Genius Manual

Jiří Lebl

   University of Illinois, Urbana-Champaign

    <jirka@5z.com>

Kai Willadsen

   University of Queensland, Australia

    <kaiw@itee.uq.edu.au>

   Copyright © 1997-2011 Jiří (George) Lebl

   Copyright © 2004 Kai Willadsen

   Permission is granted to copy, distribute and/or modify this
   document under the terms of the GNU Free Documentation License
   (GFDL), Version 1.1 or any later version published by the Free
   Software Foundation with no Invariant Sections, no Front-Cover
   Texts, and no Back-Cover Texts. You can find a copy of the GFDL
   at this link or in the file COPYING-DOCS distributed with this
   manual.

   This manual is part of a collection of GNOME manuals
   distributed under the GFDL. If you want to distribute this
   manual separately from the collection, you can do so by adding
   a copy of the license to the manual, as described in section 6
   of the license.

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   GNOME Documentation Project are made aware of those trademarks,
   then the names are in capital letters or initial capital
   letters.

   DOCUMENT AND MODIFIED VERSIONS OF THE DOCUMENT ARE PROVIDED
   UNDER THE TERMS OF THE GNU FREE DOCUMENTATION LICENSE WITH THE
   FURTHER UNDERSTANDING THAT:

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   Feedback

   To report a bug or make a suggestion regarding the Genius
   Mathematics Tool application or this manual, follow the
   directions in the GNOME Feedback Page.
   This manual describes version 1.0.13 of Genius.
     __________________________________________________________

   Table of Contents
   Introduction
   Getting Started

        To Start Genius Mathematics Tool
        When You Start Genius

   Basic Usage

        Using the Work Area
        To Create a New Program
        To Open and Run a Program

   Plotting

        Line Plots
        Parametric Plots
        Slopefield Plots
        Vectorfield Plots
        Surface Plots

   GEL Basics

        Values

              Numbers
              Booleans
              Strings
              Null

        Using Variables

              Setting Variables
              Built-in Variables
              Previous Result Variable

        Using Functions

              Defining Functions
              Variable Argument Lists
              Passing Functions to Functions
              Operations on Functions

        Absolute Value / Modulus
        Separator
        Modular Evaluation
        List of GEL Operators

   Programming with GEL

        Conditionals
        Loops

              While Loops
              For Loops
              Foreach Loops
              Break and Continue

        Sums and Products
        Comparison Operators
        Global Variables and Scope of Variables
        Parameter variables
        Returning
        References
        Lvalues

   Advanced Programming with GEL

        Error Handling
        Toplevel Syntax
        Returning Functions
        True Local Variables
        GEL Startup Procedure
        Loading Programs

   Matrices in GEL

        Entering Matrices
        Conjugate Transpose and Transpose Operator
        Linear Algebra

   Polynomials in GEL

        Using Polynomials

   Set Theory in GEL

        Using Sets

   List of GEL functions

        Commands
        Basic
        Parameters
        Constants
        Numeric
        Trigonometry
        Number Theory
        Matrix Manipulation
        Linear Algebra
        Combinatorics
        Calculus
        Functions
        Equation Solving
        Statistics
        Polynomials
        Set Theory
        Miscellaneous
        Symbolic Operations
        Plotting

   Example Programs in GEL
   Settings

        Output
        Precision
        Terminal
        Memory

   About Genius Mathematics Tool
     __________________________________________________________

Introduction

   The Genius Mathematics Tool application is a general calculator
   for use as a desktop calculator, an educational tool in
   mathematics, and is useful even for research. The language used
   in Genius Mathematics Tool is designed to be ‘mathematical’ in
   the sense that it should be ‘what you mean is what you get’. Of
   course that is not an entirely attainable goal. Genius
   Mathematics Tool features rationals, arbitrary precision
   integers and multiple precision floats using the GMP library.
   It handles complex numbers using cartesian notation. It has
   good vector and matrix manipulation and can handle basic linear
   algebra. The programming language allows user defined
   functions, variables and modification of parameters.

   Genius Mathematics Tool comes in two versions. One version is
   the graphical GNOME version, which features an IDE style
   interface and the ability to plot functions of one or two
   variables. The command line version does not require GNOME, but
   of course does not implement any feature that requires the
   graphical interface.

   This manual describes mostly the graphical version of the
   calculator, but the language is of course the same. The command
   line only version lacks the graphing capabilities and all other
   capabilities that require the graphical user interface.
     __________________________________________________________

Getting Started

To Start Genius Mathematics Tool

   You can start Genius Mathematics Tool in the following ways:

   Applications menu
          Depending on your operating system and version, the menu
          item for Genius Mathematics Tool could appear in a
          number of different places. It can be in the Education,
          Accessories, Office, Science, or similar submenu,
          depending on your particular setup. The menu item name
          you are looking for is Genius Math Tool. Once you locate
          this menu item click on it to start Genius Mathematics
          Tool.

   Run dialog
          Depending on your system installation the menu item may
          not be available. If it is not, you can open the Run
          dialog and execute gnome-genius.

   Command line
          To start the GNOME version of Genius Mathematics Tool
          execute gnome-genius from the command line.

          To start the command line only version, execute the
          following command: genius. This version does not include
          the graphical environment and some functionality such as
          plotting will not be available.
     __________________________________________________________

When You Start Genius

   When you start the GNOME edition of Genius Mathematics Tool,
   the window pictured in Figure 1 is displayed.

   [genius_window.png]

   Figure 1. Genius Mathematics Tool Window

   The Genius Mathematics Tool window contains the following
   elements:

   Menubar.
          The menus on the menubar contain all of the commands
          that you need to work with files in Genius Mathematics
          Tool. The File menu contains items for loading and
          saving items and creating new programs. The Load and
          Run... command does not open a new window for the
          program, but just executes the program directly. It is
          equivalent to the load command.

          The Calculator menu controls the calculator engine. It
          allows you to run the currently selected program or to
          interrupt the current calculation. You can also look at
          the full expression of the last answer (useful if the
          last answer was too large to fit onto the console), or
          you can view a listing of the values of all user defined
          variables. Finally it allows plotting functions using a
          user friendly dialog box.

          The other menus have same familiar functions as in other
          applications.

   Toolbar.
          The toolbar contains a subset of the commands that you
          can access from the menubar.

   Working area
          The working area is the primary method of interacting
          with the application.

          The working area initially has just the Console tab
          which is the main way of interacting with the
          calculator. Here you type expressions and the results
          are immediately returned after you hit the Enter key.

          Alternatively you can write longer programs and those
          can appear in separate tabs and can be stored in files
          for later retrieval.
     __________________________________________________________

Basic Usage

Using the Work Area

   Normally you interact with the calculator in the Console tab of
   the work area. If you are running the text only version then
   the console will be the only thing that is available to you. If
   you want to use Genius Mathematics Tool as a calculator only,
   just type in your expression here and it will be evaluated.

   Type your expression into the Console work area and press enter
   and the expression will be evaluated. Expressions are written
   in a language called GEL. The most simple GEL expression just
   looks like mathematics. For example
genius> 30*70 + 67^3.0 + ln(7) * (88.8/100)

   or
genius> 62734 + 812634 + 77^4 mod 5

   or
genius> | sin(37) - e^7 |

   or
genius> sum n=1 to 70 do 1/n

   (Last is the harmonic sum from 1 to 70)

   To get a list of functions and commands, type:
genius> help

   If you wish to get more help on a specific function, type:
genius> help FunctionName

   To view this manual, type:
genius> manual

   Suppose you have previously saved some GEL commands as a
   program to a file and you now want to execute them. To load
   this program from the file path/to/program.gel, type
genius> load path/to/program.gel

   Genius Mathematics Tool keeps track of the current directory.
   To list files in the current directory type ls, to change
   directory do cd directory as in the unix command shell.
     __________________________________________________________

To Create a New Program

   To start writing a new program, choose File->New Program. A new
   tab will appear in the work area. You can write a GEL program
   in this work area. Once you have written your program you can
   run it by Calculator->Run. This will execute your program and
   will display any output on the Console tab. Executing a program
   is equivalent of taking the text of the program and typing it
   into the console. The only difference is that this input is
   done independent of the console and just the output goes onto
   the console. Calculator->Run will always run the currently
   selected program even if you are on the Console tab. The
   currently selected program has its tab in bold type. To select
   a program, just click on its tab.

   To save the program you've just written, choose File->Save
   As...
     __________________________________________________________

To Open and Run a Program

   To open a file, choose File->Open. A new tab containing the
   file will appear in the work area. You can use this to edit the
   file.

   To run a program from a file, choose File->Load and Run....
   This will run the program without opening it in a separate tab.
   This is equivalent to the load command.
     __________________________________________________________

Plotting

   Plotting support is only available in the graphical GNOME
   version. All plotting accessible from the graphical interface
   is available from the Create Plot window. You can access this
   window by either clicking on the Plot button on the toolbar or
   selecting Plot from the Calculator menu. You can also access
   the plotting functionality by using the plotting functions of
   the GEL language. See the Chapter called GEL Basics to find out
   how to enter expressions that Genius understands.
     __________________________________________________________

Line Plots

   To graph real valued functions of one variable open the Create
   Plot window. You can also use the LinePlot function on the
   command line (see its documentation).

   Once you click the Plot button, a window opens up with some
   notebooks in it. You want to be in the Function line plot
   notebook tab, and inside you want to be on the Functions /
   Expressions notebook tab. See Figure 1.

   [line_plot.png]

   Figure 1. Create Plot Window

   Into the text boxes just type in expressions where x is the
   independent variable. You can also just give names of functions
   such as cos rather then having to type cos(x). You can graph up
   to ten functions. If you make a mistake and Genius cannot parse
   the input it will signify this with a warning icon on the right
   of the text input box where the error occurred, as well as
   giving you an error dialog. You can change the ranges of the
   dependent and independent variables in the bottom part of the
   dialog. Pressing the Plot button produces the graph shown in
   Figure 2.

   [line_plot_graph.png]

   Figure 2. Plot Window

   From here you can print out the plot, create encapsulated
   postscript or a PNG version of the plot or change the zoom. If
   the dependent axis was not set correctly you can have Genius
   fit it by finding out the extrema of the graphed functions.

   For plotting using the command line see the documentation of
   the LinePlot function.
     __________________________________________________________

Parametric Plots

   In the create plot window, you can also choose the Parametric
   notebook tab to create two dimensional parametric plots. This
   way you can plot a single parametric function. You can either
   specify the points as x and y, or giving a single complex
   number. See Figure 3.

   [parametric.png]

   Figure 3. Parametric Plot Tab

   An example of a parametric plot is given in Figure 3. Similar
   operations can be done on such graphs as can be done on the
   other line plots. For plotting using the command line see the
   documentation of the LinePlotParametric or LinePlotCParametric
   function.

   [parametric_graph.png]

   Figure 4. Parametric Plot
     __________________________________________________________

Slopefield Plots

   In the create plot window, you can also choose the Slope field
   notebook tab to create a two dimensional slope field plot.
   Similar operations can be done on such graphs as can be done on
   the other line plots. For plotting using the command line see
   the documentation of the SlopefieldPlot function.

   When a slope field is active, there is an extra Solver menu
   available, through which you can bring up the solver dialog.
   Here you can have Genius plot specific solutions for the given
   initial conditions. You can either specify initial conditions
   in the dialog, or you can click on the plot directly to specify
   the initial point. While the solver dialog is active, the
   zooming by clicking and dragging does not work. You have to
   close the dialog first if you want to zoom using the mouse.

   The solver uses the standard Runge-Kutta method. The plots will
   stay on the screen until cleared. The solver will stop whenever
   it reaches the boundary of the plot window. Zooming does not
   change the limits or parameters of the solutions, you will have
   to clear and redraw them with appropriate parameters. You can
   also use the SlopefieldDrawSolution function to draw solutions
   from the command line or programs.
     __________________________________________________________

Vectorfield Plots

   In the create plot window, you can also choose the Vector field
   notebook tab to create a two dimensional vector field plot.
   Similar operations can be done on such graphs as can be done on
   the other line plots. For plotting using the command line see
   the documentation of the VectorfieldPlot function.

   By default the direction and magnitude of the vector field is
   shown. To only show direction and not the magnitude, check the
   appropriate checkbox to normalize the arrow lengths.

   When a vector field is active, there is an extra Solver menu
   available, through which you can bring up the solver dialog.
   Here you can have Genius plot specific solutions for the given
   initial conditions. You can either specify initial conditions
   in the dialog, or you can click on the plot directly to specify
   the initial point. While the solver dialog is active, the
   zooming by clicking and dragging does not work. You have to
   close the dialog first if you want to zoom using the mouse.

   The solver uses the standard Runge-Kutta method. The plots will
   stay on the screen until cleared. Zooming does not change the
   limits or parameters of the solutions, you will have to clear
   and redraw them with appropriate parameters. You can also use
   the VectorfieldDrawSolution function to draw solutions from the
   command line or programs.
     __________________________________________________________

Surface Plots

   Genius can also plot surfaces. Select the Surface plot tab in
   the main notebook of the Create Plot window. Here you can
   specify a single expression which should use either x and y as
   real independent variables or z as a complex variable (where x
   is the real part of z and y is the imaginary part). For example
   to plot the modulus of the cosine function for complex
   parameters, you could enter |cos(z)|. This would be equivalent
   to |cos(x+1i*y)|. See Figure 5. For plotting using the command
   line see the documentation of the SurfacePlot function.

   [surface_graph.png]

   Figure 5. Surface Plot
     __________________________________________________________

GEL Basics

   GEL stands for Genius Extension Language. It is the language
   you use to write programs in Genius. A program in GEL is simply
   an expression that evaluates to a number. Genius Mathematics
   Tool can therefore be used as a simple calculator, or as a
   powerful theoretical research tool. The syntax is meant to have
   as shallow of a learning curve as possible, especially for use
   as a calculator.
     __________________________________________________________

Values

   Values in GEL can be numbers, Booleans or strings. Values can
   be used in calculations, assigned to variables and returned
   from functions, among other uses.
     __________________________________________________________

Numbers

   Integers are the first type of number in GEL. Integers are
   written in the normal way.
1234

   Hexadecimal and octal numbers can be written using C notation.
   For example:
0x123ABC
01234

   Or you can type numbers in an arbitrary base using
   <base>\<number>. Digits higher than 10 use letters in a similar
   way to hexadecimal. For example, a number in base 23 could be
   written:
23\1234ABCD

   The second type of GEL number is rationals. Rationals are
   simply achieved by dividing two integers. So one could write:
3/4

   to get three quarters. Rationals also accept mixed fraction
   notation. So in order to get one and three tenths you could
   write:
1 3/10

   The next type if number is floating point. These are entered in
   a similar fashion to C notation. You can use E, e or @ as the
   exponent delimiter. Note that using the exponent delimiter
   gives a float even if there is no decimal point in the number.
   Examples:
1.315
7.887e77
7.887e-77
.3
0.3
77e5

   When Genius prints a floating point number it will always
   append a .0 even if the number is whole. This is to indicate
   that floating point numbers are taken as imprecise quantities.
   When a number is written in the scientific notation, it is
   always a floating point number and thus Genius does not print
   the .0.

   The final type of number in gel is the complex numbers. You can
   enter a complex number as a sum of real and imaginary parts.
   The imaginary part ends with an i. Here are examples of
   entering complex numbers:
1+2i
8.01i
77*e^(1.3i)

   Important

   When entering imaginary numbers, a number must be in front of
   the i. If you use i by itself, Genius will interpret this as
   referring to the variable i. If you need to refer to i by
   itself, use 1i instead.

   In order to use mixed fraction notation with imaginary numbers
   you must have the mixed fraction in parentheses. (i.e., (1
   2/5)i)
     __________________________________________________________

Booleans

   Genius also supports native Boolean values. The two Boolean
   constants are defined as true and false; these identifiers can
   be used like any other variable. You can also use the
   identifiers True, TRUE, False and FALSE as aliases for the
   above.

   At any place where a Boolean expression is expected, you can
   use a Boolean value or any expression that produces either a
   number or a Boolean. If Genius needs to evaluate a number as a
   Boolean it will interpret 0 as false and any other number as
   true.

   In addition, you can do arithmetic with Boolean values. For
   example:
( (1 + true) - false ) * true

   is the same as:
( (true or true) or not false ) and true

   Only addition, subtraction and multiplication are supported. If
   you mix numbers with Booleans in an expression then the numbers
   are converted to Booleans as described above. This means that,
   for example:
1 == true

   always evaluates to true since 1 will be converted to true
   before being compared to true.
     __________________________________________________________

Strings

   Like numbers and Booleans, strings in GEL can be stored as
   values inside variables and passed to functions. You can also
   concatenate a string with another value using the plus
   operator. For example:
a=2+3;"The result is: "+a

   will create the string:
The result is: 5

   You can also use C-like escape sequences such as \n,\t,\b,\a
   and \r. To get a \ or " into the string you can quote it with a
   \. For example:
"Slash: \\ Quotes: \" Tabs: \t1\t2\t3"

   will make a string:
Slash: \ Quotes: " Tabs:        1       2       3

   In addition, you can use the library function string to convert
   anything to a string. For example:
string(22)

   will return
"22"

   Strings can also be compared with == (equal), != (not equal)
   and <=> (comparison) operators
     __________________________________________________________

Null

   There is a special value called null. No operations can be
   performed on it, and nothing is printed when it is returned.
   Therefore, null is useful when you do not want output from an
   expression. The value null can be obtained as an expression
   when you type ., the contant null or nothing. By nothing we
   mean that if you end an expression with a separator ;, it is
   equivalent to ending it with a separator followed by a null.

   Example:
x=5;.
x=5;

   Some functions return null when no value can be returned or an
   error happened. Also null is used as an empty vector or matrix,
   or an empty reference.
     __________________________________________________________

Using Variables

   Syntax:
VariableName

   Example:
genius> e
= 2.71828182846

   To evaluate a variable by itself, just enter the name of the
   variable. This will return the value of the variable. You can
   use a variable anywhere you would normally use a number or
   string. In addition, variables are necessary when defining
   functions that take arguments (see the Section called Defining
   Functions).

   Tip Using Tab completion


   You can use Tab completion to get Genius to complete variable
   names for you. Try typing the first few letters of the name and
   pressing Tab.

   Important Variable names are case sensitive


   The names of variables are case sensitive. That means that
   variables named hello, HELLO and Hello are all different
   variables.
     __________________________________________________________

Setting Variables

   Syntax:
<identifier> = <value>
<identifier> := <value>

   Example:
x = 3
x := 3

   To assign to a variable, use the = or := operators. These
   operators set the value of the variable and return the number
   you set, so you can do things like
a = b = 5

   The = and := operators can both be used to set variables. The
   difference between them is that the := operator always acts as
   an assignment operator, whereas the = operator may be
   interpreted as testing for equality when used in a context
   where a Boolean expression is expected.

   For issues regarding the scope of variables, see the Section
   called Global Variables and Scope of Variables in the Chapter
   called Programming with GEL.
     __________________________________________________________

Built-in Variables

   GEL has a number of built-in ‘variables’, such as e, pi or
   GoldenRatio. These are widely used constants with a preset
   value, and they cannot be assigned new values. There are a
   number of other built-in variables. See the Section called
   Constants in the Chapter called List of GEL functions for a
   full list.
     __________________________________________________________

Previous Result Variable

   The Ans and ans variables can be used to get the result of the
   last expression. For example, if you had performed some
   calculation, to add 389 to the result you could do:
Ans+389
     __________________________________________________________

Using Functions

   Syntax:
FunctionName(argument1, argument2, ...)

   Example:
Factorial(5)
cos(2*pi)
gcd(921,317)

   To evaluate a function, enter the name of the function,
   followed by the arguments (if any) to the function in
   parentheses. This will return the result of applying the
   function to its arguments. The number of arguments to the
   function is, of course, different for each function.

   There are many built-in functions, such as sin, cos and tan.
   You can use the help built-in function to get a list of
   available functions, or see the Chapter called List of GEL
   functions for a full listing.

   Tip Using Tab completion


   You can use Tab completion to get Genius to complete function
   names for you. Try typing the first few letters of the name and
   pressing Tab.

   Important Function names are case sensitive


   The names of functions are case sensitive. That means that
   functions named dosomething, DOSOMETHING and DoSomething are
   all different functions.
     __________________________________________________________

Defining Functions

   Syntax:
function <identifier>(<comma separated arguments>) = <function body>
<identifier> = (`() = <function body>)

   The ` is the backquote character, and signifies an anonymous
   function. By setting it to a variable name you effectively
   define a function.

   A function takes zero or more comma separated arguments, and
   returns the result of the function body. Defining your own
   functions is primarily a matter of convenience; one possible
   use is to have sets of functions defined in GEL files which
   Genius can load in order to make available. Example:
function addup(a,b,c) = a+b+c

   then addup(1,4,9) yields 14
     __________________________________________________________

Variable Argument Lists

   If you include ... after the last argument name in the function
   declaration, then Genius will allow any number of arguments to
   be passed in place of that argument. If no arguments were
   passed then that argument will be set to null. Otherwise, it
   will be a horizontal vector containing all the arguments. For
   example:
function f(a,b...) = b

   Then f(1,2,3) yields [2,3], while f(1) yields a null.
     __________________________________________________________

Passing Functions to Functions

   In Genius, it is possible to pass a function as an argument to
   another function. This can be done using either ‘function
   nodes’ or anonymous functions.

   If you do not enter the parentheses after a function name,
   instead of being evaluated, the function will instead be
   returned as a ‘function node’. The function node can then be
   passed to another function. Example:
function f(a,b) = a(b)+1;
function b(x) = x*x;
f(b,2)

   To pass functions which are not defined, you can use an
   anonymous function (see the Section called Defining Functions).
   That is, you want to pass a function without giving it a name.
   Syntax:
function(<comma separated arguments>) = <function body>
`(<comma separated arguments>) = <function body>

   Example:
function f(a,b) = a(b)+1;
f(`(x) = x*x,2)

   This will return 5.
     __________________________________________________________

Operations on Functions

   Some functions allow arithmetic operations, and some single
   argument functions such as exp or ln, to operate on the
   function. For example,
exp(sin*cos+4)

   will return a function that takes x and returns
   exp(sin(x)*cos(x)+4). It is functionally equivalent to typing
`(x) = exp(sin(x)*cos(x)+4)

   This operation can be useful when quickly defining functions.
   For example to create a function called f to perform the above
   operation, you can just type:
f = exp(sin*cos+4)

   It can also be used in plotting. For example, to plot sin
   squared you can enter:
LinePlot(sin^2)

   Warning

   Not all functions can be used in this way. For example, when
   you use a binary operation the functions must take the same
   number of arguments.
     __________________________________________________________

Absolute Value / Modulus

   You can make an absolute value of something by putting the |'s
   around it. For example:
   |a-b|

   In case the expression is a complex number the result will be
   the modulus (distance from the origin). For example: |3 *
   e^(1i*pi)| returns 3.
     __________________________________________________________

Separator

   In GEL if you want to type more than one command you have to
   use the ; operator, which is a way to separate expressions,
   such a combined expression will return whatever is the result
   of the last one, so suppose you type the following:
3 ; 5

   This expression will yield 5.

   This will require some parenthesizing to make it unambiguous
   sometimes, especially if the ; is not the top most primitive.
   This slightly differs from other programming languages where
   the ; is a terminator of statements, whereas in GEL it’s
   actually a binary operator. If you are familiar with pascal
   this should be second nature. However genius can let you
   pretend it is a terminator somewhat, if a ; is found at the end
   of a parenthesis or a block, genius will itself append a null
   node to it as if you would have written ;null. This is useful
   in case you do not want to return a value from say a loop, or
   if you handle the return differently. Note that it will
   slightly slow down the code if it is executed too often as
   there is one more operator involved.
     __________________________________________________________

Modular Evaluation

   Genius implements modular arithmetic. To use it you just add
   "mod <integer>" after the expression. Example:
   2^(5!) * 3^(6!) mod 5

   It could be possible to do modular arithmetic by computing with
   integers and then modding in the end with the % operator, but
   that may be time consuming if not impossible when working with
   larger numbers. For example 10^(10^10) % 6 will simply not work
   (the exponent will be too large), while 10^(10^10) mod 6 is
   instanteneous.

   You can calculate the inverses of numbers mod some integer by
   just using rational numbers (of course the inverse has to
   exist). Examples:
10^-1 mod 101
1/10 mod 101

   You can also do modular evaluation with matrices including
   taking inverses, powers and dividing. Example:
A = [1,2;3,4]
B = A^-1 mod 5
A*B mod 5

   This should yield the identity matrix as B will be the inverse
   of A mod 5.

   Some functions such as sqrt or log work in a different way when
   in modulo mode. These will then work like their discrete
   versions working within the ring of integers you selected. For
   example:
genius> sqrt(4) mod 7
=
[2, 5]
genius> 2*2 mod 7
= 4

   sqrt will actually return all the possible square roots.
     __________________________________________________________

List of GEL Operators

   Everything in gel is really just an expression. Expressions are
   stringed together with different operators. As we have seen,
   even the separator is simply a binary operator in GEL. Here is
   a list of the operators in GEL.

a;b

          The separator, just evaluates both a and b, but returns
          only the result of b.

a=b

          The assignment operator. This assigns b to a (a must be
          a valid lvalue) (note however that this operator may be
          translated to == if used in a place where boolean
          expression is expected)

a:=b

          The assignment operator. Assigns b to a (a must be a
          valid lvalue). This is different from = because it never
          gets translated to a ==.

|a|

          Absolute value or modulus (if a is a complex number).

          See Mathworld for more information.

a^b

          Exponentiation, raises a to the bth power.

a.^b

          Element by element exponentiation. Raise each element of
          a matrix a to the bth power. Or if b is a matrix of the
          same size as a, then do the operation element by
          element. If a is a number and b is a matrix then it
          creates matrix of the same size as b with a raised to
          all the different powers in b.

a+b

          Addition. Adds two numbers, matrices, functions or
          strings. If you add a string to anything the result will
          just be a string.

a-b

          Subtraction. Subtract two numbers, matrices or
          functions.

a*b

          Multiplication. This is the normal matrix
          multiplication.

a.*b

          Element by element multiplication if a and b are
          matrices.

a/b

          Division.

a./b

          Element by element division.

a\b

          Back division. That is this is the same as b/a.

a.\b

          Element by element back division.

a%b

          The mod operator. This does not turn on the modular
          mode, but just returns the remainder of a/b.

a.%b

          Element by element the mod operator. Returns the
          remainder after element by element integer a./b.

a mod b

          Modular evaluation operator. The expression a is
          evaluated modulo b. See the Section called Modular
          Evaluation. Some functions and operators behave
          differently modulo an integer.

a!

          Factorial operator. This is like 1*...*(n-2)*(n-1)*n.

a!!

          Double factorial operator. This is like
          1*...*(n-4)*(n-2)*n.

a==b

          Equality operator (returns true or false).

a!=b

          Inequality operator, returns true if a does not equal b
          else returns false.

a<>b

          Alternative inequality operator, returns true if a does
          not equal b else returns false.

a<=b

          Less than or equal operator, returns true if a is less
          than or equal to b else returns false.

a>=b

          Greater than or equal operator, returns true if a is
          greater than or equal to b else returns false.

a<=>b

          Comparison operator. If a is equal to b it returns 0, if
          a is less than b it returns -1 and if a is greater than
          b it returns 1.

a and b

          Logical and.

a or b

          Logical or.

a xor b

          Logical xor.

not a

          Logical not.

-a

          Negation operator.

&a

          Variable referencing (to pass a reference to something).
          See the Section called References in the Chapter called
          Programming with GEL.

*a

          Variable dereferencing (to access a referenced varible).
          See the Section called References in the Chapter called
          Programming with GEL.

a'

          Matrix conjugate transpose.

a.'

          Matrix transpose, does not conjugate the entries.

a@(b,c)

          Get element of a matrix in row b and column c. If b, c
          are vectors, then this gets the corresponding rows
          columns or submatrices.

a@(b,)

          Get row of a matrix (or rows if b is a vector).

a@(b,:)

          Same as above.

a@(,c)

          Get column of a matrix (or columns if c is a vector).

a@(:,c)

          Same as above.

a@(b)

          Get an element from a matrix treating it as a vector.
          This will traverse the matrix row-wise.

a:b

          Build a vector from a to b (or specify a row, column
          region for the @ operator). For example to get rows 2 to
          4 of matrix A we could do

A@(2:4,)

          as 2:4 will return a vector [2,3,4].

a:b:c

          Build a vector from a to c with b as a step. That is for
          example

genius> 1:2:9
=
`[1, 3, 5, 7, 9]

(a)i

          Make a imaginary number (multiply a by the imaginary).
          Note that normally the number i is written as 1i. So the
          above is equal to

(a)*1i

`a

          Quote an identifier so that it doesn't get evaluated. Or
          quote a matrix so that it doesn't get expanded.

a swapwith b

          Swap value of a with the value of b. Currently does not
          operate on ranges of matrix elements. It returns null.
          Available from version 1.0.13.

increment a

          Increment the variable a by 1. If a is a matrix, then
          increment each element. This is equivalent to a=a+1, but
          it is somewhat faster. It returns null. Available from
          version 1.0.13.

increment a by b

          Increment the variable a by b. If a is a matrix, then
          increment each element. This is equivalent to a=a+b, but
          it is somewhat faster. It returns null. Available from
          version 1.0.13.

   Note

   The @() operator makes the : operator most useful. With this
   you can specify regions of a matrix. So that a@(2:4,6) is the
   rows 2,3,4 of the column 6. Or a@(,1:2) will get you the first
   two columns of a matrix. You can also assign to the @()
   operator, as long as the right value is a matrix that matches
   the region in size, or if it is any other type of value.

   Note

   The comparison operators (except for the <=> operator which
   behaves normally), are not strictly binary operators, they can
   in fact be grouped in the normal mathematical way, e.g.:
   (1<x<=y<5) is a legal boolean expression and means just what it
   should, that is (1<x and x≤y and y<5)

   Note

   The unitary minus operates in a different fashion depending on
   where it appears. If it appears before a number it binds very
   closely, if it appears in front of an expression it binds less
   than the power and factorial operators. So for example -1^k is
   really (-1)^k, but -foo(1)^k is really -(foo(1)^k). So be
   careful how you use it and if in doubt, add parentheses.
     __________________________________________________________

Programming with GEL

Conditionals

   Syntax:
if <expression1> then <expression2> [else <expression3>]

   If else is omitted, then if the expression1 yields false or 0,
   NULL is returned.

   Examples:
if(a==5)then(a=a-1)
if b<a then b=a
if c>0 then c=c-1 else c=0
a = ( if b>0 then b else 1 )

   Note that = will be translated to == if used inside the
   expression for if, so
if a=5 then a=a-1

   will be interpreted as:
if a==5 then a:=a-1
     __________________________________________________________

Loops

While Loops

   Syntax:
while <expression1> do <expression2>
until <expression1> do <expression2>
do <expression2> while <expression1>
do <expression2> until <expression1>

   These are similar to other languages. However, as in gel it is
   simply an expression that must have some return value, these
   constructs will simply return the result of the last iteration
   or NULL if no iteration was done. In the boolean expression, =
   is translated into == just as for the if statement.
     __________________________________________________________

For Loops

   Syntax:
for <identifier> = <from> to <to> do <body>
for <identifier> = <from> to <to> by <increment> do <body>

   Loop with identifier being set to all values from <from> to
   <to>, optionally using an increment other than 1. These are
   faster, nicer and more compact than the normal loops such as
   above, but less flexible. The identifier must be an identifier
   and can't be a dereference. The value of identifier is the last
   value of identifier, or <from> if body was never evaluated. The
   variable is guaranteed to be initialized after a loop, so you
   can safely use it. Also the <from>, <to> and <increment> must
   be non complex values. The <to> is not guaranteed to be hit,
   but will never be overshot, for example the following prints
   out odd numbers from 1 to 19:
for i = 1 to 20 by 2 do print(i)
     __________________________________________________________

Foreach Loops

   Syntax:
   for <identifier> in <matrix> do <body>

   For each element in the matrix, going row by row from left to
   right we execute the body with the identifier set to the
   current element. To print numbers 1,2,3 and 4 in this order you
   could do:
for n in [1,2:3,4] do print(n)

   If you wish to run through the rows and columns of a matrix,
   you can use the RowsOf and ColumnsOf functions which return a
   vector of the rows or columns of the matrix. So,
for n in RowsOf ([1,2:3,4]) do print(n)

   will print out [1,2] and then [3,4].
     __________________________________________________________

Break and Continue

   You can also use the break and continue commands in loops. The
   continue continue command will restart the current loop at its
   next iteration, while the break command exits the current loop.
while(<expression1>) do (
  if(<expression2>) break
  else if(<expression3>) continue;
  <expression4>
)
     __________________________________________________________

Sums and Products

   Syntax:
sum <identifier> = <from> to <to> do <body>
sum <identifier> = <from> to <to> by <increment> do <body>
sum <identifier> in <matrix> do <body>
prod <identifier> = <from> to <to> do <body>
prod <identifier> = <from> to <to> by <increment> do <body>
prod <identifier> in <matrix> do <body>

   If you substitute for with sum or prod, then you will get a sum
   or a product instead of a for loop. Instead of returning the
   last value, these will return the sum or the product of the
   values respectively.

   If no body is executed (for example sum i=1 to 0 do ...) then
   sum returns 0 and prod returns 1 as is the standard convention.
     __________________________________________________________

Comparison Operators

   The following standard comparison operators are supported in
   GEL and have the obvious meaning: ==, >=, <=, !=, <>, <, >.
   They return true or false. The operators != and <> are the same
   thing and mean "is not equal to". GEL also supports the
   operator <=>, which returns -1 if left side is smaller, 0 if
   both sides are equal, 1 if left side is larger.

   Normally = is translated to == if it happens to be somewhere
   where GEL is expecting a condition such as in the if condition.
   For example
if a=b then c
if a==b then c

   are the same thing in GEL. However you should really use == or
   := when you want to compare or assign respectively if you want
   your code to be easy to read and to avoid mistakes.

   All the comparison operators (except for the <=> operator which
   behaves normally), are not strictly binary operators, they can
   in fact be grouped in the normal mathematical way, e.g.:
   (1<x<=y<5) is a legal boolean expression and means just what it
   should, that is (1<x and x≤y and y<5)

   To build up logical expressions use the words not, and, or,
   xor. The operators or and and are special beasts as they
   evaluate their arguments one by one, so the usual trick for
   conditional evaluation works here as well. For example, 1 or
   a=1 will not set a=1 since the first argument was true.
     __________________________________________________________

Global Variables and Scope of Variables

   GEL is a dynamically scoped language. We will explain what this
   means below. That is, normal variables and functions are
   dynamically scoped. The exception are parameter variables,
   which are always global.

   Like most programming languages, GEL has different types of
   variables. Normally when a variable is defined in a function,
   it is visible from that function and from all functions that
   are called (all higher contexts). For example, suppose a
   function f defines a variable a and then calls function g. Then
   function g can reference a. But once f returns, the variable a
   goes out of scope. For example, the following code will print
   out 5. The function g cannot be called on the top level
   (outside f as a will not be defined).
function f() = (a:=5; g());
function g() = print(a);
f();

   If you define a variable inside a function it will override any
   variables defined in calling functions. For example, we modify
   the above code and write:
function f() = (a:=5; g());
function g() = print(a);
a:=10;
f();

   This code will still print out 5. But if you call g outside of
   f then you will get a printout of 10. Note that setting a to 5
   inside f does not change the value of a at the top (global)
   level, so if you now check the value of a it will still be 10.

   Function arguments are exactly like variables defined inside
   the function, except that they are initialized with the value
   that was passed to the function. Other than this point, they
   are treated just like all other variables defined inside the
   function.

   Functions are treated exactly like variables. Hence you can
   locally redefine functions. Normally (on the top level) you
   cannot redefine protected variables and functions. But locally
   you can do this. Consider the following session:
genius> function f(x) = sin(x)^2
= (`(x)=(sin(x)^2))
genius> function f(x) = sin(x)^2
= (`(x)=(sin(x)^2))
genius> function g(x) = ((function sin(x)=x^10);f(x))
= (`(x)=((sin:=(`(x)=(x^10)));f(x)))
genius> g(10)
= 1e20

   Functions and variables defined at the top level are considered
   global. They are visible from anywhere. As we said the
   following function f will not change the value of a to 5.
a=6;
function f() = (a:=5);
f();

   Sometimes, however, it is necessary to set a global variable
   from inside a function. When this behaviour is needed, use the
   set function. Passing a string or a quoted identifier to this
   function sets the variable globally (on the top level). For
   example, to set a to the value 3 you could call:
set(`a,3)

   or:
set("a",3)

   The set function always sets the toplevel global. There is no
   way to set a local variable in some function from a subroutine.
   If this is required, must use passing by reference.

   So to recap in a more technical language: Genius operates with
   different numbered contexts. The top level is the context 0
   (zero). Whenever a function is entered, the context is raised,
   and when the function returns the context is lowered. A
   function or a variable is always visible from all higher
   numbered contexts. When a variable was defined in a lower
   numbered context, then setting this variable has the effect of
   creating a new local variable in the current context number and
   this variable will now be visible from all higher numbered
   contexts.

   There are also true local variables, which are not seen from
   anywhere but the current context. Also when returning functions
   by value it may reference variables not visible from higher
   context and this may be a problem. See the sections True Local
   Variables and Returning Functions.
     __________________________________________________________

Parameter variables

   As we said before, there exist special variables called
   parameters that exist in all scopes. To declare a parameter
   called foo with the initial value 1, we write
parameter foo = 1

   From then on, foo is a strictly global variable. Setting foo
   inside any function will modify the variable in all contexts,
   that is, functions do not have a private copy of parameters.

   When you undefine a parameter using the undefine function, it
   stops being a parameter.

   Some parameters are built-in and modify the behaviour of
   genius.
     __________________________________________________________

Returning

   Normally a function is one or several expressions separated by
   a semicolon, and the value of the last expression is returned.
   This is fine for simple functions, but sometimes you do not
   want a function to return the last thing calculated. You may,
   for example, want to return from a middle of a function. In
   this case, you can use the return keyword. return takes one
   argument, which is the value to be returned.

   Example:
function f(x) = (
  y=1;
  while true do (
    if x>50 then return y;
    y=y+1;
    x=x+1
  )
)
     __________________________________________________________

References

   It may be necessary for some functions to return more than one
   value. This may be accomplished by returning a vector of
   values, but many times it is convenient to use passing a
   reference to a variable. You pass a reference to a variable to
   a function, and the function will set the variable for you
   using a dereference. You do not have to use references only for
   this purpose, but this is their main use.

   When using functions that return values through references in
   the argument list, just pass the variable name with an
   ampersand. For example the following code will compute an
   eigenvalue of a matrix A with initial eigenvector guess x, and
   store the computed eigenvector into the variable named v:
RayleighQuotientIteration (A,x,0.001,100,&v)

   The details of how references work and the syntax is similar to
   the C language. The operator & references a variable and *
   dereferences a variable. Both can only be applied to an
   identifier, so **a is not a legal expression in GEL.

   References are best explained by an example:
a=1;
b=&a;
*b=2;

   now a contains 2. You can also reference functions:
function f(x) = x+1;
t=&f;
*t(3)

   gives us 4.
     __________________________________________________________

Lvalues

   An lvalue is the left hand side of an assignment. In other
   words, an lvalue is what you assign something to. Valid lvalues
   are:

   a
          Identifier. Here we would be setting the variable of
          name a.

   *a
          Dereference of an identifier. This will set whatever
          variable a points to.

   a@(<region>)
          A region of a matrix. Here the region is specified
          normally as with the regular @() operator, and can be a
          single entry, or an entire region of the matrix.

   Examples:
a:=4
*tmp := 89
a@(1,1) := 5
a@(4:8,3) := [1,2,3,4,5]'

   Note that both := and = can be used interchangably. Except if
   the assignment appears in a condition. It is thus always safer
   to just use := when you mean assignment, and == when you mean
   comparison.
     __________________________________________________________

Advanced Programming with GEL

Error Handling

   If you detect an error in your function, you can bail out of
   it. For normal errors, such as wrong types of arguments, you
   can fail to compute the function by adding the statement
   bailout. If something went really wrong and you want to
   completely kill the current computation, you can use exception.

   For example if you want to check for arguments in your
   function. You could use the following code.
function f(M) = (
  if not IsMatrix (M) then (
    error ("M not a matrix!");
    bailout
  );
  ...
)
     __________________________________________________________

Toplevel Syntax

   The synatax is slightly different if you enter statements on
   the top level versus when they are inside parentheses or inside
   functions. On the top level, enter acts the same as if you
   press return on the command line. Therefore think of programs
   as just sequence of lines as if were entered on the command
   line. In particular, you do not need to enter the separator at
   the end of the line (unless it is of course part of several
   statements inside parentheses).

   The following code will produce an error when entered on the
   top level of a program, while it will work just fine in a
   function.
if Something() then
  DoSomething()
else
  DoSomethingElse()

   The problem is that after Genius Mathematics Tool sees the end
   of line after the second line, it will decide that we have
   whole statement and it will execute it. After the execution is
   done, Genius Mathematics Tool will go on to the next line, it
   will see else, and it will produce a parsing error. To fix
   this, use parentheses. Genius Mathematics Tool will not be
   satisfied until it has found that all parentheses are closed.
if Something() then (
  DoSomething()
) else (
  DoSomethingElse()
)
     __________________________________________________________

Returning Functions

   It is possible to return functions as value. This way you can
   build functions that construct special purpose functions
   according to some parameters. The tricky bit is what variables
   does the function see. The way this works in GEL is that when a
   function returns another function, all identifiers referenced
   in the function body that went out of scope are prepended a
   private dictionary of the returned function. So the function
   will see all variables that were in scope when it was defined.
   For example, we define a function that returns a function that
   adds 5 to its argument.
function f() = (
  k = 5;
  `(x) = (x+k)
)

   Notice that the function adds k to x. You could use this as
   follows.
g = f();
g(5)

   And g(5) should return 10.

   One thing to note is that the value of k that is used is the
   one that's in effect when the f returns. For example:
function f() = (
  k := 5;
  function r(x) = (x+k);
  k := 10;
  r
)

   will return a function that adds 10 to its argument rather than
   5. This is because the extra dictionary is created only when
   the context in which the function was defined ends, which is
   when the function f returns. This is consistent with how you
   would expect the function r to work inside the function f
   according to the rules of scope of variables in GEL. Only those
   variables are added to the extra dictionary that are in the
   context that just ended and no longer exists. Variables used in
   the function that are in still valid contexts will work as
   usual, using the current value of the variable. The only
   difference is with global variables and functions. All
   identifiers that referenced global variables at time of the
   function definition are not added to the private dictionary.
   This is to avoid much unnecessary work when returning functions
   and would rarely be a problem. For example, suppose that you
   delete the "k=5" from the function f, and at the top level you
   define k to be say 5. Then when you run f, the function r will
   not put k into the private dictionary because it was global
   (toplevel) at the time of definition of r.

   Sometimes it is better to have more control over how variables
   are copied into the private dictionary. Since version 1.0.7,
   you can specify which variables are copied into the private
   dictionary by putting extra square brackets after the arguments
   with the list of variables to be copied separated by commas. If
   you do this, then variables are copied into the private
   dictionary at time of the function definition, and the private
   dictionary is not touched afterwards. For example
function f() = (
  k := 5;
  function r(x) [k] = (x+k);
  k := 10;
  r
)

   will return a function that when called will add 5 to its
   argument. The local copy of k was created when the function was
   defined.

   When you want the function to not have any private dictionary
   when put empty square brackets after the argument list. Then no
   private dictionary will be created at all. Doing this is good
   to increase efficiency when a private dictionary is not needed
   or when you want the function to lookup all variables as it
   sees them when called. For example suppose you want the
   function returned from f to see the value of k from the
   toplevel despite there being a local variable of the same name
   during definition. So the code
function f() = (
  k := 5;
  function r(x) [] = (x+k);
  r
);
k := 10;
g = f();
g(10)

   will return 20 and not 15, which would happen if k with a value
   of 5 was added to the private dictionary.
     __________________________________________________________

True Local Variables

   When passing functions into other functions, the normal scoping
   of variables might be undesired. For example:
k := 10;
function r(x) = (x+k);
function f(g,x) = (
  k := 5;
  g(x)
);
f(r,1)

   you probably want the function r when passed as g into f to see
   k as 10 rather than 5, so that the code returns 11 and not 6.
   However, as written, the function when executed will see the k
   that is equal to 5. There are two ways to solve this. One would
   be to have r get k in a private dictionary using the square
   bracket notation section Returning Functions.

   But there is another solution. Since version 1.0.7 there are
   true local variables. These are variables that are visible only
   from the current context and not from any called functions. We
   could define k as a local variable in the function f. To do
   this add a local statement as the first statement in the
   function (it must always be the first statement in the
   function). You can also make any arguments be local variables
   as well. That is,
function f(g,x) = (
  local g,x,k;
  k := 5;
  g(x)
);

   Then the code will work as expected and prints out 11. Note
   that the local statement initializes all the refereced
   variables (except for function arguments) to a null.

   If all variables are to be created as locals you can just pass
   an asterix instead of a list of variables. In this case the
   variables will not be initialized until they are actually set
   of course. So the following definition of f will also work:
function f(g,x) = (
  local *;
  k := 5;
  g(x)
);

   It is good practice that all functions that take other
   functions as arguments use local variables. This way the passed
   function does not see implementation details and get confused.
     __________________________________________________________

GEL Startup Procedure

   First the program looks for the installed library file (the
   compiled version lib.cgel) in the installed directory, then it
   looks into the current directory, and then it tries to load an
   uncompiled file called ~/.geniusinit.

   If you ever change the the library its installed place, you’ll
   have to first compile it with genius --compile loader.gel >
   lib.cgel
     __________________________________________________________

Loading Programs

   Sometimes you have a larger program that you wrote into a file
   and want to read in that file. In these situations, you have
   two options. You can keep the functions you use most inside the
   ~/.geniusinit file. Or if you want to load up a file in a
   middle of a session (or from within another file), you can type
   load <list of filenames> at the prompt. This has to be done on
   the top level and not inside any function or whatnot, and it
   cannot be part of any expression. It also has a slightly
   different syntax than the rest of genius, more similiar to a
   shell. You can enter the file in quotes. If you use the ''
   quotes, you will get exactly the string that you typed, if you
   use the "" quotes, special characters will be unescaped as they
   are for strings. Example:
load program1.gel program2.gel
load "Weird File Name With SPACES.gel"

   There are also cd, pwd and ls commands built in. cd will take
   one argument, ls will take an argument that is like the glob in
   the unix shell (i.e., you can use wildcards). pwd takes no
   arguments. For example:
cd directory_with_gel_programs
ls *.gel
     __________________________________________________________

Matrices in GEL

   Genius has support for vectors and matrices and a sizable
   library of matrix manipulation and linear algebra functions.
     __________________________________________________________

Entering Matrices

   To enter matrixes, you can use one of the following two
   syntaxes. You can either enter the matrix on one line,
   separating values by commas and rows by semicolons. Or you can
   enter each row on one line, separating values by commas. You
   can also just combine the two methods. So to enter a 3x3 matrix
   of numbers 1-9 you could do
[1,2,3;4,5,6;7,8,9]

   or
[1, 2, 3
 4, 5, 6
 7, 8, 9]

   Do not use both ';' and return at once on the same line though.

   You can also use the matrix expansion functionality to enter
   matricies. For example you can do:
a = [ 1, 2, 3
      4, 5, 6
      7, 8, 9]
b = [ a,  10
      11, 12]

   and you should get
[1,   2,  3, 10
 4,   5,  6, 10
 7,   8,  9, 10
 11, 11, 11, 12]

   similiarly you can build matricies out of vectors and other
   stuff like that.

   Another thing is that non-specified spots are initialized to 0,
   so
[1, 2, 3
 4, 5
 6]

   will end up being
[1, 2, 3
 4, 5, 0
 6, 0, 0]

   When matrices are evaluated, they are evaluated and traversed
   row-wise. This is just like the M@(j) operator which traverses
   the matrix row-wise.

   Note

   Be careful about using returns for expressions inside the [ ]
   brackets, as they have a slightly different meaning there. You
   will start a new row.
     __________________________________________________________

Conjugate Transpose and Transpose Operator

   You can conjugate transpose a matrix by using the ' operator.
   That is the entry in the ith column and the jth row will be the
   complex conjugate of the entry in the jth column and the ith
   row of the original matrix. For example:
[1,2,3]*[4,5,6]'

   We transpose the second vector to make matrix multiplication
   possible. If you just want to transpose a matrix without
   conjugating it, you would use the .' operator. For example:
[1,2,3]*[4,5,6i].'

   Note that normal transpose, that is the .' operator, is much
   faster and will not create a new copy of the matrix in memory.
   The conjugate transpose does create a new copy unfortunately.
   It is recommended to always use the .' operator when working
   with real matrices and vectors.
     __________________________________________________________

Linear Algebra

   Genius implements many useful linear algebra and matrix
   manipulation routines. See the Linear Algebra and Matrix
   Manipulation sections of the GEL function listing.

   The linear algebra routines implemented in GEL do not currently
   come from a well tested numerical package, and thus should not
   be used for critical numerical computation. On the other hand,
   Genius implements very well many linear algebra operations with
   rational and integer coefficients. These are inherently exact
   and in fact will give you much better results than common
   double precision routines for linear algebra.

   For example, it is pointless to compute the rank and nullspace
   of a floating point matrix since for all practical purposes, we
   need to consider the matrix as having some slight errors. You
   are likely to get a different result than you expect. The
   problem is that under a small perturbation every matrix is of
   full rank and invertible. If the matrix however is of rational
   numbers, then the rank and nullspace are always exact.

   In general when Genius computes the basis of a certain
   vectorspace (for example with the NullSpace) it will give the
   basis as a matrix, in which the columns are the vectors of the
   basis. That is, when Genius talks of a linear subspace it means
   a matrix whose column space is the given linear subspace.

   It should be noted that Genius can remember certain properties
   of a matrix. For example, it will remember that a matrix is in
   row reduced form. If many calls are made to functions that
   internally use row reduced form of the matrix, we can just row
   reduce the matrix beforehand once. Successive calls to rref
   will be very fast.
     __________________________________________________________

Polynomials in GEL

   Currently Genius can handle polynomials of one variable written
   out as vectors, and do some basic operations with these. It is
   planned to expand this support further.
     __________________________________________________________

Using Polynomials

   Currently polynomials in one variable are just horizontal
   vectors with value only nodes. The power of the term is the
   position in the vector, with the first position being 0. So,
[1,2,3]

   translates to a polynomial of
1 + 2*x + 3*x^2

   You can add, subtract and multiply polynomials using the
   AddPoly, SubtractPoly, and MultiplyPoly functions respectively.
   You can print a polynomial using the PolyToString function. For
   example,
PolyToString([1,2,3],"y")

   gives
3*y^2 + 2*y + 1

   You can also get a function representation of the polynomial so
   that you can evaluate it. This is done by using PolyToFunction,
   which returns an anonymous function.
f = PolyToFunction([0,1,1])
f(2)

   It is also possible to find roots of polynomials of degrees 1
   through 4 by using the function PolynomialRoots, which calls
   the appropriate formula function. Higher degree polynomials
   must be converted to functions and solved numerically using a
   function such as FindRootBisection, FindRootFalsePosition,
   FindRootMullersMethod, or FindRootSecant.

   See the Section called Polynomials in the Chapter called List
   of GEL functions in the function list for the rest of functions
   acting on polynomials.
     __________________________________________________________

Set Theory in GEL

   Genius has some basic set theoretic functionality built in.
   Currently a set is just a vector (or a matrix). Every distinct
   object is treated as a different element.
     __________________________________________________________

Using Sets

   Just like vectors, objects in sets can include numbers,
   strings, null, matrices and vectors. It is planned in the
   future to have a dedicated type for sets, rather than using
   vectors. Note that floating point numbers are distinct from
   integers, even if they appear the same. That is, Genius will
   treat 0 and 0.0 as two distinct elements. The null is treated
   as an empty set.

   To build a set out of a vector, use the MakeSet function.
   Currently, it will just return a new vector where every element
   is unique.
genius> MakeSet([1,2,2,3])
= [1, 2, 3]

   Similarly there are functions Union, Intersection, SetMinus,
   which are rather self explanatory. For example:
genius> Union([1,2,3], [1,2,4])
= [1, 2, 4, 3]

   Note that no order is guaranteed for the return values. If you
   wish to sort the vector you should use the SortVector function.

   For testing membership, there are functions IsIn and IsSubset,
   which return a boolean value. For example:
genius> IsIn (1, [0,1,2])
= true

   The input IsIn(x,X) is of course equivalent to IsSubset([x],X).
   Note that since the empty set is a subset of every set,
   IsSubset(null,X) is always true.
     __________________________________________________________

List of GEL functions

   To get help on a specific function from the console type:
help FunctionName
     __________________________________________________________

Commands

   help

help

help FunctionName

          Print help (or help on a function/command).

   load

load "file.gel"

          Load a file into the interpretor. The file will execute
          as if it were typed onto the command line.

   cd

cd /directory/name

          Change working directory to /directory/name.

   pwd

pwd

          Print the current working directory.

   ls

ls

          List files in the current directory.

   plugin

plugin plugin_name

          Load a plugin. Plugin of that name must be installed on
          the system in the proper directory.
     __________________________________________________________

Basic

   AskButtons

AskButtons (query)

AskButtons (query, button1, ...)

          Asks a question and presents a list of buttons to the
          user (or a menu of options in text mode). Returns the
          1-based index of the button pressed. That is, returns 1
          if the first button was pressed, 2 if the second button
          was pressed, and so on. If the user closes the window
          (or simply hits enter in text mode), then null is
          returned. The execution of the program is blocked until
          the user responds.

   AskString

AskString (query)

AskString (query, default)

          Asks a question and lets the user enter a string which
          it then returns. If the user cancels or closes the
          window, then null is returned. The execution of the
          program is blocked until the user responds. If default
          is given, then it is pre-typed in for the user to just
          press enter on.

   Compose

Compose (f,g)

          Compose two functions and return a function that is the
          composition of f and g.

   ComposePower

ComposePower (f,n,x)

          Compose and execute a function with itself n times,
          passing x as argument. Returning x if n equals 0.
          Example:

genius> function f(x) = x^2 ;
genius> ComposePower (f,3,7)
= 5764801
genius> f(f(f(7)))
= 5764801

   Evaluate

Evaluate (str)

          Parses and evaluates a string.

   GetCurrentModulo

GetCurrentModulo

          Get current modulo from the context outside the
          function. That is, if outside of the function was
          executed in modulo (using mod) then this returns what
          this modulo was. Normally the body of the function
          called is not executed in modular arithmetic, and this
          builtin function makes it possible to make GEL functions
          aware of modular arithmetic.

   Identity

Identity (x)

          Identity function, returns its argument.

   IntegerFromBoolean

IntegerFromBoolean (bval)

          Make integer (0 for false or 1 for true) from a boolean
          value. bval can also be a number in which case a
          non-zero value will be interpreted as true and zero will
          be interpreted as false.

   IsBoolean

IsBoolean (arg)

          Check if argument is a boolean (and not a number).

   IsDefined

IsDefined (id)

          Check if an id is defined. You should pass a string or
          and identifier. If you pass a matrix, each entry will be
          evaluated separately and the matrix should contain
          strings or identifiers.

   IsFunction

IsFunction (arg)

          Check if argument is a function.

   IsFunctionOrIdentifier

IsFunctionOrIdentifier (arg)

          Check if argument is a function or an identifier.

   IsFunctionRef

IsFunctionRef (arg)

          Check if argument is a function reference. This includes
          variable references.

   IsMatrix

IsMatrix (arg)

          Check if argument is a matrix. Even though null is
          sometimes considered an empty matrix, the function
          IsMatrix does not consider null a matrix.

   IsNull

IsNull (arg)

          Check if argument is a null.

   IsString

IsString (arg)

          Check if argument is a text string.

   IsValue

IsValue (arg)

          Check if argument is a number.

   Parse

Parse (str)

          Parses but does not evaluate a string. Note that certain
          pre-computation is done during the parsing stage.

   SetFunctionFlags

SetFunctionFlags (id,flags...)

          Set flags for a function, currently "PropagateMod" and
          "NoModuloArguments". If "PropagateMod" is set, then the
          body of the function is evaluated in modular arithmetic
          when the function is called inside a block that was
          evaluated using modular arithmetic (using mod). If
          "NoModuloArguments", then the arguments of the function
          are never evaluated using modular arithmetic.

   SetHelp

SetHelp (id,category,desc)

          Set the category and help description line for a
          function.

   SetHelpAlias

SetHelpAlias (id,alias)

          Sets up a help alias.

   chdir

chdir (dir)

          Changes current directory, same as the cd.

   display

display (str,expr)

          Display a string and an expression with a colon to
          separate them.

   error

error (str)

          Prints a string to the error stream (onto the console).

   exit

exit

          Aliases: quit

          Exits the program.

   false

false

          Aliases: False FALSE

          The false boolean value.

   manual

manual

          Displays the user manual.

   print

print (str)

          Prints an expression and then print a newline. The
          argument str can be any expression. It is made into a
          string before being printed.

   printn

printn (str)

          Prints an expression without a trailing newline. The
          argument str can be any expression. It is made into a
          string before being printed.

   protect

protect (id)

          Protect a variable from being modified. This is used on
          the internal GEL functions to avoid them being
          accidentally overridden.

   ProtectAll

ProtectAll ()

          Protect all currently defined variables, parameters and
          functions from being modified. This is used on the
          internal GEL functions to avoid them being accidentally
          overridden. Normally Genius Mathematics Tool considers
          unprotected variables as user defined.

   set

set (id,val)

          Set a global variable. The id can be either a string or
          a quoted identifier as follows. For example:

set(`x,1)

          will set the global variable x to the value 1.

   string

string (s)

          Make a string. This will make a string out of any
          argument.

   true

true

          Aliases: True TRUE

          The true boolean value.

   undefine

undefine (id)

          Alias: Undefine

          Undefine a variable. This includes locals and globals,
          every value on all context levels is wiped. This
          function should really not be used on local variables. A
          vector of identifiers can also be passed to undefine
          several variables.

   UndefineAll

UndefineAll ()

          Undefine all unprotected global variables (including
          functions and parameters). Normally Genius Mathematics
          Tool considers protected variables as system defined
          functions and variables. Note that UndefineAll only
          removes the global definition of symbols not local ones,
          so that it may be run from inside other functions
          safely.

   unprotect

unprotect (id)

          Unprotect a variable from being modified.

   UserVariables

UserVariables ()

          Return a vector of identifiers of user defined
          (unprotected) global variables.

   wait

wait (secs)

          Waits a specified number of seconds. secs must be
          non-negative. Zero is accepted and nothing happens in
          this case, except possibly user interface events are
          processed.

   version

version

          Returns the version of Genius as a horizontal 3-vector
          with major version first, then minor version and finally
          the patch level.

   warranty

warranty

          Gives the warranty information.
     __________________________________________________________

Parameters

   ChopTolerance

ChopTolerance = number

          Tolerance of the Chop function.

   ContinuousNumberOfTries

ContinuousNumberOfTries = number

          How many iterations to try to find the limit for
          continuity and limits.

   ContinuousSFS

ContinuousSFS = number

          How many successive steps to be within tolerance for
          calculation of continuity.

   ContinuousTolerance

ContinuousTolerance = number

          Tolerance for continuity of functions and for
          calculating the limit.

   DerivativeNumberOfTries

DerivativeNumberOfTries = number

          How many iterations to try to find the limit for
          derivative.

   DerivativeSFS

DerivativeSFS = number

          How many successive steps to be within tolerance for
          calculation of derivative.

   DerivativeTolerance

DerivativeTolerance = number

          Tolerance for calculating the derivatives of functions.

   ErrorFunctionTolerance

ErrorFunctionTolerance = number

          Tolerance of the ErrorFunction.

   FloatPrecision

FloatPrecision = number

          Floating point precision.

   FullExpressions

FullExpressions = boolean

          Print full expressions, even if more than a line.

   GaussDistributionTolerance

GaussDistributionTolerance = number

          Tolerance of the GaussDistribution function.

   IntegerOutputBase

IntegerOutputBase = number

          Integer output base.

   IsPrimeMillerRabinReps

IsPrimeMillerRabinReps = number

          Number of extra Miller-Rabin tests to run on a number
          before declaring it a prime in IsPrime.

   LinePlotDrawLegends

LinePlotDrawLegends = true

          Tells genius to draw the legends for line plotting
          functions such as LinePlot.

   LinePlotVariableNames

LinePlotVariableNames = ["x","y","z","t"]

          Tells genius which variable names are used as default
          names for line plotting functions such as LinePlot and
          friends.

   LinePlotWindow

LinePlotWindow = [x1,x2,y1,y2]

          Sets the limits for line plotting functions such as
          LinePlot.

   MaxDigits

MaxDigits = number

          Maximum digits to display.

   MaxErrors

MaxErrors = number

          Maximum errors to display.

   MixedFractions

MixedFractions = boolean

          If true, mixed fractions are printed.

   NumericalIntegralFunction

NumericalIntegralFunction = function

          The function used for numerical integration in
          NumericalIntegral.

   NumericalIntegralSteps

NumericalIntegralSteps = number

          Steps to perform in NumericalIntegral.

   OutputChopExponent

OutputChopExponent = number

          When another number in the object being printed (a
          matrix or a value) is greater than
          10^-OutputChopWhenExponent, and the number being printed
          is less than 10^-OutputChopExponent, then display 0.0
          instead of the number.

          Output is never chopped if OutputChopExponent is zero.
          It must be a non-negative integer.

          If you want output always chopped according to
          OutputChopExponent, then set OutputChopWhenExponent, to
          something greater than or equal to OutputChopExponent.

   OutputChopWhenExponent

OutputChopWhenExponent = number

          When to chop output. See OutputChopExponent.

   OutputStyle

OutputStyle = string

          Output style, this can be normal, latex, mathml or
          troff.

          This affects mostly how matrices and fractions are
          printed out and is useful for pasting into documents.
          For example you can set this to the latex by:

OutputStyle = "latex"

   ResultsAsFloats

ResultsAsFloats = boolean

          Convert all results to floats before printing.

   ScientificNotation

ScientificNotation = boolean

          Use scientific notation.

   SlopefieldTicks

SlopefieldTicks = [vertical,horizontal]

          Sets the number of vertical and horizontal ticks in a
          slopefield plot. (See SlopefieldPlot).

   SumProductNumberOfTries

SumProductNumberOfTries = number

          How many iterations to try for InfiniteSum and
          InfiniteProduct.

   SumProductSFS

SumProductSFS = number

          How many successive steps to be within tolerance for
          InfiniteSum and InfiniteProduct.

   SumProductTolerance

SumProductTolerance = number

          Tolerance for InfiniteSum and InfiniteProduct.

   SurfacePlotVariableNames

SurfacePlotVariableNames = ["x","y","z"]

          Tells genius which variable names are used as default
          names for surface plotting functions using SurfacePlot.
          Note that the z does not refer to the dependent
          (vertical) axis, but to the indepent complex variable
          z=x+iy.

   SurfacePlotWindow

SurfacePlotWindow = [x1,x2,y1,y2,z1,z2]

          Sets the limits for surface plotting (See SurfacePlot).

   VectorfieldNormalized

VectorfieldNormalized = true

          Should the vectorfield plotting have normalized arrow
          length. If true, vector fields will only show direction
          and not magnitude. (See VectorfieldPlot).

   VectorfieldTicks

VectorfieldTicks = [vertical,horizontal]

          Sets the number of vertical and horizontal ticks in a
          vectorfield plot. (See VectorfieldPlot).
     __________________________________________________________

Constants

   CatalanConstant

CatalanConstant

          Catalan's Constant, approximately 0.915... It is defined
          to be the series where terms are (-1^k)/((2*k+1)^2),
          where k ranges from 0 to infinity.

          See Mathworld for more information.

   EulerConstant

EulerConstant

          Aliases: gamma

          Euler's Constant gamma. Sometimes called the
          Euler-Mascheroni constant.

          See Wikipedia or Planetmath or Mathworld for more
          information.

   GoldenRatio

GoldenRatio

          The Golden Ratio.

          See Wikipedia or Planetmath or Mathworld for more
          information.

   Gravity

Gravity

          Free fall acceleration at sea level.

          See Wikipedia for more information.

   e

e

          The base of the natural logarithm. e^x is the
          exponential function exp. This is the number
          approximately 2.71828182846...

          See Wikipedia or Planetmath or Mathworld for more
          information.

   pi

pi

          The number pi, that is the ratio of a circle's
          circumference to its diameter. This is approximately
          3.14159265359...

          See Wikipedia or Planetmath or Mathworld for more
          information.
     __________________________________________________________

Numeric

   AbsoluteValue

AbsoluteValue (x)

          Aliases: abs

          Absolute value of a number and if x is a complex value
          the modulus of x. I.e. this the distance of x to the
          origin.

          See Wikipedia, Planetmath (absolute value), Planetmath
          (modulus), Mathworld (absolute value) or Mathworld
          (complex modulus) for more information.

   Chop

Chop (x)

          Replace very small number with zero.

   ComplexConjugate

ComplexConjugate (z)

          Aliases: conj Conj

          Calculates the complex conjugate of the complex number
          z. If z is a vector or matrix, all its elements are
          conjugated.

          See Wikipedia for more information.

   Denominator

Denominator (x)

          Get the denominator of a rational number.

          See Wikipedia for more information.

   FractionalPart

FractionalPart (x)

          Return the fractional part of a number.

          See Wikipedia for more information.

   Im

Im (z)

          Aliases: ImaginaryPart

          Get the imaginary part of a complex number.

          See Wikipedia for more information.

   IntegerQuotient

IntegerQuotient (m,n)

          Division without remainder.

   IsComplex

IsComplex (num)

          Check if argument is a complex (non-real) number.

   IsComplexRational

IsComplexRational (num)

          Check if argument is a possibly complex rational number.
          That is, if both real and imaginary parts are given as
          rational numbers. Of course rational simply means "not
          stored as a floating point number."

   IsFloat

IsFloat (num)

          Check if argument is a floating point number
          (non-complex).

   IsGaussInteger

IsGaussInteger (num)

          Aliases: IsComplexInteger

          Check if argument is a possibly complex integer.

   IsInteger

IsInteger (num)

          Check if argument is an integer (non-complex).

   IsNonNegativeInteger

IsNonNegativeInteger (num)

          Check if argument is a non-negative real integer.

   IsPositiveInteger

IsPositiveInteger (num)

          Aliases: IsNaturalNumber

          Check if argument is a positive real integer. Note that
          we accept the convention that 0 is not a natural number.

   IsRational

IsRational (num)

          Check if argument is a rational number (non-complex). Of
          course rational simply means "not stored as a floating
          point number."

   IsReal

IsReal (num)

          Check if argument is a real number.

   Numerator

Numerator (x)

          Get the numerator of a rational number.

          See Wikipedia for more information.

   Re

Re (z)

          Aliases: RealPart

          Get the real part of a complex number.

          See Wikipedia for more information.

   Sign

Sign (x)

          Aliases: sign

          Return the sign of a number. That is returns -1 if value
          is negative, 0 if value is zero and 1 if value is
          positive. If x is a complex value then Sign returns the
          direction or 0.

   ceil

ceil (x)

          Aliases: Ceiling

          Get the lowest integer more than or equal to n.

   exp

exp (x)

          The exponential function. This is the function e^x where
          e is the base of the natural logarithm.

          See Wikipedia or Planetmath or Mathworld for more
          information.

   float

float (x)

          Make number a floating point value. That is returns the
          floating point representation of the number x.

   floor

floor (x)

          Aliases: Floor

          Get the highest integer less than or equal to n.

   ln

ln (x)

          The natural logarithm, the logarithm to base e.

   log

log (x)

log (x,b)

          Logarithm of x base b (calls DiscreteLog if in modulo
          mode), if base is not given, e is used.

   log10

log10 (x)

          Logarithm of x base 10.

   log2

log2 (x)

          Aliases: lg

          Logarithm of x base 2.

   max

max (a,args...)

          Aliases: Max Maximum

          Returns the maximum of arguments or matrix.

   min

min (a,args...)

          Aliases: Min Minimum

          Returns the minimum of arguments or matrix.

   rand

rand (size...)

          Generate random float in the range [0,1). If size is
          given then a matrix (if two numbers are specified) or
          vector (if one number is specified) of the given size
          returned.

   randint

randint (max,size...)

          Generate random integer in the range [0,max). If size is
          given then a matrix (if two numbers are specified) or
          vector (if one number is specified) of the given size
          returned. For example,

genius> randint(4)
= 3
genius> randint(4,2)
=
[0      1]
genius> randint(4,2,3)
=
[2      2       1
 0      0       3]

   round

round (x)

          Aliases: Round

          Round a number.

   sqrt

sqrt (x)

          Aliases: SquareRoot

          The square root. When operating modulo some integer will
          return either a null or a vector of the square roots.
          Examples:

genius> sqrt(2)
= 1.41421356237
genius> sqrt(-1)
= 1i
genius> sqrt(4) mod 7
=
[2      5]
genius> 2*2 mod 7
= 4

          See Planetmath for more information.

   trunc

trunc (x)

          Aliases: Truncate IntegerPart

          Truncate number to an integer (return the integer part).
     __________________________________________________________

Trigonometry

   acos

acos (x)

          Aliases: arccos

          The arccos (inverse cos) function.

   acosh

acosh (x)

          Aliases: arccosh

          The arccosh (inverse cosh) function.

   acot

acot (x)

          Aliases: arccot

          The arccot (inverse cot) function.

   acoth

acoth (x)

          Aliases: arccoth

          The arccoth (inverse coth) function.

   acsc

acsc (x)

          Aliases: arccsc

          The inverse cosecant function.

   acsch

acsch (x)

          Aliases: arccsch

          The inverse hyperbolic cosecant function.

   asec

asec (x)

          Aliases: arcsec

          The inverse secant function.

   asech

asech (x)

          Aliases: arcsech

          The inverse hyperbolic secant function.

   asin

asin (x)

          Aliases: arcsin

          The arcsin (inverse sin) function.

   asinh

asinh (x)

          Aliases: arcsinh

          The arcsinh (inverse sinh) function.

   atan

atan (x)

          Aliases: arctan

          Calculates the arctan (inverse tan) function.

          See Wikipedia or Mathworld for more information.

   atanh

atanh (x)

          Aliases: arctanh

          The arctanh (inverse tanh) function.

   atan2

atan2 (y, x)

          Aliases: arctan2

          Calculates the arctan2 function. If x>0 then it returns
          atan(y/x). If x<0 then it returns sign(y) * (pi -
          atan(|y/x|). When x=0 it returns sign(y) * pi/2.
          atan2(0,0) returns 0 rather then failing.

          See Wikipedia or Mathworld for more information.

   cos

cos (x)

          Calculates the cosine function.

          See Planetmath for more information.

   cosh

cosh (x)

          Calculates the hyperbolic cosine function.

          See Planetmath for more information.

   cot

cot (x)

          The cotangent function.

          See Planetmath for more information.

   coth

coth (x)

          The hyperbolic cotangent function.

          See Planetmath for more information.

   csc

csc (x)

          The cosecant function.

          See Planetmath for more information.

   csch

csch (x)

          The hyperbolic cosecant function.

          See Planetmath for more information.

   sec

sec (x)

          The secant function.

          See Planetmath for more information.

   sech

sech (x)

          The hyperbolic secant function.

          See Planetmath for more information.

   sin

sin (x)

          Calculates the sine function.

          See Planetmath for more information.

   sinh

sinh (x)

          Calculates the hyperbolic sine function.

          See Planetmath for more information.

   tan

tan (x)

          Calculates the tan function.

          See Planetmath for more information.

   tanh

tanh (x)

          The hyperbolic tangent function.

          See Planetmath for more information.
     __________________________________________________________

Number Theory

   AreRelativelyPrime

AreRelativelyPrime (a,b)

          Are the real integers a and b relatively prime? Returns
          true or false.

          See Planetmath or Mathworld for more information.

   BernoulliNumber

BernoulliNumber (n)

          Return the nth Bernoulli number.

          See Wikipedia or Mathworld for more information.

   ChineseRemainder

ChineseRemainder (a,m)

          Aliases: CRT

          Find the x that solves the system given by the vector a
          and modulo the elements of m, using the Chinese
          Remainder Theorem.

          See Wikipedia or Planetmath or Mathworld for more
          information.

   CombineFactorizations

CombineFactorizations (a,b)

          Given two factorizations, give the factorization of the
          product.

          See Factorize.

   ConvertFromBase

ConvertFromBase (v,b)

          Convert a vector of values indicating powers of b to a
          number.

   ConvertToBase

ConvertToBase (n,b)

          Convert a number to a vector of powers for elements in
          base b.

   DiscreteLog

DiscreteLog (n,b,q)

          Find discrete log of n base b in F[q], the finite field
          of order q, where q is a prime, using the
          Silver-Pohlig-Hellman algorithm.

          See Wikipedia or Planetmath or Mathworld for more
          information.

   Divides

Divides (m,n)

          Checks divisibility (if m divides n).

   EulerPhi

EulerPhi (n)

          Compute the Euler phi function for n, that is the number
          of integers between 1 and n relatively prime to n.

          See Wikipedia or Planetmath or Mathworld for more
          information.

   ExactDivision

ExactDivision (n,d)

          Return n/d but only if d divides n. If d does not divide
          n then this function returns garbage. This is a lot
          faster for very large numbers than the operation n/d,
          but of course only useful if you know that the division
          is exact.

   Factorize

Factorize (n)

          Return factorization of a number as a matrix. The first
          row is the primes in the factorization (including 1) and
          the second row are the powers. So for example:

genius> Factorize(11*11*13)
=
[1      11      13
 1      2       1]

          See Wikipedia for more information.

   Factors

Factors (n)

          Return all factors of n in a vector. This includes all
          the non-prime factors as well. It includes 1 and the
          number itself. So for example to print all the perfect
          numbers (those that are sums of their factors) up to the
          number 1000 you could do (this is of course very
          inefficient)

for n=1 to 1000 do (
    if MatrixSum (Factors(n)) == 2*n then
        print(n)
)

   FermatFactorization

FermatFactorization (n,tries)

          Attempt fermat factorization of n into (t-s)*(t+s),
          returns t and s as a vector if possible, null otherwise.
          tries specifies the number of tries before giving up.

          This is a fairly good factorization if your number is
          the product of two factors that are very close to each
          other.

          See Wikipedia for more information.

   FindPrimitiveElementMod

FindPrimitiveElementMod (q)

          Find the first primitive element in F[q], the finite
          group of order q. Of course q must be a prime.

   FindRandomPrimitiveElementMod

FindRandomPrimitiveElementMod (q)

          Find a random primitive element in F[q], the finite
          group of order q (q must be a prime).

   IndexCalculus

IndexCalculus (n,b,q,S)

          Compute discrete log base b of n in F[q], the finite
          group of order q (q a prime), using the factor base S. S
          should be a column of primes possibly with second column
          precalculated by IndexCalculusPrecalculation.

   IndexCalculusPrecalculation

IndexCalculusPrecalculation (b,q,S)

          Run the precalculation step of IndexCalculus for
          logarithms base b in F[q], the finite group of order q
          (q a prime), for the factor base S (where S is a column
          vector of primes). The logs will be precalculated and
          returned in the second column.

   IsEven

IsEven (n)

          Tests if an integer is even.

   IsMersennePrimeExponent

IsMersennePrimeExponent (p)

          Tests if a positive integer p is a Mersenne prime
          exponent. That is if 2^p-1 is a prime. It does this by
          looking it up in a table of known values which is
          relatively short. See also MersennePrimeExponents and
          LucasLehmer.

          See Wikipedia, Planetmath, Mathworld or GIMPS for more
          information.

   IsNthPower

IsNthPower (m,n)

          Tests if a rational number m is a perfect nth power. See
          also IsPerfectPower and IsPerfectSquare.

   IsOdd

IsOdd (n)

          Tests if an integer is odd.

   IsPerfectPower

IsPerfectPower (n)

          Check an integer is any perfect power, a^b.

   IsPerfectSquare

IsPerfectSquare (n)

          Check an integer for being a perfect square of an
          integer. The number must be a real integer. Negative
          integers are of course never perfect squares of real
          integers.

   IsPrime

IsPrime (n)

          Tests primality of integers, for numbers less than
          2.5e10 the answer is deterministic (if Riemann
          hypothesis is true). For numbers larger, the probability
          of a false positive depends on IsPrimeMillerRabinReps.
          That is the probability of false positive is 1/4 to the
          power IsPrimeMillerRabinReps. The default value of 22
          yields a probability of about 5.7e-14.

          If false is returned, you can be sure that the number is
          a composite. If you want to be absolutely sure that you
          have a prime you can use MillerRabinTestSure but it may
          take a lot longer.

          See Planetmath or Mathworld for more information.

   IsPrimitiveMod

IsPrimitiveMod (g,q)

          Check if g is primitive in F[q], the finite group of
          order q, where q is a prime. If q is not prime results
          are bogus.

   IsPrimitiveModWithPrimeFactors

IsPrimitiveModWithPrimeFactors (g,q,f)

          Check if g is primitive in F[q], the finite group of
          order q, where q is a prime and f is a vector of prime
          factors of q-1. If q is not prime results are bogus.

   IsPseudoprime

IsPseudoprime (n,b)

          If n is a pseudoprime base b but not a prime, that is if
          b^(n-1) == 1 mod n. This calles the PseudoprimeTest

   IsStrongPseudoprime

IsStrongPseudoprime (n,b)

          Test if n is a strong pseudoprime to base b but not a
          prime.

   Jacobi

Jacobi (a,b)

          Aliases: JacobiSymbol

          Calculate the Jacobi symbol (a/b) (b should be odd).

   JacobiKronecker

JacobiKronecker (a,b)

          Aliases: JacobiKroneckerSymbol

          Calculate the Jacobi symbol (a/b) with the Kronecker
          extension (a/2)=(2/a) when a odd, or (a/2)=0 when a
          even.

   LeastAbsoluteResidue

LeastAbsoluteResidue (a,n)

          Return the residue of a mod n with the least absolute
          value (in the interval -n/2 to n/2).

   Legendre

Legendre (a,p)

          Aliases: LegendreSymbol

          Calculate the Legendre symbol (a/p).

          See Planetmath or Mathworld for more information.

   LucasLehmer

LucasLehmer (p)

          Test if 2^p-1 is a Mersenne prime using the Lucas-Lehmer
          test. See also MersennePrimeExponents and
          IsMersennePrimeExponent.

          See Wikipedia or Planetmath or Mathworld for more
          information.

   LucasNumber

LucasNumber (n)

          Returns the nth Lucas number.

          See Wikipedia or Planetmath or Mathworld for more
          information.

   MaximalPrimePowerFactors

MaximalPrimePowerFactors (n)

          Return all maximal prime power factors of a number.

   MersennePrimeExponents

MersennePrimeExponents

          A vector of known Mersenne prime exponents, that is a
          list of positive integers p such that 2^p-1 is a prime.
          See also IsMersennePrimeExponent and LucasLehmer.

          See Wikipedia, Planetmath, Mathworld or GIMPS for more
          information.

   MillerRabinTest

MillerRabinTest (n,reps)

          Use the Miller-Rabin primality test on n, reps number of
          times. The probability of false positive is (1/4)^reps.
          It is probably usually better to use IsPrime since that
          is faster and better on smaller integers.

          See Wikipedia or Planetmath or Mathworld for more
          information.

   MillerRabinTestSure

MillerRabinTestSure (n)

          Use the Miller-Rabin primality test on n with enough
          bases that assuming the Generalized Reimann Hypothesis
          the result is deterministic.

          See Wikipedia or Planetmath or Mathworld for more
          information.

   ModInvert

ModInvert (n,m)

          Returns inverse of n mod m.

          See Mathworld for more information.

   MoebiusMu

MoebiusMu (n)

          Return the Moebius mu function evaluated in n. That is,
          it returns 0 if n is not a product of distinct primes
          and (-1)^k if it is a product of k distinct primes.

          See Planetmath or Mathworld for more information.

   NextPrime

NextPrime (n)

          Returns the least prime greater than n. Negatives of
          primes are considered prime and so to get the previous
          prime you can use -NextPrime(-n).

          This function uses the GMP's mpz_nextprime which in turn
          uses the probabilistic Miller-Rabin test (See also
          MillerRabinTest). The probability of false positive is
          not tunable, but is low enough for all practical
          purposes.

          See Planetmath or Mathworld for more information.

   PadicValuation

PadicValuation (n,p)

          Returns the p-adic valuation (number of trailing zeros
          in base p).

          See Planetmath for more information.

   PowerMod

PowerMod (a,b,m)

          Compute a^b mod m. The b's power of a modulo m. It is
          not neccessary to use this function as it is
          automatically used in modulo mode. Hence a^b mod m is
          just as fast.

   Prime

Prime (n)

          Aliases: prime

          Return the nth prime (up to a limit).

          See Planetmath or Mathworld for more information.

   PrimeFactors

PrimeFactors (n)

          Return all prime factors of a number as a vector.

          See Mathworld for more information.

   PseudoprimeTest

PseudoprimeTest (n,b)

          Pseudoprime test, returns true if and only if b^(n-1) ==
          1 mod n

          See Planetmath or Mathworld for more information.

   RemoveFactor

RemoveFactor (n,m)

          Removes all instances of the factor m from the number n.
          That is divides by the largest power of m, that divides
          n.

          See Planetmath or Mathworld for more information.

   SilverPohligHellmanWithFactorization

SilverPohligHellmanWithFactorization (n,b,q,f)

          Find discrete log of n base b in F[q], the finite group
          of order q, where q is a prime using the
          Silver-Pohlig-Hellman algorithm, given f being the
          factorization of q-1.

   SqrtModPrime

SqrtModPrime (n,p)

          Find square root of n modulo p (where p is a prime).
          Null is returned if not a quadratic residue.

          See Planetmath or Mathworld for more information.

   StrongPseudoprimeTest

StrongPseudoprimeTest (n,b)

          Run the strong pseudoprime test base b on n.

          See Planetmath or Mathworld for more information.

   gcd

gcd (a,args...)

          Aliases: GCD

          Greatest common divisor of integers. You can enter as
          many integers in the argument list, or you can give a
          vector or a matrix of integers. If you give more than
          one matrix of the same size then GCD is done element by
          element.

          See Planetmath or Mathworld for more information.

   lcm

lcm (a,args...)

          Aliases: LCM

          Least common multiplier of integers. You can enter as
          many integers in the argument list, or you can give a
          vector or a matrix of integers. If you give more than
          one matrix of the same size then LCM is done element by
          element.

          See Planetmath or Mathworld for more information.
     __________________________________________________________

Matrix Manipulation

   ApplyOverMatrix

ApplyOverMatrix (a,func)

          Apply a function over all entries of a matrix and return
          a matrix of the results.

   ApplyOverMatrix2

ApplyOverMatrix2 (a,b,func)

          Apply a function over all entries of 2 matrices (or 1
          value and 1 matrix) and return a matrix of the results.

   ColumnsOf

ColumnsOf (M)

          Gets the columns of a matrix as a horizontal vector.

   ComplementSubmatrix

ComplementSubmatrix (m,r,c)

          Remove column(s) and row(s) from a matrix.

   CompoundMatrix

CompoundMatrix (k,A)

          Calculate the kth compound matrix of A.

   CountZeroColumns

CountZeroColumns (M)

          Count the number of zero columns in a matrix. For
          example Once you column reduce a matrix you can use this
          to find the nullity. See cref and Nullity.

   DeleteColumn

DeleteColumn (M,col)

          Delete a column of a matrix.

   DeleteRow

DeleteRow (M,row)

          Delete a row of a matrix.

   DiagonalOf

DiagonalOf (M)

          Gets the diagonal entries of a matrix as a column
          vector.

          See Wikipedia for more information.

   DotProduct

DotProduct (u,v)

          Get the dot product of two vectors. The vectors must be
          of the same size. No conjugates are taken so this is a
          bilinear form even if working over the complex numbers.

          See Planetmath for more information.

   ExpandMatrix

ExpandMatrix (M)

          Expands a matrix just like we do on unquoted matrix
          input. That is we expand any internal matrices as
          blocks. This is a way to construct matrices out of
          smaller ones and this is normally done automatically on
          input unless the matrix is quoted.

   HermitianProduct

HermitianProduct (u,v)

          Aliases: InnerProduct

          Get the Hermitian product of two vectors. The vectors
          must be of the same size. This is a sesquilinear form
          using the identity matrix.

          See Mathworld for more information.

   I

I (n)

          Aliases: eye

          Return an identity matrix of a given size, that is n by
          n. If n is zero, returns null.

          See Planetmath for more information.

   IndexComplement

IndexComplement (vec,msize)

          Return the index complement of a vector of indexes.
          Everything is one based. For example for vector [2,3]
          and size 5, we return [1,4,5]. If msize is 0, we always
          return null.

   IsDiagonal

IsDiagonal (M)

          Is a matrix diagonal.

          See Wikipedia or Planetmath for more information.

   IsIdentity

IsIdentity (x)

          Check if a matrix is the identity matrix. Automatically
          returns false if the matrix is not square. Also works on
          numbers, in which case it is equivalent to x==1. When x
          is null (we could think of that as a 0 by 0 matrix), no
          error is generated and false is returned.

   IsLowerTriangular

IsLowerTriangular (M)

          Is a matrix lower triangular. That is, are all the
          entries above the diagonal zero.

   IsMatrixInteger

IsMatrixInteger (M)

          Check if a matrix is a matrix of an integers
          (non-complex).

   IsMatrixNonnegative

IsMatrixNonnegative (M)

          Check if a matrix is non-negative, that is if each
          element is non-negative. Do not confuse positive
          matrices with positive semi-definite matrices.

          See Wikipedia for more information.

   IsMatrixPositive

IsMatrixPositive (M)

          Check if a matrix is positive, that is if each element
          is positive (and hence real). In particular, no element
          is 0. Do not confuse positive matrices with positive
          definite matrices.

          See Wikipedia for more information.

   IsMatrixRational

IsMatrixRational (M)

          Check if a matrix is a matrix of rational (non-complex)
          numbers.

   IsMatrixReal

IsMatrixReal (M)

          Check if a matrix is a matrix of real (non-complex)
          numbers.

   IsMatrixSquare

IsMatrixSquare (M)

          Check if a matrix is square, that is its width is equal
          to its height.

   IsUpperTriangular

IsUpperTriangular (M)

          Is a matrix upper triangular? That is, a matrix is upper
          triangular if all all the entries below the diagonal are
          zero.

   IsValueOnly

IsValueOnly (M)

          Check if a matrix is a matrix of numbers only. Many
          internal functions make this check. Values can be any
          number including complex numbers.

   IsVector

IsVector (v)

          Is argument a horizontal or a vertical vector. Genius
          does not distinguish between a matrix and a vector and a
          vector is just a 1 by n or n by 1 matrix.

   IsZero

IsZero (x)

          Check if a matrix is composed of all zeros. Also works
          on numbers, in which case it is equivalent to x==0. When
          x is null (we could think of that as a 0 by 0 matrix),
          no error is generated and true is returned as the
          condition is vacuous.

   LowerTriangular

LowerTriangular (M)

          Returns a copy of the matrix M with all the entries
          above the diagonal set to zero.

   MakeDiagonal

MakeDiagonal (v,arg...)

          Aliases: diag

          Make diagonal matrix from a vector.

          See Wikipedia or Planetmath for more information.

   MakeVector

MakeVector (A)

          Make column vector out of matrix by putting columns
          above each other. Returns null when given null.

   MatrixProduct

MatrixProduct (A)

          Calculate the product of all elements in a matrix or
          vector. That is we multiply all the elements and return
          a number that is the product of all the elements.

   MatrixSum

MatrixSum (A)

          Calculate the sum of all elements in a matrix or vector.
          That is we add all the elements and return a number that
          is the sum of all the elements.

   MatrixSumSquares

MatrixSumSquares (A)

          Calculate the sum of squares of all elements in a matrix
          or vector.

   OuterProduct

OuterProduct (u,v)

          Get the outer product of two vectors. That is, suppose
          that u and v are vertical vectors, then the outer
          product is v * u.'.

   ReverseVector

ReverseVector (v)

          Reverse elements in a vector. Return null if given null

   RowSum

RowSum (m)

          Calculate sum of each row in a matrix and return a
          vertical vector with the result.

   RowSumSquares

RowSumSquares (m)

          Calculate sum of squares of each row in a matrix.

   RowsOf

RowsOf (M)

          Gets the rows of a matrix as a vertical vector. Each
          element of the vector is a horizontal vector which is
          the corresponding row of M. This function is useful if
          you wish to loop over the rows of a matrix. For example,
          as for r in RowsOf(M) do something(r).

   SetMatrixSize

SetMatrixSize (M,rows,columns)

          Make new matrix of given size from old one. That is, a
          new matrix will be returned to which the old one is
          copied. Entries that don't fit are clipped and extra
          space is filled with zeros. if rows or columns are zero
          then null is returned.

   ShuffleVector

ShuffleVector (v)

          Shuffle elements in a vector. Return null if given null

   SortVector

SortVector (v)

          Sort vector elements in an increasing order.

   StripZeroColumns

StripZeroColumns (M)

          Removes any all-zero columns of M.

   StripZeroRows

StripZeroRows (M)

          Removes any all-zero rows of M.

   Submatrix

Submatrix (m,r,c)

          Return column(s) and row(s) from a matrix. This is just
          equivalent to m@(r,c). r and c should be vectors of rows
          and columns (or single numbers if only one row or column
          is needed).

   SwapRows

SwapRows (m,row1,row2)

          Swap two rows in a matrix.

   UpperTriangular

UpperTriangular (M)

          Returns a copy of the matrix M with all the entries
          below the diagonal set to zero.

   columns

columns (M)

          Get the number of columns of a matrix.

   elements

elements (M)

          Get the total number of elements of a matrix. This is
          the number of columns times the number of rows.

   ones

ones (rows,columns...)

          Make an matrix of all ones (or a row vector if only one
          argument is given). Returns null if either rows or
          columns are zero.

   rows

rows (M)

          Get the number of rows of a matrix.

   zeros

zeros (rows,columns...)

          Make a matrix of all zeros (or a row vector if only one
          argument is given). Returns null if either rows or
          columns are zero.
     __________________________________________________________

Linear Algebra

   AuxiliaryUnitMatrix

AuxiliaryUnitMatrix (n)

          Get the auxiliary unit matrix of size n. This is a
          square matrix matrix with that is all zero except the
          superdiagonal being all ones. It is the Jordan block
          matrix of one zero eigenvalue.

          See Planetmath or Mathworld for more information on
          Jordan Cannonical Form.

   BilinearForm

BilinearForm (v,A,w)

          Evaluate (v,w) with respect to the bilinear form given
          by the matrix A.

   BilinearFormFunction

BilinearFormFunction (A)

          Return a function that evaluates two vectors with
          respect to the bilinear form given by A.

   CharacteristicPolynomial

CharacteristicPolynomial (M)

          Aliases: CharPoly

          Get the characteristic polynomial as a vector. That is,
          return the coefficients of the polynomial starting with
          the constant term. This is the polynomial defined by
          det(M-xI). The roots of this polynomial are the
          eigenvalues of M. See also
          CharacteristicPolynomialFunction.

          See Planetmath for more information.

   CharacteristicPolynomialFunction

CharacteristicPolynomialFunction (M)

          Get the characteristic polynomial as a function. This is
          the polynomial defined by det(M-xI). The roots of this
          polynomial are the eigenvalues of M. See also
          CharacteristicPolynomial.

          See Planetmath for more information.

   ColumnSpace

ColumnSpace (M)

          Get a basis matrix for the columnspace of a matrix. That
          is, return a matrix whose columns are the basis for the
          column space of M. That is the space spanned by the
          columns of M.

   CommutationMatrix

CommutationMatrix (m, n)

          Return the commutation matrix K(m,n) which is the unique
          m*n by m*n matrix such that K(m,n) * MakeVector(A) =
          MakeVector(A.') for all m by n matrices A.

   CompanionMatrix

CompanionMatrix (p)

          Companion matrix of a polynomial (as vector).

   ConjugateTranspose

ConjugateTranspose (M)

          Conjugate transpose of a matrix (adjoint). This is the
          same as the ' operator.

          See Planetmath for more information.

   Convolution

Convolution (a,b)

          Aliases: convol

          Calculate convolution of two horizontal vectors.

   ConvolutionVector

ConvolutionVector (a,b)

          Calculate convolution of two horizontal vectors. Return
          result as a vector and not added together.

   CrossProduct

CrossProduct (v,w)

          CrossProduct of two vectors in R^3 as a column vector.

   DeterminantalDivisorsInteger

DeterminantalDivisorsInteger (M)

          Get the determinantal divisors of an integer matrix (not
          its characteristic).

   DirectSum

DirectSum (M,N...)

          Direct sum of matrices.

   DirectSumMatrixVector

DirectSumMatrixVector (v)

          Direct sum of a vector of matrices.

   Eigenvalues

Eigenvalues (M)

          Aliases: eig

          Get the eigenvalues of a square matrix. Currently only
          works for matrices of size up to 4 by 4, or for
          triangular matrices (for which the eigenvalues are on
          the diagonal).

          See Wikipedia or Planetmath or Mathworld for more
          information.

   Eigenvectors

Eigenvectors (M)

Eigenvectors (M, &eigenvalues)

Eigenvectors (M, &eigenvalues, &multiplicities)

          Get the eigenvectors of a square matrix. Optionally get
          also the eigenvalues and their algebraic multiplicities.
          Currently only works for matrices of size up to 2 by 2.

          See Wikipedia or Planetmath or Mathworld for more
          information.

   GramSchmidt

GramSchmidt (v,B...)

          Apply the Gram-Schmidt process (to the columns) with
          respect to inner product given by B. If B is not given
          then the standard Hermitian product is used. B can
          either be a sesquilinear function of two arguments or it
          can be a matrix giving a sesquilinear form. The vectors
          will be made orthonormal with respect to B.

          See Planetmath for more information.

   HankelMatrix

HankelMatrix (c,r)

          Hankel matrix.

   HilbertMatrix

HilbertMatrix (n)

          Hilbert matrix of order n.

          See Planetmath for more information.

   Image

Image (T)

          Get the image (columnspace) of a linear transform.

   InfNorm

InfNorm (v)

          Get the Inf Norm of a vector, sometimes called the sup
          norm or the max norm.

   InvariantFactorsInteger

InvariantFactorsInteger (M)

          Get the invariant factors of a square integer matrix
          (not its characteristic).

   InverseHilbertMatrix

InverseHilbertMatrix (n)

          Inverse Hilbert matrix of order n.

          See Planetmath for more information.

   IsHermitian

IsHermitian (M)

          Is a matrix Hermitian. That is, is it equal to its
          conjugate transpose.

          See Planetmath for more information.

   IsInSubspace

IsInSubspace (v,W)

          Test if a vector is in a subspace.

   IsInvertible

IsInvertible (n)

          Is a matrix (or number) invertible (Integer matrix is
          invertible iff it's invertible over the integers).

   IsInvertibleField

IsInvertibleField (n)

          Is a matrix (or number) invertible over a field.

   IsNormal

IsNormal (M)

          Is M a normal matrix. That is, does M*M' == M'*M.

          See Planetmath or Mathworld for more information.

   IsPositiveDefinite

IsPositiveDefinite (M)

          Is M a Hermitian positive definite matrix. That is if
          HermitianProduct(M*v,v) is always strictly positive for
          any vector v. M must be square and Hermitian to be
          positive definite. The check that is performed is that
          every principal submatrix has a non-negative
          determinant. (See HermitianProduct)

          Note that some authors (for example Mathworld) do not
          require that M be Hermitian, and then the condition is
          on the real part of the inner product, but we do not
          take this view. If you wish to perform this check, just
          check the Hermitian part of the matrix M as follows:
          IsPositiveDefinite(M+M').

          See Planetmath or Mathworld for more information.

   IsPositiveSemidefinite

IsPositiveSemidefinite (M)

          Is M a Hermitian positive semidefinite matrix. That is
          if HermitianProduct(M*v,v) is always non-negative for
          any vector v. M must be square and Hermitian to be
          positive semidefinite. The check that is performed is
          that every principal submatrix has a non-negative
          determinant. (See HermitianProduct)

          Note that some authors do not require that M be
          Hermitian, and then the condition is on the real part of
          the inner product, but we do not take this view. If you
          wish to perform this check, just check the Hermitian
          part of the matrix M as follows:
          IsPositiveSemidefinite(M+M').

          See Planetmath or Mathworld for more information.

   IsSkewHermitian

IsSkewHermitian (M)

          Is a matrix skew-Hermitian. That is, is the conjugate
          transpose equal to negative of the matrix.

          See Planetmath for more information.

   IsUnitary

IsUnitary (M)

          Is a matrix unitary? That is, does M'*M and M*M' equal
          the identity.

          See Planetmath or Mathworld for more information.

   JordanBlock

JordanBlock (n,lambda)

          Aliases: J

          Get the Jordan block corresponding to the eigenvalue
          lambda with multiplicity n.

          See Planetmath or Mathworld for more information.

   Kernel

Kernel (T)

          Get the kernel (nullspace) of a linear transform.

          (See NullSpace)

   LUDecomposition

LUDecomposition (A, L, U)

          Get the LU decomposition of A and store the result in
          the L and U which should be references. It returns true
          if successful. For example suppose that A is a square
          matrix, then after running:

genius> LUDecomposition(A,&L,&U)

          You will have the lower matrix stored in a variable
          called L and the upper matrix in a variable called U.

          This is the LU decomposition of a matrix aka Crout
          and/or Cholesky reduction. (ISBN 0-201-11577-8
          pp.99-103) The upper triangular matrix features a
          diagonal of values 1 (one). This is not Doolittle's
          Method which features the 1's diagonal on the lower
          matrix.

          Not all matrices have LU decompositions, for example
          [0,1;1,0] does not and this function returns false in
          this case and sets L and U to null.

          See Planetmath or Mathworld for more information.

   Minor

Minor (M,i,j)

          Get the i-j minor of a matrix.

          See Planetmath for more information.

   NonPivotColumns

NonPivotColumns (M)

          Return the columns that are not the pivot columns of a
          matrix.

   Norm

Norm (v,p...)

          Aliases: norm

          Get the p Norm (or 2 Norm if no p is supplied) of a
          vector.

   NullSpace

NullSpace (T)

          Get the nullspace of a matrix. That is the kernel of the
          linear mapping that the matrix represents. This is
          returned as a matrix whose column space is the nullspace
          of T.

          See Planetmath for more information.

   Nullity

Nullity (M)

          Aliases: nullity

          Get the nullity of a matrix. That is, return the
          dimension of the nullspace; the dimension of the kernel
          of M.

          See Planetmath for more information.

   OrthogonalComplement

OrthogonalComplement (M)

          Get the orthogonal complement of the columnspace.

   PivotColumns

PivotColumns (M)

          Return pivot columns of a matrix, that is columns which
          have a leading 1 in row reduced form. Also returns the
          row where they occur.

   Projection

Projection (v,W,B...)

          Projection of vector v onto subspace W with respect to
          inner product given by B. If B is not given then the
          standard Hermitian product is used. B can either be a
          sesquilinear function of two arguments or it can be a
          matrix giving a sesquilinear form.

   QRDecomposition

QRDecomposition (A, Q)

          Get the QR decomposition of a square matrix A, returns
          the upper triangular matrix R and sets Q to the
          orthogonal (unitary) matrix. Q should be a reference or
          null if you don't want any return. For example:

genius> R = QRDecomposition(A,&Q)

          You will have the upper triangular matrix stored in a
          variable called R and the orthogonal (unitary) matrix
          stored in Q.

          See Planetmath or Mathworld for more information.

   RayleighQuotient

RayleighQuotient (A,x)

          Return the Rayleigh quotient (also called the
          Rayleigh-Ritz quotient or ratio) of a matrix and a
          vector.

          See Planetmath for more information.

   RayleighQuotientIteration

RayleighQuotientIteration (A,x,epsilon,maxiter,vecref)

          Find eigenvalues of A using the Rayleigh quotient
          iteration method. x is a guess at a eigenvector and
          could be random. It should have nonzero imaginary part
          if it will have any chance at finding complex
          eigenvalues. The code will run at most maxiter
          iterations and return null if we cannot get within an
          error of epsilon. vecref should either be null or a
          reference to a variable where the eigenvector should be
          stored.

          See Planetmath for more information on Rayleigh
          quotient.

   Rank

Rank (M)

          Aliases: rank

          Get the rank of a matrix.

          See Planetmath for more information.

   RosserMatrix

RosserMatrix ()

          Rosser matrix, a classic symmetric eigenvalue test
          problem.

   Rotation2D

Rotation2D (angle)

          Aliases: RotationMatrix

          Return the matrix corresponding to rotation around
          origin in R^2.

   Rotation3DX

Rotation3DX (angle)

          Return the matrix corresponding to rotation around
          origin in R^3 about the x-axis.

   Rotation3DY

Rotation3DY (angle)

          Return the matrix corresponding to rotation around
          origin in R^3 about the y-axis.

   Rotation3DZ

Rotation3DZ (angle)

          Return the matrix corresponding to rotation around
          origin in R^3 about the z-axis.

   RowSpace

RowSpace (M)

          Get a basis matrix for the rowspace of a matrix.

   SesquilinearForm

SesquilinearForm (v,A,w)

          Evaluate (v,w) with respect to the sesquilinear form
          given by the matrix A.

   SesquilinearFormFunction

SesquilinearFormFunction (A)

          Return a function that evaluates two vectors with
          respect to the sesquilinear form given by A.

   SmithNormalFormField

SmithNormalFormField (A)

          Smith Normal Form for fields (will end up with 1's on
          the diagonal).

   SmithNormalFormInteger

SmithNormalFormInteger (M)

          Smith Normal Form for square integer matrices (not its
          characteristic).

   SolveLinearSystem

SolveLinearSystem (M,V,args...)

          Solve linear system Mx=V, return solution V if there is
          a unique solution, null otherwise. Extra two reference
          parameters can optionally be used to get the reduced M
          and V.

   ToeplitzMatrix

ToeplitzMatrix (c, r...)

          Return the Toeplitz matrix constructed given the first
          column c and (optionally) the first row r. If only the
          column c is given then it is conjugated and the
          nonconjugated version is used for the first row to give
          a Hermitian matrix (if the first element is real of
          course).

          See Planetmath for more information.

   Trace

Trace (M)

          Aliases: trace

          Calculate the trace of a matrix. That is the sum of the
          diagonal elements.

          See Planetmath for more information.

   Transpose

Transpose (M)

          Transpose of a matrix. This is the same as the .'
          operator.

          See Planetmath for more information.

   VandermondeMatrix

VandermondeMatrix (v)

          Aliases: vander

          Return the Vandermonde matrix.

   VectorAngle

VectorAngle (v,w,B...)

          The angle of two vectors with respect to inner product
          given by B. If B is not given then the standard
          Hermitian product is used. B can either be a
          sesquilinear function of two arguments or it can be a
          matrix giving a sesquilinear form.

   VectorSpaceDirectSum

VectorSpaceDirectSum (M,N)

          The direct sum of the vector spaces M and N.

   VectorSubspaceIntersection

VectorSubspaceIntersection (M,N)

          Intersection of the subspaces given by M and N.

   VectorSubspaceSum

VectorSubspaceSum (M,N)

          The sum of the vector spaces M and N, that is {w |
          w=m+n, m in M, n in N}.

   adj

adj (m)

          Aliases: Adjugate

          Get the classical adjoint (adjugate) of a matrix.

   cref

cref (M)

          Aliases: CREF ColumnReducedEchelonForm

          Compute the Column Reduced Echelon Form.

   det

det (M)

          Aliases: Determinant

          Get the determinant of a matrix.

          See Wikipedia or Planetmath for more information.

   ref

ref (M)

          Aliases: REF RowEchelonForm

          Get the row echelon form of a matrix. That is, apply
          gaussian elimination but not backaddition to M. The
          pivot rows are divided to make all pivots 1.

          See Wikipedia or Planetmath for more information.

   rref

rref (M)

          Aliases: RREF ReducedRowEchelonForm

          Get the reduced row echelon form of a matrix. That is,
          apply gaussian elimination together with backaddition to
          M.

          See Wikipedia or Planetmath for more information.
     __________________________________________________________

Combinatorics

   Catalan

Catalan (n)

          Get n'th catalan number.

          See Planetmath for more information.

   Combinations

Combinations (k,n)

          Get all combinations of k numbers from 1 to n as a
          vector of vectors. (See also NextCombination)

   DoubleFactorial

DoubleFactorial (n)

          Double factorial: n(n-2)(n-4)...

          See Planetmath for more information.

   Factorial

Factorial (n)

          Factorial: n(n-1)(n-2)...

          See Planetmath for more information.

   FallingFactorial

FallingFactorial (n,k)

          Falling factorial: (n)_k = n(n-1)...(n-(k-1))

          See Planetmath for more information.

   Fibonacci

Fibonacci (x)

          Aliases: fib

          Calculate nth Fibonacci number. That is the number
          defined recursively by Fibonacci(n) = Fibonacci(n-1) +
          Fibonacci(n-2) and Fibonacci(1) = Fibonacci(2) = 1.

          See Wikipedia or Planetmath or Mathworld for more
          information.

   FrobeniusNumber

FrobeniusNumber (v,arg...)

          Calculate the Frobenius number. That is calculate
          smallest number that cannot be given as a non-negative
          integer linear combination of a given vector of
          non-negative integers. The vector can be given as
          separate numbers or a single vector. All the numbers
          given should have GCD of 1.

          See Mathworld for more information.

   GaloisMatrix

GaloisMatrix (combining_rule)

          Galois matrix given a linear combining rule
          (a_1*x_+...+a_n*x_n=x_(n+1)).

   GreedyAlgorithm

FrobeniusNumber (n,v)

          Find the vector c of non-negative integers such that
          taking the dot product with v is equal to n. If not
          possible returns null. v should be given sorted in
          increasing order and should consist of non-negative
          integers.

          See Mathworld for more information.

   HarmonicNumber

HarmonicNumber (n,r)

          Aliases: HarmonicH

          Harmonic Number, the n'th harmonic number of order r.

   Hofstadter

Hofstadter (n)

          Hofstadter's function q(n) defined by q(1)=1, q(2)=1,
          q(n)=q(n-q(n-1))+q(n-q(n-2)).

   LinearRecursiveSequence

LinearRecursiveSequence (seed_values,combining_rule,n)

          Compute linear recursive sequence using galois stepping.

   Multinomial

Multinomial (v,arg...)

          Calculate multinomial coefficients. Takes a vector of k
          non-negative integers and computes the multinomial
          coefficient. This corresponds to the coefficient in the
          homogeneous polynomial in k variables with the
          corresponding powers.

          The formula for Multinomial(a,b,c) can be written as:

(a+b+c)! / (a!b!c!)

          In other words, if we would have only two elements, then
          Multinomial(a,b) is the same thing as Binomial(a+b,a) or
          Binomial(a+b,b).

          See Planetmath, Mathworld, or Wikipedia for more
          information.

   NextCombination

NextCombination (v,n)

          Get combination that would come after v in call to
          combinations, first combination should be [1:k]. This
          function is useful if you have many combinations to go
          through and you don't want to waste memory to store them
          all.

          For example with Combination you would normally write a
          loop like:

for n in Combinations (4,6) do (
  SomeFunction (n)
);

          But with NextCombination you would write something like:

n:=[1:4];
do (
  SomeFunction (n)
) while not IsNull(n:=NextCombination(n,6));

          See also Combinations.

   Pascal

Pascal (i)

          Get the Pascal's triangle as a matrix. This will return
          an i+1 by i+1 lower diagonal matrix which is the
          Pascal's triangle after i iterations.

          See Planetmath for more information.

   Permutations

Permutations (k,n)

          Get all permutations of k numbers from 1 to n as a
          vector of vectors.

          See Mathworld or Wikipedia for more information.

   RisingFactorial

RisingFactorial (n,k)

          Aliases: Pochhammer

          (Pochhammer) Rising factorial: (n)_k =
          n(n+1)...(n+(k-1)).

          See Planetmath for more information.

   StirlingNumberFirst

StirlingNumberFirst (n,m)

          Aliases: StirlingS1

          Stirling number of the first kind.

          See Planetmath or Mathworld for more information.

   StirlingNumberSecond

StirlingNumberSecond (n,m)

          Aliases: StirlingS2

          Stirling number of the second kind.

          See Planetmath or Mathworld for more information.

   Subfactorial

Subfactorial (n)

          Subfactorial: n! times sum_{k=1}^n (-1)^k/k!.

   Triangular

Triangular (nth)

          Calculate the n'th triangular number.

          See Planetmath for more information.

   nCr

nCr (n,r)

          Aliases: Binomial

          Calculate combinations, that is, the binomial
          coefficient. n can be any real number.

          See Planetmath for more information.

   nPr

nPr (n,r)

          Calculate the number of permutations of size rof numbers
          from 1 to n.

          See Mathworld or Wikipedia for more information.
     __________________________________________________________

Calculus

   CompositeSimpsonsRule

CompositeSimpsonsRule (f,a,b,n)

          Integration of f by Composite Simpson's Rule on the
          interval [a,b] with n subintervals with error of
          max(f'''')*h^4*(b-a)/180, note that n should be even.

          See Planetmath for more information.

   CompositeSimpsonsRuleTolerance

CompositeSimpsonsRuleTolerance (f,a,b,FourthDerivativeBound,Tolerance)

          Integration of f by Composite Simpson's Rule on the
          interval [a,b] with the number of steps calculated by
          the fourth derivative bound and the desired tolerance.

          See Planetmath for more information.

   Derivative

Derivative (f,x0)

          Attempt to calculate derivative by trying first
          symbolically and then numerically.

   EvenPeriodicExtension

EvenPeriodicExtension (f,L)

          Return a function which is even periodic extension of f
          with half period L. That is a function defined on the
          interval [0,L] extended to be even on [-L,L] and then
          extended to be periodic with period 2*L.

          See also OddPeriodicExtension and PeriodicExtension.

   FourierSeriesFunction

FourierSeriesFunction (a,b,L)

          Return a function which is a Fourier series with the
          coefficients given by the vectors a (sines) and b
          (cosines). Note that a@(1) is the constant coefficient!
          That is, a@(n) refers to the term cos(x*(n-1)*pi/L),
          while b@(n) refers to the term sin(x*n*pi/L). Either a
          or b can be null.

          See Wikipedia or Mathworld for more information.

   InfiniteProduct

InfiniteProduct (func,start,inc)

          Try to calculate an infinite product for a single
          parameter function.

   InfiniteProduct2

InfiniteProduct2 (func,arg,start,inc)

          Try to calculate an infinite product for a double
          parameter function with func(arg,n).

   InfiniteSum

InfiniteSum (func,start,inc)

          Try to calculate an infinite sum for a single parameter
          function.

   InfiniteSum2

InfiniteSum2 (func,arg,start,inc)

          Try to calculate an infinite sum for a double parameter
          function with func(arg,n).

   IsContinuous

IsContinuous (f,x0)

          Try and see if a real-valued function is continuous at
          x0 by calculating the limit there.

   IsDifferentiable

IsDifferentiable (f,x0)

          Test for differentiability by approximating the left and
          right limits and comparing.

   LeftLimit

LeftLimit (f,x0)

          Calculate the left limit of a real-valued function at
          x0.

   Limit

Limit (f,x0)

          Calculate the limit of a real-valued function at x0.
          Tries to calculate both left and right limits.

   MidpointRule

MidpointRule (f,a,b,n)

          Integration by midpoint rule.

   NumericalDerivative

NumericalDerivative (f,x0)

          Aliases: NDerivative

          Attempt to calculate numerical derivative.

   NumericalFourierSeriesCoefficients

NumericalFourierSeriesCoefficients (f,L,N)

          Return a vector of vectors [a,b] where a are the cosine
          coefficients and b are the sine coefficients of the
          Fourier series of f with half-period L (that is defined
          on [-L,L] and extended periodically) with coefficients
          up to Nth harmonic computed numerically. The
          coefficients are computed by numerical integration using
          NumericalIntegral.

          See Wikipedia or Mathworld for more information.

   NumericalFourierSeriesFunction

NumericalFourierSeriesFunction (f,L,N)

          Return a function which is the Fourier series of f with
          half-period L (that is defined on [-L,L] and extended
          periodically) with coefficients up to Nth harmonic
          computed numerically. This is the trigonometric real
          series composed of sines and cosines. The coefficients
          are computed by numerical integration using
          NumericalIntegral.

          See Wikipedia or Mathworld for more information.

   NumericalFourierCosineSeriesCoefficients

NumericalFourierCosineSeriesCoefficients (f,L,N)

          Return a vector of coefficients of the the cosine
          Fourier series of f with half-period L. That is, we take
          f defined on [0,L] take the even periodic extension and
          compute the Fourier series, which only has cosine terms.
          The series is computed up to the Nth harmonic. The
          coefficients are computed by numerical integration using
          NumericalIntegral. Note that a@(1) is the constant
          coefficient! That is, a@(n) refers to the term
          cos(x*(n-1)*pi/L).

          See Wikipedia or Mathworld for more information.

   NumericalFourierCosineSeriesFunction

NumericalFourierCosineSeriesFunction (f,L,N)

          Return a function which is the cosine Fourier series of
          f with half-period L. That is, we take f defined on
          [0,L] take the even periodic extension and compute the
          Fourier series, which only has cosine terms. The series
          is computed up to the Nth harmonic. The coefficients are
          computed by numerical integration using
          NumericalIntegral.

          See Wikipedia or Mathworld for more information.

   NumericalFourierSineSeriesCoefficients

NumericalFourierSineSeriesCoefficients (f,L,N)

          Return a vector of coefficients of the the sine Fourier
          series of f with half-period L. That is, we take f
          defined on [0,L] take the odd periodic extension and
          compute the Fourier series, which only has sine terms.
          The series is computed up to the Nth harmonic. The
          coefficients are computed by numerical integration using
          NumericalIntegral.

          See Wikipedia or Mathworld for more information.

   NumericalFourierSineSeriesFunction

NumericalFourierSineSeriesFunction (f,L,N)

          Return a function which is the sine Fourier series of f
          with half-period L. That is, we take f defined on [0,L]
          take the odd periodic extension and compute the Fourier
          series, which only has sine terms. The series is
          computed up to the Nth harmonic. The coefficients are
          computed by numerical integration using
          NumericalIntegral.

          See Wikipedia or Mathworld for more information.

   NumericalIntegral

NumericalIntegral (f,a,b)

          Integration by rule set in NumericalIntegralFunction of
          f from a to b using NumericalIntegralSteps steps.

   NumericalLeftDerivative

NumericalLeftDerivative (f,x0)

          Attempt to calculate numerical left derivative.

   NumericalLimitAtInfinity

NumericalLimitAtInfinity (_f,step_fun,tolerance,successive_for_success,N
)

          Attempt to calculate the limit of f(step_fun(i)) as i
          goes from 1 to N.

   NumericalRightDerivative

NumericalRightDerivative (f,x0)

          Attempt to calculate numerical right derivative.

   OddPeriodicExtension

OddPeriodicExtension (f,L)

          Return a function which is odd periodic extension of f
          with half period L. That is a function defined on the
          interval [0,L] extended to be odd on [-L,L] and then
          extended to be periodic with period 2*L.

          See also EvenPeriodicExtension and PeriodicExtension.

   OneSidedFivePointFormula

OneSidedFivePointFormula (f,x0,h)

          Compute one-sided derivative using five point formula.

   OneSidedThreePointFormula

OneSidedThreePointFormula (f,x0,h)

          Compute one-sided derivative using three-point formula.

   PeriodicExtension

PeriodicExtension (f,a,b)

          Return a function which is the periodic extension of f
          defined on the interval [a,b] and has period b-a.

          See also OddPeriodicExtension and EvenPeriodicExtension.

   RightLimit

RightLimit (f,x0)

          Calculate the right limit of a real-valued function at
          x0.

   TwoSidedFivePointFormula

TwoSidedFivePointFormula (f,x0,h)

          Compute two-sided derivative using five-point formula.

   TwoSidedThreePointFormula

TwoSidedThreePointFormula (f,x0,h)

          Compute two-sided derivative using three-point formula.
     __________________________________________________________

Functions

   Argument

Argument (z)

          Aliases: Arg arg

          argument (angle) of complex number.

   DirichletKernel

DirichletKernel (n,t)

          Dirichlet kernel of order n.

   DiscreteDelta

DiscreteDelta (v)

          Returns 1 iff all elements are zero.

   ErrorFunction

ErrorFunction (x)

          Aliases: erf

          The error function, 2/sqrt(pi) * int_0^x e^(-t^2) dt.

          See Planetmath for more information.

   FejerKernel

FejerKernel (n,t)

          Fejer kernel of order n evaluated at t

          See Planetmath for more information.

   GammaFunction

GammaFunction (x)

          Aliases: Gamma

          The Gamma function. Currently only implemented for real
          values.

          See Planetmath for more information.

   KroneckerDelta

KroneckerDelta (v)

          Returns 1 iff all elements are equal.

   MinimizeFunction

MinimizeFunction (func,x,incr)

          Find the first value where f(x)=0.

   MoebiusDiskMapping

MoebiusDiskMapping (a,z)

          Moebius mapping of the disk to itself mapping a to 0.

          See Planetmath for more information.

   MoebiusMapping

MoebiusMapping (z,z2,z3,z4)

          Moebius mapping using the cross ratio taking z2,z3,z4 to
          1,0, and infinity respectively.

          See Planetmath for more information.

   MoebiusMappingInftyToInfty

MoebiusMappingInftyToInfty (z,z2,z3)

          Moebius mapping using the cross ratio taking infinity to
          infinity and z2,z3 to 1 and 0 respectively.

          See Planetmath for more information.

   MoebiusMappingInftyToOne

MoebiusMappingInftyToOne (z,z3,z4)

          Moebius mapping using the cross ratio taking infinity to
          1 and z3,z4 to 0 and infinity respectively.

          See Planetmath for more information.

   MoebiusMappingInftyToZero

MoebiusMappingInftyToZero (z,z2,z4)

          Moebius mapping using the cross ratio taking infinity to
          0 and z2,z4 to 1 and infinity respectively.

          See Planetmath for more information.

   PoissonKernel

PoissonKernel (r,sigma)

          Poisson kernel on D(0,1) (not normalized to 1, that is
          integral of this is 2pi).

   PoissonKernelRadius

PoissonKernelRadius (r,sigma)

          Poisson kernel on D(0,R) (not normalized to 1).

   RiemannZeta

RiemannZeta (x)

          Aliases: zeta

          The Riemann zeta function. Currently only implemented
          for real values.

          See Planetmath for more information.

   UnitStep

UnitStep (x)

          The unit step function is 0 for x<0, 1 otherwise. This
          is the integral of the Dirac Delta function. Also called
          the Heaviside function.

          See Wikipedia for more information.

   cis

cis (x)

          The cis function, that is the same as cos(x)+1i*sin(x)

   deg2rad

deg2rad (x)

          Convert degrees to radians.

   rad2deg

rad2deg (x)

          Convert radians to degrees.
     __________________________________________________________

Equation Solving

   CubicFormula

CubicFormula (p)

          Compute roots of a cubic (degree 3) polynomial using the
          cubic formula. The polynomial should be given as a
          vector of coefficients. That is 4*x^3 + 2*x + 1
          corresponds to the vector [1,2,0,4]. Returns a column
          vector of the three solutions. The first solution is
          always the real one as a cubic always has one real
          solution.

          See Planetmath, Mathworld, or Wikipedia for more
          information.

   EulersMethod

EulersMethod (f,x0,y0,x1,n)

          Use classical Euler's method to numerically solve
          y'=f(x,y) for initial x0, y0 going to x1 with n
          increments, returns y at x1.

          Systems can be solved by just having y be a (column)
          vector everywhere. That is, y0 can be a vector in which
          case f should take a number x and a vector of the same
          size for the second argument and should return a vector
          of the same size.

          See Mathworld, or Wikipedia for more information.

   EulersMethodFull

EulersMethodFull (f,x0,y0,x1,n)

          Use classical Euler's method to numerically solve
          y'=f(x,y) for initial x0, y0 going to x1 with n
          increments, returns a 2 by n+1 matrix with the x and y
          values. Suitable for plugging into LinePlotDrawLine.

          Systems can be solved by just having y be a (column)
          vector everywhere. That is, y0 can be a vector in which
          case f should take a number x and a vector of the same
          size for the second argument and should return a vector
          of the same size.

          See Mathworld, or Wikipedia for more information.

   FindRootBisection

FindRootBisection (f,a,b,TOL,N)

          Find root of a function using the bisection method. TOL
          is the desired tolerance and N is the limit on the
          number of iterations to run, 0 means no limit. The
          function returns a vector [success,value,iteration],
          where success is a boolean indicating success, value is
          the last value computed, and iteration is the number of
          iterations done.

   FindRootFalsePosition

FindRootFalsePosition (f,a,b,TOL,N)

          Find root of a function using the method of false
          position. TOL is the desired tolerance and N is the
          limit on the number of iterations to run, 0 means no
          limit. The function returns a vector
          [success,value,iteration], where success is a boolean
          indicating success, value is the last value computed,
          and iteration is the number of iterations done.

   FindRootMullersMethod

FindRootMullersMethod (f,x1,x2,x3,TOL,N)

          Find root of a function using the Muller's method. TOL
          is the desired tolerance and N is the limit on the
          number of iterations to run, 0 means no limit. The
          function returns a vector [success,value,iteration],
          where success is a boolean indicating success, value is
          the last value computed, and iteration is the number of
          iterations done.

   FindRootSecant

FindRootSecant (f,a,b,TOL,N)

          Find root of a function using the secant method. TOL is
          the desired tolerance and N is the limit on the number
          of iterations to run, 0 means no limit. The function
          returns a vector [success,value,iteration], where
          success is a boolean indicating success, value is the
          last value computed, and iteration is the number of
          iterations done.

   PolynomialRoots

PolynomialRoots (p)

          Compute roots of a polynomial (degrees 1 through 4)
          using one of the formulas for such polynomials. The
          polynomial should be given as a vector of coefficients.
          That is 4*x^3 + 2*x + 1 corresponds to the vector
          [1,2,0,4]. Returns a column vector of the solutions.

          The function calls QuadraticFormula, CubicFormula, and
          QuarticFormula.

   QuadraticFormula

QuadraticFormula (p)

          Compute roots of a quadratic (degree 2) polynomial using
          the quadratic formula. The polynomial should be given as
          a vector of coefficients. That is 3*x^2 + 2*x + 1
          corresponds to the vector [1,2,3]. Returns a column
          vector of the two solutions.

          See Planetmath or Mathworld for more information.

   QuarticFormula

QuarticFormula (p)

          Compute roots of a quartic (degree 4) polynomial using
          the quartic formula. The polynomial should be given as a
          vector of coefficients. That is 5*x^4 + 2*x + 1
          corresponds to the vector [1,2,0,0,5]. Returns a column
          vector of the four solutions.

          See Planetmath, Mathworld, or Wikipedia for more
          information.

   RungeKutta

RungeKutta (f,x0,y0,x1,n)

          Use classical non-adaptive fourth order Runge-Kutta
          method to numerically solve y'=f(x,y) for initial x0, y0
          going to x1 with n increments, returns y at x1.

          Systems can be solved by just having y be a (column)
          vector everywhere. That is, y0 can be a vector in which
          case f should take a number x and a vector of the same
          size for the second argument and should return a vector
          of the same size.

          See Mathworld, or Wikipedia for more information.

   RungeKuttaFull

RungeKuttaFull (f,x0,y0,x1,n)

          Use classical non-adaptive fourth order Runge-Kutta
          method to numerically solve y'=f(x,y) for initial x0, y0
          going to x1 with n increments, returns a 2 by n+1 matrix
          with the x and y values. Suitable for plugging into
          LinePlotDrawLine.

          Systems can be solved by just having y be a (column)
          vector everywhere. That is, y0 can be a vector in which
          case f should take a number x and a vector of the same
          size for the second argument and should return a vector
          of the same size.

          See Mathworld, or Wikipedia for more information.
     __________________________________________________________

Statistics

   Average

Average (m)

          Aliases: average Mean mean

          Calculate average of an entire matrix.

          See Mathworld for more information.

   GaussDistribution

GaussDistribution (x,sigma)

          Integral of the GaussFunction from 0 to x (area under
          the normal curve).

          See Mathworld for more information.

   GaussFunction

GaussFunction (x,sigma)

          The normalized Gauss distribution function (the normal
          curve).

          See Mathworld for more information.

   Median

Median (m)

          Aliases: median

          Calculate median of an entire matrix.

          See Mathworld for more information.

   PopulationStandardDeviation

PopulationStandardDeviation (m)

          Aliases: stdevp

          Calculate the population standard deviation of a whole
          matrix.

   RowAverage

RowAverage (m)

          Aliases: RowMean

          Calculate average of each row in a matrix.

          See Mathworld for more information.

   RowMedian

RowMedian (m)

          Calculate median of each row in a matrix and return a
          column vector of the medians.

          See Mathworld for more information.

   RowPopulationStandardDeviation

RowPopulationStandardDeviation (m)

          Aliases: rowstdevp

          Calculate the population standard deviations of rows of
          a matrix and return a vertical vector.

   RowStandardDeviation

RowStandardDeviation (m)

          Aliases: rowstdev

          Calculate the standard deviations of rows of a matrix
          and return a vertical vector.

   StandardDeviation

StandardDeviation (m)

          Aliases: stdev

          Calculate the standard deviation of a whole matrix.
     __________________________________________________________

Polynomials

   AddPoly

AddPoly (p1,p2)

          Add two polynomials (vectors).

   DividePoly

DividePoly (p,q,&r)

          Divide two polynomials (as vectors) using long division.
          Returns the quotient of the two polynomials. The
          optional argument r is used to return the remainder. The
          remainder will have lower degree than q.

          See Planetmath for more information.

   IsPoly

IsPoly (p)

          Check if a vector is usable as a polynomial.

   MultiplyPoly

MultiplyPoly (p1,p2)

          Multiply two polynomials (as vectors).

   NewtonsMethodPoly

NewtonsMethodPoly (poly,guess,epsilon,maxn)

          Run newton's method on a polynomial to attempt to find a
          root, returns after two successive values are within
          epsilon or after maxn tries (then returns null).

   Poly2ndDerivative

Poly2ndDerivative (p)

          Take second polynomial (as vector) derivative.

   PolyDerivative

PolyDerivative (p)

          Take polynomial (as vector) derivative.

   PolyToFunction

PolyToFunction (p)

          Make function out of a polynomial (as vector).

   PolyToString

PolyToString (p,var...)

          Make string out of a polynomial (as vector).

   SubtractPoly

SubtractPoly (p1,p2)

          Subtract two polynomials (as vectors).

   TrimPoly

TrimPoly (p)

          Trim zeros from a polynomial (as vector).
     __________________________________________________________

Set Theory

   Intersection

Intersection (X,Y)

          Returns a set theoretic intersection of X and Y (X and Y
          are vectors pretending to be sets).

   IsIn

IsIn (x,X)

          Returns true if the element x is in the set X (where X
          is a vector pretending to be a set).

   IsSubset

IsSubset (X, Y)

          Returns true if X is a subset of Y (X and Y are vectors
          pretending to be sets).

   MakeSet

MakeSet (X)

          Returns a vector where every element of X appears only
          once.

   SetMinus

SetMinus (X,Y)

          Returns a set theoretic difference X-Y (X and Y are
          vectors pretending to be sets).

   Union

Union (X,Y)

          Returns a set theoretic union of X and Y (X and Y are
          vectors pretending to be sets).
     __________________________________________________________

Miscellaneous

   ASCIIToString

ASCIIToString (vec)

          Convert a vector of ASCII values to a string.

   AlphabetToString

AlphabetToString (vec,alphabet)

          Convert a vector of 0-based alphabet values (positions
          in the alphabet string) to a string.

   StringToASCII

StringToASCII (str)

          Convert a string to a vector of ASCII values.

   StringToAlphabet

StringToAlphabet (str,alphabet)

          Convert a string to a vector of 0-based alphabet values
          (positions in the alphabet string), -1's for unknown
          letters.
     __________________________________________________________

Symbolic Operations

   SymbolicDerivative

SymbolicDerivative (f)

          Attempt to symbolically differentiate the function f,
          where f is a function of one variable.

          Examples:

genius> SymbolicDerivative(sin)
= (`(x)=cos(x))
genius> SymbolicDerivative(`(x)=7*x^2)
= (`(x)=(7*(2*x)))

   SymbolicDerivativeTry

SymbolicDerivativeTry (f)

          Attempt to symbolically differentiate the function f,
          where f is a function of one variable, returns null if
          unsuccessful but is silent. (See SymbolicDerivative)

   SymbolicNthDerivative

SymbolicNthDerivative (f,n)

          Attempt to symbolically differentiate a function n
          times. (See SymbolicDerivative)

   SymbolicNthDerivativeTry

SymbolicNthDerivativeTry (f,n)

          Attempt to symbolically differentiate a function n times
          quietly and return null on failure (See
          SymbolicNthDerivative)

   SymbolicTaylorApproximationFunction

SymbolicTaylorApproximationFunction (f,x0,n)

          Attempt to construct the taylor approximation function
          around x0 to the nth degree. (See SymbolicDerivative)
     __________________________________________________________

Plotting

   LinePlot

LinePlot (func1,func2,func3,...)

LinePlot (func1,func2,func3,x1,x2,y1,y2)

          Plot a function (or several functions) with a line.
          First up to 10 arguments are functions, then optionally
          you can specify the limits of the plotting window as x1,
          x2, y1, y2. If limits are not specified, then the
          currently set limits apply (See LinePlotWindow)

          The parameter LinePlotDrawLegends controls the drawing
          of the legend.

          Examples:

genius> LinePlot(sin,cos)
genius> LinePlot(`(x)=x^2,-1,1,0,1)

   LinePlotClear

LinePlotClear ()

          Show the line plot window and clear out functions and
          any other lines that were drawn.

   LinePlotDrawLine

LinePlotDrawLine (x1,y1,x2,y2,...)

LinePlotDrawLine (v,...)

          Draw a line from x1,y1 to x2,y2. x1,y1, x2,y2 can be
          replaced by an n by 2 matrix for a longer line.

          Extra parameters can be added to specify line color,
          thickness, arrows, and the plotting window. You can do
          this by adding a string "color", "thickness", "window",
          or "arrow", and after it either the color string, the
          thicknes as an integer, the window as 4-vector, and for
          arrow either "origin", "end", "both", or "none". For
          "window" we can specify "fit" rather than a vector in
          which case, the x range will be set precisely and the y
          range will be set with five percent borders around the
          line. Finally, the legend can be specified by adding
          "legend" and the string with the legend.

          Examples:

genius> LinePlotDrawLine(0,0,1,1,"color","blue","thickness",3)
genius> LinePlotDrawLine([0,0;1,-1;-1,-1])
genius> LinePlotDrawLine([0,0;1,1],"arrow","end")
genius> LinePlotDrawLine(EulersMethodFull(`(x,y)=y,0,3,100),"color","blu
e","legend","The Solution")

   LinePlotParametric

LinePlotParametric (xfunc,yfunc,...)

LinePlotParametric (xfunc,yfunc,t1,t2,tinc)

LinePlotParametric (xfunc,yfunc,t1,t2,tinc,x1,x2,y1,y2)

          Plot a parametric function with a line. First come the
          functions for x and y then optionally the t limits as
          t1,t2,tinc, then optionally the limits as x1,x2,y1,y2.

          If limits are not specified, then the currently set
          limits apply (See LinePlotWindow).

          The parameter LinePlotDrawLegends controls the drawing
          of the legend.

   LinePlotCParametric

LinePlotCParametric (func,...)

LinePlotCParametric (func,t1,t2,tinc)

LinePlotCParametric (func,t1,t2,tinc,x1,x2,y1,y2)

          Plot a parametric complex valued function with a line.
          First comes the function that returns x+iy, then
          optionally the t limits as t1,t2,tinc, then optionally
          the limits as x1,x2,y1,y2.

          If limits are not specified, then the currently set
          limits apply (See LinePlotWindow).

          The parameter LinePlotDrawLegends controls the drawing
          of the legend.

   SlopefieldClearSolutions

SlopefieldClearSolutions ()

          Clears the solutions drawn by the SlopefieldDrawSolution
          function.

   SlopefieldDrawSolution

SlopefieldDrawSolution (x, y, dx)

          When a slope field plot is active, draw a solution with
          the specified initial condition. The standard
          Runge-Kutta method is used with increment dx. Solutions
          stay on the graph until a different plot is shown or
          until you call SlopefieldClearSolutions. You can also
          use the graphical interface to draw solutions and
          specify initial conditions with the mouse.

   SlopefieldPlot

SlopefieldPlot (func)

SlopefieldPlot (func,x1,x2,y1,y2)

          Plot a slope field. The function func should take two
          real numbers x and y, or a single complex number.
          Optionally you can specify the limits of the plotting
          window as x1, x2, y1, y2. If limits are not specified,
          then the currently set limits apply (See
          LinePlotWindow).

          The parameter LinePlotDrawLegends controls the drawing
          of the legend.

          Examples:

genius> Slopefield(`(x,y)=sin(x-y),-5,5,-5,5)

   SurfacePlot

SurfacePlot (func)

SurfacePlot (func,x1,x2,y1,y2,z1,z2)

          Plot a surface function which takes either two arguments
          or a complex number. First comes the function then
          optionally limits as x1, x2, y1, y2, z1, z2. If limits
          are not specified, then the currently set limits apply
          (See SurfacePlotWindow). Genius can only plot a single
          surface function at this time.

          Examples:

genius> SurfacePlot(|sin|,-1,1,-1,1,0,1.5)
genius> SurfacePlot(`(x,y)=x^2+y,-1,1,-1,1,-2,2)
genius> SurfacePlot(`(z)=|z|^2,-1,1,-1,1,0,2)

   VectorfieldClearSolutions

VectorfieldClearSolutions ()

          Clears the solutions drawn by the
          VectorfieldDrawSolution function.

   VectorfieldDrawSolution

VectorfieldDrawSolution (x, y, dt, tlen)

          When a vector field plot is active, draw a solution with
          the specified initial condition. The standard
          Runge-Kutta method is used with increment dt for an
          interval of length tlen. Solutions stay on the graph
          until a different plot is shown or until you call
          VectorfieldClearSolutions. You can also use the
          graphical interface to draw solutions and specify
          initial conditions with the mouse.

   VectorfieldPlot

VectorfieldPlot (funcx, funcy)

VectorfieldPlot (funcx, funcy, x1, x2, y1, y2)

          Plot a two dimensional vector field. The function funcx
          should be the dx/dt of the vectorfield and the function
          funcy should be the dy/dt of the vectorfield. The
          functions should take two real numbers x and y, or a
          single complex number. When the parameter
          VectorfieldNormalized is true, then the magnitude of the
          vectors is normalized. That is, only the direction and
          not the magnitude is shown.

          Optionally you can specify the limits of the plotting
          window as x1, x2, y1, y2. If limits are not specified,
          then the currently set limits apply (See
          LinePlotWindow).

          The parameter LinePlotDrawLegends controls the drawing
          of the legend.

          Examples:

genius> VectorfieldPlot(`(x,y)=x^2-y, `(x,y)=y^2-x, -1, 1, -1, 1)
     __________________________________________________________

Example Programs in GEL

   Here is a function that calculates factorials:
function f(x) = if x <= 1 then 1 else (f(x-1)*x)

   With indentation it becomes:
function f(x) = (
  if x <= 1 then
    1
  else
    (f(x-1)*x)
)

   This is a direct port of the factorial function from the bc
   manpage. The syntax seems similar to bc, but different in that
   in GEL, the last expression is the one that is returned. Using
   the return function instead, it would be:
function f(x) = (
  if (x <= 1) then return (1);
  return (f(x-1) * x)
)

   By far the easiest way to define a factorial function would be
   using the product loop as follows. This is not only the
   shortest and fastest, but also probably the most readable
   version.
function f(x) = prod k=1 to x do k

   Here is a larger example, this basically redefines the internal
   ref function to calculate the row echelon form of a matrix. The
   function ref is built in and much faster, but this example
   demonstrates some of the more complex features of GEL.
# Calculate the row-echelon form of a matrix
function MyOwnREF(m) = (
  if not IsMatrix(m) or not IsValueOnly(m) then
    (error("ref: argument not a value only matrix");bailout);
  s := min(rows(m), columns(m));
  i := 1;
  d := 1;
  while d <= s and i <= columns(m) do (

    # This just makes the anchor element non-zero if at
    # all possible
    if m@(d,i) == 0 then (
      j := d+1;
      while j <= rows(m) do (
        if m@(j,i) == 0 then
          (j=j+1;continue);
        a := m@(j,);
        m@(j,) := m@(d,);
        m@(d,) := a;
        j := j+1;
        break
      )
    );
    if m@(d,i) == 0 then
      (i:=i+1;continue);

    # Here comes the actual zeroing of all but the anchor
    # element rows
    j := d+1;
    while j <= rows(m)) do (
      if m@(j,i) != 0 then (
        m@(j,) := m@(j,)-(m@(j,i)/m@(d,i))*m@(d,)
      );
      j := j+1
    );
    m@(d,) := m@(d,) * (1/m@(d,i));
    d := d+1;
    i := i+1
  );
  m
)
     __________________________________________________________

Settings

   To configure Genius Mathematics Tool, choose
   Settings->Preferences. There are several basic parameters
   provided by the calculator in addition to the ones provided by
   the standard library. These control how the calculator behaves.

   Note Changing Settings with GEL


   Many of the settings in Genius are simply global variables, and
   can be evaluated and assigned to in the same way as normal
   variables. See the Section called Using Variables in the
   Chapter called GEL Basics about evaluating and assigning to
   variables, and the Section called Parameters in the Chapter
   called List of GEL functions for a list of settings that can be
   modified in this way.

   As an example, you can set the maximum number of digits in a
   result to 12 by typing:
MaxDigits = 12
     __________________________________________________________

Output

   Maximum digits to output
          The maximum digits in a result (MaxDigits)

   Results as floats
          If the results should be always printed as floats
          (ResultsAsFloats)

   Floats in scientific notation
          If floats should be in scientific notation
          (ScientificNotation)

   Always print full expressions
          Should we print out full expressions for non-numeric
          return values (longer than a line) (FullExpressions)

   Use mixed fractions
          If fractions should be printed as mixed fractions such
          as "1 1/3" rather than "4/3". (MixedFractions)

   Display 0.0 when floating point number is less than 10^-x
          (0=never chop)
          How to chop output. But only when other numbers nearby
          are large. See the documentation of the paramter
          OutputChopExponent.

   Only chop numbers when another number is greater than 10^-x
          When to chop output. This is set by the paramter
          OutputChopWhenExponent. See the documentation of the
          paramter OutputChopExponent.

   Remember output settings across sessions
          Should the output settings in the Number/Expression
          output options frame be remembered for next session.
          Does not apply to the Error/Info output options frame.

          If unchecked, either the default or any previously saved
          settings are used each time Genius starts up. Note that
          settings are saved at the end of the session, so if you
          wish to change the defaults check this box, restart
          Genius Mathematics Tool and then uncheck it again.

   Display errors in a dialog
          If set the errors will be displayed in a seprate dialog,
          if unset the errors will be printed on the console.

   Display information messages in a dialog
          If set the information messages will be displayed in a
          seprate dialog, if unset the information messages will
          be printed on the console.

   Maximum errors to display
          The maximum number of errors to return on one evaluation
          (MaxErrors). If you set this to 0 then all errors are
          always returned. Usually if some loop causes many
          errors, then it is unlikely that you will be able to
          make sense out of more than a few of these, so seeing a
          long list of errors is usually not helpful.

   In addition to these preferences, there are some preferences
   that can only be changed by setting them in the workspace
   console. For others that may affect the output see the Section
   called Parameters in the Chapter called List of GEL functions.

   IntegerOutputBase
          The base that will be used to output integers

   OutputStyle
          A string, can be "normal", "latex", "mathml" or "troff"
          and it will effect how matrices (and perhaps other
          stuff) is printed, useful for pasting into documents.
          Normal style is the default human readable printing
          style of Genius Mathematics Tool. The other styles are
          for typsetting in LaTeX, MathML (XML), or in Troff.
     __________________________________________________________

Precision

   Floating point precision
          The floating point precision in bits (FloatPrecision).
          Note that changing this only affects newly computed
          quantities. Old values stored in variables are obviously
          still in the old precision and if you want to have them
          more precise you will have to recompute them. Exceptions
          to this are the system constants such as pi or e.

   Remember precision setting across sessions
          Should the precision setting be remembered for the next
          session. If unchecked, either the default or any
          previously saved setting is used each time Genius starts
          up. Note that settings are saved at the end of the
          session, so if you wish to change the default check this
          box, restart genius and then uncheck it again.
     __________________________________________________________

Terminal

   Terminal refers to the console in the work area.

   Scrollback lines
          Lines of scrollback in the terminal.

   Font
          The font to use on the terminal.

   Black on white
          If to use black on white on the terminal.

   Blinking cursor
          If the cursor in the terminal should blink when the
          terminal is in focus. This can sometimes be annoying and
          it generates idle traffic if you are using Genius
          remotely.
     __________________________________________________________

Memory

   Maximum number of nodes to allocate
          Internally all data is put onto small nodes in memory.
          This gives a limit on the maximum number of nodes to
          allocate for computations. This avoids the problem of
          running out of memory if you do something by mistake
          which uses too much memory, such as a recursion without
          end. This could slow your computer and make it hard to
          even interrupt the program.

          Once the limit is reached, Genius Mathematics Tool asks
          if you wish to interrupt the computation or if you wish
          to continue. If you continue, no limit is applied and it
          will be possible to run your computer out of memory. The
          limit will be applied again next time you execute a
          program or an expression on the Console regardless of
          how you answered the question.

          Setting the limit to zero means there is no limit to the
          amount of memory that genius uses.
     __________________________________________________________

About Genius Mathematics Tool

   Genius Mathematics Tool was written by Jiří (George) Lebl
   (<jirka@5z.com>). The history of Genius Mathematics Tool goes
   back to late 1997. It was the first calculator program for
   GNOME, but it then grew beyond being just a desktop calculator.
   To find more information about Genius Mathematics Tool, please
   visit the Genius Web page.

   To report a bug or make a suggestion regarding this application
   or this manual, follow the directions in this document.

   This program is distributed 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. A copy of this license can be found at this link, or
   in the file COPYING included with the source code of this
   program.

   Jiří Lebl was during various parts of the development partially
   supported for the work by NSF grant DMS 0900885 and the
   University of Illinois at Urbana-Champaign. The software has
   been used for both teaching and research.
