment

##### Internal function

##### Calling format:

    Set(var, exp)

##### Parameters:

`var` -- variable which should be assigned

`exp` -- expression to assign to the variable

##### Description:

The expression "exp" is evaluated and assigned it to the variable named "var". The first argument is not evaluated. The value True is returned.

The statement `Set(var, exp)` is equivalent to `var := exp`, but the `:=` operator has more uses, e.g. changing individual entries in a list.

##### Examples:

    In> Set(a, Sin(x)+3);
    Out> True;
    In> a;
    Out> Sin(x)+3;

##### See also:

Clear , := .

###

Clear -- undo an assignment

##### Internal function

##### Calling format:

    Clear(var, ...)

##### Parameters:

`var` -- name of variable to be cleared

##### Description:

All assignments made to the variables listed as arguments are undone. From now on, all these variables remain unevaluated (until a subsequent assignment is made). The result of the expression is True.

##### Examples:

    In> a := 5;
    Out> 5;
    In> a^2;
    Out> 25;

    In> Clear(a);
    Out> True;
    In> a^2;
    Out> a^2;

##### See also:

Set , := .

###

Local -- declare new local variables

##### Internal function

##### Calling format:

    Local(var, ...)

##### Parameters:

`var` -- name of variable to be declared as local

##### Description:

All variables in the argument list are declared as local variables. The arguments are not evaluated. The value True is returned.

By default, all variables in Yacas are global. This means that the variable has the same value everywhere. But sometimes it is useful to have a private copy of some variable, either to prevent the outside world from changing it or to prevent accidental changes to the outside world. This can be achieved by declaring the variable local. Now only expressions within the `Prog` block (or its syntactic equivalent, the `[ ]` block) can access and change it. Functions called within this block cannot access the local copy unless this is specifically allowed with `UnFence`.

##### Examples:

    In> a := 3;
    Out> 3;

    In> [ a := 4; a; ];
    Out> 4;
    In> a;
    Out> 4;

    In> [ Local(a); a := 5; a; ];
    Out> 5;
    In> a;
    Out> 4;

In the first block, `a` is not declared local and hence defaults to be a global variable. Indeed, changing the variable inside the block also changes the value of `a` outside the block. However, in the second block `a` is defined to be local and now the value outside the block stays the same, even though `a` is assigned the value 5 inside the block.

##### See also:

LocalSymbols , Prog , [] , UnFence .

###

++ -- increment variable

##### Standard library

##### Calling format:

    var++

##### Parameters:

`var` -- variable to increment

##### Description:

The variable with name "var" is incremented, i.e. the number 1 is added to it. The expression `x++` is equivalent to the assignment `x := x + 1`, except that the assignment returns the new value of `x` while `x++` always returns true. In this respect, Yacas' `++` differs from the corresponding operator in the programming language C.

##### Examples:

    In> x := 5;
    Out> 5;
    In> x++;
    Out> True;
    In> x;
    Out> 6;

##### See also:

-- , := .

###

-- -- decrement variable

##### Standard library

##### Calling format:

    var--

##### Parameters:

`var` -- variable to decrement

##### Description:

The variable with name "var" is decremented, i.e. the number 1 is subtracted from it. The expression `x--` is equivalent to the assignment `x := x - 1`, except that the assignment returns the new value of `x` while `x--` always returns true. In this respect, Yacas' `--` differs from the corresponding operator in the programming language C.

##### Examples:

    In> x := 5;
    Out> 5;
    In> x--;
    Out> True;
    In> x;
    Out> 4;

##### See also:

++ , := .

###

Object -- create an incomplete type

##### Standard library

##### Calling format:

    Object("pred", exp)

##### Parameters:

`pred` -- name of the predicate to apply

`exp` -- expression on which "pred" should be applied

##### Description:

This function returns "obj" as soon as "pred" returns `True` when applied on "obj". This is used to declare so-called incomplete types.

##### Examples:

    In> a := Object("IsNumber", x);
    Out> Object("IsNumber",x);
    In> Eval(a);
    Out> Object("IsNumber",x);
    In> x := 5;
    Out> 5;
    In> Eval(a);
    Out> 5;

##### See also:

IsNonObject .

###

LazyGlobal -- global variable is to be evaluated lazily

##### Internal function

##### Calling format:

    LazyGlobal(var)

##### Parameters:

`var` -- variable (held argument)

##### Description:

`LazyGlobal` enforces that a global variable will re-evaluate when used. The global variable needs to exist for this function to work. Also, this functionality doesn't survive if `Clear(var)` is called afterwards.

Places where this is used include the global variables `%` and `I`.

The use of lazy in the name stems from the concept of lazy evaluation. The object the global variable is bound to will only be evaluated when called. The `LazyGlobal` property only holds once: after that, the result of evaluation is stored in the global variable, and it won't be reevaluated again:

    In> a:=Hold(Taylor(x,0,30)Sin(x))
    Out> Taylor(x,0,30)Sin(x);
    In> LazyGlobal(a)

Then the first time you call `a` it evaluates `Taylor(...)` and assigns the result to `a`. The next time you call `a` it immediately returns the result. `LazyGlobal` is called for `%` each time `%` changes.

##### Examples:

    In> a:=Hold(2+3)
    Out> 2+3;
    In> a
    Out> 2+3;
    In> LazyGlobal(a)
    Out> True;
    In> a
    Out> 5;

##### See also:

Set , Clear , Local , % , I .

###

UniqueConstant -- create a unique identifier

##### Standard library

##### Calling format:

    UniqueConstant()

##### Description:

This function returns a unique constant atom each time you call it. The atom starts with a C character, and a unique number is appended to it.

##### Examples:

    In> UniqueConstant()
    Out> C9
    In>  UniqueConstant()
    Out> C10

##### See also:

LocalSymbols .

FullForm , Echo , PrettyForm , EvalFormula , TeXForm , CForm , IsCFormable , Write , WriteString , Space , NewLine , FromFile , FromString , ToFile , ToString , Read , ToStdout , ReadCmdLineString , LispRead, LispReadListed , ReadToken , Load , Use , DefLoad , FindFile , PatchLoad , Nl , V, Verbose , Plot2D , Plot3DS , XmlExplodeTag , DefaultTokenizer, XmlTokenizer, CTokenizer , OMForm, OMRead , OMDef .

# Input/output and plotting

This chapter contains commands to use for input and output and plotting. All output commands write to the same destination stream, called the "current output". This is initially the screen, but may be redirected by some commands. Similarly, most input commands read from the "current input" stream, which can also be redirected. The exception to this rule are the commands for reading script files, which simply read a specified file.

###

FullForm -- print an expression in LISP-format

##### Internal function

##### Calling format:

    FullForm(expr)

##### Parameters:

`expr` -- expression to be printed in LISP-format

##### Description:

Evaluates "expr", and prints it in LISP-format on the current output. It is followed by a newline. The evaluated expression is also returned.

This can be useful if you want to study the internal representation of a certain expression.

##### Examples:

    In> FullForm(a+b+c);
    (+ (+ a b )c )
    Out> a+b+c;
    In> FullForm(2*I*b^2);
    (* (Complex 0 2 )(^ b 2 ))
    Out> Complex(0,2)*b^2;

The first example shows how the expression `a+b+c` is internally represented. In the second example, `2*I` is first evaluated to `Complex(0,2)` before the expression is printed.

##### See also:

LispRead , Listify , Unlist .

###

Echo -- high-level printing routine

##### Standard library

##### Calling format:

    Echo(item)
    Echo(list)
    Echo(item,item,item,...)

##### Parameters:

`item` -- the item to be printed

`list` -- a list of items to be printed

##### Description:

If passed a single item, `Echo` will evaluate it and print it to the current output, followed by a newline. If `item` is a string, it is printed without quotation marks.

If there is one argument, and it is a list, `Echo` will print all the entries in the list subsequently to the current output, followed by a newline. Any strings in the list are printed without quotation marks. All other entries are followed by a space.

`Echo` can be called with a variable number of arguments, they will all be printed, followed by a newline.

`Echo` always returns `True`.

##### Examples:

    In> Echo(5+3);
     8
    Out> True;
    In> Echo({"The square of two is ", 2*2});
    The square of two is 4
    Out> True;
    In> Echo("The square of two is ", 2*2);
    The square of two is 4
    Out> True;

Note that one must use the second calling format if one wishes to print a list:

    In> Echo({a,b,c});
    a b c
    Out> True;
    In> Echo({{a,b,c}});
    {a,b,c}
    Out> True;

##### See also:

PrettyForm , Write , WriteString , RuleBaseListed .

###

PrettyForm -- print an expression nicely with ASCII art

##### Standard library

##### Calling format:

    PrettyForm(expr)

##### Parameters:

`expr` -- an expression

##### Description:

`PrettyForm` renders an expression in a nicer way, using ascii art. This is generally useful when the result of a calculation is more complex than a simple number.

##### Examples:

    In> Taylor(x,0,9)Sin(x)
    Out> x-x^3/6+x^5/120-x^7/5040+x^9/362880;
    In> PrettyForm(%)

         3    5      7       9
        x    x      x       x
    x - -- + --- - ---- + ------
        6    120   5040   362880

    Out> True;

##### See also:

EvalFormula , PrettyPrinter .

###

EvalFormula -- print an evaluation nicely with ASCII art

##### Standard library

##### Calling format:

    EvalFormula(expr)

##### Parameters:

`expr` -- an expression

##### Description:

Show an evaluation in a nice way, using PrettyPrinter to show 'input = output'.

##### Examples:

    In> EvalFormula(Taylor(x,0,7)Sin(x))

                                          3    5
                                         x    x
    Taylor( x , 0 , 5 , Sin( x ) ) = x - -- + ---
                                         6    120

##### See also:

PrettyForm .

###

TeXForm -- export expressions to LaTeX

##### Standard library

##### Calling format:

    TeXForm(expr)

##### Parameters:

`expr` -- an expression to be exported

##### Description:

`TeXForm` returns a string containing a LaTeX representation of the Yacas expression `expr`. Currently the exporter handles most expression types but not all.

##### Example:

    In> TeXForm(Sin(a1)+2*Cos(b1))
    Out> "$\sin a_{1} + 2 \cos b_{1}$";

##### See also:

PrettyForm , CForm , ShowPS .

###

CForm -- export expression to C++ code

##### Standard library

##### Calling format:

    CForm(expr)

##### Parameters:

`expr` -- expression to be exported

##### Description:

`CForm` returns a string containing C++ code that attempts to implement the Yacas expression `expr`. Currently the exporter handles most expression types but not all.

##### Example:

    In> CForm(Sin(a1)+2*Cos(b1));
    Out> "sin(a1) + 2 * cos(b1)";

##### See also:

PrettyForm , TeXForm , IsCFormable .

###

IsCFormable -- check possibility to export expression to C++ code

##### Standard library

##### Calling format:

    IsCFormable(expr)
    IsCFormable(expr, funclist)

##### Parameters:

`expr` -- expression to be exported (this argument is not evaluated)

`funclist` -- list of "allowed" function atoms

##### Description:

`IsCFormable` returns `True` if the Yacas expression `expr` can be exported into C++ code. This is a check whether the C++ exporter `CForm` can be safely used on the expression.

A Yacas expression is considered exportable if it contains only functions that can be translated into C++ (e.g. `UnList` cannot be exported). All variables and constants are considered exportable.

The verbose option prints names of functions that are not exportable.

The second calling format of `IsCFormable` can be used to "allow" certain function names that will be available in the C++ code.

##### Examples:

    In> IsCFormable(Sin(a1)+2*Cos(b1))
    Out> True;
    In> V(IsCFormable(1+func123(b1)))
    IsCFormable: Info: unexportable function(s):
      func123
    Out> False;

This returned `False` because the function `func123` is not available in C++. We can explicitly allow this function and then the expression will be considered exportable:

    In> IsCFormable(1+func123(b1), {func123})
    Out> True;

##### See also:

CForm , V .

###

Write -- low-level printing routine

##### Internal function

##### Calling format:

    Write(expr, ...)

##### Parameters:

`expr` -- expression to be printed

##### Description:

The expression "expr" is evaluated and written to the current output. Note that Write accept an arbitrary number of arguments, all of which are written to the current output (see second example). `Write` always returns `True`.

##### Examples:

    In> Write(1);
    1Out> True;
    In> Write(1,2);
     1 2Out> True;

Write does not write a newline, so the `Out>` prompt immediately follows the output of `Write`.

##### See also:

Echo , WriteString .

###

WriteString -- low-level printing routine for strings

##### Internal function

##### Calling format:

    WriteString(string)

##### Parameters:

`string` -- the string to be printed

##### Description:

The expression "string" is evaluated and written to the current output without quotation marks. The argument should be a string. WriteString always returns True.

##### Examples:

    In> Write("Hello, world!");
    "Hello, world!"Out> True;
    In> WriteString("Hello, world!");
    Hello, world!Out> True;

This example clearly shows the difference between Write and WriteString. Note that Write and WriteString do not write a newline, so the `Out>` prompt immediately follows the output.

##### See also:

Echo , Write .

###

Space -- print one or more spaces

##### Standard library

##### Calling format:

    Space()
    Space(nr)

##### Parameters:

`nr` -- the number of spaces to print

##### Description:

The command `Space()` prints one space on the current output. The second form prints `nr` spaces on the current output. The result is always True.

##### Examples:

    In> Space(5);
         Out> True;

##### See also:

Echo , Write , NewLine .

###

NewLine -- print one or more newline characters

##### Standard library

##### Calling format:

    NewLine()
    NewLine(nr)

##### Parameters:

`nr` -- the number of newline characters to print

##### Description:

The command `NewLine()` prints one newline character on the current output. The second form prints "nr" newlines on the current output. The result is always True.

##### Examples:

    In> NewLine();

    Out> True;

##### See also:

Echo , Write , Space .

###

FromFile -- connect current input to a file

##### Internal function

##### Calling format:

    FromFile(name) body

##### Parameters:

`name` - string, the name of the file to read

`body` - expression to be evaluated

##### Description:

The current input is connected to the file "name". Then the expression "body" is evaluated. If some functions in "body" try to read from current input, they will now read from the file "name". Finally, the file is closed and the result of evaluating "body" is returned.

##### Examples:

Suppose that the file `foo` contains

    2 + 5;

Then we can have the following dialogue:

    In> FromFile("foo") res := Read();
    Out> 2+5;
    In> FromFile("foo") res := ReadToken();
    Out> 2;

##### See also:

ToFile , FromString , Read , ReadToken .

###

FromString -- connect current input to a string

##### Internal function

##### Calling format:

    FromString(str) body;

##### Parameters:

`str` -- a string containing the text to parse

`body` -- expression to be evaluated

##### Description:

The commands in "body" are executed, but everything that is read from the current input is now read from the string "str". The result of "body" is returned.

##### Examples:

    In> FromString("2+5; this is never read") \
      res := Read();
    Out> 2+5;
    In> FromString("2+5; this is never read") \
      res := Eval(Read());
    Out> 7;

##### See also:

ToString , FromFile , Read , ReadToken .

###

ToFile -- connect current output to a file

##### Internal function

##### Calling format:

    ToFile(name) body

##### Parameters:

`name` -- string, the name of the file to write the result to

`body` -- expression to be evaluated

##### Description:

The current output is connected to the file "name". Then the expression "body" is evaluated. Everything that the commands in "body" print to the current output, ends up in the file "name". Finally, the file is closed and the result of evaluating "body" is returned.

If the file is opened again, the old contents will be overwritten. This is a limitation of `ToFile`: one cannot append to a file that has already been created.

##### Examples:

Here is how one can create a file with C code to evaluate an expression:

    In> ToFile("expr1.c") WriteString(
      CForm(Sqrt(x-y)*Sin(x)) );
    Out> True;

The file `expr1.c` was created in the current working directory and it contains the line

    sqrt(x-y)*sin(x)

As another example, take a look at the following command:

    In> [ Echo("Result:");  \
      PrettyForm(Taylor(x,0,9) Sin(x)); ];
    Result:

         3    5      7       9
        x    x      x       x
    x - -- + --- - ---- + ------
        6    120   5040   362880

    Out> True;

Now suppose one wants to send the output of this command to a file. This can be achieved as follows:

    In> ToFile("out") [ Echo("Result:");  \
      PrettyForm(Taylor(x,0,9) Sin(x)); ];
    Out> True;

After this command the file `out` contains:

    Result:

         3    5      7       9
        x    x      x       x
    x - -- + --- - ---- + ------
        6    120   5040   362880

##### See also:

FromFile , ToString , Echo , Write , WriteString , PrettyForm , Taylor .

###

ToString -- connect current output to a string

##### Internal function

##### Calling format:

    ToString() body

##### Parameters:

`body` -- expression to be evaluated

##### Description:

The commands in "body" are executed. Everything that is printed on the current output, by `Echo` for instance, is collected in a string and this string is returned.

##### Examples:

    In> str := ToString() [ WriteString(  \
      "The square of 8 is "); Write(8^2); ];
    Out> "The square of 8 is  64";

##### See also:

FromFile , ToString , Echo , Write , WriteString .

###

Read -- read an expression from current input

##### Internal function

##### Calling format:

    Read()

##### Description:

Read an expression from the current input, and return it unevaluated. When the end of an input file is encountered, the token atom `EndOfFile` is returned.

##### Examples:

    In> FromString("2+5;") Read();
    Out> 2+5;
    In> FromString("") Read();
    Out> EndOfFile;

##### See also:

FromFile , FromString , LispRead , ReadToken , Write .

###

ToStdout -- select initial output stream for output

##### Internal function

##### Calling format:

    ToStdout() body

##### Parameters:

`body` -- expression to be evaluated

##### Description:

When using `ToString` or `ToFile`, it might happen that something needs to be written to the standard default initial output (typically the screen). `ToStdout` can be used to select this stream.

##### Example:

    In> ToString()[Echo("aaaa");ToStdout()Echo("bbbb");];
    bbbb
    Out> "aaaa
    "

##### See also:

ToString , ToFile .

###

ReadCmdLineString -- read an expression from command line and return in string

##### Internal function

##### Calling format:

    ReadCmdLineString(prompt)

##### Parameters:

`prompt` -- string representing the prompt shown on screen

##### Description:

This function allows for interactive input similar to the command line. When using this function, the history from the command line is also available.

The result is returned in a string, so it still needs to be parsed.

This function will typically be used in situations where one wants a custom read-eval-print loop.

##### Examples:

The following defines a function that when invoked keeps asking for an expression (the read step), and then takes the derivative of it (the eval step) and then uses PrettyForm to display the result (the print step).

    In> ReEvPr() := \
    In>   While(True) [ \
    In>     PrettyForm(Deriv(x) \
    In>      FromString(ReadCmdLineString("Deriv> "):";")Read()); \
    In> ];
    Out> True;

Then one can invoke the command, from which the following interaction might follow:

    In> ReEvPr()
    Deriv> Sin(a^2*x/b)

       /  2     \
       | a  * x |    2
    Cos| ------ | * a  * b
       \   b    /
    ----------------------
               2
              b

    Deriv> Sin(x)

    Cos( x )

    Deriv>

##### See also:

Read , LispRead , LispReadListed .

###

LispRead, LispReadListed -- read expressions in LISP syntax

##### Internal function

##### Calling format:

    LispRead()
    LispReadListed()

##### Description:

The function `LispRead` reads an expression in the LISP syntax from the current input, and returns it unevaluated. When the end of an input file is encountered, the special token atom `EndOfFile` is returned.

The Yacas expression `a+b` is written in the LISP syntax as `(+ a b)`. The advantage of this syntax is that it is less ambiguous than the infix operator grammar that Yacas uses by default.

The function `LispReadListed` reads a LISP expression and returns it in a list, instead of the form usual to Yacas (expressions). The result can be thought of as applying `Listify` to `LispRead`. The function `LispReadListed` is more useful for reading arbitrary LISP expressions, because the first object in a list can be itself a list (this is never the case for Yacas expressions where the first object in a list is always a function atom).

##### Examples:

    In> FromString("(+ a b)") LispRead();
    Out> a+b;
    In> FromString("(List (Sin x) (- (Cos x)))") \
      LispRead();
    Out> {Sin(x),-Cos(x)};
    In> FromString("(+ a b)")LispRead()
    Out> a+b;
    In> FromString("(+ a b)")LispReadListed()
    Out> {+,a,b};

##### See also:

FromFile , FromString , Read , ReadToken , FullForm .

###

ReadToken -- read a token from current input

##### Internal function

##### Calling format:

    ReadToken()

##### Description:

Read a token from the current input, and return it unevaluated. The returned object is a Yacas atom (not a string). When the end of an input file is encountered, the token atom `EndOfFile` is returned.

A token is for computer languages what a word is for human languages: it is the smallest unit in which a command can be divided, so that the semantics (that is the meaning) of the command is in some sense a combination of the semantics of the tokens. Hence `a := foo` consists of three tokens, namely `a`, `:=`, and `foo`.

The parsing of the string depends on the syntax of the language. The part of the kernel that does the parsing is the "tokenizer". Yacas can parse its own syntax (the default tokenizer) or it can be instructed to parse XML or C++ syntax using the directives `DefaultTokenizer`, `XmlTokenizer`, or `CTokenizer`. Setting a tokenizer is a global action that affects all `ReadToken` calls.

##### Examples:

    In> FromString("a := Sin(x)") While \
      ((tok := ReadToken()) != EndOfFile) \
      Echo(tok);
    a
    :=
    Sin
    (
    x
    )
    Out> True;

We can read some junk too:

    In> FromString("-$3")ReadToken();
    Out> -$;

The result is an atom with the string representation `-$`. Yacas assumes that `-$` is an operator symbol yet to be defined. The "`3`" will be in the next token. (The results will be different if a non-default tokenizer is selected.)

##### See also:

FromFile , FromString , Read , LispRead , DefaultTokenizer .

###

Load -- evaluate all expressions in a file

##### Internal function

##### Calling format:

    Load(name)

##### Parameters:

`name` -- string, name of the file to load

##### Description:

The file "name" is opened. All expressions in the file are read and evaluated. `Load` always returns `true`.

##### See also:

Use , DefLoad , DefaultDirectory , FindFile .

###

Use -- load a file, but not twice

##### Internal function

##### Calling format:

    Use(name)

##### Parameters:

`name` -- string, name of the file to load

##### Description:

If the file "name" has been loaded before, either by an earlier call to `Use` or via the `DefLoad` mechanism, nothing happens. Otherwise all expressions in the file are read and evaluated. `Use` always returns `true`.

The purpose of this function is to make sure that the file will at least have been loaded, but is not loaded twice.

##### See also:

Load , DefLoad , DefaultDirectory .

###

DefLoad -- load a `.def` file

##### Internal function

##### Calling format:

    DefLoad(name)

##### Parameters:

`name` -- string, name of the file (without `.def` suffix)

##### Description:

The suffix `.def` is appended to "name" and the file with this name is loaded. It should contain a list of functions, terminated by a closing brace \} (the end-of-list delimiter). This tells the system to load the file "name" as soon as the user calls one of the functions named in the file (if not done so already). This allows for faster startup times, since not all of the rules databases need to be loaded, just the descriptions on which files to load for which functions.

##### See also:

Load , Use , DefaultDirectory .

###

FindFile -- find a file in the current path

##### Internal function

##### Calling format:

    FindFile(name)

##### Parameters:

`name` -- string, name of the file or directory to find

##### Description:

The result of this command is the full path to the file that would be opened when the command `Load(name)` would be invoked. This means that the input directories are subsequently searched for a file called "name". If such a file is not found, `FindFile` returns an empty string.

`FindFile("")` returns the name of the default directory (the first one on the search path).

##### See also:

Load , DefaultDirectory .

###

PatchLoad -- execute commands between `<?` and `?>` in file

##### Internal function

##### Calling format:

    PatchLoad(name)

##### Parameters:

`name` -- string, name of the file to "patch"

##### Description:

`PatchLoad` loads in a file and outputs the contents to the current output. The file can contain blocks delimited by `<?` and `?>` (meaning "Yacas Begin" and "Yacas End"). The piece of text between such delimiters is treated as a separate file with Yacas instructions, which is then loaded and executed. All output of write statements in that block will be written to the same current output.

This is similar to the way PHP works. You can have a static text file with dynamic content generated by Yacas.

##### See also:

PatchString , Load .

###

Nl -- the newline character

##### Standard library

##### Calling format:

    Nl()

##### Description:

This function returns a string with one element in it, namely a newline character. This may be useful for building strings to send to some output in the end.

Note that the second letter in the name of this command is a lower case `L` (from "line").

##### Examples:

    In> WriteString("First line" : Nl() : "Second line" : Nl());
    First line
    Second line
    Out> True;

##### See also:

NewLine .

###

V, Verbose -- set verbose output mode

##### Standard library

##### Calling format:

    V(expression)

##### Parameters:

`expression` -- expression to be evaluated in verbose mode

##### Description:

The function `V(expression)` will evaluate the expression in verbose mode. Various parts of Yacas can show extra information about the work done while doing a calculation when using `V`.

The function is currently implemented using a global variable `Verbose` which can be set to `True` or `False`.

##### Examples:

    In> OldSolve({x+2==0},{x})
    Out> {{-2}};
    In> V(OldSolve({x+2==0},{x}))
    Entering OldSolve
    From  x+2==0  it follows that  x  = -2
       x+2==0  simplifies to  True
    Leaving OldSolve
    Out> {{-2}};

##### See also:

Echo , N , OldSolve .

###

Plot2D -- adaptive two-dimensional plotting

##### Standard library

##### Calling format:

    Plot2D(f(x))
    Plot2D(f(x), a:b)
    Plot2D(f(x), a:b, option=value)
    Plot2D(f(x), a:b, option=value, ...)
    Plot2D(list, ...)

##### Parameters:

`f(x)` -- unevaluated expression containing one variables (function to be plotted)

`list` -- list of functions to plot

`a`, `b` -- numbers, plotting range in the x coordinate

`option` -- atom, option name

`value` -- atom, number or string (value of option)

##### Description:

The routine `Plot2D` performs adaptive plotting of one or several functions of one variable in the specified range. The result is presented as a line given by the equation y=f(x). Several functions can be plotted at once. Various plotting options can be specified. Output can be directed to a plotting program (the default is to use `gnuplot`), to a data file, or to a list of values.

The function parameter `f(x)` must evaluate to a Yacas expression containing at most one variable. (The variable does not have to be called `x`.) Also, `N(f(x))` must evaluate to a real (not complex) numerical value when given a numerical value of the argument `x`. If the function `f(x)` does not satisfy these requirements, an error is raised.

Several functions may be specified as a list and they do not have to depend on the same variable, for example, `{f(x), g(y)}`. The functions will be plotted on the same graph using the same coordinate ranges.

If you have defined a function which accepts a number but does not accept an undefined variable, `Plot2D` will fail to plot it. Use `NFunction` to overcome this difficulty.

Data files are created in a temporary directory `/tmp/plot.tmp/` unless otherwise requested. File names and other information is printed if the `Verbose` option is switched on using `V()`.

The current algorithm uses Newton-Cotes quadratures and some heuristics for error estimation (see The Yacas book of algorithms, Chapter 3, Section 1 ). The initial grid of `points+1` points is refined between any grid points a, b if the integral Integrate(x,a,b)f(x) is not approximated to the given precision by the existing grid.

Default plotting range is `-5:5`. Range can also be specified as `x= -5:5` (note the mandatory space separating "`=`" and "`-`"); currently the variable name `x` is ignored in this case.

Options are of the form `option=value`. Currently supported option names are: "points", "precision", "depth", "output", "filename", "yrange". Option values are either numbers or special unevaluated atoms such as `data`. If you need to use the names of these atoms in your script, strings can be used (e.g. `output="gnuplot"`). Several option/value pairs may be specified (the function `Plot2D` has a variable number of arguments).

- `yrange`: the range of ordinates to use for plotting, e.g. `yrange=0:20`. If no range is specified, the default is usually to leave the choice to the plotting backend. `gnuplot` and `plotutils` will choose the range automatically based on actual values (which may give visually inadequate plots if the function has a singularity).
- `points`: initial number of points (default 23) -- at least that many points will be plotted. The initial grid of this many points will be adaptively refined.
- `precision`: graphing precision (default 10^(-6)). This is interpreted as the relative precision of computing the integral of f(x)-Min(f(x)) using the grid points. For a smooth, non-oscillating function this value should be roughly 1/(number of screen pixels in the plot).
- `depth`: max. refinement depth, logarithmic (default 5) -- means there will be at most 2^depth extra points per initial grid point.
- `output`: name of the plotting backend. Supported names: `gnuplot` (default), `plotutils`, `gmt`, `datafile`, `data`. The `gnuplot`, `gmt` and `plotutils` backends will display the plot on the screen (this requires either the `gnuplot` program or the GNU `plotutils` suite or the `gmt` suite). The support of `gmt` ("Generic Mapping Tools") is currently Unix-only. The `datafile` backend will write the numerical data to a text file with specified name. The `data` backend will return the data as a list of pairs such as `{{x1,y1}, {x2,y2}, ...}`.
- `filename`: specify name of the created data file. For example: `filename="data1.txt"`. The default is the name `"output.data"`. Note that if several functions are plotted, the data files will have a number appended to the given name, for example `data.txt1`, `data.txt2`.

Other options may be supported in the future.

The current implementation can deal with a singularity within the plotting range only if the function `f(x)` returns `Infinity`, `-Infinity` or `Undefined` at the singularity. If the function `f(x)` generates a numerical error and fails at a singularity, `Plot2D` will fail if one of the grid points falls on the singularity. (All grid points are generated by bisection so in principle the endpoints and the `points` parameter could be chosen to avoid numerical singularities.)

*WIN32

On the Wind*ws platforms, there are some limitations:

- The `gnuplot` program must be installed on the path, or in the same directory as Yacas is started. The executable must be called `wgnupl32.exe` and the file `wgnuplot.mnu` must be there as well.
- The `plotutils` backend is not yet supported.
- The `gmt` backend is not yet supported.
- Some extra messages are printed to the console.
- All files are created in the subdirectory `plot.tmp` under the current directory; also the system cannot determine whether the current directory is writable and whether the `plot.tmp` subdirectory has been successfully created.

##### Examples:

    In> Plot2D({(2+x)/(2-x),Exp(y)},-1.8:1.5)
    Out> True;
    In> V(Plot2D(Sin(1/x),0:1, depth=4, \
    In>   precision=0.001, output=plotutils))
    Plot2D: using 65 points for function Sin(1/x)
    Plot2D'datafile: created file '/tmp/plot.tmp/data1'
    Out> True;

If a function takes a long time to evaluate, one can use `MakeFunctionPlugin` (a Unix-specific tool) to compile the function and plot it faster:

    In> f(x) := Cos( (Abs(Pi/x))^1.5 )
    Out> True;
    In> Time( Plot2D(f(x),0:1) )
    94.93 seconds taken
    Out> True;
    In> MakeFunctionPlugin("f1", f(x))
    Function f1(x) loaded from plugins.tmp/
      libf1_plugin_cc.so
    Out> True;
    In> Time( Plot2D(f1(x),0:1) )
    6.97 seconds taken
    Out> True;

##### See also:

V , NFunction , MakeFunctionPlugin , Plot3DS .

###

Plot3DS -- three-dimensional (surface) plotting

##### Standard library

##### Calling format:

    Plot3DS(f(x,y))
    Plot3DS(f(x,y), a:b, c:d)
    Plot3DS(f(x,y), a:b, c:d, option=value)
    Plot3DS(f(x,y), a:b, c:d, option=value, ...)
    Plot3DS(list, ...)

##### Parameters:

`f(x,y)` -- unevaluated expression containing two variables (function to be plotted)

`list` -- list of functions to plot

`a`, `b`, `c`, `d` -- numbers, plotting ranges in the x and y coordinates

`option` -- atom, option name

`value` -- atom, number or string (value of option)

##### Description:

The routine `Plot3DS` performs adaptive plotting of a function of two variables in the specified ranges. The result is presented as a surface given by the equation z=f(x,y). Several functions can be plotted at once, by giving a list of functions. Various plotting options can be specified. Output can be directed to a plotting program (the default is to use `gnuplot`), to a data file, or to a list of values.

The function parameter `f(x,y)` must evaluate to a Yacas expression containing at most two variables. (The variables do not have to be called `x` and `y`.) Also, `N(f(x,y))` must evaluate to a real (not complex) numerical value when given numerical values of the arguments `x`, `y`. If the function `f(x,y)` does not satisfy these requirements, an error is raised.

Several functions may be specified as a list but they have to depend on the same symbolic variables, for example, `{f(x,y), g(y,x)}`, but not `{f(x,y), g(a,b)}`. The functions will be plotted on the same graph using the same coordinate ranges.

If you have defined a function which accepts a number but does not accept an undefined variable, `Plot3DS` will fail to plot it. Use `NFunction` to overcome this difficulty.

Data files are created in a temporary directory `/tmp/plot.tmp/` unless otherwise requested. File names and other information is printed if the `Verbose` option is switched on using `V()`.

The current algorithm uses Newton-Cotes cubatures and some heuristics for error estimation (see The Yacas book of algorithms, Chapter 3, Section 1 ). The initial rectangular grid of `xpoints+1`*`ypoints+1` points is refined within any rectangle where the integral of f(x,y) is not approximated to the given precision by the existing grid.

Default plotting range is `-5:5` in both coordinates. A range can also be specified with a variable name, e.g. `x= -5:5` (note the mandatory space separating "`=`" and "`-`"). The variable name `x` should be the same as that used in the function `f(x,y)`. If ranges are not given with variable names, the first variable encountered in the function `f(x,y)` is associated with the first of the two ranges.

Options are of the form `option=value`. Currently supported option names are "points", "xpoints", "ypoints", "precision", "depth", "output", "filename", "xrange", "yrange", "zrange". Option values are either numbers or special unevaluated atoms such as `data`. If you need to use the names of these atoms in your script, strings can be used (e.g. `output="gnuplot"`). Several option/value pairs may be specified (the function `Plot3DS` has a variable number of arguments).

- `xrange`, `yrange`: optionally override coordinate ranges. Note that `xrange` is always the first variable and `yrange` the second variable, regardless of the actual variable names.
- `zrange`: the range of the z axis to use for plotting, e.g. `zrange=0:20`. If no range is specified, the default is usually to leave the choice to the plotting backend. Automatic choice based on actual values may give visually inadequate plots if the function has a singularity.
- `points`, `xpoints`, `ypoints`: initial number of points (default 10 each) -- at least that many points will be plotted in each coordinate. The initial grid of this many points will be adaptively refined. If `points` is specified, it serves as a default for both `xpoints` and `ypoints`; this value may be overridden by `xpoints` and `ypoints` values.
- `precision`: graphing precision (default 0.01). This is interpreted as the relative precision of computing the integral of f(x,y)-Min(f(x,y)) using the grid points. For a smooth, non-oscillating function this value should be roughly 1/(number of screen pixels in the plot).
- `depth`: max. refinement depth, logarithmic (default 3) -- means there will be at most 2^depth extra points per initial grid point (in each coordinate).
- `output`: name of the plotting backend. Supported names: `gnuplot` (default), `gmt`, `datafile`, `data`. The `gnuplot` and `gmt` backends will display the plot on the screen (this requires the `gnuplot` or `gmt` suite). The support of `gmt` ("Generic Mapping Tools") is currently Unix-only. The `datafile` backend will write the numerical data to a text file with specified name. The `data` backend will return the data as a list of triples such as `{{x1, y1, z1}, {x2, y2, z2}, ...}`.
- `hidden`: boolean value, whether to use the hidden-line visualization. Default is `True`. Currently used only for `gnuplot` backend.
- `filename`: specify name of the created data file. For example: `filename="data1.txt"`. The default is the name `"output.data"`. Note that if several functions are plotted, the data files will have a number appended to the given name, for example `data.txt1`, `data.txt2`.

Other options may be supported in the future.

The current implementation can deal with a singularity within the plotting range only if the function `f(x,y)` returns `Infinity`, `-Infinity` or `Undefined` at the singularity. If the function `f(x,y)` generates a numerical error and fails at a singularity, `Plot3DS` will fail only if one of the grid points falls on the singularity. (All grid points are generated by bisection so in principle the endpoints and the `xpoints`, `ypoints` parameters could be chosen to avoid numerical singularities.)

The `filename` option is optional if using graphical backends, but can be used to specify the location of the created data file.

*WIN32

Same limitations as `Plot2D`.

##### Examples:

    In> Plot3DS(a*b^2)
    Out> True;
    In> V(Plot3DS(Sin(x)*Cos(y),x=0:20, y=0:20,depth=3))
    CachedConstant: Info: constant Pi is being
      recalculated at precision 10
    CachedConstant: Info: constant Pi is being
      recalculated at precision 11
    Plot3DS: using 1699  points for function Sin(x)*Cos(y)
    Plot3DS: max. used 8 subdivisions for Sin(x)*Cos(y)
    Plot3DS'datafile: created file '/tmp/plot.tmp/data1'
    Out> True;

##### See also:

V , NFunction , MakeFunctionPlugin , Plot2D .

###

XmlExplodeTag -- convert XML strings to tag objects

##### Internal function

##### Calling format:

    XmlExplodeTag(xmltext)

##### Parameters:

`xmltext` -- string containing some XML tokens

##### Description:

`XmlExplodeTag` parses the first XML token in `xmltext` and returns a Yacas expression.

The following subset of XML syntax is supported currently:

- `<TAG [options]>` -- an opening tag
- `</TAG [options]>` -- a closing tag
- `<TAG [options] />` -- an open/close tag
- plain (non-tag) text

The tag options take the form `paramname="value"`.

If given an XML tag, `XmlExplodeTag` returns a structure of the form `XmlTag(name,params,type)`. In the returned object, `name` is the (capitalized) tag name, `params` is an assoc list with the options (key fields capitalized), and type can be either "Open", "Close" or "OpenClose".

If given a plain text string, the same string is returned.

##### Examples:

    In> XmlExplodeTag("some plain text")
    Out> "some plain text";
    In> XmlExplodeTag("<a name=\"blah blah\"
      align=\"left\">")
    Out> XmlTag("A",{{"ALIGN","left"},
      {"NAME","blah blah"}},"Open");
    In> XmlExplodeTag("</p>")
    Out> XmlTag("P",{},"Close");
    In> XmlExplodeTag("<br/>")
    Out> XmlTag("BR",{},"OpenClose");

##### See also:

XmlTokenizer .

###

DefaultTokenizer, XmlTokenizer, CTokenizer -- select a syntax for parsing the input

##### Internal function

##### Calling format:

    DefaultTokenizer()
    XmlTokenizer()
    CTokenizer()

##### Description:

A "tokenizer" is an internal routine in the kernel that parses the input into Yacas expressions. This affects all input typed in by a user at the prompt and also the input redirected from files or strings using `FromFile` and `FromString` and read using `Read` or `ReadToken`.

The Yacas environment currently supports some experimental tokenizers for various syntaxes. `DefaultTokenizer` switches to the tokenizer used for default Yacas syntax. `XmlTokenizer` switches to an XML syntax and `CTokenizer` to a C syntax. Note that setting the tokenizer is a global side effect. One typically needs to switch back to the default tokenizer when finished reading the special syntax.

Care needs to be taken when kernel errors are raised during a non-default tokenizer operation (as with any global change in the environment). Errors need to be caught with the `TrapError` function. The error handler code should re-instate the default tokenizer, or else the user will be unable to continue the session (everything a user types will be parsed using a non-default tokenizer).

When reading XML syntax, the supported formats are the same as those of `XmlExplodeTag`. The parser does not validate anything in the XML input. After an XML token has been read in, it can be converted into an Yacas expression with `XmlExplodeTag`. Note that when reading XML, any plain text between tags is returned as one token. Any malformed XML will be treated as plain text.

`CTokenizer` is an implementation of a C/C++ tokenizer. It is still work in progress.

##### Example:

    In> [XmlTokenizer(); q:=ReadToken(); \
      DefaultTokenizer();q;]
    <a>Out> <a>;

Note that:

- after switching to `XmlTokenizer` the `In>` prompt disappeared; the user typed `<a>` and the `Out>` prompt with the resulting expression appeared.
- The resulting expression is an atom with the string representation `<a>`; it is not a string.

##### See also:

OMRead , TrapError , XmlExplodeTag , ReadToken , FromFile , FromString .

###

OMForm, OMRead -- convert expressions to and from OpenMath

##### Standard library

##### Calling format:

    OMForm(expression)
    OMRead()

##### Parameters:

`expression` -- expression to convert

##### Description:

`OMForm` prints an OpenMath representation of the input parameter `expression` to standard output. `OMRead` reads an OpenMath expression from standard input and returns a normal Yacas expression that matches the input OpenMath expression.

If a Yacas symbol does not have a mapping defined by `OMDef`, it is translated to and from OpenMath as the OpenMath symbol in the CD "yacas" with the same name as it has in Yacas.

##### Example:

    In> str:=ToString()OMForm(2+Sin(a*3))
    Out> "<OMOBJ>
      <OMA>
        <OMS cd="arith1" name="plus"/>
        <OMI>2</OMI>
        <OMA>
          <OMS cd="transc1" name="sin"/>
          <OMA>
            <OMS cd="arith1" name="times"/>
            <OMV name="a"/>
            <OMI>3</OMI>
          </OMA>
        </OMA>
      </OMA>
    </OMOBJ>
    ";
    In> FromString(str)OMRead()
    Out> 2+Sin(a*3);

    In> OMForm(NotDefinedInOpenMath(2+3))
    <OMOBJ>
      <OMA>
        <OMS cd="yacas" name="NotDefinedInOpenMath"/>
        <OMA>
          <OMS cd="arith1" name="plus"/>
          <OMI>2</OMI>
          <OMI>3</OMI>
        </OMA>
      </OMA>
    </OMOBJ>
    Out> True

##### See also:

XmlTokenizer , XmlExplodeTag , OMDef .

###

OMDef -- define translations from Yacas to OpenMath and vice-versa.

##### Standard library

##### Calling format:

    OMDef(yacasSymbol, cd, name)
    OMDef(yacasSymbol, cd, name, yacasToOM)
    OMDef(yacasSymbol, cd, name, yacasToOM, omToYacas)

##### Parameters:

`yacasSymbol` -- string with the name of a Yacas symbol

`cd` -- OpenMath Content Dictionary for the symbol

`name` -- OpenMath name for the symbol

`yacasToOM` -- rule for translating an application of that symbol in Yacas into an OpenMath expression

`omToYacas` -- rule for translating an OpenMath expression into an application of this symbol in Yacas

##### Description:

`OMDef` defines the translation rules for symbols between the Yacas representation and `OpenMath`. The definitions for the symbols in the Yacas library are in the `*.rep` script subdirectories. In those modules for which the mappings are defined, there is a file called `om.ys` that contains the `OMDef` calls. Those files are loaded in `openmath.rep/om.ys`, so any new file must be added to the list there, at the end of the file.

A rule is represented as a list of expressions. Since both OM and Yacas expressions are actually lists, the syntax is the same in both directions. There are two template forms that are expanded before the translation:

- `$`: this symbol stands for the translation of the symbol applied in the original expression.

- `_path`: a path into the original expression (list) to extract an element, written as an underscore applied to an integer or a list of integers. Those integers are indexes into expressions, and integers in a list are applied recursively starting at the original expression. For example, `_2` means the second parameter of the expression, while `_{3,2,1}` means the first parameter of the second parameter of the third parameter of the original expression.

They can appear anywhere in the rule as expressions or subexpressions.

Finally, several alternative rules can be specified by joining them with the `|` symbol, and each of them can be annotated with a post-predicate applied with the underscore `_` symbol, in the style of Yacas' simplification rules. Only the first alternative rule that matches is applied, so the more specific rules must be written first.

There are special symbols recognized by `OMForm` to output `OpenMath` constructs that have no specific parallel in Yacas, such as an OpenMath symbol having a `CD` and `name`: Yacas symbols have only a name. Those special symbols are:

- `OMS(cd, name)`: `<OMS cd="cd" name="name">`
- `OMBIND(binderSymbol, bvars, expression)`: `<OMBIND>binderSymbol bvars expression</OMBIND>`, where `bvars` must be produced by using `OMBVAR(...)`.
- `OMBVAR(x y ...)`: `<OMBVAR>x y ...</OMBVAR>`
- `OME(...)`: `<OME>...</OME>`

When translating from OpenMath to Yacas, we just store unknown symbols as `OMS("cd", "name")`. This way we don't have to bother defining bogus symbols for concepts that Yacas does not handle, and we can evaluate expressions that contain them.

##### Examples:

    In> OMDef( "Sqrt" ,  "arith1", "root", { $, _1, 2 }, $(_1)_(_2=2) | (_1^(1/_2)) );
    Out> True
    In> OMForm(Sqrt(3))
    <OMOBJ>
      <OMA>
        <OMS cd="arith1" name="root"/>
        <OMI>3</OMI>
        <OMI>2</OMI>
      </OMA>
    </OMOBJ>
    Out> True
    In> FromString("<OMOBJ><OMA><OMS cd=\"arith1\" name=\"root\"/><OMI>16</OMI><OMI>2</OMI></OMA></OMOBJ> ")OMRead()
    Out> Sqrt(16)
    In> FromString("<OMOBJ><OMA><OMS cd=\"arith1\" name=\"root\"/><OMI>16</OMI><OMI>3</OMI></OMA></OMOBJ> ")OMRead()
    Out> 16^(1/3)

    In> OMDef("Limit", "limit1", "limit", \
          {  $, _2, OMS("limit1", "under"), OMBIND(OMS("fns1", "lambda"), OMBVAR(_1), _4) }_(_3=Left)  \
          |{ $, _2, OMS("limit1", "above"), OMBIND(OMS("fns1", "lambda"), OMBVAR(_1), _4) }_(_3=Right) \
          |{ $, _2, OMS("limit1", "both_sides"), OMBIND(OMS("fns1", "lambda"), OMBVAR(_1), _3) },      \
          { $, _{3,2,1}, _1, Left,  _{3,3}}_(_2=OMS("limit1", "below")) \
          |{$, _{3,2,1}, _1, Right, _{3,3}}_(_2=OMS("limit1", "above")) \
          |{$, _{3,2,1}, _1, _{3,3}}                                    \
         );
    In> OMForm(Limit(x,0) Sin(x)/x)
    <OMOBJ>
      <OMA>
        <OMS cd="limit1" name="limit"/>
        <OMI>0</OMI>
        <OMS cd="limit1" name="both_sides"/>
        <OMBIND>
          <OMS cd="fns1" name="lambda"/>
          <OMBVAR>
            <OMV name="x"/>
          </OMBVAR>
          <OMA>
            <OMS cd="arith1" name="divide"/>
            <OMA>
              <OMS cd="transc1" name="sin"/>
              <OMV name="x"/>
            </OMA>
            <OMV name="x"/>
          </OMA>
        </OMBIND>
      </OMA>
    </OMOBJ>
    Out> True
    In> OMForm(Limit(x,0,Right) 1/x)
    <OMOBJ>
      <OMA>
        <OMS cd="limit1" name="limit"/>
        <OMI>0</OMI>
        <OMS cd="limit1" name="above"/>
        <OMBIND>
          <OMS cd="fns1" name="lambda"/>
          <OMBVAR>
            <OMV name="x"/>
          </OMBVAR>
          <OMA>
            <OMS cd="arith1" name="divide"/>
            <OMI>1</OMI>
            <OMV name="x"/>
          </OMA>
        </OMBIND>
      </OMA>
    </OMOBJ>
    Out> True
    In> FromString(ToString()OMForm(Limit(x,0,Right) 1/x))OMRead()
    Out> Limit(x,0,Right)1/x
    In> %
    Out> Infinity

##### See also:

OMForm , OMRead .

SetStringMid , StringMid , String, Atom , ConcatStrings , LocalSymbols , PatchString .

# String manipulation

###

SetStringMid -- change a substring

##### Internal function

##### Calling format:

    SetStringMid(index,substring,string)

##### Parameters:

`index` -- index of substring to get

`substring` -- substring to store

`string` -- string to store substring in.

##### Description:

Set (change) a part of a string. It leaves the original alone, returning a new changed copy.

##### Examples:

    In> SetStringMid(3,"XY","abcdef")
    Out> "abXYef";

##### See also:

StringMid , Length .

###

StringMid -- retrieve a substring

##### Internal function

##### Calling format:

    StringMid(index,length,string)

##### Parameters:

`index` -- index of substring to get

`length` -- length of substring to get

`string` -- string to get substring from

##### Description:

`StringMid` returns a part of a string. Substrings can also be accessed using the `[]` operator.

##### Examples:

    In> StringMid(3,2,"abcdef")
    Out> "cd";
    In> "abcdefg"[2 .. 4]
    Out> "bcd";

##### See also:

SetStringMid , Length .

###

String, Atom -- convert atom to string and vice versa

##### Internal function

##### Calling format:

    Atom("string")
    String(atom)

##### Parameters:

`atom` -- an atom

`"string"` -- a string

##### Description:

Returns an atom with the string representation given as the evaluated argument. Example: `Atom("foo");` returns `foo`.

`String` is the inverse of `Atom`: turns `atom` into `"atom"`.

##### Examples:

    In> String(a)
    Out> "a";
    In> Atom("a")
    Out> a;

###

ConcatStrings -- concatenate strings

##### Internal function

##### Calling format:

    ConcatStrings(strings)

##### Parameters:

`strings` -- one or more strings

##### Description:

Concatenates strings.

##### Examples:

    In> ConcatStrings("a","b","c")
    Out> "abc";

##### See also:

: .

###

LocalSymbols -- create unique local symbols with given prefix

##### Standard library

##### Calling format:

    LocalSymbols(var1, var2, ...) body

##### Parameters:

`var1`, `var2`, ... -- atoms, symbols to be made local

`body` -- expression to execute

##### Description:

Given the symbols passed as the first arguments to LocalSymbols a set of local symbols will be created, and creates unique ones for them, typically of the form `$<symbol><number>`, where `symbol` was the symbol entered by the user, and `number` is a unique number. This scheme was used to ensure that a generated symbol can not accidentally be entered by a user.

This is useful in cases where a guaranteed free variable is needed, for example, in the macro-like functions (`For`, `While`, etc.).

##### Examples:

    In> LocalSymbols(a,b)a+b
    Out> $a6+ $b6;

##### See also:

UniqueConstant .

###

PatchString -- execute commands between `<?` and `?>` in strings

##### Internal function

##### Calling format:

    PatchString(string)

##### Parameters:

`string` -- a string to patch

##### Description:

This function does the same as PatchLoad, but it works on a string in stead of on the contents of a text file. See PatchLoad for more details.

##### Examples:

    In> PatchString("Two plus three \
      is <? Write(2+3); ?> ");
    Out> "Two plus three is 5 ";

##### See also:

PatchLoad .

BernoulliDistribution , BinomialDistribution , tDistribution , PDF , ChiSquareTest .

# Probability and Statistics

Each distribution is represented as an entity. For each distribution known to the system the consistency of parameters is checked. If the parameters for a distribution are invalid, the functions return `Undefined`. For example, `NormalDistribution``(a,-1)` evaluates to `Undefined`, because of negative variance.

##

Probability

###

BernoulliDistribution -- Bernoulli distribution

##### Standard library

##### Calling format:

    BernoulliDistribution(p)

##### Parameters:

`p` -- number, probability of an event in a single trial

##### Description:

A random variable has a Bernoulli distribution with probability `p` if it can be interpreted as an indicator of an event, where `p` is the probability to observe the event in a single trial.

Numerical value of `p` must satisfy 0<p<1.

##### See also:

BinomialDistribution .

###

BinomialDistribution -- binomial distribution

##### Standard library

##### Calling format:

    BinomialDistribution(p,n)

##### Parameters:

`p` -- number, probability to observe an event in single trial

`n` -- number of trials

##### Description:

Suppose we repeat a trial `n` times, the probability to observe an event in a single trial is `p` and outcomes in all trials are mutually independent. Then the number of trials when the event occurred is distributed according to the binomial distribution. The probability of that is `BinomialDistribution``(p,n)`.

Numerical value of `p` must satisfy 0<p<1. Numerical value of `n` must be a positive integer.

##### See also:

BernoulliDistribution .

###

tDistribution -- Student's t distribution

##### Standard library

##### Calling format:

    {tDistribution}(m)

##### Parameters:

`m` -- integer, number of degrees of freedom

##### Description:

Let Y and Z be independent random variables, Y have the NormalDistribution(0,1), `Z` have ChiSquareDistribution(m). Then Y/Sqrt(Z/m) has tDistribution(m).

Numerical value of `m` must be positive integer.

###

PDF -- probability density function

##### Standard library

##### Calling format:

    PDF(dist,x)

##### Parameters:

`dist` -- a distribution type

`x` -- a value of random variable

##### Description:

If `dist` is a discrete distribution, then `PDF` returns the probability for a random variable with distribution `dist` to take a value of `x`. If `dist` is a continuous distribution, then `PDF` returns the density function at point x.

##### See also:

CDF .

##

Statistics

###

ChiSquareTest -- Pearson's ChiSquare test

##### Standard library

##### Calling format:

    ChiSquareTest(observed,expected)
    ChiSquareTest(observed,expected,params)

##### Parameters:

`observed` -- list of observed frequencies

`expected` -- list of expected frequencies

`params` -- number of estimated parameters

##### Description:

`ChiSquareTest` is intended to find out if our sample was drawn from a given distribution or not. To find this out, one has to calculate observed frequencies into certain intervals and expected ones. To calculate expected frequency the formula n[i]:=n*p[i] must be used, where p[i] is the probability measure of i-th interval, and n is the total number of observations. If any of the parameters of the distribution were estimated, this number is given as `params`.

The function returns a list of three local substitution rules. First of them contains the test statistic, the second contains the value of the parameters, and the last one contains the degrees of freedom.

The test statistic is distributed as ChiSquareDistribution.

GetYacasPID , ShowPS , MakeFunctionPlugin , Version , Vi , PlatformOS, OSVersion, FilePathSeparator , CopyFile, DeleteFile, DeleteDir, MakeDir, MakeFilePath, TemporaryDir, TemporaryFile , SystemCallBg .

# Platform-dependent packages

Certain facilities have been developed for use on Unix-like platforms, which is currently the main development target for Yacas. Other facilities have limited support on the Windows platform as well. These functions are described in this chapter.

###

GetYacasPID -- obtain Yacas process number

##### Unix-specific add-on

##### Calling format:

    GetYacasPID()

##### Description:

Returns an integer containing the process number (PID) of the Yacas session. This number can be used to create unique file names.

Requires: a Unix shell.

##### Example:

    In> GetYacasPID()
    Out> 26456;

##### See also:

SystemCall .

###

ShowPS -- view equations graphically

##### Unix-specific add-on

##### Calling format:

    ShowPS(expr)

##### Parameters:

`expr` -- any expression (not evaluated)

##### Description:

Exports a Yacas expression to LaTeX, generates a Postscript file and shows it in a viewer. The free Postscript viewer `gv` must be available on the Unix shell path. An alternative viewer can be specified by assigning to the global variable PSViewCommand.

Requires: a Unix shell, `latex`, `dvips`, `gv` or another Postscript viewer.

##### Example:

    In> [ PSViewCommand := "ghostview"; \
      ShowPS(x+2*Sin(x)); ]
    Expression exported as /tmp/yacas-tmp
    file-28802.tex
    Out> True;

##### See also:

TeXForm .

###

MakeFunctionPlugin -- compile numerical functions into plugins

##### Unix-specific add-on

##### Calling format:

    MakeFunctionPlugin("name", body)
    MakeFunctionPlugin()

##### Parameters:

`"name"` -- string, name of a new function

`body` -- expression, function of arguments, must evaluate to a function of some variables.

##### Description:

The first form of the function compiles an external plugin library that computes a user-defined numerical function and dynamically loads it into Yacas, enabling a new function called `"name"`.

The second form of the function is a predicate that checks that the function plugin facility is supported. (It may not be available on all platforms.)

Requires: a Unix shell, a compiler named `c++` with ELF `.so` support, Yacas headers in `FindFile("")/include`; current directory must be writable. Error messages will be printed otherwise.

The `body` expression must be a `CForm()`-exportable function of the arguments and may contain numerical constants. `Pi` is allowed and will be converted to floating-point.

All arguments and the return value of the function are assumed to be double precision real numbers. The result of passing a non-numerical argument will be an unevaluated expression.

The function creates the following files in subdirectory `plugins.tmp/` of current directory:

- `f1_plugin.h`, `f1_plugin.cc` -- C++ code of the plugin; the function in the above example is implemented in C++ as

      double f1_plugin_cc(double x, double y)
        { return sin(x/y); }

- `f1_plugin.stub` -- Yacas-language stub
- `f1_plugin_api.cc` -- Yacas-generated C++ stub
- `f1_plugin_api.description` - Yacas-generated documentation
- `f1_plugin.compile` -- command line to compile
- `libf1_plugin_cc.so` -- compiled plugin Note that all files have names matching "`*_plugin*`".

After creating these files, `MakeFunctionPlugin()` will:

- Run a `c++` compiler command; if compilation fails, all error messages will appear in the log file `f1_plugin.log`;
- Load resulting .so object with `DllLoad()`;
- Print an information message on success.

If you call `MakeFunctionPlugin()` repeatedly to define a function with the same name, old files will be overwritten and old libraries will be unloaded with `DllUnload()`.

If the numerical calculation does not return a number (for example, it might return the atom `nan`, "not a number", for some arguments), then the new function will return `Undefined`. This is the behavior of `NFunction` which is used to wrap the numerical routine.

##### Example:

    In> MakeFunctionPlugin("f1", Sin(x/y))
    Function f1(x,y) loaded from
    plugins.tmp/libf1_plugin_cc.so
    Out> True;
    In> f1(2,3)
    Out> 0.618369803069736989620253;
    In> f1(x,5)
    Out> f1(x,5);

##### See also:

DllLoad , DllUnload , DllEnumerate , CForm , NFunction .

###

Version -- show version of Yacas

##### Internal function

##### Calling format:

    Version()

##### Description:

The function `Version()` returns a string representing the version of the currently running Yacas interpreter.

##### Examples:

    In> Version()
    Out> "1.0.48rev3";
    In> LessThan(Version(), "1.0.47")
    Out> False;
    In> GreaterThan(Version(), "1.0.47")
    Out> True;

The last two calls show that the `LessThan` and `GreaterThan` functions can be used for comparing version numbers. This method is only guaranteed, however, if the version is always expressed in the form `d.d.dd` as above.

##### See also:

LessThan , GreaterThan .

###

Vi -- edit a file or function

##### Unix-specific add-on

##### Calling format:

    Vi(filename);
    Vi(functionname);

##### Parameters:

`filename` - name of a file to edit

`functionname` - name of a function to find for editing

##### Description:

`Vi` will try to edit a file, or if the argument passed is a function, it will try to edit the file the function is defined in. It will try to do so by invoking the editor `vi`.

It finds the function by scanning the `*.def` files that have been reported to the system. (`Vi` calls `FindFunction` for this.) If editing a function, the command will jump directly to the first occurrence of the name of the function in the file (usually the beginning of a definition of a function).

If you would like to use `Vi()` to actually edit the Yacas library source file where the function is defined, you need to start Yacas from the `scripts/` directory in the development tree. In that case, `FindFunction()` will return the filename under that directory. Otherwise, `FindFunction()` will return a name in the systemwide installation directory (or directory specified in the `--rootdir` option).

##### Examples:

    In> Vi("yacasinit.ys")
    Out> True;
    In> Vi("Sum")
    Out> True;

##### See also:

FindFunction .

###

PlatformOS, OSVersion, FilePathSeparator -- OS-dependent constants

##### Standard library

##### Calling format:

    FilePathSeparator
    PlatformOS()
    OSVersion()

##### Description:

The functions `OSVersion()` and `PlatformOS()` return strings with an abbreviated description of the platform for which the current Yacas interpreter was compiled. Possible values are for instance `"linux-gnu"`, `"Win32"` or `"solaris2.6"`. These constants can be used in scripts if one needs to know the platform.

The value of `OSVersion()` is normally determined when Yacas is compiled. It is usually the operating system name and version as reported by the `config.guess` script. However, it may be overridden at build time (on non-Unix systems, the script cannot be run and the value must be specified by hand).

The value of `PlatformOS()` is defined in the library (in `osdep.rep/`). Thus the library is able to override the platform-specific value. The scripts should use `PlatformOS()` to distinguish between broad classes of systems. The currently supported values are `"Unix"` and `"Win32"`.

The constant `FilePathSeparator` is the string that separates directories in the file system tree. This is the forward slash "`/`" on Unix and the backslash "`\`" on Windows.

##### Examples:

    In> Check(StringMid(1, 5, OSVersion())="linux", \
              "BAAA! I WANT LINUX!")
    Out> True;
    In> PlatformOS()
    Out> "Unix";

##### See also:

MakeFilePath .

###

CopyFile, DeleteFile, DeleteDir, MakeDir, MakeFilePath, TemporaryDir, TemporaryFile -- manipulate files and directories

##### Standard library

##### Calling format:

    CopyFile("src","dest")
    DeleteFile("file", ...)
    DeleteDir("dir", ...)
    MakeDir("dir1", ...)
    MakeFilePath("dir1", "dir2", ..., "file")
    TemporaryDir()
    TemporaryFile()

##### Parameters:

`"src"`, `"dest"`, `"file"`, `"dir"` -- strings, names of files and directories

##### Description:

These functions provide a platform-independent interface to file operations on the underlying operating system (OS):

- `CopyFile` will copy a file `"src"` to another file or directory `"dest"`.
- `DeleteFile` will delete one or more files. If a file does not exist, no error is generated on Unix.
- `DeleteDir` will delete one or more directories with all their contents. (No warnings or questions are printed,---use with caution!) If a directory does not exist, no error is generated on Unix.
- `MakeDir` will create new directories. Under Windows, all intermediate subdirectories must exist for this to work. Under Unix, any intermediate subdirectories will be created automatically if needed.
- `MakeFilePath` returns a string which is the file path to the given file under given subdirectories. It uses the global variable `FilePathSeparator`.
- `TemporaryDir` and `TemporaryFile` return the names of a temporary directory and a temporary file. (These names are OS-dependent.)

Absolute file paths or relative file paths can be given (the current directory is the directory where Yacas was started).

File and directory names may contain spaces, although the behavior on Windows platform may be incorrect due to its broken filesystem and command interpreter. File and directory names should not contain double quotes (`"`). (On Unix, double quotes may be escaped by a backslash.)

When operating on many files or directories at once, there may be errors associated with some files but not others. The functions return `True` only if all operations succeeded. Error messages from the OS are printed on the console as usual for `SystemCall`.

##### Examples:

    In> MakeFilePath("", "usr", "bin", "yacas")
    Out> "/usr/bin/yacas";
    In> TemporaryDir()
    Out> "/tmp";

If the file `f1.txt` exists, it can be copied:

    In> CopyFile("f1.txt", "f2.txt")
    Out> True;

Create a subdirectory `subdir` under the temporary directory:

    In> MakeDir(d:=MakeFilePath(TemporaryDir(), \
      "subdir"))
    Out> True;
    In> d
    Out> "/tmp/subdir";

Copy a file into the newly created temporary location:

    In> CopyFile("f2.txt", d)
    Out> True;

On Unix, spaces can be included in a file name and quotes can be escaped like this:

    In> MakeDir("a \\\"")
    Out> True;

Now a directory with the name "`a "`" is created.

An unsuccessful attempt to delete a directory:

    In> DeleteDir("C:\\WINDOWS")
    Out> False;

##### See also:

SystemCall , FilePathSeparator .

###

SystemCallBg -- execute a system command in the background

##### Standard library

##### Calling format:

    SystemCallBg("command")

##### Parameters:

`"command"` -- string, command line to execute

##### Description:

This function is similar to `SystemCall` except the command is executed in the background. This is useful, for example, to start a GUI program while not interrupting the Yacas session.

Unlike the `SystemCall` function, the return status of the command is not available.

Error messages from the OS are printed on the console as usual for `SystemCall`.

##### Examples:

    In> SystemCallBg("mozilla")
    Out> True;

(The Yacas session continues right away, but the disk starts to churn until finally the Mozilla window appears.)

##### See also:

SystemCall .

# Yacas plugins

Plugins are dynamic libraries loaded at runtime. They are normally used to provide access to external libraries or applications. `Yacas` comes with a few plugins, such as the GSL plugin. This chapter contains the documentation generated for each plugin.

##

The file scanner plugin

The file scanner plugin allows the programmer to scan files and directories in the file system. It is currently supported for Unix-like and Windows systems.

To use the plugin, the plugin has to be loaded through:

    Use("filescanner");

The file scanner plugin offers one principal new function; `ScanFiles`. The prototype for this function is:

    ScanFiles(BaseDirectory,SubDirectory,CallFunction);

The arguments:

- `BaseDirectory` - the base directory to scan in
- `SubDirectory` - the sub-directory to scan in
- `CallFunction` - the call-back function

The call-back function should accept three arguments: the base directory, the file name and a boolean specifying if the file is a sub-directory.

The full file name can be obtained by concatenating the file name to the base directory. If the file is a sub-directory, the file name can be used as the name of the sub-directory in subsequent recursive calls to `ScanFiles`.

For example, the following definition of the function `tst` can be used as a third argument to `ScanFiles`, in order to scan directories recursively, printing file names of files in the directories.

    // File, print
    10 # tst(_a,_b,False) <-- Echo("File: ",b)
    // Directory, scan recursively
    20 # tst(_a,_b,True) <-- ScanFiles(a,b,"tst")

Then, if the `Yacas` source is in `/Users/ayalpink/yacas/`, one can call:

    ScanFiles("/Users/someone/yacas/","plugins","tst")

The first lines of output (in `Yacas` version 1.0.55) is:

    File: plugins/.cvsignore
    File: plugins/CVS/Entries
    File: plugins/CVS/Repository
    File: plugins/CVS/Root
    File: plugins/doc.txt

Here the scan went into the `CVS` sub-directory. Appending these file names to the base directory, `/Users/someone/yacas/`, would give the full path to the file.

##

The regular expressions plugin

This plugin allows for text processing using regular expressions. The GPL'ed package `pcre` is included for this purpose (pcre stands for Perl compatible regular expressions).

To use the plugin, the plugin has to be loaded through:

    Use("pcre");

The `pcre` plugin offers two principal functions: `PcreLexer` and `PcreNextToken`. `PcreLexer` sets up and initializes the tokenizer to scan for a set of regular expressions. A call to `PcreNextToken` returns the result of the scan in the current file, starting from the current position.

`PcreLexer` does not interfere with normal reading of a file. When normal read calls are made, the default tokenizer is used. The tokenizer set up by `PcreLexer` is only used when `PcreNextToken` is called.

The syntax for `PcreLexer` is:

    PcreLexer(tokens);

The argument `tokens` is a list of token definitions. A token definition in turn is a list, with the first element a string containing the regular expression, and the second containing an atom that can describe the type of the token (for easy recognition later).

Calling `PcreNextToken()` will then return the first matching token from the current input.

The following example sets up the tokenizer to recognize either an integer, or a word consisting only of lowercase letters.

    In> DllLoad("pcre")
    Out> True
    In> PcreLexer({{"[0-9]+",Integer},{"[a-z]+",Word}})
    Out> True
    In> FromString("123abc")PcreNextToken()
    Out> {"123",Integer}
    In> FromString("===abc")PcreNextToken()
    Out> {"abc",Word}

##

The GSL library plugin

This plugin provides access to functions from the GNU Scientific Library (GSL). See http://www.gnu.org/software/gsl/ for more information and for documentation on specific library functions.

##

The example plugin

This servers as an illustration of the plugin capability of Yacas. The plugin defines some functions operating on integers, floats, and on a custom structure type "`Bla`".

This is an example mini-API.

Function `AddTwoIntegers`(`arg1,arg2`), calls `int add_integers` (`int` `arg1`, `int` `arg2`)

Function `AddTwoDoubles`(`x,y`), calls `double add_doubles` (`double` `x`, `double` `y`)

Declared struct `Bla*`

Function `CreateBla`(`arg1,arg2`), calls `Bla* CreateBla` (`int` `arg1`, `int` `arg2`)

Function `BlaSetA`(`arg1,arg2`), calls `void BlaSetA` (`Bla*` `arg1`, `int` `arg2`)

Function `BlaGetA`(`arg1`), calls `int BlaGetA` (`Bla*` `arg1`)

##

The Forth plugin

This is the forth api.

# GNU Free Documentation License

##### Version 1.1, March 2000

Copyright (C) 2000 Free Software Foundation, Inc.

    59 Temple Place, Suite 330
    Boston, MA, 02111-1307
    USA

Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.

###

Preamble

The purpose of this License is to make a manual, textbook, or other written document ``free'' in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others.

This License is a kind of ``copyleft'', which means that derivative works of the document must themselves be free in the same sense. It complements the GNU General Public License, which is a copyleft license designed for free software.

We have designed this License in order to use it for manuals for free software, because free software needs free documentation: a free program should come with manuals providing the same freedoms that the software does. But this License is not limited to software manuals; it can be used for any textual work, regardless of subject matter or whether it is published as a printed book. We recommend this License principally for works whose purpose is instruction or reference.

###

Applicability and Definitions

This License applies to any manual or other work that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. The ``Document'', below, refers to any such manual or work. Any member of the public is a licensee, and is addressed as ``you''.

A ``Modified Version'' of the Document means any work containing the Document or a portion of it, either copied verbatim, or with modifications and/or translated into another language.

A ``Secondary Section'' is a named appendix or a front-matter section of the Document that deals exclusively with the relationship of the publishers or authors of the Document to the Document's overall subject (or to related matters) and contains nothing that could fall directly within that overall subject. (For example, if the Document is in part a textbook of mathematics, a Secondary Section may not explain any mathematics.) The relationship could be a matter of historical connection with the subject or with related matters, or of legal, commercial, philosophical, ethical or political position regarding them.

The ``Invariant Sections'' are certain Secondary Sections whose titles are designated, as being those of Invariant Sections, in the notice that says that the Document is released under this License.

The ``Cover Texts'' are certain short passages of text that are listed, as Front-Cover Texts or Back-Cover Texts, in the notice that says that the Document is released under this License.

A ``Transparent'' copy of the Document means a machine-readable copy, represented in a format whose specification is available to the general public, whose contents can be viewed and edited directly and straightforwardly with generic text editors or (for images composed of pixels) generic paint programs or (for drawings) some widely available drawing editor, and that is suitable for input to text formatters or for automatic translation to a variety of formats suitable for input to text formatters. A copy made in an otherwise Transparent file format whose markup has been designed to thwart or discourage subsequent modification by readers is not Transparent. A copy that is not ``Transparent'' is called ``Opaque''.

Examples of suitable formats for Transparent copies include plain ASCII without markup, Texinfo input format, `LaTeX` input format, SGML or XML using a publicly available DTD, and standard-conforming simple HTML designed for human modification. Opaque formats include PostScript, PDF, proprietary formats that can be read and edited only by proprietary word processors, SGML or XML for which the DTD and/or processing tools are not generally available, and the machine-generated HTML produced by some word processors for output purposes only.

The ``Title Page'' means, for a printed book, the title page itself, plus such following pages as are needed to hold, legibly, the material this License requires to appear in the title page. For works in formats which do not have any title page as such, ``Title Page'' means the text near the most prominent appearance of the work's title, preceding the beginning of the body of the text.

###

Verbatim Copying

You may copy and distribute the Document in any medium, either commercially or noncommercially, provided that this License, the copyright notices, and the license notice saying this License applies to the Document are reproduced in all copies, and that you add no other conditions whatsoever to those of this License. You may not use technical measures to obstruct or control the reading or further copying of the copies you make or distribute. However, you may accept compensation in exchange for copies. If you distribute a large enough number of copies you must also follow the conditions in section 3.

You may also lend copies, under the same conditions stated above, and you may publicly display copies.

###

Copying in Quantity

If you publish printed copies of the Document numbering more than 100, and the Document's license notice requires Cover Texts, you must enclose the copies in covers that carry, clearly and legibly, all these Cover Texts: Front-Cover Texts on the front cover, and Back-Cover Texts on the back cover. Both covers must also clearly and legibly identify you as the publisher of these copies. The front cover must present the full title with all words of the title equally prominent and visible. You may add other material on the covers in addition. Copying with changes limited to the covers, as long as they preserve the title of the Document and satisfy these conditions, can be treated as verbatim copying in other respects.

If the required texts for either cover are too voluminous to fit legibly, you should put the first ones listed (as many as fit reasonably) on the actual cover, and continue the rest onto adjacent pages.

If you publish or distribute Opaque copies of the Document numbering more than 100, you must either include a machine-readable Transparent copy along with each Opaque copy, or state in or with each Opaque copy a publicly-accessible computer-network location containing a complete Transparent copy of the Document, free of added material, which the general network-using public has access to download anonymously at no charge using public-standard network protocols. If you use the latter option, you must take reasonably prudent steps, when you begin distribution of Opaque copies in quantity, to ensure that this Transparent copy will remain thus accessible at the stated location until at least one year after the last time you distribute an Opaque copy (directly or through your agents or retailers) of that edition to the public.

It is requested, but not required, that you contact the authors of the Document well before redistributing any large number of copies, to give them a chance to provide you with an updated version of the Document.

###

Modifications

You may copy and distribute a Modified Version of the Document under the conditions of sections 2 and 3 above, provided that you release the Modified Version under precisely this License, with the Modified Version filling the role of the Document, thus licensing distribution and modification of the Modified Version to whoever possesses a copy of it. In addition, you must do these things in the Modified Version:

- Use in the Title Page (and on the covers, if any) a title distinct from that of the Document, and from those of previous versions (which should, if there were any, be listed in the History section of the Document). You may use the same title as a previous version if the original publisher of that version gives permission.
- List on the Title Page, as authors, one or more persons or entities responsible for authorship of the modifications in the Modified Version, together with at least five of the principal authors of the Document (all of its principal authors, if it has less than five).
- State on the Title page the name of the publisher of the Modified Version, as the publisher.
- Preserve all the copyright notices of the Document.
- Add an appropriate copyright notice for your modifications adjacent to the other copyright notices.
- Include, immediately after the copyright notices, a license notice giving the public permission to use the Modified Version under the terms of this License, in the form shown in the Addendum below.
- Preserve in that license notice the full lists of Invariant Sections and required Cover Texts given in the Document's license notice.
- Include an unaltered copy of this License.
- Preserve the section entitled ``History'', and its title, and add to it an item stating at least the title, year, new authors, and publisher of the Modified Version as given on the Title Page. If there is no section entitled ``History'' in the Document, create one stating the title, year, authors, and publisher of the Document as given on its Title Page, then add an item describing the Modified Version as stated in the previous sentence.
- Preserve the network location, if any, given in the Document for public access to a Transparent copy of the Document, and likewise the network locations given in the Document for previous versions it was based on. These may be placed in the ``History'' section. You may omit a network location for a work that was published at least four years before the Document itself, or if the original publisher of the version it refers to gives permission.
- In any section entitled ``Acknowledgements'' or ``Dedications'', preserve the section's title, and preserve in the section all the substance and tone of each of the contributor acknowledgements and/or dedications given therein.
- Preserve all the Invariant Sections of the Document, unaltered in their text and in their titles. Section numbers or the equivalent are not considered part of the section titles.
- Delete any section entitled ``Endorsements''. Such a section may not be included in the Modified Version.
- Do not retitle any existing section as ``Endorsements'' or to conflict in title with any Invariant Section.

If the Modified Version includes new front-matter sections or appendices that qualify as Secondary Sections and contain no material copied from the Document, you may at your option designate some or all of these sections as invariant. To do this, add their titles to the list of Invariant Sections in the Modified Version's license notice. These titles must be distinct from any other section titles.

You may add a section entitled ``Endorsements'', provided it contains nothing but endorsements of your Modified Version by various parties -- for example, statements of peer review or that the text has been approved by an organization as the authoritative definition of a standard.

You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through arrangements made by) any one entity. If the Document already includes a cover text for the same cover, previously added by you or by arrangement made by the same entity you are acting on behalf of, you may not add another; but you may replace the old one, on explicit permission from the previous publisher that added the old one.

The author(s) and publisher(s) of the Document do not by this License give permission to use their names for publicity for or to assert or imply endorsement of any Modified Version.

###

Combining Documents

You may combine the Document with other documents released under this License, under the terms defined in section 4 above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them all as Invariant Sections of your combined work in its license notice.

The combined work need only contain one copy of this License, and multiple identical Invariant Sections may be replaced with a single copy. If there are multiple Invariant Sections with the same name but different contents, make the title of each such section unique by adding at the end of it, in parentheses, the name of the original author or publisher of that section if known, or else a unique number. Make the same adjustment to the section titles in the list of Invariant Sections in the license notice of the combined work.

In the combination, you must combine any sections entitled ``History'' in the various original documents, forming one section entitled ``History''; likewise combine any sections entitled ``Acknowledgements'', and any sections entitled ``Dedications''. You must delete all sections entitled ``Endorsements.''

###

Collections of Documents

You may make a collection consisting of the Document and other documents released under this License, and replace the individual copies of this License in the various documents with a single copy that is included in the collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects.

You may extract a single document from such a collection, and distribute it individually under this License, provided you insert a copy of this License into the extracted document, and follow this License in all other respects regarding verbatim copying of that document.

###

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/*, */, // , Prog, [, ] , Bodied, Infix, Postfix, Prefix , IsBodied, IsInfix, IsPostfix, IsPrefix , OpPrecedence, OpLeftPrecedence, OpRightPrecedence , RightAssociative , LeftPrecedence, RightPrecedence , RuleBase , RuleBaseListed , Rule , HoldArg , Retract , UnFence , HoldArgNr , RuleBaseArgList , MacroSet, MacroClear, MacroLocal, MacroRuleBase, MacroRuleBaseListed, MacroRule , Backquoting , DefMacroRuleBase , DefMacroRuleBaseListed , SetExtraInfo, GetExtraInfo , GarbageCollect , FindFunction , Secure .

# Programming

This chapter describes functions useful for writing Yacas scripts.

###

/*, */, // -- comments

##### Internal function

##### Calling format:

    /* comment */
    // comment

##### Description:

Introduce a comment block in a source file, similar to C++ comments. `//` makes everything until the end of the line a comment, while `/*` and `*/` may delimit a multi-line comment.

##### Examples:

    a+b; // get result
    a + /* add them */ b;

###

Prog, [, ] -- block of statements

##### Internal function

##### Calling format:

    Prog(statement1, statement2, ...)
    [ statement1; statement2; ... ]

##### Parameters:

`statement1`, `statement2` -- expressions

##### Description:

The `Prog` and the `[ ... ]` construct have the same effect: they evaluate all arguments in order and return the result of the last evaluated expression.

`Prog(a,b);` is the same as typing `[a;b;];` and is very useful for writing out function bodies. The `[ ... ]` construct is a syntactically nicer version of the `Prog` call; it is converted into `Prog(...)` during the parsing stage.

###

Bodied, Infix, Postfix, Prefix -- define function syntax

##### Internal function

##### Calling format:

    Bodied("op", precedence)
    Infix("op")
    Infix("op", precedence)
    Postfix("op")
    Postfix("op", precedence)
    Prefix("op")
    Prefix("op", precedence)

##### Parameters:

`"op"` -- string, the name of a function

`precedence` -- nonnegative integer (evaluated)

##### Description:

Declares a special syntax for the function to be parsed as a bodied, infix, postfix, or prefix operator.

"Bodied" functions have all arguments except the first one inside parentheses and the last argument outside, for example:

    For(pre, condition, post) statement;

Here the function `For` has 4 arguments and the last argument is placed outside the parentheses. The `precedence` of a "bodied" function refers to how tightly the last argument is bound to the parentheses. This makes a difference when the last argument contains other operators. For example, when taking the derivative

    D(x) Sin(x)+Cos(x)

both `Sin` and `Cos` are under the derivative because the bodied function `D` binds less tightly than the infix operator "`+`".

"Infix" functions must have two arguments and are syntactically placed between their arguments. Names of infix functions can be arbitrary, although for reasons of readability they are usually made of non-alphabetic characters.

"Prefix" functions must have one argument and are syntactically placed before their argument. "Postfix" functions must have one argument and are syntactically placed after their argument.

Function name can be any string but meaningful usage and readability would require it to be either made up entirely of letters or entirely of non-letter characters (such as "+", ":" etc.). Precedence is optional (will be set to 0 by default).

##### Examples:

    In> YY x := x+1;
    CommandLine(1) : Error parsing expression

    In> Prefix("YY", 2)
    Out> True;
    In> YY x := x+1;
    Out> True;
    In> YY YY 2*3
    Out> 12;
    In> Infix("##", 5)
    Out> True;
    In> a ## b ## c
    Out> a##b##c;

Note that, due to a current parser limitation, a function atom that is declared prefix cannot be used by itself as an argument.

    In> YY
    CommandLine(1) : Error parsing expression

##### See also:

IsBodied , OpPrecedence .

###

IsBodied, IsInfix, IsPostfix, IsPrefix -- check for function syntax

##### Internal function

##### Calling format:

    IsBodied("op")
    IsInfix("op")
    IsPostfix("op")
    IsPrefix("op")

##### Parameters:

`"op"` -- string, the name of a function

##### Description:

Check whether the function with given name `"op"` has been declared as a "bodied", infix, postfix, or prefix operator, and return `True` or `False`.

##### Examples:

    In> IsInfix("+");
    Out> True;
    In> IsBodied("While");
    Out> True;
    In> IsBodied("Sin");
    Out> False;
    In> IsPostfix("!");
    Out> True;

##### See also:

Bodied , OpPrecedence .

###

OpPrecedence, OpLeftPrecedence, OpRightPrecedence -- get operator precedence

##### Internal function

##### Calling format:

    OpPrecedence("op")
    OpLeftPrecedence("op")
    OpRightPrecedence("op")

##### Parameters:

`"op"` -- string, the name of a function

##### Description:

Returns the precedence of the function named "op" which should have been declared as a bodied function or an infix, postfix, or prefix operator. Generates an error message if the string str does not represent a type of function that can have precedence.

For infix operators, right precedence can differ from left precedence. Bodied functions and prefix operators cannot have left precedence, while postfix operators cannot have right precedence; for these operators, there is only one value of precedence.

##### Examples:

    In> OpPrecedence("+")
    Out> 6;
    In> OpLeftPrecedence("!")
    Out> 0;

###

RightAssociative -- declare associativity

##### Internal function

##### Calling format:

    RightAssociative("op")

##### Parameters:

`"op"` -- string, the name of a function

##### Description:

This makes the operator right-associative. For example:

    RightAssociative("*")

would make multiplication right-associative. Take care not to abuse this function, because the reverse, making an infix operator left-associative, is not implemented. (All infix operators are by default left-associative until they are declared to be right-associative.)

##### See also:

OpPrecedence .

###

LeftPrecedence, RightPrecedence -- set operator precedence

##### Internal function

##### Calling format:

    LeftPrecedence("op",precedence)
    RightPrecedence("op",precedence)

##### Parameters:

`"op"` -- string, the name of a function

`precedence` -- nonnegative integer

##### Description:

`"op"` should be an infix operator. This function call tells the infix expression printer to bracket the left or right hand side of the expression if its precedence is larger than precedence.

This functionality was required in order to display expressions like `a-(b-c)` correctly. Thus, `a+b+c` is the same as `a+(b+c)`, but `a-(b-c)` is not the same as `a-b-c`.

Note that the left and right precedence of an infix operator does not affect the way Yacas interprets expressions typed by the user. You cannot make Yacas parse `a-b-c` as `a-(b-c)` unless you declare the operator "`-`" to be right-associative.

##### See also:

OpPrecedence , OpLeftPrecedence , OpRightPrecedence , RightAssociative .

###

RuleBase -- define function with a fixed number of arguments

##### Internal function

##### Calling format:

    RuleBase(name,params)

##### Parameters:

`name` -- string, name of function

`params` -- list of arguments to function

##### Description:

Define a new rules table entry for a function "name", with `params` as the parameter list. Name can be either a string or simple atom.

In the context of the transformation rule declaration facilities this is a useful function in that it allows the stating of argument names that can he used with HoldArg.

Functions can be overloaded: the same function can be defined with different number of arguments.

##### See also:

MacroRuleBase , RuleBaseListed , MacroRuleBaseListed , HoldArg , Retract .

###

RuleBaseListed -- define function with variable number of arguments

##### Internal function

##### Calling format:

    RuleBaseListed("name", params)

##### Parameters:

`"name"` -- string, name of function

`params` -- list of arguments to function

##### Description:

The command `RuleBaseListed` defines a new function. It essentially works the same way as `RuleBase`, except that it declares a new function with a variable number of arguments. The list of parameters `params` determines the smallest number of arguments that the new function will accept. If the number of arguments passed to the new function is larger than the number of parameters in `params`, then the last argument actually passed to the new function will be a list containing all the remaining arguments.

A function defined using `RuleBaseListed` will appear to have the arity equal to the number of parameters in the `param` list, and it can accept any number of arguments greater or equal than that. As a consequence, it will be impossible to define a new function with the same name and with a greater arity.

The function body will know that the function is passed more arguments than the length of the `param` list, because the last argument will then be a list. The rest then works like a `RuleBase`-defined function with a fixed number of arguments. Transformation rules can be defined for the new function as usual.

##### Examples:

The definitions

    RuleBaseListed("f",{a,b,c})
    10 # f(_a,_b,{_c,_d}) <--
      Echo({"four args",a,b,c,d});
    20 # f(_a,_b,c_IsList) <--
      Echo({"more than four args",a,b,c});
    30 # f(_a,_b,_c) <-- Echo({"three args",a,b,c});

give the following interaction:

    In> f(A)
    Out> f(A);
    In> f(A,B)
    Out> f(A,B);
    In> f(A,B,C)
    three args A B C
    Out> True;
    In> f(A,B,C,D)
    four args A B C D
    Out> True;
    In> f(A,B,C,D,E)
    more than four args A B {C,D,E}
    Out> True;
    In> f(A,B,C,D,E,E)
    more than four args A B {C,D,E,E}
    Out> True;

The function `f` now appears to occupy all arities greater than 3:

    In> RuleBase("f", {x,y,z,t});
    CommandLine(1) : Rule base with this arity
      already defined

##### See also:

RuleBase , Retract , Echo .

###

Rule -- define a rewrite rule

##### Internal function

##### Calling format:

    Rule("operator", arity,
      precedence, predicate) body

##### Parameters:

`"operator"` -- string, name of function

`arity`, `precedence` -- integers

`predicate` -- function returning boolean

`body` -- expression, body of rule

##### Description:

Define a rule for the function "operator" with "arity", "precedence", "predicate" and "body". The "precedence" goes from low to high: rules with low precedence will be applied first.

The arity for a rules database equals the number of arguments. Different rules data bases can be built for functions with the same name but with a different number of arguments.

Rules with a low precedence value will be tried before rules with a high value, so a rule with precedence 0 will be tried before a rule with precedence 1.

###

HoldArg -- mark argument as not evaluated

##### Internal function

##### Calling format:

    HoldArg("operator",parameter)

##### Parameters:

`"operator"` -- string, name of a function

`parameter` -- atom, symbolic name of parameter

##### Description:

Specify that parameter should not be evaluated before used. This will be declared for all arities of "operator", at the moment this function is called, so it is best called after all `RuleBase` calls for this operator. "operator" can be a string or atom specifying the function name.

The `parameter` must be an atom from the list of symbolic arguments used when calling `RuleBase`.

##### See also:

RuleBase , HoldArgNr , RuleBaseArgList .

###

Retract -- erase rules for a function

##### Internal function

##### Calling format:

    Retract("function",arity)

##### Parameters:

`"function"` -- string, name of function

`arity` -- positive integer

##### Description:

Remove a rulebase for the function named `"function"` with the specific `arity`, if it exists at all. This will make Yacas forget all rules defined for a given function. Rules for functions with the same name but different arities are not affected.

Assignment `:=` of a function does this to the function being (re)defined.

##### See also:

RuleBaseArgList , RuleBase , := .

###

UnFence -- change local variable scope for a function

##### Internal function

##### Calling format:

    UnFence("operator",arity)

##### Parameters:

`"operator"` -- string, name of function

`arity` -- positive integers

##### Description:

When applied to a user function, the bodies defined for the rules for "operator" with given arity can see the local variables from the calling function. This is useful for defining macro-like procedures (looping and such).

The standard library functions `For` and `ForEach` use `UnFence`.

###

HoldArgNr -- specify argument as not evaluated

##### Standard library

##### Calling format:

    HoldArgNr("function", arity, argNum)

##### Parameters:

`"function"` -- string, function name

`arity`, `argNum` -- positive integers

##### Description:

Declares the argument numbered `argNum` of the function named `"function"` with specified `arity` to be unevaluated ("held"). Useful if you don't know symbolic names of parameters, for instance, when the function was not declared using an explicit `RuleBase` call. Otherwise you could use `HoldArg`.

##### See also:

HoldArg , RuleBase .

###

RuleBaseArgList -- obtain list of arguments

##### Internal function

##### Calling format:

    RuleBaseArgList("operator", arity)

##### Parameters:

`"operator"` -- string, name of function

`arity` -- integer

##### Description:

Returns a list of atoms, symbolic parameters specified in the `RuleBase` call for the function named `"operator"` with the specific `arity`.

##### See also:

RuleBase , HoldArgNr , HoldArg .

###

MacroSet, MacroClear, MacroLocal, MacroRuleBase, MacroRuleBaseListed, MacroRule -- define rules in functions

##### Internal function

##### Description:

These functions have the same effect as their non-macro counterparts, except that their arguments are evaluated before the required action is performed. This is useful in macro-like procedures or in functions that need to define new rules based on parameters.

Make sure that the arguments of `Macro`... commands evaluate to expressions that would normally be used in the non-macro versions!

##### See also:

Set , Clear , Local , RuleBase , Rule , Backquoting .

###

Backquoting -- macro expansion (LISP-style backquoting)

##### Internal function

##### Calling format:

    `(expression)

##### Parameters:

`expression` -- expression containing "`@var`" combinations to substitute the value of variable "`var`"

##### Description:

Backquoting is a macro substitution mechanism. A backquoted `expression` is evaluated in two stages: first, variables prefixed by `@` are evaluated inside an expression, and second, the new expression is evaluated.

To invoke this functionality, a backquote ``` needs to be placed in front of an expression. Parentheses around the expression are needed because the backquote binds tighter than other operators.

The expression should contain some variables (assigned atoms) with the special prefix operator `@`. Variables prefixed by `@` will be evaluated even if they are inside function arguments that are normally not evaluated (e.g. functions declared with `HoldArg`). If the `@var` pair is in place of a function name, e.g. "`@f(x)`", then at the first stage of evaluation the function name itself is replaced, not the return value of the function (see example); so at the second stage of evaluation, a new function may be called.

One way to view backquoting is to view it as a parametric expression generator. `@var` pairs get substituted with the value of the variable `var` even in contexts where nothing would be evaluated. This effect can be also achieved using `UnList` and `Hold` but the resulting code is much more difficult to read and maintain.

This operation is relatively slow since a new expression is built before it is evaluated, but nonetheless backquoting is a powerful mechanism that sometimes allows to greatly simplify code.

##### Examples:

This example defines a function that automatically evaluates to a number as soon as the argument is a number (a lot of functions do this only when inside a `N(...)` section).

    In> Decl(f1,f2) := \
    In>   `(@f1(x_IsNumber) <-- N(@f2(x)));
    Out> True;
    In> Decl(nSin,Sin)
    Out> True;
    In> Sin(1)
    Out> Sin(1);
    In> nSin(1)
    Out> 0.8414709848;

This example assigns the expression `func(value)` to variable `var`. Normally the first argument of `Set` would be unevaluated.

    In> SetF(var,func,value) := \
    In>     `(Set(@var,@func(@value)));
    Out> True;
    In> SetF(a,Sin,x)
    Out> True;
    In> a
    Out> Sin(x);

##### See also:

MacroSet , MacroLocal , MacroRuleBase , Hold , HoldArg , DefMacroRuleBase .

###

DefMacroRuleBase -- define a function as a macro

##### Standard library

##### Calling format:

    DefMacroRuleBase(name,params)

##### Parameters:

`name` -- string, name of a function

`params` -- list of arguments

##### Description:

`DefMacroRuleBase` is similar to `RuleBase`, with the difference that it declares a macro, instead of a function. After this call, rules can be defined for the function "`name`", but their interpretation will be different.

With the usual functions, the evaluation model is that of the applicative-order model of substitution, meaning that first the arguments are evaluated, and then the function is applied to the result of evaluating these arguments. The function is entered, and the code inside the function can not access local variables outside of its own local variables.

With macros, the evaluation model is that of the normal-order model of substitution, meaning that all occurrences of variables in an expression are first substituted into the body of the macro, and only then is the resulting expression evaluated in its calling environment. This is important, because then in principle a macro body can access the local variables from the calling environment, whereas functions can not do that.

As an example, suppose there is a function `square`, which squares its argument, and a function `add`, which adds its arguments. Suppose the definitions of these functions are:

    add(x,y) <-- x+y;

and

    square(x) <-- x*x;

In applicative-order mode (the usual way functions are evaluated), in the following expression

    add(square(2),square(3))

first the arguments to `add` get evaluated. So, first `square(2)` is evaluated. To evaluate this, first `2` is evaluated, but this evaluates to itself. Then the `square` function is applied to it, `2*2`, which returns 4. The same is done for `square(3)`, resulting in `9`. Only then, after evaluating these two arguments, `add` is applied to them, which is equivalent to

    add(4,9)

resulting in calling `4+9`, which in turn results in `13`.

In contrast, when `add` is a macro, the arguments to `add` are first expanded. So

    add(square(2),square(3))

first expands to

    square(2) + square(3)

and then this expression is evaluated, as if the user had written it directly. In other words, `square(2)` is not evaluated before the macro has been fully expanded.

Macros are useful for customizing syntax, and compilers can potentially greatly optimize macros, as they can be inlined in the calling environment, and optimized accordingly.

There are disadvantages, however. In interpreted mode, macros are slower, as the requirement for substitution means that a new expression to be evaluated has to be created on the fly. Also, when one of the parameters to the macro occur more than once in the body of the macro, it is evaluated multiple times.

When defining transformation rules for macros, the variables to be substituted need to be preceded by the `@` operator, similar to the back-quoting mechanism. Apart from that, the two are similar, and all transformation rules can also be applied to macros.

Macros can co-exist with functions with the same name but different arity. For instance, one can have a function `foo(a,b)` with two arguments, and a macro `foo(a,b,c)` with three arguments.

##### Example:

The following example defines a function `myfor`, and shows one use, referencing a variable `a` from the calling environment.

    In> DefMacroRuleBase("myfor",{init,pred,inc,body})
    Out> True;
    In> myfor(_init,_pred,_inc,_body)<--[@init;While(@pred)[@body;@inc;];True;];
    Out> True;
    In> a:=10
    Out> 10;
    In> myfor(i:=1,i<10,i++,Echo(a*i))
    10
    20
    30
    40
    50
    60
    70
    80
    90
    Out> True;
    In> i
    Out> 10;

##### See also:

RuleBase , Backquoting , DefMacroRuleBaseListed .

###

DefMacroRuleBaseListed -- define macro with variable number of arguments

##### Internal function

##### Calling format:

    DefMacroRuleBaseListed("name", params)

##### Parameters:

`"name"` -- string, name of function

`params` -- list of arguments to function

##### Description:

This does the same as `DefMacroRuleBase` (define a macro), but with a variable number of arguments, similar to `RuleBaseListed`.

##### See also:

RuleBase , RuleBaseListed , Backquoting , DefMacroRuleBase .

###

SetExtraInfo, GetExtraInfo -- annotate objects with additional information

##### Internal function

##### Calling format:

    SetExtraInfo(expr,tag)
    GetExtraInfo(expr)

##### Parameters:

`expr` -- any expression

`tag` -- tag information (any other expression)

##### Description:

Sometimes it is useful to be able to add extra tag information to "annotate" objects or to label them as having certain "properties". The functions `SetExtraInfo` and `GetExtraInfo` enable this.

The function `SetExtraInfo` returns the tagged expression, leaving the original expression alone. This means there is a common pitfall: be sure to assign the returned value to a variable, or the tagged expression is lost when the temporary object is destroyed.

The original expression is left unmodified, and the tagged expression returned, in order to keep the atomic objects small. To tag an object, a new type of object is created from the old object, with one added property (the tag). The tag can be any expression whatsoever.

The function `GetExtraInfo(x)` retrieves this tag expression from an object `x`. If an object has no tag, it looks the same as if it had a tag with value `False`.

No part of the Yacas core uses tags in a way that is visible to the outside world, so for specific purposes a programmer can devise a format to use for tag information. Association lists (hashes) are a natural fit for this, although it is not required and a tag can be any object (except the atom `False` because it is indistinguishable from having no tag information). Using association lists is highly advised since it is most likely to be the format used by other parts of the library, and one needs to avoid clashes with other library code. Typically, an object will either have no tag or a tag which is an associative list (perhaps empty). A script that uses tagged objects will check whether an object has a tag and if so, will add or modify certain entries of the association list, preserving any other tag information.

Note that `FlatCopy` currently does not copy the tag information (see examples).

##### Examples:

    In> a:=2*b
    Out> 2*b;
    In> a:=SetExtraInfo(a,{{"type","integer"}})
    Out> 2*b;
    In> a
    Out> 2*b;
    In> GetExtraInfo(a)
    Out> {{"type","integer"}};
    In> GetExtraInfo(a)["type"]
    Out> "integer";
    In> c:=a
    Out> 2*b;
    In> GetExtraInfo(c)
    Out> {{"type","integer"}};
    In> c
    Out> 2*b;
    In> d:=FlatCopy(a);
    Out> 2*b;
    In> GetExtraInfo(d)
    Out> False;

##### See also:

Assoc , := .

###

GarbageCollect -- do garbage collection on unused memory

##### Internal function

##### Calling format:

    GarbageCollect()

##### Description:

`GarbageCollect` garbage-collects unused memory. The Yacas system uses a reference counting system for most objects, so this call is usually not necessary.

Reference counting refers to bookkeeping where in each object a counter is held, keeping track of the number of parts in the system using that object. When this count drops to zero, the object is automatically removed. Reference counting is not the fastest way of doing garbage collection, but it can be implemented in a very clean way with very little code.

Among the most important objects that are not reference counted are the strings. `GarbageCollect` collects these and disposes of them when they are not used any more.

`GarbageCollect` is useful when doing a lot of text processing, to clean up the text buffers. It is not highly needed, but it keeps memory use low.

###

FindFunction -- find the library file where a function is defined

##### Internal function

##### Calling format:

    FindFunction(function)

##### Parameters:

`function` -- string, the name of a function

##### Description:

This function is useful for quickly finding the file where a standard library function is defined. It is likely to only be useful for developers. The function `FindFunction` scans the `.def` files that were loaded at start-up. This means that functions that are not listed in `.def` files will not be found with `FindFunction`.

##### Examples:

    In> FindFunction("Sum")
    Out> "sums.rep/code.ys";
    In> FindFunction("Integrate")
    Out> "integrate.rep/code.ys";

##### See also:

Vi .

###

Secure -- guard the host OS

##### Internal function

##### Calling format:

    Secure(body)

##### Parameters:

`body` -- expression

##### Description:

`Secure` evaluates `body` in a "safe" environment, where files cannot be opened and system calls are not allowed. This can help protect the system when e.g. a script is sent over the Internet to be evaluated on a remote computer, which is potentially unsafe.

##### See also:

SystemCall .

MultiplyNum , CachedConstant , NewtonNum , SumTaylorNum , IntPowerNum , BinSplitNum, BinSplitData, BinSplitFinal , MathSetExactBits, MathGetExactBits .

# Arbitrary-precision numerical programming

This chapter contains functions that help programming numerical calculations with arbitrary precision.

###

MultiplyNum -- optimized numerical multiplication

##### Standard library

##### Calling format:

    MultiplyNum(x,y)
    MultiplyNum(x,y,z,...)
    MultiplyNum({x,y,z,...})

##### Parameters:

`x`, `y`, `z` -- integer, rational or floating-point numbers to multiply

##### Description:

The function `MultiplyNum` is used to speed up multiplication of floating-point numbers with rational numbers. Suppose we need to compute p/q*x where p, q are integers and x is a floating-point number. At high precision, it is faster to multiply x by an integer p and divide by an integer q than to compute p/q to high precision and then multiply by x. The function `MultiplyNum` performs this optimization.

The function accepts any number of arguments (not less than two) or a list of numbers. The result is always a floating-point number (even if `Numeric` is not set).

##### See also:

MathMultiply .

###

CachedConstant -- precompute multiple-precision constants

##### Standard library

##### Calling format:

    CachedConstant(cache, Cname, Cfunc)

##### Parameters:

`cache` -- atom, name of the cache

`Cname` -- atom, name of the constant

`Cfunc` -- expression that evaluates the constant

##### Description:

This function is used to create precomputed multiple-precision values of constants. Caching these values will save time if they are frequently used.

The call to `CachedConstant` defines a new function named `Cname()` that returns the value of the constant at given precision. If the precision is increased, the value will be recalculated as necessary, otherwise calling `Cname()` will take very little time.

The parameter `Cfunc` must be an expression that can be evaluated and returns the value of the desired constant at the current precision. (Most arbitrary-precision mathematical functions do this by default.)

The associative list `cache` contains elements of the form `{Cname, prec, value}`, as illustrated in the example. If this list does not exist, it will be created.

This mechanism is currently used by `N()` to precompute the values of Pi and gamma. The name of the cache for `N()` is `CacheOfConstantsN`. The code in the function `N()` assigns unevaluated calls to `Pi()` and `gamma()` to the atoms `Pi` and `gamma` and declares them `LazyGlobal`. The result is that the constants will be recalculated only when they are used in the expression under `N()`. In other words, the code in `N()` does the equivalent of

    Pi := Hold(Pi());
    gamma := Hold(gamma());
    LazyGlobal(Pi);
    LazyGlobal(gamma);

After this, evaluating an expression such as `1/2+gamma` will call the function `gamma()` but not the function `Pi()`.

##### Example:

    In> CachedConstant( my'cache, Ln2, LnNum(2) )
    Out> 0.6931471806;
    In> Ln2
    Out> Ln2;
    In> Ln2()
    Out> 0.6931471806;
    In> [ Precision(20); V( Ln2() ); ]
    CachedConstant: Info: constant Ln2 is being
      recalculated at precision 20
    Out> 0.69314718055994530942;
    In> my'cache
    Out> {{"Ln2",20,0.69314718055994530942}};

##### See also:

N , Pi() , Precision , gamma .

###

NewtonNum -- low-level optimized Newton's iterations

##### Standard library

##### Calling format:

    NewtonNum(func, x0, prec0, order)
    NewtonNum(func, x0, prec0)
    NewtonNum(func, x0)

##### Parameters:

`func` -- a function specifying the iteration sequence

`x0` -- initial value (must be close enough to the root)

`prec0` -- initial precision (at least 4, default 5)

`order` -- convergence order (typically 2 or 3, default 2)

##### Description:

This function is an optimized interface for computing Newton's iteration sequences for numerical solution of equations in arbitrary precision.

`NewtonNum` will iterate the given function starting from the initial value, until the sequence converges within current precision. Initially, up to 5 iterations at the initial precision `prec0` is performed (the low precision is set for speed). The initial value `x0` must be close enough to the root so that the initial iterations converge. If the sequence does not produce even a single correct digit of the root after these initial iterations, an error message is printed. The default value of the initial precision is 5.

The `order` parameter should give the convergence order of the scheme. Normally, Newton iteration converges quadratically (so the default value is `order`=2) but some schemes converge faster and you can speed up this function by specifying the correct order. (Caution: if you give `order`=3 but the sequence is actually quadratic, the result will be silently incorrect. It is safe to use `order`=2.)

##### Example:

    In> Precision(20)
    Out> True;
    In> NewtonNum({{x}, x+Sin(x)}, 3, 5, 3)
    Out> 3.14159265358979323846;

##### See also:

Newton .

###

SumTaylorNum -- optimized numerical evaluation of Taylor series

##### Standard library

##### Calling format:

    SumTaylorNum(x, NthTerm, order)
    SumTaylorNum(x, NthTerm, TermFactor, order)
    SumTaylorNum(x, ZerothTerm, TermFactor, order)

##### Parameters:

`NthTerm` -- a function specifying n-th coefficient of the series

`ZerothTerm` -- value of the 0-th coefficient of the series

`x` -- number, value of the expansion variable

`TermFactor` -- a function specifying the ratio of n-th term to the previous one

`order` -- power of x in the last term

##### Description:

`SumTaylorNum` computes a Taylor series Sum(k,0,n,a[k]*x^k) numerically. This function allows very efficient computations of functions given by Taylor series, although some tweaking of the parameters is required for good results.

The coefficients a[k] of the Taylor series are given as functions of one integer variable (k). It is convenient to pass them to `SumTaylorNum` as closures. For example, if a function `a(k)` is defined, then

    SumTaylorNum(x, {{k}, a(k)}, n)

computes the series Sum(k,0,n,a(k)*x^k).

Often a simple relation between successive coefficients a[k-1], a[k] of the series is available; usually they are related by a rational factor. In this case, the second form of `SumTaylorNum` should be used because it will compute the series faster. The function `TermFactor` applied to an integer k>=1 must return the ratio a[k]/a[k-1]. (If possible, the function `TermFactor` should return a rational number and not a floating-point number.) The function `NthTerm` may also be given, but the current implementation only calls `NthTerm(0)` and obtains all other coefficients by using `TermFactor`. Instead of the function `NthTerm`, a number giving the 0-th term can be given.

The algorithm is described elsewhere in the documentation. The number of terms `order`+1 must be specified and a sufficiently high precision must be preset in advance to achieve the desired accuracy. (The function `SumTaylorNum` does not change the current precision.)

##### Examples:

To compute 20 digits of Exp(1) using the Taylor series, one needs 21 digits of working precision and 21 terms of the series.

    In> Precision(21)
    Out> True;
    In> SumTaylorNum(1, {{k},1/k!}, 21)
    Out> 2.718281828459045235351;
    In> SumTaylorNum(1, 1, {{k},1/k}, 21)
    Out> 2.71828182845904523535;
    In> SumTaylorNum(1, {{k},1/k!}, {{k},1/k}, 21)
    Out> 2.71828182845904523535;
    In> RoundTo(N(Ln(%)),20)
    Out> 1;

##### See also:

Taylor .

###

IntPowerNum -- optimized computation of integer powers

##### Standard library

##### Calling format:

    IntPowerNum(x, n, mult, unity)

##### Parameters:

`x` -- a number or an expression

`n` -- a non-negative integer (power to raise `x` to)

`mult` -- a function that performs one multiplication

`unity` -- value of the unity with respect to that multiplication

##### Description:

`IntPowerNum` computes the power x^n using the fast binary algorithm. It can compute integer powers with n>=0 in any ring where multiplication with unity is defined. The multiplication function and the unity element must be specified. The number of multiplications is no more than 2*Ln(n)/Ln(2).

Mathematically, this function is a generalization of `MathPower` to rings other than that of real numbers.

In the current implementation, the `unity` argument is only used when the given power `n` is zero.

##### Examples:

For efficient numerical calculations, the `MathMultiply` function can be passed:

    In> IntPowerNum(3, 3, MathMultiply,1)
    Out> 27;

Otherwise, the usual `*` operator suffices:

    In> IntPowerNum(3+4*I, 3, *,1)
    Out> Complex(-117,44);
    In> IntPowerNum(HilbertMatrix(2), 4, *,
      Identity(2))
    Out> {{289/144,29/27},{29/27,745/1296}};

Compute Mod(3^100,7):

    In> IntPowerNum(3,100,{{x,y},Mod(x*y,7)},1)
    Out> 4;

##### See also:

MultiplyNum , MathPower , MatrixPower .

###

BinSplitNum, BinSplitData, BinSplitFinal -- computations of series by the binary splitting method

##### Standard library

##### Calling format:

    BinSplitNum(n1, n2, a, b, c, d)
    BinSplitData(n1,n2, a, b, c, d)
    BinSplitFinal({P,Q,B,T})

##### Parameters:

`n1`, `n2` -- integers, initial and final indices for summation

`a`, `b`, `c`, `d` -- functions of one argument, coefficients of the series

`P`, `Q`, `B`, `T` -- numbers, intermediate data as returned by `BinSplitData`

##### Description:

The binary splitting method is an efficient way to evaluate many series when fast multiplication is available and when the series contains only rational numbers. The function `BinSplitNum` evaluates a series of the form

S(n[1],n[2])=Sum(k,n[1],n[2],a(k)/b(k)*p(0)/q(0)*...*p(k)/q(k)).
Most series for elementary and special functions at rational points are of this form when the functions a(k), b(k), p(k), q(k) are chosen appropriately.

The last four arguments of `BinSplitNum` are functions of one argument that give the coefficients a(k), b(k), p(k), q(k). In most cases these will be short integers that are simple to determine. The binary splitting method will work also for non-integer coefficients, but the calculation will take much longer in that case.

Note: the binary splitting method outperforms the straightforward summation only if the multiplication of integers is faster than quadratic in the number of digits. See the algorithm documentation for more information.

The two other functions are low-level functions that allow a finer control over the calculation. The use of the low-level routines allows checkpointing or parallelization of a binary splitting calculation.

The binary splitting method recursively reduces the calculation of S(n[1],n[2]) to the same calculation for the two halves of the interval [n[1], n[2]]. The intermediate results of a binary splitting calculation are returned by `BinSplitData` and consist of four integers P, Q, B, T. These four integers are converted into the final answer S by the routine `BinSplitFinal` using the relation

S=T/(B*Q).

##### Examples:

Compute the series for e=Exp(1) using binary splitting. (We start from n=1 to simplify the coefficient functions.)

    In> Precision(21)
    Out> True;
    In>  BinSplitNum(1,21, {{k},1},
      {{k},1},{{k},1},{{k},k})
    Out> 1.718281828459045235359;
    In> N(Exp(1)-1)
    Out> 1.71828182845904523536;
    In>  BinSplitData(1,21, {{k},1},
      {{k},1},{{k},1},{{k},k})
    Out> {1,51090942171709440000,1,
      87788637532500240022};
    In> BinSplitFinal(%)
    Out> 1.718281828459045235359;

##### See also:

SumTaylorNum .

###

MathSetExactBits, MathGetExactBits -- manipulate precision of floating-point numbers

##### Internal function

##### Calling format:

    MathGetExactBits(x)
    MathSetExactBits(x,bits)

##### Parameters:

`x` -- an expression evaluating to a floating-point number

`bits` -- integer, number of bits

##### Description:

Each floating-point number in Yacas has an internal precision counter that stores the number of exact bits in the mantissa. The number of exact bits is automatically updated after each arithmetic operation to reflect the gain or loss of precision due to round-off. The functions `MathGetExactBits`, `MathSetExactBits` allow to query or set the precision flags of individual number objects.

`MathGetExactBits(x)` returns an integer number n such that `x` represents a real number in the interval [ x*(1-2^(-n)), x*(1+2^(-n))] if x!=0 and in the interval [-2^(-n), 2^(-n)] if x=0. The integer n is always nonnegative unless `x` is zero (a "floating zero"). A floating zero can have a negative value of the number n of exact bits.

These functions are only meaningful for floating-point numbers. (All integers are always exact.) For integer `x`, the function `MathGetExactBits` returns the bit count of `x` and the function `MathSetExactBits` returns the unmodified integer `x`.

##### Examples:

The default precision of 10 decimals corresponds to 33 bits:

    In> MathGetExactBits(1000.123)
    Out> 33;
    In> x:=MathSetExactBits(10., 20)
    Out> 10.;
    In> MathGetExactBits(x)
    Out> 20;

Prepare a "floating zero" representing an interval [-4, 4]:

    In> x:=MathSetExactBits(0., -2)
    Out> 0.;
    In> x=0
    Out> True;

##### See also:

Precision , GetPrecision .

Check , TrapError , GetCoreError , Assert , DumpErrors, ClearErrors , IsError , GetError, ClearError , CurrentFile, CurrentLine .

# Error reporting

This chapter contains commands useful for reporting errors to the user.

###

Check -- report "hard" errors

###

TrapError -- trap "hard" errors

###

GetCoreError -- get "hard" error string

##### Internal function

##### Calling format:

    Check(predicate,"error text")
    TrapError(expression,errorHandler)
    GetCoreError()

##### Parameters:

`predicate` -- expression returning `True` or `False`

`"error text"` -- string to print on error

`expression` -- expression to evaluate (causing potential error)

`errorHandler` -- expression to be called to handle error

##### Description:

If `predicate` does not evaluate to `True`, the current operation will be stopped, the string `"error text"` will be printed, and control will be returned immediately to the command line. This facility can be used to assure that some condition is satisfied during evaluation of expressions (guarding against critical internal errors).

A "soft" error reporting facility that does not stop the execution is provided by the function `Assert`.

##### Example:

    In> [Check(1=0,"bad value"); Echo(OK);]
    In function "Check" :
    CommandLine(1) : "bad value"

Note that `OK` is not printed.

TrapError evaluates its argument `expression`, returning the result of evaluating `expression`. If an error occurs, `errorHandler` is evaluated, returning its return value in stead.

GetCoreError returns a string describing the core error. TrapError and GetCoreError can be used in combination to write a custom error handler.

##### See also:

Assert .

###

Assert -- signal "soft" custom error

##### Standard library

##### Calling format:

    Assert("str", expr) pred
    Assert("str") pred
    Assert() pred

Precedence: 60000

##### Parameters:

`pred` -- predicate to check

`"str"` -- string to classify the error

`expr` -- expression, error object

##### Description:

`Assert` is a global error reporting mechanism. It can be used to check for errors and report them. An error is considered to occur when the predicate `pred` evaluates to anything except `True`. In this case, the function returns `False` and an error object is created and posted to the global error tableau. Otherwise the function returns `True`.

Unlike the "hard" error function `Check`, the function `Assert` does not stop the execution of the program.

The error object consists of the string `"str"` and an arbitrary expression `expr`. The string should be used to classify the kind of error that has occurred, for example "domain" or "format". The error object can be any expression that might be useful for handling the error later; for example, a list of erroneous values and explanations. The association list of error objects is currently the global variable `ErrorTableau`.

If the parameter `expr` is missing, `Assert` substitutes `True`. If both optional parameters `"str"` and `expr` are missing, `Assert` creates an error of class `"generic"`.

Errors can be handled by a custom error handler in the portion of the code that is able to handle a certain class of errors. The functions `IsError`, `GetError` and `ClearError` can be used.

Normally, all errors posted to the error tableau during evaluation of an expression should be eventually printed to the screen. This is the behavior of prettyprinters `DefaultPrint`, `Print`, `PrettyForm` and `TeXForm` (but not of the inline prettyprinter, which is enabled by default); they call `DumpErrors` after evaluating the expression.

##### Examples:

    In> Assert("bad value", "must be zero") 1=0
    Out> False;
    In> Assert("bad value", "must be one") 1=1
    Out> True;
    In> IsError()
    Out> True;
    In> IsError("bad value")
    Out> True;
    In> IsError("bad file")
    Out> False;
    In> GetError("bad value");
    Out> "must be zero";
    In> DumpErrors()
    Error: bad value: must be zero
    Out> True;

No more errors left:

    In> IsError()
    Out> False;
    In> DumpErrors()
    Out> True;

##### See also:

IsError , DumpErrors , Check , GetError , ClearError , ClearErrors .

###

DumpErrors, ClearErrors -- simple error handlers

##### Standard library

##### Calling format:

    DumpErrors()
    ClearErrors()

##### Description:

`DumpErrors` is a simple error handler for the global error reporting mechanism. It prints all errors posted using `Assert` and clears the error tableau.

`ClearErrors` is a trivial error handler that does nothing except it clears the tableau.

##### See also:

Assert , IsError .

###

IsError -- check for custom error

##### Standard library

##### Calling format:

    IsError()
    IsError("str")

##### Parameters:

`"str"` -- string to classify the error

##### Description:

`IsError()` returns `True` if any custom errors have been reported using `Assert`. The second form takes a parameter `"str"` that designates the class of the error we are interested in. It returns `True` if any errors of the given class `"str"` have been reported.

##### See also:

GetError , ClearError , Assert , Check .

###

GetError, ClearError -- custom errors handlers

##### Standard library

##### Calling format:

    GetError("str")
    ClearError("str")

##### Parameters:

`"str"` -- string to classify the error

##### Description:

These functions can be used to create a custom error handler.

`GetError` returns the error object if a custom error of class `"str"` has been reported using `Assert`, or `False` if no errors of this class have been reported.

`ClearError("str")` deletes the same error object that is returned by `GetError("str")`. It deletes at most one error object. It returns `True` if an object was found and deleted, and `False` otherwise.

##### Examples:

    In> x:=1
    Out> 1;
    In> Assert("bad value", {x,"must be zero"}) x=0
    Out> False;
    In> GetError("bad value")
    Out> {1, "must be zero"};
    In> ClearError("bad value");
    Out> True;
    In> IsError()
    Out> False;

##### See also:

IsError , Assert , Check , ClearErrors .

###

CurrentFile, CurrentLine -- show current file and line of input

##### Internal function

##### Calling format:

    CurrentFile()
    CurrentLine()

##### Description:

The functions `CurrentFile` and `CurrentLine` return a string with the file name of the current file and the current line of input respectively.

These functions are most useful in batch file calculations, where there is a need to determine at which line an error occurred. One can define a function

    tst() := Echo({CurrentFile(),CurrentLine()});

which can then be inserted into the input file at various places, to see how far the interpreter reaches before an error occurs.

##### See also:

Echo .

MathNot , MathAnd , MathOr , BitAnd, BitOr, BitXor , Equals , GreaterThan, LessThan , Math... , Fast... , ShiftLeft, ShiftRight , IsPromptShown , MathLibrary , GetTime .

# Built-in (core) functions

Yacas comes with a small core of built-in functions and a large library of user-defined functions. Some of these core functions are documented in this chapter.

It is important for a developer to know which functions are built-in and cannot be redefined or `Retract`-ed. Also, core functions may be somewhat faster to execute than functions defined in the script library. All core functions are listed in the file `corefunctions.h` in the `src/` subdirectory of the Yacas source tree. The declarations typically look like this:

    SetCommand(LispSubtract, "MathSubtract");

Here `LispSubtract` is the Yacas internal name for the function and `MathSubtract` is the name visible to the Yacas language. Built-in bodied functions and infix operators are declared in the same file.

###

MathNot -- built-in logical "not"

##### Internal function

##### Calling format:

    MathNot(expression)

##### Description:

Returns "False" if "expression" evaluates to "True", and vice versa.

###

MathAnd -- built-in logical "and"

##### Calling format:

    MathAnd(...)

##### Description:

Lazy logical `And`: returns `True` if all args evaluate to `True`, and does this by looking at first, and then at the second argument, until one is `False`. If one of the arguments is `False`, `And` immediately returns `False` without evaluating the rest. This is faster, but also means that none of the arguments should cause side effects when they are evaluated.

###

MathOr -- built-in logical "or"

##### Internal function

##### Calling format:

    MathOr(...)

`MathOr` is the basic logical "or" function. Similarly to `And`, it is lazy-evaluated. `And(...)` and `Or(...)` do also exist, defined in the script library. You can redefine them as infix operators yourself, so you have the choice of precedence. In the standard scripts they are in fact declared as infix operators, so you can write `expr1 And expr`.

###

BitAnd, BitOr, BitXor -- bitwise arithmetic

##### Internal function

##### Calling format:

    BitAnd(n,m)
    BitOr(n,m)
    BitXor(n,m)

##### Description:

These functions return bitwise "and", "or" and "xor" of two numbers.

###

Equals -- check equality

##### Internal function

##### Calling format:

    Equals(a,b)

##### Description:

Compares evaluated `a` and `b` recursively (stepping into expressions). So "Equals(a,b)" returns "True" if the expressions would be printed exactly the same, and "False" otherwise.

###

GreaterThan, LessThan -- comparison predicates

##### Internal function

##### Calling format:

    GreaterThan(a,b)
    LessThan(a,b)

##### Parameters:

`a`, `b` -- numbers or strings

##### Description:

Comparing numbers or strings (lexicographically).

##### Example:

    In> LessThan(1,1)
    Out> False;
    In> LessThan("a","b")
    Out> True;

###

Math... -- arbitrary-precision math functions

##### Internal function

##### Calling format:

    MathGcd(n,m)      (Greatest Common Divisor)
    MathAdd(x,y)      (add two numbers)
    MathSubtract(x,y) (subtract two numbers)
    MathMultiply(x,y) (multiply two numbers)
    MathDivide(x,y)   (divide two numbers)
    MathSqrt(x)    (square root, must be x>=0)
    MathFloor(x)   (largest integer not larger than x)
    MathCeil(x)    (smallest integer not smaller than x)
    MathAbs(x)     (absolute value of x, or |x| )
    MathExp(x)     (exponential, base 2.718...)
    MathLog(x)     (natural logarithm, for x>0)
    MathPower(x,y) (power, x ^ y)
    MathSin(x)     (sine)
    MathCos(x)     (cosine)
    MathTan(x)     (tangent)
    MathSinh(x)     (hyperbolic sine)
    MathCosh(x)     (hyperbolic cosine)
    MathTanh(x)     (hyperbolic tangent)
    MathArcSin(x)   (inverse sine)
    MathArcCos(x)   (inverse cosine)
    MathArcTan(x)   (inverse tangent)
    MathArcSinh(x)  (inverse hyperbolic sine)
    MathArcCosh(x)  (inverse hyperbolic cosine)
    MathArcTanh(x)  (inverse hyperbolic tangent)
    MathDiv(x,y)    (integer division, result is an integer)
    MathMod(x,y)    (remainder of division, or x mod y)

##### Description:

These commands perform the calculation of elementary mathematical functions. The arguments must be numbers. The reason for the prefix `Math` is that the library needs to define equivalent non-numerical functions for symbolic computations, such as `Exp`, `Sin` and so on.

Note that all functions, such as the `MathPower`, `MathSqrt`, `MathAdd` etc., accept integers as well as floating-point numbers. The resulting values may be integers or floats. If the mathematical result is an exact integer, then the integer is returned. For example, `MathSqrt(25)` returns the integer `5`, and `MathPower(2,3)` returns the integer `8`. In such cases, the integer result is returned even if the calculation requires more digits than set by `Precision`. However, when the result is mathematically not an integer, the functions return a floating-point result which is correct only to the current precision.

##### Example:

    In> Precision(10)
    Out> True
    In> Sqrt(10)
    Out> Sqrt(10)
    In> MathSqrt(10)
    Out> 3.16227766
    In> MathSqrt(490000*2^150)
    Out> 26445252304070013196697600
    In> MathSqrt(490000*2^150+1)
    Out> 0.264452523e26
    In> MathPower(2,3)
    Out> 8
    In> MathPower(2,-3)
    Out> 0.125

###

Fast... -- double-precision math functions

##### Internal function

##### Calling format:

FastExp(x), FastLog(x) (natural logarithm), FastPower(x,y), FastSin(x), FastCos(x), FastTan(x), FastArcSin(x), FastArcCos(x), FastArcTan(x)

##### Description:

Versions of these functions using the C++ library. These should then at least be faster than the arbitrary precision versions.

###

ShiftLeft, ShiftRight -- built-in bit shifts

##### Internal function

##### Calling format:

    ShiftLeft(expr,bits)
    ShiftRight(expr,bits)

##### Description:

Shift bits to the left or to the right.

###

IsPromptShown -- test for the Yacas prompt option

##### Internal function

##### Calling format:

    IsPromptShown()

##### Description:

Returns `False` if Yacas has been started with the option to suppress the prompt, and `True` otherwise.

###

MathLibrary -- obtain current math library name

##### Internal function

##### Calling format:

    MathLibrary()

##### Description:

Returns a string that describes the currently used arbitrary-precision arithmetic library.

Possible names supported at the moment are `"Internal"` and `"Gmp"`, indicating the internal math library `libyacasnumbers` and the GNU Multiple Precision library `libgmp`.

##### Example:

    In>  MathLibrary()
    Out> "Internal";

###

GetTime -- measure the time taken by an evaluation

##### Internal function

##### Calling format:

    GetTime(expr)

##### Parameters:

`expr` -- any expression

##### Description:

The function `GetTime(expr)` evaluates the expression `expr` and returns the time needed for the evaluation. The result is returned as a floating-point number of seconds. The value of the expression `expr` is lost.

The result is the "user time" as reported by the OS, not the real ("wall clock") time. Therefore, any CPU-intensive processes running alongside Yacas will not significantly affect the result of `GetTime`.

##### Example:

    In> GetTime(Simplify((a*b)/(b*a)))
    Out> 0.09;

##### See also:

Time .

##

Full listing of core functions

The following Yacas functions are currently declared in `corefunctions.h` as core functions. The list indicates whether a function is a real function (evaluating its arguments) or a macro (not evaluating its arguments). Also a function can either take a fixed number of arguments or a variable number (say, 2 or more). Some kernel functions are additionally declared as operators (bodied, prefix, infix, postfix) with the given precedence.

- `=` -- function, 2 arguments
- `@` -- prefix operator, prec. `0`
- `And` -- macro, 1 or more arguments
- `ApplyPure` -- function, 2 arguments
- `ArrayCreate` -- function, 2 arguments
- `ArrayGet` -- function, 2 arguments
- `ArraySet` -- function, 3 arguments
- `ArraySize` -- function, 1 argument
- `Atom` -- function, 1 argument
- `BackQuote` -- bodied operator, prec. `KMaxPrecedence`
- `BitAnd` -- function, 2 arguments
- `BitOr` -- function, 2 arguments
- `BitXor` -- function, 2 arguments
- `BitsToDigits` -- function, 2 arguments
- `Bodied` -- function, 2 arguments
- `CTokenizer` -- function, 0 arguments
- `CharString` -- function, 1 argument
- `Check` -- macro, 2 arguments
- `Clear` -- macro, 1 or more arguments
- `CommonLispTokenizer` -- function, 0 arguments
- `ConcatStrings` -- function, 1 or more arguments
- `Concat` -- function, 1 or more arguments
- `CurrentFile` -- function, 0 arguments
- `CurrentLine` -- function, 0 arguments
- `CustomEval'Expression` -- function, 0 arguments
- `CustomEval'Locals` -- function, 0 arguments
- `CustomEval'Result` -- function, 0 arguments
- `CustomEval'Stop` -- function, 0 arguments
- `CustomEval` -- macro, 4 arguments
- `DebugFile` -- function, 1 argument
- `DebugLine` -- function, 1 argument
- `DefLoadFunction` -- function, 1 argument
- `DefLoad` -- function, 1 argument
- `DefMacroRuleBaseListed` -- macro, 2 arguments
- `DefMacroRuleBase` -- macro, 2 arguments
- `DefaultDirectory` -- function, 1 argument
- `DefaultTokenizer` -- function, 0 arguments
- `Delete` -- function, 2 arguments
- `DestructiveDelete` -- function, 2 arguments
- `DestructiveInsert` -- function, 3 arguments
- `DestructiveReplace` -- function, 3 arguments
- `DestructiveReverse` -- function, 1 argument
- `DigitsToBits` -- function, 2 arguments
- `DllDirectory` -- function, 1 argument
- `DllEnumerate` -- function, 0 arguments
- `DllLoad` -- function, 1 argument
- `DllUnload` -- function, 1 argument
- `Equals` -- function, 2 arguments
- `Eval` -- function, 1 argument
- `Exit` -- function, 0 arguments
- `FastAbs` -- function, 1 argument
- `FastArcCos` -- function, 1 argument
- `FastArcSin` -- function, 1 argument
- `FastArcTan` -- function, 1 argument
- `FastAssoc` -- function, 2 arguments
- `FastCeil` -- function, 1 argument
- `FastCos` -- function, 1 argument
- `FastExp` -- function, 1 argument
- `FastFloor` -- function, 1 argument
- `FastIsPrime` -- function, 1 argument
- `FastLog` -- function, 1 argument
- `FastMod` -- function, 2 arguments
- `FastPi` -- function, 0 arguments
- `FastPower` -- function, 2 arguments
- `FastSin` -- function, 1 argument
- `FastSqrt` -- function, 1 argument
- `FastTan` -- function, 1 argument
- `FileSize` -- function, 1 argument
- `FindFile` -- function, 1 argument
- `FindFunction` -- function, 1 argument
- `FlatCopy` -- function, 1 argument
- `FromBase` -- function, 2 arguments
- `FromFile` -- bodied operator, prec. `KMaxPrecedence`
- `FromFile` -- macro, 2 arguments
- `FromString` -- bodied operator, prec. `KMaxPrecedence`
- `FromString` -- macro, 2 arguments
- `FullForm` -- function, 1 argument
- `GarbageCollect` -- function, 0 arguments
- `GenericTypeName` -- function, 1 argument
- `GetCoreError` -- function, 0 arguments
- `GetExtraInfo` -- function, 1 argument
- `GetPrecision` -- function, 0 arguments
- `GetPrettyPrinter` -- function, 0 arguments
- `GetPrettyReader` -- function, 0 arguments
- `GetTime` -- macro, 1 argument
- `GreaterThan` -- function, 2 arguments
- `Head` -- function, 1 argument
- `HistorySize` -- function, 1 argument
- `HoldArg` -- macro, 2 arguments
- `Hold` -- macro, 1 argument
- `If` -- macro, 2 or more arguments
- `InDebugMode` -- function, 0 arguments
- `Infix` -- function, 2 arguments
- `Insert` -- function, 3 arguments
- `IsAtom` -- function, 1 argument
- `IsBodied` -- function, 1 argument
- `IsBound` -- macro, 1 argument
- `IsExitRequested` -- function, 0 arguments
- `IsFunction` -- function, 1 argument
- `IsGeneric` -- function, 1 argument
- `IsInfix` -- function, 1 argument
- `IsInteger` -- function, 1 argument
- `IsList` -- function, 1 argument
- `IsNumber` -- function, 1 argument
- `IsPostfix` -- function, 1 argument
- `IsPrefix` -- function, 1 argument
- `IsPromptShown` -- function, 0 arguments
- `IsString` -- function, 1 argument
- `LazyGlobal` -- macro, 1 argument
- `LeftPrecedence` -- function, 2 arguments
- `Length` -- function, 1 argument
- `LessThan` -- function, 2 arguments
- `LispReadListed` -- function, 0 arguments
- `LispRead` -- function, 0 arguments
- `Listify` -- function, 1 argument
- `List` -- macro, 1 or more arguments
- `Load` -- function, 1 argument
- `LocalSymbols` -- bodied operator, prec. `KMaxPrecedence`
- `LocalSymbols` -- macro, 1 or more arguments
- `Local` -- macro, 1 or more arguments
- `MacroClear` -- function, 1 or more arguments
- `MacroLocal` -- function, 1 or more arguments
- `MacroRuleBaseListed` -- function, 2 arguments
- `MacroRuleBase` -- function, 2 arguments
- `MacroRulePattern` -- bodied operator, prec. `KMaxPrecedence`
- `MacroRulePattern` -- function, 5 arguments
- `MacroRule` -- bodied operator, prec. `KMaxPrecedence`
- `MacroRule` -- function, 5 arguments
- `MacroSet` -- macro, 2 arguments
- `MathAbs` -- function, 1 argument
- `MathAdd` -- function, 2 arguments
- `MathAnd` -- macro, 1 or more arguments
- `MathArcCos` -- function, 1 argument
- `MathArcTan` -- function, 1 argument
- `MathBitCount` -- function, 1 argument
- `MathCeil` -- function, 1 argument
- `MathDebugInfo` -- function, 1 argument
- `MathDivide` -- function, 2 arguments
- `MathDiv` -- function, 2 arguments
- `MathFac` -- function, 1 argument
- `MathFloor` -- function, 1 argument
- `MathGcd` -- function, 2 arguments
- `MathGetExactBits` -- function, 1 argument
- `MathIsSmall` -- function, 1 argument
- `MathLibrary` -- function, 0 arguments
- `MathMod` -- function, 2 arguments
- `MathMultiply` -- function, 2 arguments
- `MathNegate` -- function, 1 argument
- `MathNot` -- function, 1 argument
- `MathNth` -- function, 2 arguments
- `MathOr` -- macro, 1 or more arguments
- `MathSetExactBits` -- function, 2 arguments
- `MathSign` -- function, 1 argument
- `MathSubtract` -- function, 2 arguments
- `MaxEvalDepth` -- function, 1 argument
- `Not` -- function, 1 argument
- `OSVersion` -- function, 0 arguments
- `OpLeftPrecedence` -- function, 1 argument
- `OpPrecedence` -- function, 1 argument
- `OpRightPrecedence` -- function, 1 argument
- `Or` -- macro, 1 or more arguments
- `PatchLoad` -- function, 1 argument
- `PatchString` -- function, 1 argument
- `PatternCreate` -- function, 2 arguments
- `PatternMatches` -- function, 2 arguments
- `PluginsCanBeLoaded` -- function, 0 arguments
- `Postfix` -- function, 2 arguments
- `Precision` -- function, 1 argument
- `Prefix` -- function, 2 arguments
- `PrettyPrinter` -- function, 1 or more arguments
- `PrettyReader` -- function, 1 or more arguments
- `Prog` -- macro, 1 or more arguments
- `ReadCmdLineString` -- function, 1 argument
- `ReadToken` -- function, 0 arguments
- `Read` -- function, 0 arguments
- `Replace` -- function, 3 arguments
- `Retract` -- function, 2 arguments
- `RightAssociative` -- function, 1 argument
- `RightPrecedence` -- function, 2 arguments
- `RuleBaseArgList` -- function, 2 arguments
- `RuleBaseDefined` -- function, 2 arguments
- `RuleBaseListed` -- macro, 2 arguments
- `RuleBase` -- macro, 2 arguments
- `RulePattern` -- bodied operator, prec. `KMaxPrecedence`
- `RulePattern` -- macro, 5 arguments
- `Rule` -- bodied operator, prec. `KMaxPrecedence`
- `Rule` -- macro, 5 arguments
- `Secure` -- macro, 1 argument
- `SetExtraInfo` -- function, 2 arguments
- `SetStringMid` -- function, 3 arguments
- `Set` -- macro, 2 arguments
- `ShiftLeft` -- function, 2 arguments
- `ShiftRight` -- function, 2 arguments
- `StaSiz` -- function, 0 arguments
- `StringMid` -- function, 3 arguments
- `String` -- function, 1 argument
- `Subst` -- bodied operator, prec. `KMaxPrecedence`
- `Subst` -- function, 3 arguments
- `SystemCall` -- function, 1 argument
- `Tail` -- function, 1 argument
- `ToBase` -- function, 2 arguments
- `ToFile` -- bodied operator, prec. `KMaxPrecedence`
- `ToFile` -- macro, 2 arguments
- `ToStdout` -- bodied operator, prec. `KMaxPrecedence`
- `ToStdout` -- macro, 1 argument
- `ToString` -- bodied operator, prec. `KMaxPrecedence`
- `ToString` -- macro, 1 argument
- `TraceRule` -- bodied operator, prec. `KMaxPrecedence`
- `TraceRule` -- macro, 2 arguments
- `TraceStack` -- macro, 1 argument
- `TrapError` -- macro, 2 arguments
- `Type` -- function, 1 argument
- `UnFence` -- function, 2 arguments
- `UnList` -- function, 1 argument
- `Use` -- function, 1 argument
- `Version` -- function, 0 arguments
- `While` -- bodied operator, prec. `KMaxPrecedence`
- `While` -- macro, 2 arguments
- `WriteString` -- function, 1 argument
- `Write` -- function, 1 or more arguments
- `XmlExplodeTag` -- function, 1 argument
- `XmlTokenizer` -- function, 0 arguments
- `_` -- infix operator, prec. `0`
- `_` -- prefix operator, prec. `0`
- ``` -- macro, 1 argument
- ``` -- prefix operator, prec. `0`

DllLoad , DllUnload , DllEnumerate , StubApiCStart , StubApiCPrettyName , StubApiCPluginPurpose , StubApiCShortIntegerConstant , StubApiCInclude , StubApiCFunction , StubApiCRemark , StubApiCSetEnv , StubApiCFile , StubApiCStruct .

# The Yacas plugin structure

Yacas supports dynamically loading libraries at runtime. This chapter describes functions for working with plugins.

The plugin feature allows Yacas to interface with other libraries that support additional functionality. For example, there could be a plugin enabling the user to script a user interface from within Yacas, or a specific powerful library to do numeric calculations.

The plugin feature is currently in an experimental stage, and it will not work on each platform. To be precise, the `libltdl` library is used to load the plugins. This will only work on platforms which are supported by `libltdl`, which includes Linux, Mac OS X, Solaris, and HP-UX.

The remainder of the section is only of interest to users who want to write plugins themselves. It is assumed that the reader is comfortable programming in C++.

In addition to the plugin structure in the Yacas engine, there is a module `cstubgen` (currently still under development) that allows a rapid creation of a plugin that interfaces to an external library. Essentially all that is required is to write a file that looks like the header file of the original library, but written in Yacas syntax. The module `cstubgen` is then able to write out a C++ file that can be compiled and linked with the original library, and then loaded from within Yacas. To include a function in the plugin typically takes one line of Yacas code. There are a few examples in the `plugins/` directory (the files ending with api.stub). The build system converts these automatically to the required C++ files.

In addition to the C++ stub file, `cstubgen` also automatically generates some documentation on the functions included in the stub. This documentation is put in a file with extension '.man.txt'.

An example describing the use of `cstubgen` can be found in the essay "Creating plugins for Yacas" . See Essays on Yacas, Chapter 5 for a description of the integration of plugins in the build system.

###

DllLoad -- load a plugin

##### Internal function

##### Calling format:

    DllLoad(file)

##### Parameters:

`file` -- file name of the plugin

##### Description:

`DllLoad` loads the plugin `file` into Yacas. Neither the directory nor the extension of `file` need to be specified. The result is `True` if the plugin is loaded successfully, and `False` otherwise.

If the argument `file` does not specify a directory, all directories in the path for Yacas plugins are searched. This path is built up by calls to `DllDirectory`. In addition, it contains the default plugin directory which is specified by the command line option `--dlldir` (see The Yacas User's Function Reference, Chapter 1 ). If this option is not present, the default plugin directory is as specified at compile time (the default value is `/usr/local/lib/yacas`).

If the argument `file` does not specify the extension, then first the extension `.la` is appended. This is the standard extension for Libtool libraries. If no plugin with this name can be found, a platform-specific extension is tried (as an example, this is `.so` on platforms with ELF libraries).

##### Example:

The following commands load the example plugin, and then call the function `AddTwoIntegers` which is defined in this plugin.

     In> DllLoad("example");
     Out> True
     In> AddTwoIntegers(2,3);
     Out> 5

Note that it suffices to specify the string `example`. Yacas knows that the plugin resides in the file `/usr/local/lib/yacas/example.la` (if plugins are installed in the default locations).

##### See also:

DllDirectory , DllUnload , DllEnumerate .

###

DllUnload -- unload a plugin

##### Internal function

##### Calling format:

    DllUnload(file)

##### Parameters:

`file` -- file name of the plugin

##### Description:

`DllUnload` unloads a plugin previously loaded with `DllLoad`. The argument `file` has to be exactly the same as specified in the call to `DllLoad`, or the system will not be able to determine which plugin to unload. It will scan all plugins which are currently loaded, and delete the first one found to exactly match.

`DllUnload` always returns `True`, even if no plugin with the name `file` is found.

##### Example:

First, we load the example plugin, and call the function `AddTwoIntegers` which is defined in this plugin.

    In> DllLoad("example");
    Out> True
    In> AddTwoIntegers(2,3);
    Out> 5

Then, we unload the plugin, and we check that the function `AddTwoIntegers` is indeed no longer defined.

    In> DllUnload("example");
    Out> True
    In> AddTwoIntegers(2,3);
    Out> AddTwoIntegers(2,3)

##### See also:

DllLoad , DllEnumerate .

###

DllEnumerate -- enumerate all loaded plugins

##### Internal function

##### Calling format:

    DllEnumerate()

##### Description:

`DllEnumerate` returns a list with the names of all the plugins which are currently loaded.

##### Example:

At startup, no plugins are loaded, so `DllEnumerate()` evaluates to the empty list.

    In> DllEnumerate();
    Out> {}

This changes after we load the example plugin.

    In> DllLoad("example");
    Out> True
    In> DllEnumerate();
    Out> {"example"}

##### See also:

DllLoad , DllUnload .

###

StubApiCStart -- begin a C++ plugin API description

##### Standard library

##### Calling format:

    StubApiCStart(name)

##### Parameters:

`name` -- string, name of the plugin

##### Description:

This function must be called to begin generating a stub file for linking a C or C++ library with Yacas. A stub specification file needs to start with this function call, to reset the internal state of Yacas for emitting a stub C++ file. The parameter `name` should identify the plugin uniquely.

##### See also:

StubApiCShortIntegerConstant , StubApiCInclude , StubApiCFunction , StubApiCFile , StubApiCSetEnv , StubApiCPrettyName .

###

StubApiCPrettyName -- give a descriptive name for documentation

##### Standard library

##### Calling format:

    StubApiCPrettyName(name)

##### Parameters:

`name` -- descriptive name of the plugin

##### Description:

This function sets a descriptive name for a plugin. It should be very short, for example, "plotting" or "XYZ library", and it should not contain the word "plugin".

This name is used for producing documentation. It will appear in the title of the documentation section describing the plugin.

##### Example:

    StubApiCPrettyName("multithreaded GUI")

The documentation section will be titled "The multithreaded GUI plugin".

##### See also:

StubApiCSStart , StubApiCPluginPurpose .

###

StubApiCPluginPurpose -- document the purpose of the plugin

##### Standard library

##### Calling format:

    StubApiCPluginPurpose(text)

##### Parameters:

`text` -- a short description of the plugin

##### Description:

The text should be in the reference manual plain text format. (See the documentation on the format .) It will be inserted after the section title in the plugin documentation.

##### See also:

StubApiCRemark , StubApiCPrettyName .

###

StubApiCShortIntegerConstant -- declare integer constant in plugin

##### Standard library

##### Calling format:

    StubApiCShortIntegerConstant(const,value)

##### Parameters:

`const` -- string representing the global variable to be bound runtime

`value` -- integer value the global should be bound to

##### Description:

define a constant 'const' to have value 'value'. The value should be short integer constant. This is useful for linking in defines and enumerated values into Yacas. If the library for instance has a define

    #define FOO 10

Then

    StubApiCShortIntegerConstant("FOO","FOO")

will bind the global variable FOO to the value for FOO defined in the library header file.

##### See also:

StubApiCStart , StubApiCInclude , StubApiCFunction , StubApiCFile , StubApiCSetEnv .

###

StubApiCInclude -- declare include file in plugin

##### Standard library

##### Calling format:

    StubApiCInclude(file)

##### Parameters:

`file` -- file to include from the library the plugin is based on

##### Description:

Declare an include file (a header file for the library, for instance) The delimiters need to be specified too. So, for a standard library like the one needed for OpenGL, you need to specify

    StubApiCInclude("\<GL/gl.h\>")

and for user include file:

    StubApiCInclude("\"GL/gl.h\"")

##### See also:

StubApiCStart , StubApiCShortIntegerConstant , StubApiCFunction , StubApiCFile , StubApiCSetEnv .

###

StubApiCFunction -- declare C++ function in plugin

##### Standard library

##### Calling format:

    StubApiCFunction(returntype,fname,args)
    StubApiCFunction(returntype,fname,
      fname2,args)

##### Parameters:

`returntype` -- return type of new function

`fname` -- function of built-in function

`fname2` -- (optional) function name to be used from within Yacas

`args` -- list of arguments to the function

##### Description:

This function declares a new library function, along with its calling sequence. `cstubgen` will then generate the C++ code required to call this function.

Return type, function name, and list of arguments should be literal strings (surrounded by quotes).

If `fname2` is not supplied, it will be assumed to be the same as fname.

The return types currently supported are "`int`", "`double`" and "`void`".

The argument values that are currently supported are "`int`", "`double`", and "`input_string`".

Argument types can be specified simply as a string referring to their type, like `"int"`, or they can be lists with an additional element stating the name of the variable: `{"int","n"}`. The variable will then show up in the automatically generated documentation as having the name "n".

##### Examples:

To define an OpenGL function `glVertex3d` that accepts three doubles and returns void:

    StubApiCFunction("void","glVertex3d",
      {"double","double","double"});

##### See also:

StubApiCStart , StubApiCShortIntegerConstant , StubApiCInclude , StubApiCFile , StubApiCSetEnv .

###

StubApiCRemark -- provide more documentation for plugin

##### Standard library

##### Calling format:

    StubApiCRemark(string)

##### Parameters:

`string` -- text to be added to the documentation

##### Description:

`StubApiCRemark` adds the text to the stub documentation file that gets generated automatically. The documentation is put in a `.man.txt` file while the input file is being processed, so adding a remark on a function just after a function declaration adds a remark on that function. The format of the text must be that of the reference manual source in plaintext, see documentation on the format .

##### See also:

StubApiCPrettyName , StubApiCPluginPurpose , StubApiCStart , StubApiCSetEnv , StubApiCFile .

###

StubApiCSetEnv -- access Yacas environment in plugin

##### Standard library

##### Calling format:

    StubApiCSetEnv(func)

##### Parameters:

`func` -- name of the function to call to set the environment variable

##### Description:

This function forces the plugin to call the function func, with as argument `LispEnvironment&` `aEnvironment`. This lets the plugin store the environment class (which is needed for almost any thing to do with Yacas), somewhere in a global variable. aEnvironment can then be used from within a callback function in the plugin that doesn't take the extra argument by design.

There needs to ba a function in the plugin somewhere of the form

    static LispEnvironment* env = NULL;
    void GlutSetEnv(LispEnvironment& aEnv)
    {
        env = &aEnv;
    }

Then calling

    StubApiCSetEnv("GlutSetEnv");

will force the plugin to call `GlutSetEnv` at load time. All functions in the plugin will then have access to the Yacas environment.

##### See also:

StubApiCStart , StubApiCShortIntegerConstant , StubApiCInclude , StubApiCFunction , StubApiCFile .

###

StubApiCFile -- set file name for plugin API

##### Standard library

##### Calling format:

    StubApiCFile(pluginname,basename)

##### Parameters:

`pluginname` -- name of the plugin, as passed to `DllLoad`

`basename` -- name for the generation of the stub file

##### Description:

Generate the C++ stub file, "basename.cc", and a documentation file named "basename.description". The descriptions are automatically generated while adding functions and constants to the stub.

`pluginname` should be the name as passed to `DllLoad`. If the plugin name is loaded with `DllLoad("example")`, then pluginname should be `"example"`.

##### See also:

StubApiCStart , StubApiCShortIntegerConstant , StubApiCInclude , StubApiCFunction , StubApiCSetEnv .

###

StubApiCStruct -- declare C struct in plugin

##### Standard library

##### Calling format:

    StubApiCStruct(name)
    StubApiCStruct(name,freefunction)

##### Parameters:

`name` -- name of structure

`freefunction` -- function that can be called to clean up the object

##### Description:

`StubApiCStruct` declares a `struct` in a specific library. The name should be followed by an asterisk (clearly showing that it is a pointer). After that, in the stub API definition, this type can be used as argument or return type to functions to the library.

By default the struct will be deleted from memory with a normal call to `free(...)`. This can be overridden with a function given as second argument, freefunction. This is needed in the case where there are additional operations that need to be performed in order to delete the object from memory.

##### Examples:

In a library header file, define:

    typedef struct SomeStruct
    {
      int a;
      int b;
    } SomeStruct;

Then in the stub file you can declare this struct by calling:

    StubApiCStruct("SomeStruct*")

##### See also:

StubApiCFunction .

IsGeneric , GenericTypeName , ArrayCreate , ArraySize , ArrayGet , ArraySet , ArrayCreateFromList , ListFromArray .

# Generic objects

Generic objects are objects that are implemented in C++, but can be accessed through the Yacas interpreter.

###

IsGeneric -- check for generic object

##### Internal function

##### Calling format:

    IsGeneric(object)

##### Description:

Returns `True` if an object is of a generic object type.

###

GenericTypeName -- get type name

##### Internal function

##### Calling format:

    GenericTypeName(object)

##### Description:

Returns a string representation of the name of a generic object.

EG

    In> GenericTypeName(ArrayCreate(10,1))
    Out> "Array";

###

ArrayCreate -- create array

##### Internal function

##### Calling format:

    ArrayCreate(size,init)

##### Description:

Creates an array with `size` elements, all initialized to the value `init`.

###

ArraySize -- get array size

##### Internal function

##### Calling format:

    ArraySize(array)

##### Description:

Returns the size of an array (number of elements in the array).

###

ArrayGet -- fetch array element

##### Internal function

##### Calling format:

    ArrayGet(array,index)

##### Description:

Returns the element at position index in the array passed. Arrays are treated as base-one, so `index` set to 1 would return the first element.

Arrays can also be accessed through the `[]` operators. So `array[index]` would return the same as `ArrayGet(array, index)`.

###

ArraySet -- set array element

##### Internal function

##### Calling format:

    ArraySet(array,index,element)

##### Description:

Sets the element at position index in the array passed to the value passed in as argument to element. Arrays are treated as base-one, so `index` set to 1 would set first element.

Arrays can also be accessed through the `[]` operators. So `array[index] := element` would do the same as `ArraySet(array, index,element)`.

###

ArrayCreateFromList -- convert list to array

##### Internal function

##### Calling format:

    ArrayCreateFromList(list)

##### Description:

Creates an array from the contents of the list passed in.

###

ListFromArray -- convert array to list

##### Internal function

##### Calling format:

    ListFromArray(array)

##### Description:

Creates a list from the contents of the array passed in.

Verify, TestYacas, LogicVerify , KnownFailure , RoundTo , VerifyArithmetic, RandVerifyArithmetic, VerifyDiv .

# The Yacas test suite

This chapter describes commands used for verifying correct performance of Yacas.

Yacas comes with a test suite which can be found in the directory `tests/`. Typing

    make test

on the command line after Yacas was built will run the test. This test can be run even before `make install`, as it only uses files in the local directory of the Yacas source tree. The default extension for test scripts is `.yts` (Yacas test script).

The verification commands described in this chapter only display the expressions that do not evaluate correctly. Errors do not terminate the execution of the Yacas script that uses these testing commands, since they are meant to be used in test scripts.

###

Verify, TestYacas, LogicVerify -- verifying equivalence of two expressions

##### Standard library

##### Calling format:

    Verify(question,answer)
    TestYacas(question,answer)
    LogicVerify(question,answer)

##### Parameters:

`question` -- expression to check for

`answer` -- expected result after evaluation

##### Description:

The commands `Verify`, `TestYacas`, `LogicVerify` can be used to verify that an expression is equivalent to a correct answer after evaluation. All three commands return `True` or `False`.

For some calculations, the demand that two expressions are identical syntactically is too stringent. The Yacas system might change at various places in the future, but 1+x would still be equivalent, from a mathematical point of view, to x+1.

The general problem of deciding that two expressions a and b are equivalent, which is the same as saying that a-b=0 , is generally hard to decide on. The following commands solve this problem by having domain-specific comparisons.

The comparison commands do the following comparison types:

- `Verify` -- verify for literal equality. This is the fastest and simplest comparison, and can be used, for example, to test that an expression evaluates to 2.
- `TestYacas` -- compare two expressions after simplification as multivariate polynomials. If the two arguments are equivalent multivariate polynomials, this test succeeds. `TestYacas` uses `Simplify`. Note: `TestYacas` currently should not be used to test equality of lists.
- `LogicVerify` -- Perform a test by using `CanProve` to verify that from `question` the expression `answer` follows. This test command is used for testing the logic theorem prover in Yacas.

##### Examples:

    In> Verify(1+2,3)
    Out> True;
    In> Verify(x*(1+x),x^2+x)
    ******************
    x*(x+1) evaluates to x*(x+1) which differs
      from x^2+x
    ******************
    Out> False;
    In> TestYacas(x*(1+x),x^2+x)
    Out> True;
    In> Verify(a And c Or b And Not c,a Or b)
    ******************
     a And c Or b And Not c evaluates to  a And c
      Or b And Not c which differs from  a Or b
    ******************
    Out> False;
    In> LogicVerify(a And c Or b And Not c,a Or b)
    Out> True;
    In> LogicVerify(a And c Or b And Not c,b Or a)
    Out> True;

##### See also:

Simplify , CanProve , KnownFailure .

###

KnownFailure -- Mark a test as a known failure

##### Standard library

##### Calling format:

    KnownFailure(test)

##### Parameters:

`test` -- expression that should return `False` on failure

##### Description:

The command `KnownFailure` marks a test as known to fail by displaying a message to that effect on screen.

This might be used by developers when they have no time to fix the defect, but do not wish to alarm users who download Yacas and type `make test`.

##### Examples:

    In> KnownFailure(Verify(1,2))
    Known failure:
    ******************
     1 evaluates to  1 which differs from  2
    ******************
    Out> False;
    In> KnownFailure(Verify(1,1))
    Known failure:
    Failure resolved!
    Out> True;

##### See also:

Verify , TestYacas , LogicVerify .

###

RoundTo -- Round a real-valued result to a set number of digits

##### Standard library

##### Calling format:

    RoundTo(number,precision)

##### Parameters:

`number` -- number to round off

`precision` -- precision to use for round-off

##### Description:

The function `RoundTo` rounds a floating point number to a specified precision, allowing for testing for correctness using the `Verify` command.

##### Examples:

    In> N(RoundTo(Exp(1),30),30)
    Out> 2.71828182110230114951959786552;
    In> N(RoundTo(Exp(1),20),20)
    Out> 2.71828182796964237096;

##### See also:

Verify , VerifyArithmetic , VerifyDiv .

###

VerifyArithmetic, RandVerifyArithmetic, VerifyDiv -- Special purpose arithmetic verifiers

##### Standard library

##### Calling format:

    VerifyArithmetic(x,n,m)
    RandVerifyArithmetic(n)
    VerifyDiv(u,v)

##### Parameters:

`x`, `n`, `m`, `u`, `v` -- integer arguments

##### Description:

The commands `VerifyArithmetic` and `VerifyDiv` test a mathematic equality which should hold, testing that the result returned by the system is mathematically correct according to a mathematically provable theorem.

`VerifyArithmetic` verifies for an arbitrary set of numbers x, n and m that

(x^n-1)*(x^m-1)=x^(n+m)-x^n-x^m+1.

The left and right side represent two ways to arrive at the same result, and so an arithmetic module actually doing the calculation does the calculation in two different ways. The results should be exactly equal.

`RandVerifyArithmetic(n)` calls `VerifyArithmetic` with random values, `n` times.

`VerifyDiv(u,v)` checks that

u=v*Div(u,v)+Mod(u,v).

##### Examples:

    In> VerifyArithmetic(100,50,60)
    Out> True;
    In> RandVerifyArithmetic(4)
    Out> True;
    In> VerifyDiv(x^2+2*x+3,x+1)
    Out> True;
    In> VerifyDiv(3,2)
    Out> True;

##### See also:

Verify .

# Glossary

This is a short glossary of terms frequently used in the `Yacas` documentation.

###

arity

"Arity" is the number of arguments of a function. For example, the function `Cos(x)` has one argument and so we say that "`Cos` has arity 1". Arity of a function can be 0, 1, 2, ...

`Yacas` allows to define functions with the same name but different arities, and different rules corresponding to these arities will be used. Also, it is possible to define a function with optional arguments, for example, `Plot2D` is one such function. Such functions can have any arity larger or equal to a certain minimum arity (the number of non-optional arguments).

##### See also:

Function , OpPrecedence , Rule .

###

array

An array is a container object that can hold a fixed number of other Yacas objects in it. Individual elements of an array can be accessed using the `[]` operation. Most list operations also work on arrays.

Arrays are faster than lists but the array size cannot be changed.

##### See also:

ArrayCreate .

###

atoms

Atoms are basic Yacas objects that are used to represent symbols, numbers, and function names. An atom has a string representation which is shown when it is displayed. For example, `3.14159`, `x`, `A123`, `+`, `"good morning"` are atoms.

Atoms can be of type string, number, or symbol. For example, `y1` is a symbolic atom, `954` is a number atom, and `""` is an (empty) string atom. Symbolic atoms are normally used in `Yacas` to denote mathematical unknowns and function names. Number and string atoms are used to denote values.

A symbolic atom can be bound to a value (in which case it becomes a variable), or to a rule or several rules (in which case it becomes a function). An atom can also have a property object.

##### See also:

Atom , String .

###

CAS

Abbreviation for "computer algebra system". `Yacas` is a CAS.

###

constants

Constants such as `I`, `Pi` or `GoldenRatio` are symbolic atoms that are specially interpreted by `Yacas`. For example, there are simplification rules that transform expressions such as `Sin(Pi)` into `0`. When requesting a numerical evaluation of a constant, the numerical value is given to the current value of `Precision`.

Some constants take a long time to compute and therefore they are cached at the highest precision computed so far. These are the "cached constants".

##### See also:

N , CachedConstant , Pi , GoldenRatio , CatalanConstant , gamma , I , Precision .

###

equations

To denote symbolic equations, the operator `==` is used. This operator does not assign or compare its sides. For example, the expression `Sin(x)==1` is kept unevaluated and can be passed as argument to functions. For example,

    In> Solve(Sin(x)==1, x)
    Out> {x==Pi/2};

The symbolic equation operator `==` is also useful to represent solutions of equations or to specify substitutions, give options, and so on.

##### See also:

Solve , Where , Plot2D .

###

functions

A function is a symbolic atom that is bound to a rule or several rules. A function can have none, one, or more arguments. Functions can also have a variable number of arguments. Arguments of functions are arbitrary Yacas objects.

Functions can be evaluated, that is, the rules bound to them may be executed. For example, `Cos(Pi+0)` is an expression that contains two functions and four atoms. The atom `Pi` is a symbolic atom which is normally not bound to anything. The atom `0` is a numeric atom.

The atoms `Cos` and `+` are symbolic atoms which are bound to appropriate simplification rules. So these two atoms are functions. Note that these functions have different syntax. `Cos` is a normal function which takes its arguments in parentheses. The atom `+` is a function with special syntax because "`+`" is placed between its arguments and no parentheses are used.

The rules to which `+` is bound are such that the expression `Pi+0` is evaluated to the symbolic atom `Pi`. The rules for `Cos` are such that the expression `Cos(Pi)` is evaluated to the numeric atom `-1`. The example `Yacas` session is:

    In> Cos(Pi+0)
    Out> -1;

Some functions are built-in and implemented in C++, while others are library functions.

The built-in functions are usually less flexible than the library functions because they cannot be left unevaluated. Given incorrect arguments, a built-in function will generate an error. However, a user-defined function may simply return unevaluated in such cases.

##### See also:

Function , Rule , <-- .

###

lists

A list is a basic `Yacas` container object. A list is written as e.g. `{a, b, c}` or `{}` (empty list). Individual elements of a list can be accessed using the `[]` operation. Lists can be concatenated, and individual elements can be removed or inserted.

Lists are ubiquitous in `Yacas`. Most data structures in the standard library is based on lists.

Lists are also used internally to represent `Yacas` expressions. For example, the expression `Cos(x+1)` is represented internally as a nested list:

    In> FullForm( Cos(x+1) )
    (Cos
        (+ x 1 ))
    Out> Cos(x+1);

##### See also:

List , Listify , UnList , Length , FullForm .

###

matrices

A matrix is represented as a list of lists. Matrices are represented in the "row-major" order: a matrix is a list of rows, and each row is a list of its elements.

Some basic linear algebra operations on matrices are supported.

##### See also:

Determinant , Identity , IsDiagonal , EigenValues .

###

operators

Operators are functions that have special syntax declared for them. An operator can be "bodied", infix, prefix or postfix. Because of this, operators must have precedence.

Apart from the syntax, operators are exactly the same as any other functions, they can have rules bound to them in the same way.

##### See also:

Bodied , Infix , Prefix , Postfix .

###

plotting

Plotting is currently supported via the `Plot2D` and `Plot3DS` functions. Functions of one or two variables can be plotted on given intervals with a given precision. `Yacas` generates all numerical data for plots, while external programs such as `gnuplot` or `plotutils` are executed to actually produce and display the plots.

##### See also:

Plot2D , Plot3DS .

###

plugin

A plugin is an compiled dynamic library of functions that can be loaded into `Yacas` at run time. After loading a plugin, the functions implemented in the library become available as new Yacas functions. Currently plugins have to be written in C++.

##### See also:

DllLoad , MakeFunctionPlugin .

###

precedence

Precedence is a property of the syntax of an operator that specifies how it is parsed. Only operators, i.e. functions with special syntax, can have precedence. Precedence values are nonnegative integers: 0, 1, ... Lower numbers bind more tightly.

For example, the operator "`+`" binds less tightly (i.e. has a higher precedence value) than the operator "`*`" and so the expression `a+b*c` is parsed as `a+(b*c)`, as one would expect.

Infix operators can have different left-side and right-side precedence. For example, the infix operator "`-`" has left precedence 70 and right precedence 40 -- this allows us to parse expressions such as `a-b+c` correctly, as a-b+c, and not as a-(b+c).

##### See also:

Bodied , OpPrecedence , OpLeftPrecedence , OpRightPrecedence .

###

properties

Properties are special additional objects (tags) that can be tied to expressions. For example, the expression `1+x` may be tagged by an expression `y` by the command

    In> a:= SetExtraInfo(1+x,y);
    Out> 1+x;

Now `a` refers to an expression `1+x` which is different from all other copies of `1+x` because it is tagged by `y`.

##### See also:

GetExtraInfo , SetExtraInfo .

###

rules

Rules are the principal mechanism of expression evaluation in `Yacas`. A rule specifies that a certain symbolic expression is to be replaced by another expression. If no rule that matches a given symbolic expression can be found, the expression is left unevaluated. This is usually the desired behavior for a CAS. For example, a user can type

    In> func1(x+0)
    Out> func1(x);

and use an undefined function `func1`. Since no rules are defined for the function `func1`, it is not evaluated, but its argument has been simplified.

Only expressions containing functions can be evaluated by rules. (Atoms are evaluated only if they are bound to a value.)

Several rules can be defined for a given function. Rules can be erased or added at run time.

##### See also:

Rule , <-- , Retract .

###

strings

A string is an atom with character string value, for example, `"abcd"`. Individual characters of the string can be accessed using the `[]` operation. Some string manipulation functions are supported.

##### See also:

String , StringMid .

###

syntax

`Yacas` uses an infix syntax similar to C or Fortran. However, the syntax is entirely user-defined and very flexible. Infix, prefix, postfix operators can be defined, as well as "bodied" functions. This allows to write mathematical expressions more comfortably, for example,

    In> D(x) Sin(x)+1
    Out> Cos(x);

Functions with special syntax can have different precedence.

##### See also:

Bodied , Infix , Prefix , Postfix , OpPrecedence .

###

variables

Variables are symbolic atoms bound to a "value". Value is any Yacas object, such as an atom or a list. For example, after executing

    In> a := 1
    Out> 1;

the symbolic atom `a` becomes a variable bound to a value, the numeric atom `1`.

##### See also:

Eval , := , Clear .

###

warranty

`Yacas` is Free Software ("logiciel libre") and comes with `NO WARRANTY`. See the appropriate section of the GPL for full information.

# GNU General Public License

This chapter contains the GNU General Public License (GPL). This information is important for everyone using or modifying the Yacas source code or binaries.

GNU GENERAL PUBLIC LICENSE Version 2, June 1991

Copyright (C) 1989, 1991 Free Software Foundation, Inc. 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.

##

Preamble

The licenses for most software are designed to take away your freedom to share and change it. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change free software--to make sure the software is free for all its users. This General Public License applies to most of the Free Software Foundation's software and to any other program whose authors commit to using it. (Some other Free Software Foundation software is covered by the GNU Library General Public License instead.) You can apply it to your programs, too.

When we speak of free software, we are referring to freedom, not price. Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software (and charge for this service if you wish), that you receive source code or can get it if you want it, that you can change the software or use pieces of it in new free programs; and that you know you can do these things.

To protect your rights, we need to make restrictions that forbid anyone to deny you these rights or to ask you to surrender the rights. These restrictions translate to certain responsibilities for you if you distribute copies of the software, or if you modify it.

For example, if you distribute copies of such a program, whether gratis or for a fee, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must show them these terms so they know their rights.

We protect your rights with two steps: (1) copyright the software, and (2) offer you this license which gives you legal permission to copy, distribute and/or modify the software.

Also, for each author's protection and ours, we want to make certain that everyone understands that there is no warranty for this free software. If the software is modified by someone else and passed on, we want its recipients to know that what they have is not the original, so that any problems introduced by others will not reflect on the original authors' reputations.

Finally, any free program is threatened constantly by software patents. We wish to avoid the danger that redistributors of a free program will individually obtain patent licenses, in effect making the program proprietary. To prevent this, we have made it clear that any patent must be licensed for everyone's free use or not licensed at all.

The precise terms and conditions for copying, distribution and modification follow.

GNU GENERAL PUBLIC LICENSE TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION

0. This License applies to any program or other work which contains a notice placed by the copyright holder saying it may be distributed under the terms of this General Public License. The "Program", below, refers to any such program or work, and a "work based on the Program" means either the Program or any derivative work under copyright law: that is to say, a work containing the Program or a portion of it, either verbatim or with modifications and/or translated into another language. (Hereinafter, translation is included without limitation in the term "modification".) Each licensee is addressed as "you".

Activities other than copying, distribution and modification are not covered by this License; they are outside its scope. The act of running the Program is not restricted, and the output from the Program is covered only if its contents constitute a work based on the Program (independent of having been made by running the Program). Whether that is true depends on what the Program does.

1. You may copy and distribute verbatim copies of the Program's source code as you receive it, in any medium, provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice and disclaimer of warranty; keep intact all the notices that refer to this License and to the absence of any warranty; and give any other recipients of the Program a copy of this License along with the Program.

You may charge a fee for the physical act of transferring a copy, and you may at your option offer warranty protection in exchange for a fee.

2. You may modify your copy or copies of the Program or any portion of it, thus forming a work based on the Program, and copy and distribute such modifications or work under the terms of Section 1 above, provided that you also meet all of these conditions:

a) You must cause the modified files to carry prominent notices stating that you changed the files and the date of any change.

b) You must cause any work that you distribute or publish, that in whole or in part contains or is derived from the Program or any part thereof, to be licensed as a whole at no charge to all third parties under the terms of this License.

c) If the modified program normally reads commands interactively when run, you must cause it, when started running for such interactive use in the most ordinary way, to print or display an announcement including an appropriate copyright notice and a notice that there is no warranty (or else, saying that you provide a warranty) and that users may redistribute the program under these conditions, and telling the user how to view a copy of this License. (Exception: if the Program itself is interactive but does not normally print such an announcement, your work based on the Program is not required to print an announcement.)

These requirements apply to the modified work as a whole. If identifiable sections of that work are not derived from the Program, and can be reasonably considered independent and separate works in themselves, then this License, and its terms, do not apply to those sections when you distribute them as separate works. But when you distribute the same sections as part of a whole which is a work based on the Program, the distribution of the whole must be on the terms of this License, whose permissions for other licensees extend to the entire whole, and thus to each and every part regardless of who wrote it.

Thus, it is not the intent of this section to claim rights or contest your rights to work written entirely by you; rather, the intent is to exercise the right to control the distribution of derivative or collective works based on the Program.

In addition, mere aggregation of another work not based on the Program with the Program (or with a work based on the Program) on a volume of a storage or distribution medium does not bring the other work under the scope of this License.

3. You may copy and distribute the Program (or a work based on it, under Section 2) in object code or executable form under the terms of Sections 1 and 2 above provided that you also do one of the following:

a) Accompany it with the complete corresponding machine-readable source code, which must be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange; or,

b) Accompany it with a written offer, valid for at least three years, to give any third party, for a charge no more than your cost of physically performing source distribution, a complete machine-readable copy of the corresponding source code, to be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange; or,

c) Accompany it with the information you received as to the offer to distribute corresponding source code. (This alternative is allowed only for noncommercial distribution and only if you received the program in object code or executable form with such an offer, in accord with Subsection b above.)

The source code for a work means the preferred form of the work for making modifications to it. For an executable work, complete source code means all the source code for all modules it contains, plus any associated interface definition files, plus the scripts used to control compilation and installation of the executable. However, as a special exception, the source code distributed need not include anything that is normally distributed (in either source or binary form) with the major components (compiler, kernel, and so on) of the operating system on which the executable runs, unless that component itself accompanies the executable.

If distribution of executable or object code is made by offering access to copy from a designated place, then offering equivalent access to copy the source code from the same place counts as distribution of the source code, even though third parties are not compelled to copy the source along with the object code.

4. You may not copy, modify, sublicense, or distribute the Program except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense or distribute the Program is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance.

5. You are not required to accept this License, since you have not signed it. However, nothing else grants you permission to modify or distribute the Program or its derivative works. These actions are prohibited by law if you do not accept this License. Therefore, by modifying or distributing the Program (or any work based on the Program), you indicate your acceptance of this License to do so, and all its terms and conditions for copying, distributing or modifying the Program or works based on it.

6. Each time you redistribute the Program (or any work based on the Program), the recipient automatically receives a license from the original licensor to copy, distribute or modify the Program subject to these terms and conditions. You may not impose any further restrictions on the recipients' exercise of the rights granted herein. You are not responsible for enforcing compliance by third parties to this License.

7. If, as a consequence of a court judgment or allegation of patent infringement or for any other reason (not limited to patent issues), conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot distribute so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not distribute the Program at all. For example, if a patent license would not permit royalty-free redistribution of the Program by all those who receive copies directly or indirectly through you, then the only way you could satisfy both it and this License would be to refrain entirely from distribution of the Program.

If any portion of this section is held invalid or unenforceable under any particular circumstance, the balance of the section is intended to apply and the section as a whole is intended to apply in other circumstances.

It is not the purpose of this section to induce you to infringe any patents or other property right claims or to contest validity of any such claims; this section has the sole purpose of protecting the integrity of the free software distribution system, which is implemented by public license practices. Many people have made generous contributions to the wide range of software distributed through that system in reliance on consistent application of that system; it is up to the author/donor to decide if he or she is willing to distribute software through any other system and a licensee cannot impose that choice.

This section is intended to make thoroughly clear what is believed to be a consequence of the rest of this License.

8. If the distribution and/or use of the Program is restricted in certain countries either by patents or by copyrighted interfaces, the original copyright holder who places the Program under this License may add an explicit geographical distribution limitation excluding those countries, so that distribution is permitted only in or among countries not thus excluded. In such case, this License incorporates the limitation as if written in the body of this License.

9. The Free Software Foundation may publish revised and/or new versions of the General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns.

Each version is given a distinguishing version number. If the Program specifies a version number of this License which applies to it and "any later version", you have the option of following the terms and conditions either of that version or of any later version published by the Free Software Foundation. If the Program does not specify a version number of this License, you may choose any version ever published by the Free Software Foundation.

10. If you wish to incorporate parts of the Program into other free programs whose distribution conditions are different, write to the author to ask for permission. For software which is copyrighted by the Free Software Foundation, write to the Free Software Foundation; we sometimes make exceptions for this. Our decision will be guided by the two goals of preserving the free status of all derivatives of our free software and of promoting the sharing and reuse of software generally.

##

NO WARRANTY

11. BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.

12. IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR REDISTRIBUTE THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.

END OF TERMS AND CONDITIONS

###

How to Apply These Terms to Your New Programs

If you develop a new program, and you want it to be of the greatest possible use to the public, the best way to achieve this is to make it free software which everyone can redistribute and change under these terms.

To do so, attach the following notices to the program. It is safest to attach them to the start of each source file to most effectively convey the exclusion of warranty; and each file should have at least the "copyright" line and a pointer to where the full notice is found.

`<`one line to give the program's name and a brief idea of what it does.`>` Copyright (C) `<`year`>` `<`name of author`>`

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; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA

Also add information on how to contact you by electronic and paper mail.

If the program is interactive, make it output a short notice like this when it starts in an interactive mode:

Gnomovision version 69, Copyright (C) year name of author Gnomovision comes with ABSOLUTELY NO WARRANTY; for details type `show w'. This is free software, and you are welcome to redistribute it under certain conditions; type `show c' for details.

The hypothetical commands `show w' and `show c' should show the appropriate parts of the General Public License. Of course, the commands you use may be called something other than `show w' and `show c'; they could even be mouse-clicks or menu items--whatever suits your program.

You should also get your employer (if you work as a programmer) or your school, if any, to sign a "copyright disclaimer" for the program, if necessary. Here is a sample; alter the names:

Yoyodyne, Inc., hereby disclaims all copyright interest in the program `Gnomovision' (which makes passes at compilers) written by James Hacker.

`<`signature of Ty Coon`>`, 1 April 1989 Ty Coon, President of Vice

This General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Library General Public License instead of this License.

# GNU Free Documentation License

##### Version 1.1, March 2000

Copyright (C) 2000 Free Software Foundation, Inc.

    59 Temple Place, Suite 330
    Boston, MA, 02111-1307
    USA

Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.

###

Preamble

The purpose of this License is to make a manual, textbook, or other written document ``free'' in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others.

This License is a kind of ``copyleft'', which means that derivative works of the document must themselves be free in the same sense. It complements the GNU General Public License, which is a copyleft license designed for free software.

We have designed this License in order to use it for manuals for free software, because free software needs free documentation: a free program should come with manuals providing the same freedoms that the software does. But this License is not limited to software manuals; it can be used for any textual work, regardless of subject matter or whether it is published as a printed book. We recommend this License principally for works whose purpose is instruction or reference.

###

Applicability and Definitions

This License applies to any manual or other work that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. The ``Document'', below, refers to any such manual or work. Any member of the public is a licensee, and is addressed as ``you''.

A ``Modified Version'' of the Document means any work containing the Document or a portion of it, either copied verbatim, or with modifications and/or translated into another language.

A ``Secondary Section'' is a named appendix or a front-matter section of the Document that deals exclusively with the relationship of the publishers or authors of the Document to the Document's overall subject (or to related matters) and contains nothing that could fall directly within that overall subject. (For example, if the Document is in part a textbook of mathematics, a Secondary Section may not explain any mathematics.) The relationship could be a matter of historical connection with the subject or with related matters, or of legal, commercial, philosophical, ethical or political position regarding them.

The ``Invariant Sections'' are certain Secondary Sections whose titles are designated, as being those of Invariant Sections, in the notice that says that the Document is released under this License.

The ``Cover Texts'' are certain short passages of text that are listed, as Front-Cover Texts or Back-Cover Texts, in the notice that says that the Document is released under this License.

A ``Transparent'' copy of the Document means a machine-readable copy, represented in a format whose specification is available to the general public, whose contents can be viewed and edited directly and straightforwardly with generic text editors or (for images composed of pixels) generic paint programs or (for drawings) some widely available drawing editor, and that is suitable for input to text formatters or for automatic translation to a variety of formats suitable for input to text formatters. A copy made in an otherwise Transparent file format whose markup has been designed to thwart or discourage subsequent modification by readers is not Transparent. A copy that is not ``Transparent'' is called ``Opaque''.

Examples of suitable formats for Transparent copies include plain ASCII without markup, Texinfo input format, `LaTeX` input format, SGML or XML using a publicly available DTD, and standard-conforming simple HTML designed for human modification. Opaque formats include PostScript, PDF, proprietary formats that can be read and edited only by proprietary word processors, SGML or XML for which the DTD and/or processing tools are not generally available, and the machine-generated HTML produced by some word processors for output purposes only.

The ``Title Page'' means, for a printed book, the title page itself, plus such following pages as are needed to hold, legibly, the material this License requires to appear in the title page. For works in formats which do not have any title page as such, ``Title Page'' means the text near the most prominent appearance of the work's title, preceding the beginning of the body of the text.

###

Verbatim Copying

You may copy and distribute the Document in any medium, either commercially or noncommercially, provided that this License, the copyright notices, and the license notice saying this License applies to the Document are reproduced in all copies, and that you add no other conditions whatsoever to those of this License. You may not use technical measures to obstruct or control the reading or further copying of the copies you make or distribute. However, you may accept compensation in exchange for copies. If you distribute a large enough number of copies you must also follow the conditions in section 3.

You may also lend copies, under the same conditions stated above, and you may publicly display copies.

###

Copying in Quantity

If you publish printed copies of the Document numbering more than 100, and the Document's license notice requires Cover Texts, you must enclose the copies in covers that carry, clearly and legibly, all these Cover Texts: Front-Cover Texts on the front cover, and Back-Cover Texts on the back cover. Both covers must also clearly and legibly identify you as the publisher of these copies. The front cover must present the full title with all words of the title equally prominent and visible. You may add other material on the covers in addition. Copying with changes limited to the covers, as long as they preserve the title of the Document and satisfy these conditions, can be treated as verbatim copying in other respects.

If the required texts for either cover are too voluminous to fit legibly, you should put the first ones listed (as many as fit reasonably) on the actual cover, and continue the rest onto adjacent pages.

If you publish or distribute Opaque copies of the Document numbering more than 100, you must either include a machine-readable Transparent copy along with each Opaque copy, or state in or with each Opaque copy a publicly-accessible computer-network location containing a complete Transparent copy of the Document, free of added material, which the general network-using public has access to download anonymously at no charge using public-standard network protocols. If you use the latter option, you must take reasonably prudent steps, when you begin distribution of Opaque copies in quantity, to ensure that this Transparent copy will remain thus accessible at the stated location until at least one year after the last time you distribute an Opaque copy (directly or through your agents or retailers) of that edition to the public.

It is requested, but not required, that you contact the authors of the Document well before redistributing any large number of copies, to give them a chance to provide you with an updated version of the Document.

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Modifications

You may copy and distribute a Modified Version of the Document under the conditions of sections 2 and 3 above, provided that you release the Modified Version under precisely this License, with the Modified Version filling the role of the Document, thus licensing distribution and modification of the Modified Version to whoever possesses a copy of it. In addition, you must do these things in the Modified Version:

- Use in the Title Page (and on the covers, if any) a title distinct from that of the Document, and from those of previous versions (which should, if there were any, be listed in the History section of the Document). You may use the same title as a previous version if the original publisher of that version gives permission.
- List on the Title Page, as authors, one or more persons or entities responsible for authorship of the modifications in the Modified Version, together with at least five of the principal authors of the Document (all of its principal authors, if it has less than five).
- State on the Title page the name of the publisher of the Modified Version, as the publisher.
- Preserve all the copyright notices of the Document.
- Add an appropriate copyright notice for your modifications adjacent to the other copyright notices.
- Include, immediately after the copyright notices, a license notice giving the public permission to use the Modified Version under the terms of this License, in the form shown in the Addendum below.
- Preserve in that license notice the full lists of Invariant Sections and required Cover Texts given in the Document's license notice.
- Include an unaltered copy of this License.
- Preserve the section entitled ``History'', and its title, and add to it an item stating at least the title, year, new authors, and publisher of the Modified Version as given on the Title Page. If there is no section entitled ``History'' in the Document, create one stating the title, year, authors, and publisher of the Document as given on its Title Page, then add an item describing the Modified Version as stated in the previous sentence.
- Preserve the network location, if any, given in the Document for public access to a Transparent copy of the Document, and likewise the network locations given in the Document for previous versions it was based on. These may be placed in the ``History'' section. You may omit a network location for a work that was published at least four years before the Document itself, or if the original publisher of the version it refers to gives permission.
- In any section entitled ``Acknowledgements'' or ``Dedications'', preserve the section's title, and preserve in the section all the substance and tone of each of the contributor acknowledgements and/or dedications given therein.
- Preserve all the Invariant Sections of the Document, unaltered in their text and in their titles. Section numbers or the equivalent are not considered part of the section titles.
- Delete any section entitled ``Endorsements''. Such a section may not be included in the Modified Version.
- Do not retitle any existing section as ``Endorsements'' or to conflict in title with any Invariant Section.

If the Modified Version includes new front-matter sections or appendices that qualify as Secondary Sections and contain no material copied from the Document, you may at your option designate some or all of these sections as invariant. To do this, add their titles to the list of Invariant Sections in the Modified Version's license notice. These titles must be distinct from any other section titles.

You may add a section entitled ``Endorsements'', provided it contains nothing but endorsements of your Modified Version by various parties -- for example, statements of peer review or that the text has been approved by an organization as the authoritative definition of a standard.

You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through arrangements made by) any one entity. If the Document already includes a cover text for the same cover, previously added by you or by arrangement made by the same entity you are acting on behalf of, you may not add another; but you may replace the old one, on explicit permission from the previous publisher that added the old one.

The author(s) and publisher(s) of the Document do not by this License give permission to use their names for publicity for or to assert or imply endorsement of any Modified Version.

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Combining Documents

You may combine the Document with other documents released under this License, under the terms defined in section 4 above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them all as Invariant Sections of your combined work in its license notice.

The combined work need only contain one copy of this License, and multiple identical Invariant Sections may be replaced with a single copy. If there are multiple Invariant Sections with the same name but different contents, make the title of each such section unique by adding at the end of it, in parentheses, the name of the original author or publisher of that section if known, or else a unique number. Make the same adjustment to the section titles in the list of Invariant Sections in the license notice of the combined work.

In the combination, you must combine any sections entitled ``History'' in the various original documents, forming one section entitled ``History''; likewise combine any sections entitled ``Acknowledgements'', and any sections entitled ``Dedications''. You must delete all sections entitled ``Endorsements.''

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Collections of Documents

You may make a collection consisting of the Document and other documents released under this License, and replace the individual copies of this License in the various documents with a single copy that is included in the collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects.

You may extract a single document from such a collection, and distribute it individually under this License, provided you insert a copy of this License into the extracted document, and follow this License in all other respects regarding verbatim copying of that document.

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Aggregation With Independent Works

A compilation of the Document or its derivatives with other separate and independent documents or works, in or on a volume of a storage or distribution medium, does not as a whole count as a Modified Version of the Document, provided no compilation copyright is claimed for the compilation. Such a compilation is called an ``aggregate'', and this License does not apply to the other self-contained works thus compiled with the Document, on account of their being thus compiled, if they are not themselves derivative works of the Document.

If the Cover Text requirement of section 3 is applicable to these copies of the Document, then if the Document is less than one quarter of the entire aggregate, the Document's Cover Texts may be placed on covers that surround only the Document within the aggregate. Otherwise they must appear on covers around the whole aggregate.

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Translation

Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section 4. Replacing Invariant Sections with translations requires special permission from their copyright holders, but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections. You may include a translation of this License provided that you also include the original English version of this License. In case of a disagreement between the translation and the original English version of this License, the original English version will prevail.

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Termination

You may not copy, modify, sublicense, or distribute the Document except as expressly provided for under this License. Any other attempt to copy, modify, sublicense or distribute the Document is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance.

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Future Revisions of This License

The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See http://www.gnu.org/copyleft/ .

Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License "or any later version" applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation.

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ADDENDUM: How to use this License for your documents

To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page:

    Copyright (C) YEAR   YOUR NAME. Permission is
    granted to copy, distribute and/or modify this
    document under the terms of the GNU Free
    Documentation License, Version 1.1 or any later
    version published by the Free Software Foundation;
    with the Invariant Sections being LIST THEIR
    TITLES, with the Front-Cover Texts being LIST, and
    with the Back-Cover Texts being LIST. A copy of
    the license is included in the section entitled
    ``GNU Free Documentation License''.

If you have no Invariant Sections, write ``with no Invariant Sections'' instead of saying which ones are invariant. If you have no Front-Cover Texts, write ``no Front-Cover Texts'' instead of ``Front-Cover Texts being LIST''; likewise for Back-Cover Texts.

If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.