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|
Genius Manual
Jiř Lebl
Oklahoma State University
<jirka@5z.com>
Kai Willadsen
University of Queensland, Australia
<kaiw@itee.uq.edu.au>
Copyright © 1997-2016 Jiř (George) Lebl
Copyright © 2004 Kai Willadsen
Manual for the Genius Math Tool.
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.
Many of the names used by companies to distinguish their
products and services are claimed as trademarks. Where those
names appear in any GNOME documentation, and the members of the
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:
1. DOCUMENT IS PROVIDED ON AN "AS IS" BASIS, WITHOUT WARRANTY
OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING,
WITHOUT LIMITATION, WARRANTIES THAT THE DOCUMENT OR
MODIFIED VERSION OF THE DOCUMENT IS FREE OF DEFECTS
MERCHANTABLE, FIT FOR A PARTICULAR PURPOSE OR
NON-INFRINGING. THE ENTIRE RISK AS TO THE QUALITY,
ACCURACY, AND PERFORMANCE OF THE DOCUMENT OR MODIFIED
VERSION OF THE DOCUMENT IS WITH YOU. SHOULD ANY DOCUMENT OR
MODIFIED VERSION PROVE DEFECTIVE IN ANY RESPECT, YOU (NOT
THE INITIAL WRITER, AUTHOR OR ANY CONTRIBUTOR) ASSUME THE
COST OF ANY NECESSARY SERVICING, REPAIR OR CORRECTION. THIS
DISCLAIMER OF WARRANTY CONSTITUTES AN ESSENTIAL PART OF
THIS LICENSE. NO USE OF ANY DOCUMENT OR MODIFIED VERSION OF
THE DOCUMENT IS AUTHORIZED HEREUNDER EXCEPT UNDER THIS
DISCLAIMER; AND
2. UNDER NO CIRCUMSTANCES AND UNDER NO LEGAL THEORY, WHETHER
IN TORT (INCLUDING NEGLIGENCE), CONTRACT, OR OTHERWISE,
SHALL THE AUTHOR, INITIAL WRITER, ANY CONTRIBUTOR, OR ANY
DISTRIBUTOR OF THE DOCUMENT OR MODIFIED VERSION OF THE
DOCUMENT, OR ANY SUPPLIER OF ANY OF SUCH PARTIES, BE LIABLE
TO ANY PERSON FOR ANY DIRECT, INDIRECT, SPECIAL,
INCIDENTAL, OR CONSEQUENTIAL DAMAGES OF ANY CHARACTER
INCLUDING, WITHOUT LIMITATION, DAMAGES FOR LOSS OF
GOODWILL, WORK STOPPAGE, COMPUTER FAILURE OR MALFUNCTION,
OR ANY AND ALL OTHER DAMAGES OR LOSSES ARISING OUT OF OR
RELATING TO USE OF THE DOCUMENT AND MODIFIED VERSIONS OF
THE DOCUMENT, EVEN IF SUCH PARTY SHALL HAVE BEEN INFORMED
OF THE POSSIBILITY OF SUCH DAMAGES.
Feedback
To report a bug or make a suggestion regarding the Genius
Mathematics Tool application or this manual, please visit the
Genius Web page or email me at <jirka@5z.com>.
__________________________________________________________
Table of Contents
1. Introduction
2. Getting Started
2.1. To Start Genius Mathematics Tool
2.2. When You Start Genius
3. Basic Usage
3.1. Using the Work Area
3.2. To Create a New Program
3.3. To Open and Run a Program
4. Plotting
4.1. Line Plots
4.2. Parametric Plots
4.3. Slopefield Plots
4.4. Vectorfield Plots
4.5. Surface Plots
5. GEL Basics
5.1. Values
5.1.1. Numbers
5.1.2. Booleans
5.1.3. Strings
5.1.4. Null
5.2. Using Variables
5.2.1. Setting Variables
5.2.2. Built-in Variables
5.2.3. Previous Result Variable
5.3. Using Functions
5.3.1. Defining Functions
5.3.2. Variable Argument Lists
5.3.3. Passing Functions to Functions
5.3.4. Operations on Functions
5.4. Separator
5.5. Comments
5.6. Modular Evaluation
5.7. List of GEL Operators
6. Programming with GEL
6.1. Conditionals
6.2. Loops
6.2.1. While Loops
6.2.2. For Loops
6.2.3. Foreach Loops
6.2.4. Break and Continue
6.3. Sums and Products
6.4. Comparison Operators
6.5. Global Variables and Scope of Variables
6.6. Parameter variables
6.7. Returning
6.8. References
6.9. Lvalues
7. Advanced Programming with GEL
7.1. Error Handling
7.2. Toplevel Syntax
7.3. Returning Functions
7.4. True Local Variables
7.5. GEL Startup Procedure
7.6. Loading Programs
8. Matrices in GEL
8.1. Entering Matrices
8.2. Conjugate Transpose and Transpose Operator
8.3. Linear Algebra
9. Polynomials in GEL
9.1. Using Polynomials
10. Set Theory in GEL
10.1. Using Sets
11. List of GEL functions
11.1. Commands
11.2. Basic
11.3. Parameters
11.4. Constants
11.5. Numeric
11.6. Trigonometry
11.7. Number Theory
11.8. Matrix Manipulation
11.9. Linear Algebra
11.10. Combinatorics
11.11. Calculus
11.12. Functions
11.13. Equation Solving
11.14. Statistics
11.15. Polynomials
11.16. Set Theory
11.17. Commutative Algebra
11.18. Miscellaneous
11.19. Symbolic Operations
11.20. Plotting
12. Example Programs in GEL
13. Settings
13.1. Output
13.2. Precision
13.3. Terminal
13.4. Memory
14. About Genius Mathematics Tool
List of Figures
2-1. Genius Mathematics Tool Window
4-1. Create Plot Window
4-2. Plot Window
4-3. Parametric Plot Tab
4-4. Parametric Plot
4-5. Surface Plot
__________________________________________________________
Chapter 1. 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.
Parts of this manual describe 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.
Generally, when some feature of the language (function,
operator, etc...) is new in some version past 1.0.5, it is
mentioned, but below 1.0.5 you would have to look at the NEWS
file.
__________________________________________________________
Chapter 2. Getting Started
2.1. 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.
__________________________________________________________
2.2. When You Start Genius
When you start the GNOME edition of Genius Mathematics Tool,
the window pictured in Figure 2-1 is displayed.
Figure 2-1. Genius Mathematics Tool Window
[genius_window.png]
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. You can also monitor user variables, which is
especially useful while a long calculation is running,
or to debug a certain program. Finally the Calculator
allows plotting functions using a user friendly dialog
box.
The Examples menu is a list of example programs or
demos. If you open the menu, it will load the example
into a new program, which you can run, edit, modify, and
save. These programs should be well documented and
generally demonstrate either some feature of Genius
Mathematics Tool or some mathematical concept.
The Programs menu lists the currently open programs and
allows you to switch between them.
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. The programs are a set of
commands or functions that can be run all at once rather
than entering them at the command line. The programs can
be saved in files for later retrieval.
__________________________________________________________
Chapter 3. Basic Usage
3.1. 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 in the console, it will be
evaluated, and the returned value will be printed.
To evaluate an expression, type it into the Console work area
and press enter. Expressions are written in a language called
GEL. The most simple GEL expressions 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.
__________________________________________________________
3.2. To Create a New Program
If you wish to enter several more complicated commands, or
perhaps write a complicated function using the GEL language,
you can 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 (or the Run toolbar button). 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.... Similarly as in other programs you can choose File->Save
to save a program that already has a filename attached to it.
If you have many opened programs you have edited and wish to
save you can also choose File->Save All Unsaved.
Programs that have unsaved changes will have a "[+]" next to
their filename. This way you can see if the file on disk and
the currently opened tab differ in content. Programs which have
not yet had a filename associated with them are always
considered unsaved and no "[+]" is printed.
__________________________________________________________
3.3. 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.
If you have made edits to a file you wish to throw away and
want to reload to the version that's on disk, you can choose
the File->Reload from Disk menuitem. This is useful for
experimenting with a program and making temporary edits, to run
a program, but that you do not intend to keep.
__________________________________________________________
Chapter 4. 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 Chapter 5 to find out how to enter
expressions that Genius understands.
__________________________________________________________
4.1. 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 4-1.
Figure 4-1. Create Plot Window
[line_plot.png]
Type expressions with x as the independent variable into the
textboxes. Alternatively you can 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. The y (dependent) range can be set automatically by
turning on the Fit dependent axis checkbox. The names of the
variables can also be changed. Pressing the Plot button
produces the graph shown in Figure 4-2.
The variables can be renamed by clicking the Change variable
names... button, which is useful if you wish to print or save
the figure and don't want to use the standard names. Finally
you can also avoid printing the legend and the axis labels
completely, which is also useful if printing or saving, when
the legend might simply be clutter.
Figure 4-2. Plot Window
[line_plot_graph.png]
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.
__________________________________________________________
4.2. 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 as a function of the variable t. The range of the
variable t is given explicitly, and the function is sampled
according to the given increment. The x and y range can be set
automatically by turning on the Fit dependent axis checkbox, or
it can be specified explicitly. See Figure 4-3.
Figure 4-3. Parametric Plot Tab
[parametric.png]
An example of a parametric plot is given in Figure 4-4. 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.
Figure 4-4. Parametric Plot
[parametric_graph.png]
__________________________________________________________
4.3. 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.
__________________________________________________________
4.4. 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.
__________________________________________________________
4.5. 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 that 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 4-5. For plotting using the
command line see the documentation of the SurfacePlot function.
The z range can be set automatically by turning on the Fit
dependent axis checkbox. The variables can be renamed by
clicking the Change variable names... button, which is useful
if you wish to print or save the figure and don't want to use
the standard names. Finally you can also avoid printing the
legend, which is also useful if printing or saving, when the
legend might simply be clutter.
Figure 4-5. Surface Plot
[surface_graph.png]
In surface mode, left and right arrow keys on your keyboard
will rotate the view along the z axis. Alternatively you can
rotate along any axis by selecting Rotate axis... in the View
menu. The View menu also has a top view mode which rotates the
graph so that the z axis is facing straight out, that is, we
view the graph from the top and get essentially just the colors
that define the values of the function getting a temperature
plot of the function. Finally you should try Start rotate
animation, to start a continuous slow rotation. This is
especially good if using Genius Mathematics Tool to present to
an audience.
__________________________________________________________
Chapter 5. 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, a matrix, or another
object in GEL. Genius Mathematics Tool can 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.
__________________________________________________________
5.1. Values
Values in GEL can be numbers, Booleans, or strings. GEL also
treats matrices as values. Values can be used in calculations,
assigned to variables and returned from functions, among other
uses.
__________________________________________________________
5.1.1. 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 of 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. To
add an imaginary part, append 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)
__________________________________________________________
5.1.2. 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.
__________________________________________________________
5.1.3. 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
Do note however that when a string is returned from a function,
escapes are quoted, so that the output can be used as input. If
you wish to print the string as it is (without escapes), use
the print or printn functions.
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
__________________________________________________________
5.1.4. 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 constant 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.
__________________________________________________________
5.2. 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 Section 5.3.1).
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.
__________________________________________________________
5.2.1. Setting Variables
Syntax:
<identifier> = <value>
<identifier> := <value>
Example:
x = 3
x := 3
To assign a value to a variable, use the = or := operators.
These operators set the value of the variable and return the
value you set, so you can do things like
a = b = 5
This will set b to 5 and then also set a to 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 more information about the scope of variables, that is when
are what variables visible, see Section 6.5.
__________________________________________________________
5.2.2. 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 Section 11.4 for a full
list. Note that i is not by default the square root of negative
one (the imaginary number), and is undefined to allow its use
as a counter. If you wish to write the imaginary number you
need to use 1i.
__________________________________________________________
5.2.3. 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
__________________________________________________________
5.3. 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 command to get a list of
available functions, or see Chapter 11 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.
__________________________________________________________
5.3.1. 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 that
Genius can load in order to make them available. Example:
function addup(a,b,c) = a+b+c
then addup(1,4,9) yields 14
__________________________________________________________
5.3.2. 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.
__________________________________________________________
5.3.3. 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 that are not defined, you can use an
anonymous function (see Section 5.3.1). 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.
__________________________________________________________
5.3.4. 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.
__________________________________________________________
5.4. Separator
GEL is somewhat different from other languages in how it deals
with multiple commands and functions. In GEL you must chain
commands together with a separator operator. That is, if you
want to type more than one expression you have to use the ;
operator in between the expressions. This is a way in which
both expressions are evaluated and the result of the second one
(or the last one if there is more than two expressions) is
returned. 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 to some degree. If a ; is found at
the end of a parenthesis or a block, genius will append a null
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.
If you are typing expressions in a program you do not have to
add a semicolon. In this case genius will simply print the
return value whenever it executes the expression. However, do
note that if you are defining a function, the body of the
function is a single expression.
__________________________________________________________
5.5. Comments
GEL is similar to other scripting languages in that # denotes a
comment, that is text that is not meant to be evaluated.
Everything beyond the pound sign till the end of line will just
be ignored. For example,
# This is just a comment
# every line in a comment must have its own pound sign
# in the next line we set x to the value 123
x=123;
__________________________________________________________
5.6. 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, which simply gives the remainder, 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
instantaneous. The first expression first tries to compute the
integer 10^(10^10) and then find remainder after division by 6,
while the second expression evaluates everything modulo 6 to
begin with.
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.
Do not chain mod operators, simply place it at the end of the
computation, all computations in the expression on the left
will be carried out in mod arithmetic. If you place a mod
inside a mod, you will get unexpected results. If you simply
want to mod a single number and control exactly when remainders
are taken, best to use the % operator. When you need to chain
several expressions in modular arithmetic with different
divisors, it may be best to just split up the expression into
several and use temporary variables to avoid a mod inside a
mod.
__________________________________________________________
5.7. 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. 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.
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. If one is a square matrix and the
other a number, then the number is multiplied by the
identity matrix.
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. When a and b are just numbers this is the
normal division. When they are matrices, then this is
equivalent to a*b^-1.
a./b
Element by element division. Same as a/b for numbers,
but operates element by element on matrices.
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 integer division
a/b.
a.%b
Element by element mod operator. Returns the remainder
after element by element integer division a./b.
a mod b
Modular evaluation operator. The expression a is
evaluated modulo b. See Section 5.6. 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 depending on a
and b being equal or not.
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. These can be
chained as in a <= b <= c (can also be combined with the
less than operator).
a>=b
Greater than or equal operator, returns true if a is
greater than or equal to b else returns false. These can
be chained as in a >= b >= c (and they can also be
combined with the greater than operator).
a<b
Less than operator, returns true if a is less than b
else returns false. These can be chained as in a < b < c
(they can also be combined with the less than or equal
to operator).
a>b
Greater than operator, returns true if a is greater than
b else returns false. These can be chained as in a > b >
c (they can also be combined with the greater than or
equal to operator).
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. Returns true if both a and b are true, else
returns false. If given numbers, nonzero numbers are
treated as true.
a or b
Logical or. Returns true if either a or b is true, else
returns false. If given numbers, nonzero numbers are
treated as true.
a xor b
Logical xor. Returns true if exactly one of a or b is
true, else returns false. If given numbers, nonzero
numbers are treated as true.
not a
Logical not. Returns the logical negation of a.
-a
Negation operator. Returns the negative of a number or a
matrix (works element-wise on a matrix).
&a
Variable referencing (to pass a reference to a
variable). See Section 6.8.
*a
Variable dereferencing (to access a referenced
variable). See Section 6.8.
a'
Matrix conjugate transpose. That is, rows and columns
get swapped and we take complex conjugate of all
entries. That is if the i,j element of a is x+iy, then
the j,i element of a' is x-iy.
a.'
Matrix transpose, does not conjugate the entries. That
is, the i,j element of a becomes the j,i element of a.'.
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 multiple 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]
When the numbers involved are floating point numbers,
for example 1.0:0.4:3.0, the output is what is expected
even though adding 0.4 to 1.0 five times is actually
just slightly more than 3.0 due to the way that floating
point numbers are stored in base 2 (there is no 0.4, the
actual number stored is just ever so slightly bigger).
The way this is handled is the same as in the for, sum,
and prod loops. If the end is within 2^-20 times the
step size of the endpoint, the endpoint is used and we
assume there were roundoff errors. This is not perfect,
but it handles the majority of the cases. This check is
done only from version 1.0.18 onwards, so execution of
your code may differ on older versions. If you want to
avoid dealing with this issue, use actual rational
numbers, possibly using the float if you wish to get
floating point numbers in the end. For example 1:2/5:3
does the right thing and float(1:2/5:3) even gives you
floating point numbers and is ever so slightly more
precise than 1.0:0.4:3.0.
(a)i
Make a into an imaginary number (multiply a by the
imaginary). Normally the imaginary 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.
__________________________________________________________
Chapter 6. Programming with GEL
6.1. 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
__________________________________________________________
6.2. Loops
6.2.1. 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.
__________________________________________________________
6.2.2. 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)
When one of the values is a floating point number, then the
final check is done to within 2^-20 of the step size. That is,
even if we overshoot by 2^-20 times the "by" above, we still
execute the last iteration. This way
for x = 0 to 1 by 0.1 do print(x)
does the expected even though adding 0.1 ten times becomes just
slightly more than 1.0 due to the way that floating point
numbers are stored in base 2 (there is no 0.1, the actual
number stored is just ever so slightly bigger). This is not
perfect but it handles the majority of the cases. If you want
to avoid dealing with this issue, use actual rational numbers
for example:
for x = 0 to 1 by 1/10 do print(x)
This check is done only from version 1.0.16 onwards, so
execution of your code may differ on older versions.
__________________________________________________________
6.2.3. 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].
__________________________________________________________
6.2.4. 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>
)
__________________________________________________________
6.3. 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.
For floating point numbers the same roundoff error protection
is done as in the for loop. See Section 6.2.2.
__________________________________________________________
6.4. 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.
__________________________________________________________
6.5. 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 behavior 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.
See also the SetElement and SetVElement functions.
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 that 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.
__________________________________________________________
6.6. 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 behavior of genius.
__________________________________________________________
6.7. 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
)
)
__________________________________________________________
6.8. 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.
__________________________________________________________
6.9. 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 interchangeably. 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.
__________________________________________________________
Chapter 7. Advanced Programming with GEL
7.1. 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
);
...
)
__________________________________________________________
7.2. Toplevel Syntax
The syntax 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 a sequence of lines as if they 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). When a statement
does not end with a separator on the top level, the result is
printed after being executed.
For example,
function f(x)=x^2
f(3)
will print first the result of setting a function (a
representation of the function, in this case (`(x)=(x^2))) and
then the expected 9. To avoid this, enter a separator after the
function definition.
function f(x)=x^2;
f(3)
If you need to put a separator into your function then you have
to surround with parenthesis. For example:
function f(x)=(
y=1;
for j=1 to x do
y = y+j;
y^2
);
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()
)
__________________________________________________________
7.3. 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
then 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.
__________________________________________________________
7.4. 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 referenced
variables (except for function arguments) to a null.
If all variables are to be created as locals you can just pass
an asterisk 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.
__________________________________________________________
7.5. 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 library in its installed place, you’ll
have to first compile it with genius --compile loader.gel >
lib.cgel
__________________________________________________________
7.6. Loading Programs
Sometimes you have a larger program you wrote into a file and
want to read that file into Genius Mathematics Tool. 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 similar 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
__________________________________________________________
Chapter 8. Matrices in GEL
Genius has support for vectors and matrices and possesses a
sizable library of matrix manipulation and linear algebra
functions.
__________________________________________________________
8.1. Entering Matrices
To enter matrices, 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
matrices. 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]
similarly you can build matrices 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.
__________________________________________________________
8.2. 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.
__________________________________________________________
8.3. 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.
__________________________________________________________
Chapter 9. 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.
__________________________________________________________
9.1. 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 Section 11.15 in the function list for the rest of
functions acting on polynomials.
__________________________________________________________
Chapter 10. 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.
__________________________________________________________
10.1. 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.
__________________________________________________________
Chapter 11. List of GEL functions
To get help on a specific function from the console type:
help FunctionName
__________________________________________________________
11.1. Commands
help
help
help FunctionName
Print help (or help on a function/command).
load
load "file.gel"
Load a file into the interpreter. 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.
__________________________________________________________
11.2. 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.
Version 1.0.10 onwards.
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 (version 1.0.6 onwards).
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. It is
equivalent to function Identity(x)=x.
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.
CurrentTime
CurrentTime
Returns the current UNIX time with microsecond precision
as a floating point number. That is, returns the number
of seconds since January 1st 1970.
Version 1.0.15 onwards.
display
display (str,expr)
Display a string and an expression with a colon to
separate them.
DisplayVariables
DisplayVariables (var1,var2,...)
Display set of variables. The variables can be given as
strings or identifiers. For example:
DisplayVariables(`x,`y,`z)
If called without arguments (must supply empty argument
list) as
DisplayVariables()
then all variables are printed including a stacktrace
similar to Show user variables in the graphical version.
Version 1.0.18 onwards.
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.
PrintTable
PrintTable (f,v)
Print a table of values for a function. The values are
in the vector v. You can use the vector building
notation as follows:
PrintTable (f,[0:10])
If v is a positive integer, then the table of integers
from 1 up to and including v will be used.
Version 1.0.18 onwards.
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.
Version 1.0.7 onwards.
set
set (id,val)
Set a global variable. The id can be either a string or
a quoted identifier. For example:
set(`x,1)
will set the global variable x to the value 1.
The function returns the val, to be usable in chaining.
SetElement
SetElement (id,row,col,val)
Set an element of a global variable which is a matrix.
The id can be either a string or a quoted identifier.
For example:
SetElement(`x,2,3,1)
will set the second row third column element of the
global variable x to the value 1. If no global variable
of the name exists, or if it is set to something that's
not a matrix, a new zero matrix of appropriate size will
be created.
The row and col can also be ranges, and the semantics
are the same as for regular setting of the elements with
an equals sign.
The function returns the val, to be usable in chaining.
Available from 1.0.18 onwards.
SetVElement
SetElement (id,elt,val)
Set an element of a global variable which is a vector.
The id can be either a string or a quoted identifier.
For example:
SetElement(`x,2,1)
will set the second element of the global vector
variable x to the value 1. If no global variable of the
name exists, or if it is set to something that's not a
vector (matrix), a new zero row vector of appropriate
size will be created.
The elt can also be a range, and the semantics are the
same as for regular setting of the elements with an
equals sign.
The function returns the val, to be usable in chaining.
Available from 1.0.18 onwards.
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.
Version 1.0.7 onwards.
unprotect
unprotect (id)
Unprotect a variable from being modified.
UserVariables
UserVariables ()
Return a vector of identifiers of user defined
(unprotected) global variables.
Version 1.0.7 onwards.
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.
Since version 1.0.18, secs can be a noninteger number,
so wait(0.1) will wait for one tenth of a second.
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.
__________________________________________________________
11.3. 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.
LinePlotDrawAxisLabels
LinePlotDrawAxisLabels = true
Tells genius to draw the axis labels for line plotting
functions such as LinePlot.
Version 1.0.16 onwards.
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.
Version 1.0.10 onwards.
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).
Version 1.0.10 onwards.
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.
SurfacePlotDrawLegends
SurfacePlotDrawLegends = true
Tells genius to draw the legends for surface plotting
functions such as SurfacePlot.
Version 1.0.16 onwards.
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 independent complex variable
z=x+iy.
Version 1.0.10 onwards.
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).
Version 1.0.10 onwards.
__________________________________________________________
11.4. 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 Wikipedia or 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 in meters per second
squared. This is the standard gravity constant 9.80665.
The gravity in your particular neck of the woods might
be different due to different altitude and the fact that
the earth is not perfectly round and uniform.
See Wikipedia for more information.
e
e
The base of the natural logarithm. e^x is the
exponential function exp. It is approximately
2.71828182846... This number is sometimes called Euler's
number, although there are several numbers that are also
called Euler's. An example is the gamma constant:
EulerConstant.
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.
__________________________________________________________
11.5. 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. This is equivalent to |x|.
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. For example
Re(3+4i) yields 4.
See Wikipedia for more information.
IntegerQuotient
IntegerQuotient (m,n)
Division without remainder.
IsComplex
IsComplex (num)
Check if argument is a complex (non-real) number. Do
note that we really mean nonreal number. That is,
IsComplex(3) yields false, while IsComplex(3-1i) yields
true.
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 real floating point number
(non-complex).
IsGaussInteger
IsGaussInteger (num)
Aliases: IsComplexInteger
Check if argument is a possibly complex integer. That is
a complex integer is a number of the form n+1i*m where n
and m are integers.
IsInteger
IsInteger (num)
Check if argument is an integer (non-complex).
IsNonNegativeInteger
IsNonNegativeInteger (num)
Check if argument is a non-negative real integer. That
is, either a positive integer or zero.
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. For example
Re(3+4i) yields 3.
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.
Examples:
genius> ceil(1.1)
= 2
genius> ceil(-1.1)
= -1
Note that you should be careful and notice that floating
point numbers are stored in binary and so may not be
what you expect. For example ceil(420/4.2) returns 101
instead of the expected 100. This is because 4.2 is
actually very slightly less than 4.2. Use rational
representation 42/10 if you want exact arithmetic.
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.
See Wikipedia or Planetmath or Mathworld for more
information.
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 Wikipedia or Planetmath for more information.
trunc
trunc (x)
Aliases: Truncate IntegerPart
Truncate number to an integer (return the integer part).
__________________________________________________________
11.6. 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 than failing.
See Wikipedia or Mathworld for more information.
cos
cos (x)
Calculates the cosine function.
See Wikipedia or Planetmath for more information.
cosh
cosh (x)
Calculates the hyperbolic cosine function.
See Wikipedia or Planetmath for more information.
cot
cot (x)
The cotangent function.
See Wikipedia or Planetmath for more information.
coth
coth (x)
The hyperbolic cotangent function.
See Wikipedia or Planetmath for more information.
csc
csc (x)
The cosecant function.
See Wikipedia or Planetmath for more information.
csch
csch (x)
The hyperbolic cosecant function.
See Wikipedia or Planetmath for more information.
sec
sec (x)
The secant function.
See Wikipedia or Planetmath for more information.
sech
sech (x)
The hyperbolic secant function.
See Wikipedia or Planetmath for more information.
sin
sin (x)
Calculates the sine function.
See Wikipedia or Planetmath for more information.
sinh
sinh (x)
Calculates the hyperbolic sine function.
See Wikipedia or Planetmath for more information.
tan
tan (x)
Calculates the tan function.
See Wikipedia or Planetmath for more information.
tanh
tanh (x)
The hyperbolic tangent function.
See Wikipedia or Planetmath for more information.
__________________________________________________________
11.7. Number Theory
AreRelativelyPrime
AreRelativelyPrime (a,b)
Are the real integers a and b relatively prime? Returns
true or false.
See Wikipedia or 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, 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, 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 for being any perfect power, a^b.
IsPerfectSquare
IsPerfectSquare (n)
Check an integer for being a perfect square of an
integer. The number must be an integer. Negative
integers are of course never perfect squares of
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 calls 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, Planetmath, or Mathworld for more
information.
LucasNumber
LucasNumber (n)
Returns the nth Lucas number.
See Wikipedia, 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 Riemann Hypothesis
the result is deterministic.
See Wikipedia, 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 GMPs 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 Wikipedia or 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 necessary 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 Wikipedia or 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 Wikipedia, Planetmath, or Mathworld for more
information.
gcd
gcd (a,args...)
Aliases: GCD
Greatest common divisor of integers. You can enter as
many integers as you want 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 Wikipedia, Planetmath, or Mathworld for more
information.
lcm
lcm (a,args...)
Aliases: LCM
Least common multiplier of integers. You can enter as
many integers as you want 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 Wikipedia, Planetmath, or Mathworld for more
information.
__________________________________________________________
11.8. 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;
This is the bilinear scalar product not the sesquilinear
scalar product. See HermitianProduct for the standard
sesquilinear inner product.
See Wikipedia or 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 Wikipedia or 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 Wikipedia or 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 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 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. Alternatively you
can pass in the values to put on the diagonal as
arguments. So MakeDiagonal([1,2,3]) is the same as
MakeDiagonal(1,2,3).
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.
NonzeroColumns
NonzeroColumns (M)
Returns a row vector of the indices of nonzero columns
in the matrix M.
Version 1.0.18 onwards.
NonzeroElements
NonzeroElements (v)
Returns a row vector of the indices of nonzero elements
in the vector v.
Version 1.0.18 onwards.
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 and
return a vertical vector with the results.
RowsOf
RowsOf (M)
Gets the rows of a matrix as a vertical vector. Each
element of the vector is a horizontal vector that 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.
Version 1.0.13 onwards.
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.
__________________________________________________________
11.9. Linear Algebra
AuxiliaryUnitMatrix
AuxiliaryUnitMatrix (n)
Get the auxiliary unit matrix of size n. This is a
square 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 Canonical 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 Wikipedia or 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 Wikipedia or 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.
See Wikipedia for more information.
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 Wikipedia or 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.
See Wikipedia for more information.
DeterminantalDivisorsInteger
DeterminantalDivisorsInteger (M)
Get the determinantal divisors of an integer matrix.
DirectSum
DirectSum (M,N...)
Direct sum of matrices.
See Wikipedia for more information.
DirectSumMatrixVector
DirectSumMatrixVector (v)
Direct sum of a vector of matrices.
See Wikipedia for more information.
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, 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, 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 Wikipedia or Planetmath for more information.
HankelMatrix
HankelMatrix (c,r)
Hankel matrix, a matrix whose skew-diagonals are
constant. c is the first row and r is the last column.
It is assumed that both arguments are vectors and the
last element of c is the same as the first element of r.
See Wikipedia for more information.
HilbertMatrix
HilbertMatrix (n)
Hilbert matrix of order n.
See Wikipedia or Planetmath for more information.
Image
Image (T)
Get the image (columnspace) of a linear transform.
See Wikipedia for more information.
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.
InverseHilbertMatrix
InverseHilbertMatrix (n)
Inverse Hilbert matrix of order n.
See Wikipedia or Planetmath for more information.
IsHermitian
IsHermitian (M)
Is a matrix Hermitian. That is, is it equal to its
conjugate transpose.
See Wikipedia or 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 if and only if it is 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 Wikipedia, 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)
KroneckerProduct
KroneckerProduct (M, N)
Aliases: TensorProduct
Compute the Kronecker product (tensor product in
standard basis) of two matrices.
See Wikipedia, Planetmath or Mathworld for more
information.
Version 1.0.18 onwards.
LUDecomposition
LUDecomposition (A, L, U)
Get the LU decomposition of A, that is find a lower
triangular matrix and upper triangular matrix whose
product is A. 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 Wikipedia, 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 that
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 Wikipedia or 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 ()
Returns the Rosser matrix, which is 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)
Returns the Smith normal form of a matrix over fields
(will end up with 1's on the diagonal).
See Wikipedia for more information.
SmithNormalFormInteger
SmithNormalFormInteger (M)
Return the Smith normal form for square integer matrices
over integers.
See Wikipedia for more information.
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 Wikipedia or Planetmath for more information.
Trace
Trace (M)
Aliases: trace
Calculate the trace of a matrix. That is the sum of the
diagonal elements.
See Wikipedia or Planetmath for more information.
Transpose
Transpose (M)
Transpose of a matrix. This is the same as the .'
operator.
See Wikipedia or Planetmath for more information.
VandermondeMatrix
VandermondeMatrix (v)
Aliases: vander
Return the Vandermonde matrix.
See Wikipedia for more information.
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.
__________________________________________________________
11.10. Combinatorics
Catalan
Catalan (n)
Get nth 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)
See Wikipedia for more information.
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
largest 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 Wikipedia or Mathworld for more information.
GaloisMatrix
GaloisMatrix (combining_rule)
Galois matrix given a linear combining rule
(a_1*x_1+...+a_n*x_n=x_(n+1)).
GreedyAlgorithm
GreedyAlgorithm (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 Wikipedia or Mathworld for more information.
HarmonicNumber
HarmonicNumber (n,r)
Aliases: HarmonicH
Harmonic Number, the nth harmonic number of order r.
That is, it is the sum of 1/k^r for k from 1 to n.
Equivalent to sum k = 1 to n do 1/k^r.
See Wikipedia for more information.
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)).
See Wikipedia for more information. The sequence is
A005185 in OEIS.
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 Wikipedia, Planetmath, or Mathworld 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 Combinations 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.
See Wikipedia for more information.
Pascal
Pascal (i)
Get the Pascal's triangle as a matrix. This will return
an i+1 by i+1 lower diagonal matrix that 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=0}^n (-1)^k/k!.
Triangular
Triangular (nth)
Calculate the nth 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 r of
numbers from 1 to n.
See Mathworld or Wikipedia for more information.
__________________________________________________________
11.11. 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.
See Wikipedia for more information.
EvenPeriodicExtension
EvenPeriodicExtension (f,L)
Return a function that is the 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.
Version 1.0.7 onwards.
FourierSeriesFunction
FourierSeriesFunction (a,b,L)
Return a function that 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.
See Wikipedia for more information.
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.
Version 1.0.7 onwards.
NumericalFourierSeriesFunction
NumericalFourierSeriesFunction (f,L,N)
Return a function that 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.
Version 1.0.7 onwards.
NumericalFourierCosineSeriesCoefficients
NumericalFourierCosineSeriesCoefficients (f,L,N)
Return a vector of coefficients of 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.
Version 1.0.7 onwards.
NumericalFourierCosineSeriesFunction
NumericalFourierCosineSeriesFunction (f,L,N)
Return a function that 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.
Version 1.0.7 onwards.
NumericalFourierSineSeriesCoefficients
NumericalFourierSineSeriesCoefficients (f,L,N)
Return a vector of coefficients of 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.
Version 1.0.7 onwards.
NumericalFourierSineSeriesFunction
NumericalFourierSineSeriesFunction (f,L,N)
Return a function that 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.
Version 1.0.7 onwards.
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 that is the 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.
Version 1.0.7 onwards.
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 that is the periodic extension of f
defined on the interval [a,b] and has period b-a.
See also OddPeriodicExtension and EvenPeriodicExtension.
Version 1.0.7 onwards.
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.
__________________________________________________________
11.12. Functions
Argument
Argument (z)
Aliases: Arg arg
argument (angle) of complex number.
BesselJ0
BesselJ0 (x)
Bessel function of the first kind of order 0. Only
implemented for real numbers.
See Wikipedia for more information.
Version 1.0.16 onwards.
BesselJ1
BesselJ1 (x)
Bessel function of the first kind of order 1. Only
implemented for real numbers.
See Wikipedia for more information.
Version 1.0.16 onwards.
BesselJn
BesselJn (n,x)
Bessel function of the first kind of order n. Only
implemented for real numbers.
See Wikipedia for more information.
Version 1.0.16 onwards.
BesselY0
BesselY0 (x)
Bessel function of the second kind of order 0. Only
implemented for real numbers.
See Wikipedia for more information.
Version 1.0.16 onwards.
BesselY1
BesselY1 (x)
Bessel function of the second kind of order 1. Only
implemented for real numbers.
See Wikipedia for more information.
Version 1.0.16 onwards.
BesselYn
BesselYn (n,x)
Bessel function of the second kind of order n. Only
implemented for real numbers.
See Wikipedia for more information.
Version 1.0.16 onwards.
DirichletKernel
DirichletKernel (n,t)
Dirichlet kernel of order n.
DiscreteDelta
DiscreteDelta (v)
Returns 1 if and only if all elements are zero.
ErrorFunction
ErrorFunction (x)
Aliases: erf
The error function, 2/sqrt(pi) * int_0^x e^(-t^2) dt.
See Wikipedia or 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 or Wikipedia for more information.
KroneckerDelta
KroneckerDelta (v)
Returns 1 if and only if all elements are equal.
LambertW
LambertW (x)
The principal branch of Lambert W function computed for
only real values greater than or equal to -1/e. That is,
LambertW is the inverse of the expression x*e^x. Even
for real x this expression is not one to one and
therefore has two branches over [-1/e,0). See LambertWm1
for the other real branch.
See Wikipedia for more information.
Version 1.0.18 onwards.
LambertWm1
LambertWm1 (x)
The minus-one branch of Lambert W function computed for
only real values greater than or equal to -1/e and less
than 0. That is, LambertWm1 is the second branch of the
inverse of x*e^x. See LambertW for the principal branch.
See Wikipedia for more information.
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 Wikipedia or 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 Wikipedia or 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 Wikipedia or 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 Wikipedia or 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 Wikipedia or 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 or Wikipedia 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.
sinc
sinc (x)
Calculates the unnormalized sinc function, that is
sin(x)/x. If you want the normalized function call
sinc(pi*x).
See Wikipedia for more information.
Version 1.0.16 onwards.
__________________________________________________________
11.13. 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. Unless you explicitly want
to use Euler's method, you should really think about
using RungeKutta for solving ODE.
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 an n+1 by 2 matrix with the x and y
values. Unless you explicitly want to use Euler's
method, you should really think about using
RungeKuttaFull for solving ODE. Suitable for plugging
into LinePlotDrawLine or LinePlotDrawPoints.
Example:
genius> LinePlotClear();
genius> line = EulersMethodFull(`(x,y)=y,0,1.0,3.0,50);
genius> LinePlotDrawLine(line,"window","fit","color","blue","legend","Ex
ponential growth");
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.
The output for a system is still a n by 2 matrix with
the second entry being a vector. If you wish to plot the
line, make sure to use row vectors, and then flatten the
matrix with ExpandMatrix, and pick out the right
columns. Example:
genius> LinePlotClear();
genius> lines = EulersMethodFull(`(x,y)=[y@(2),-y@(1)],0,[1.0,1.0],10.0,
500);
genius> lines = ExpandMatrix(lines);
genius> firstline = lines@(,[1,2]);
genius> secondline = lines@(,[1,3]);
genius> LinePlotWindow = [0,10,-2,2];
genius> LinePlotDrawLine(firstline,"color","blue","legend","First");
genius> LinePlotDrawPoints(secondline,"color","red","thickness",3,"legen
d","Second");
See Mathworld or Wikipedia for more information.
Version 1.0.10 onwards.
FindRootBisection
FindRootBisection (f,a,b,TOL,N)
Find root of a function using the bisection method. a
and b are the initial guess interval, f(a) and f(b)
should have opposite signs. 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. a and b are the initial guess interval, f(a)
and f(b) should have opposite signs. 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,x0,x1,x2,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. a and b
are the initial guess interval, f(a) and f(b) should
have opposite signs. 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.
HalleysMethod
HalleysMethod (f,df,ddf,guess,epsilon,maxn)
Find zeros using Halley's method. f is the function, df
is the derivative of f, and ddf is the second derivative
of f. guess is the initial guess. The function returns
after two successive values are within epsilon of each
other, or after maxn tries, in which case the function
returns null indicating failure.
See also NewtonsMethod and SymbolicDerivative.
Example to find the square root of 10:
genius> HalleysMethod(`(x)=x^2-10,`(x)=2*x,`(x)=2,3,10^-10,100)
See Wikipedia for more information.
Version 1.0.18 onwards.
NewtonsMethod
NewtonsMethod (f,df,guess,epsilon,maxn)
Find zeros using Newton's method. f is the function and
df is the derivative of f. guess is the initial guess.
The function returns after two successive values are
within epsilon of each other, or after maxn tries, in
which case the function returns null indicating failure.
See also NewtonsMethodPoly and SymbolicDerivative.
Example to find the square root of 10:
genius> NewtonsMethod(`(x)=x^2-10,`(x)=2*x,3,10^-10,100)
See Wikipedia for more information.
Version 1.0.18 onwards.
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, or Wikipedia 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 an n+1 by 2
matrix with the x and y values. Suitable for plugging
into LinePlotDrawLine or LinePlotDrawPoints.
Example:
genius> LinePlotClear();
genius> line = RungeKuttaFull(`(x,y)=y,0,1.0,3.0,50);
genius> LinePlotDrawLine(line,"window","fit","color","blue","legend","Ex
ponential growth");
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.
The output for a system is still a n by 2 matrix with
the second entry being a vector. If you wish to plot the
line, make sure to use row vectors, and then flatten the
matrix with ExpandMatrix, and pick out the right
columns. Example:
genius> LinePlotClear();
genius> lines = RungeKuttaFull(`(x,y)=[y@(2),-y@(1)],0,[1.0,1.0],10.0,10
0);
genius> lines = ExpandMatrix(lines);
genius> firstline = lines@(,[1,2]);
genius> secondline = lines@(,[1,3]);
genius> LinePlotWindow = [0,10,-2,2];
genius> LinePlotDrawLine(firstline,"color","blue","legend","First");
genius> LinePlotDrawPoints(secondline,"color","red","thickness",3,"legen
d","Second");
See Mathworld or Wikipedia for more information.
Version 1.0.10 onwards.
__________________________________________________________
11.14. Statistics
Average
Average (m)
Aliases: average Mean mean
Calculate average (the arithmetic mean) of an entire
matrix.
See Wikipedia or Mathworld for more information.
GaussDistribution
GaussDistribution (x,sigma)
Integral of the GaussFunction from 0 to x (area under
the normal curve).
See Wikipedia or Mathworld for more information.
GaussFunction
GaussFunction (x,sigma)
The normalized Gauss distribution function (the normal
curve).
See Wikipedia or Mathworld for more information.
Median
Median (m)
Aliases: median
Calculate median of an entire matrix.
See Wikipedia or 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. That is,
compute the arithmetic mean.
See Wikipedia or Mathworld for more information.
RowMedian
RowMedian (m)
Calculate median of each row in a matrix and return a
column vector of the medians.
See Wikipedia or 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.
__________________________________________________________
11.15. 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)
Find a root of a polynomial using Newton's method. poly
is the polynomial as a vector and guess is the initial
guess. The function returns after two successive values
are within epsilon of each other, or after maxn tries,
in which case the function returns null indicating
failure.
See also NewtonsMethod.
Example to find the square root of 10:
genius> NewtonsMethodPoly([-10,0,1],3,10^-10,100)
See Wikipedia for more information.
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).
__________________________________________________________
11.16. 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).
__________________________________________________________
11.17. Commutative Algebra
MacaulayBound
MacaulayBound (c,d)
For a Hilbert function that is c for degree d, given the
Macaulay bound for the Hilbert function of degree d+1
(The c^<d> operator from Green's proof).
Version 1.0.15 onwards.
MacaulayLowerOperator
MacaulayLowerOperator (c,d)
The c_<d> operator from Green's proof of Macaulay's
Theorem.
Version 1.0.15 onwards.
MacaulayRep
MacaulayRep (c,d)
Return the dth Macaulay representation of a positive
integer c.
Version 1.0.15 onwards.
__________________________________________________________
11.18. Miscellaneous
ASCIIToString
ASCIIToString (vec)
Convert a vector of ASCII values to a string. See also
StringToASCII.
Example:
genius> ASCIIToString([97,98,99])
= "abc"
See Wikipedia for more information.
AlphabetToString
AlphabetToString (vec,alphabet)
Convert a vector of 0-based alphabet values (positions
in the alphabet string) to a string. A null vector
results in an empty string. See also StringToAlphabet.
Examples:
genius> AlphabetToString([1,2,3,0,0],"abcd")
= "bcdaa"
genius> AlphabetToString(null,"abcd")
= ""
StringToASCII
StringToASCII (str)
Convert a string to a (row) vector of ASCII values. See
also ASCIIToString.
Example:
genius> StringToASCII("abc")
= [97, 98, 99]
See Wikipedia for more information.
StringToAlphabet
StringToAlphabet (str,alphabet)
Convert a string to a (row) vector of 0-based alphabet
values (positions in the alphabet string), -1's for
unknown letters. An empty string results in a null. See
also AlphabetToString.
Examples:
genius> StringToAlphabet("cca","abcd")
= [2, 2, 0]
genius> StringToAlphabet("ccag","abcd")
= [2, 2, 0, -1]
__________________________________________________________
11.19. 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)))
See Wikipedia for more information.
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)
See Wikipedia for more information.
SymbolicNthDerivative
SymbolicNthDerivative (f,n)
Attempt to symbolically differentiate a function n
times. (See SymbolicDerivative)
See Wikipedia for more information.
SymbolicNthDerivativeTry
SymbolicNthDerivativeTry (f,n)
Attempt to symbolically differentiate a function n times
quietly and return null on failure (See
SymbolicNthDerivative)
See Wikipedia for more information.
SymbolicTaylorApproximationFunction
SymbolicTaylorApproximationFunction (f,x0,n)
Attempt to construct the Taylor approximation function
around x0 to the nth degree. (See SymbolicDerivative)
__________________________________________________________
11.20. Plotting
ExportPlot
ExportPlot (file,type)
ExportPlot (file)
Export the contents of the plotting window to a file.
The type is a string that specifies the file type to
use, "png", "eps", or "ps". If the type is not
specified, then it is taken to be the extension, in
which case the extension must be ".png", ".eps", or
".ps".
Note that files are overwritten without asking.
On successful export, true is returned. Otherwise error
is printed and exception is raised.
Examples:
genius> ExportPlot("file.png")
genius> ExportPlot("/directory/file","eps")
Version 1.0.16 onwards.
LinePlot
LinePlot (func1,func2,func3,...)
LinePlot (func1,func2,func3,x1,x2)
LinePlot (func1,func2,func3,x1,x2,y1,y2)
LinePlot (func1,func2,func3,[x1,x2])
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)
If the y limits are not specified, then the functions
are computed and then the maxima and minima are used.
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.
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). If instead the string
"fit" is given for the x and y limits, then the limits
are the maximum extent of the graph
The parameter LinePlotDrawLegends controls the drawing
of the legend.
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 polyline.
Alternatively the vector v may be a column vector of
complex numbers, that is an n by 1 matrix and each
complex number is then considered a point in the plane.
Extra parameters can be added to specify line color,
thickness, arrows, the plotting window, or legend. You
can do this by adding an argument string "color",
"thickness", "window", "arrow", or "legend", and after
it specify the color, the thickness, the window as
4-vector, type of arrow, or the legend. (Arrow and
window are from version 1.0.6 onwards.)
If the line is to be treated as a filled polygon, filled
with the given color, you can specify the argument
"filled". Since version 1.0.22 onwards.
The color should be either a string indicating the
common English word for the color that GTK will
recognize such as "red", "blue", "yellow", etc...
Alternatively the color can be specified in RGB format
as "#rgb", "#rrggbb", or "#rrrrggggbbbb", where the r,
g, or b are hex digits of the red, green, and blue
components of the color. Finally, since version 1.0.18,
the color can also be specified as a real vector
specifying the red green and blue components where the
components are between 0 and 1, e.g. [1.0,0.5,0.1].
The window should be given as usual as [x1,x2,y1,y2], or
alternatively can be given as a string "fit" in which
case, the x range will be set precisely and the y range
will be set with five percent borders around the line.
Arrow specification should be "origin", "end", "both",
or "none".
Finally, legend should be a string that can be used as
the legend in the graph. That is, if legends are being
printed.
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(RungeKuttaFull(`(x,y)=y,0,0.001,10,100),"color"
,"blue","legend","The Solution")
genius> for r=0.0 to 1.0 by 0.1 do LinePlotDrawLine([0,0;1,r],"color",[r
,(1-r),0.5],"window",[0,1,0,1])
genius> LinePlotDrawLine([0,0;10,0;10,10;0,10],"filled","color","green")
Unlike many other functions that do not care if they
take a column or a row vector, if specifying points as a
vector of complex values, due to possible ambiguities,
it must always be given as a column vector.
Specifying v as a column vector of complex numbers is
implemented from version 1.0.22 and onwards.
LinePlotDrawPoints
LinePlotDrawPoints (x,y,...)
LinePlotDrawPoints (v,...)
Draw a point at x,y. The input can be an n by 2 matrix
for n different points. This function has essentially
the same input as LinePlotDrawLine. Alternatively the
vector v may be a column vector of complex numbers, that
is an n by 1 matrix and each complex number is then
considered a point in the plane.
Extra parameters can be added to specify color,
thickness, the plotting window, or legend. You can do
this by adding an argument string "color", "thickness",
"window", or "legend", and after it specify the color,
the thickness, the window as 4-vector, or the legend.
The color should be either a string indicating the
common English word for the color that GTK will
recognize such as "red", "blue", "yellow", etc...
Alternatively the color can be specified in RGB format
as "#rgb", "#rrggbb", or "#rrrrggggbbbb", where the r,
g, or b are hex digits of the red, green, and blue
components of the color. Finally the color can also be
specified as a real vector specifying the red green and
blue components where the components are between 0 and
1.
The window should be given as usual as [x1,x2,y1,y2], or
alternatively can be given as a string "fit" 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, legend should be a string that can be used as
the legend in the graph. That is, if legends are being
printed.
Examples:
genius> LinePlotDrawPoints(0,0,"color","blue","thickness",3)
genius> LinePlotDrawPoints([0,0;1,-1;-1,-1])
genius> LinePlotDrawPoints(RungeKuttaFull(`(x,y)=y,0,0.001,10,100),"colo
r","blue","legend","The Solution")
genius> LinePlotDrawPoints([1;1+1i;1i;0],"thickness",5)
genius> LinePlotDrawPoints(ApplyOverMatrix((0:6)',`(k)=exp(k*2*pi*1i/7))
,"thickness",3,"legend","The 7th roots of unity")
Unlike many other functions that do not care if they
take a column or a row vector, if specifying points as a
vector of complex values, due to possible ambiguities,
it must always be given as a column vector. Therefore,
notice in the last example the transpose of the vector
0:6 to make it into a column vector.
Available from version 1.0.18 onwards. Specifying v as a
column vector of complex numbers is implemented from
version 1.0.22 and onwards.
LinePlotMouseLocation
LinePlotMouseLocation ()
Returns a row vector of a point on the line plot
corresponding to the current mouse location. If the line
plot is not visible, then prints an error and returns
null. In this case you should run LinePlot or
LinePlotClear to put the graphing window into the line
plot mode. See also LinePlotWaitForClick.
LinePlotParametric
LinePlotParametric (xfunc,yfunc,...)
LinePlotParametric (xfunc,yfunc,t1,t2,tinc)
LinePlotParametric (xfunc,yfunc,t1,t2,tinc,x1,x2,y1,y2)
LinePlotParametric (xfunc,yfunc,t1,t2,tinc,[x1,x2,y1,y2])
LinePlotParametric (xfunc,yfunc,t1,t2,tinc,"fit")
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 x and y limits are not specified, then the currently
set limits apply (See LinePlotWindow). If instead the
string "fit" is given for the x and y limits, then the
limits are the maximum extent of the graph
The parameter LinePlotDrawLegends controls the drawing
of the legend.
LinePlotWaitForClick
LinePlotWaitForClick ()
If in line plot mode, waits for a click on the line plot
window and returns the location of the click as a row
vector. If the window is closed the function returns
immediately with null. If the window is not in line plot
mode, it is put in it and shown if not shown. See also
LinePlotMouseLocation.
PlotCanvasFreeze
PlotCanvasFreeze ()
Freeze drawing of the canvas plot temporarily. Useful if
you need to draw a bunch of elements and want to delay
drawing everything to avoid flicker in an animation.
After everything has been drawn you should call
PlotCanvasThaw.
The canvas is always thawed after end of any execution,
so it will never remain frozen. The moment a new command
line is shown for example the plot canvas is thawed
automatically. Also note that calls to freeze and thaw
may be safely nested.
Version 1.0.18 onwards.
PlotCanvasThaw
PlotCanvasThaw ()
Thaw the plot canvas frozen by PlotCanvasFreeze and
redraw the canvas immediately. The canvas is also always
thawed after end of execution of any program.
Version 1.0.18 onwards.
PlotWindowPresent
PlotWindowPresent ()
Show and raise the plot window, creating it if
necessary. Normally the window is created when one of
the plotting functions is called, but it is not always
raised if it happens to be below other windows. So this
function is good to call in scripts where the plot
window might have been created before, and by now is
hidden behind the console or other windows.
Version 1.0.19 onwards.
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> SlopefieldPlot(`(x,y)=sin(x-y),-5,5,-5,5)
SurfacePlot
SurfacePlot (func)
SurfacePlot (func,x1,x2,y1,y2,z1,z2)
SurfacePlot (func,x1,x2,y1,y2)
SurfacePlot (func,[x1,x2,y1,y2,z1,z2])
SurfacePlot (func,[x1,x2,y1,y2])
Plot a surface function that 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.
If the z limits are not specified then the maxima and
minima of the function are used.
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)
SurfacePlotClear
SurfacePlotClear ()
Show the surface plot window and clear out functions and
any other lines that were drawn.
Available in version 1.0.19 and onwards.
SurfacePlotData
SurfacePlotData (data)
SurfacePlotData (data,label)
SurfacePlotData (data,x1,x2,y1,y2,z1,z2)
SurfacePlotData (data,label,x1,x2,y1,y2,z1,z2)
SurfacePlotData (data,[x1,x2,y1,y2,z1,z2])
SurfacePlotData (data,label,[x1,x2,y1,y2,z1,z2])
Plot a surface from data. The data is an n by 3 matrix
whose rows are the x, y and z coordinates. The data can
also be simply a vector whose length is a multiple of 3
and so contains the triples of x, y, z. The data should
contain at least 3 points.
Optionally we can give the label and also optionally the
limits. If limits are not given, they are computed from
the data, SurfacePlotWindow is not used, if you want to
use it, pass it in explicitly. If label is not given
then empty label is used.
Examples:
genius> SurfacePlotData([0,0,0;1,0,1;0,1,1;1,1,3])
genius> SurfacePlotData(data,"My data")
genius> SurfacePlotData(data,-1,1,-1,1,0,10)
genius> SurfacePlotData(data,SurfacePlotWindow)
Here's an example of how to plot in polar coordinates,
in particular how to plot the function -r^2 * theta:
genius> d:=null; for r=0 to 1 by 0.1 do for theta=0 to 2*pi by pi/5 do d
=[d;[r*cos(theta),r*sin(theta),-r^2*theta]];
genius> SurfacePlotData(d)
Version 1.0.16 onwards.
SurfacePlotDataGrid
SurfacePlotDataGrid (data,[x1,x2,y1,y2])
SurfacePlotDataGrid (data,[x1,x2,y1,y2,z1,z2])
SurfacePlotDataGrid (data,[x1,x2,y1,y2],label)
SurfacePlotDataGrid (data,[x1,x2,y1,y2,z1,z2],label)
Plot a surface from regular rectangular data. The data
is given in a n by m matrix where the rows are the x
coordinate and the columns are the y coordinate. The x
coordinate is divided into equal n-1 subintervals and y
coordinate is divided into equal m-1 subintervals. The
limits x1 and x2 give the interval on the x-axis that we
use, and the limits y1 and y2 give the interval on the
y-axis that we use. If the limits z1 and z2 are not
given they are computed from the data (to be the extreme
values from the data).
Optionally we can give the label, if label is not given
then empty label is used.
Examples:
genius> SurfacePlotDataGrid([1,2;3,4],[0,1,0,1])
genius> SurfacePlotDataGrid(data,[-1,1,-1,1],"My data")
genius> d:=null; for i=1 to 20 do for j=1 to 10 do d@(i,j) = (0.1*i-1)^2
-(0.1*j)^2;
genius> SurfacePlotDataGrid(d,[-1,1,0,1],"half a saddle")
Version 1.0.16 onwards.
SurfacePlotDrawLine
SurfacePlotDrawLine (x1,y1,z1,x2,y2,z2,...)
SurfacePlotDrawLine (v,...)
Draw a line from x1,y1,z1 to x2,y2,z2. x1,y1,z1,
x2,y2,z2 can be replaced by an n by 3 matrix for a
longer polyline.
Extra parameters can be added to specify line color,
thickness, the plotting window, or legend. You can do
this by adding an argument string "color", "thickness",
"window", or "legend", and after it specify the color,
the thickness, the window as 6-vector, or the legend.
The color should be either a string indicating the
common English word for the color that GTK will
recognize such as "red", "blue", "yellow", etc...
Alternatively the color can be specified in RGB format
as "#rgb", "#rrggbb", or "#rrrrggggbbbb", where the r,
g, or b are hex digits of the red, green, and blue
components of the color. Finally, since version 1.0.18,
the color can also be specified as a real vector
specifying the red green and blue components where the
components are between 0 and 1, e.g. [1.0,0.5,0.1].
The window should be given as usual as
[x1,x2,y1,y2,z1,z2], or alternatively can be given as a
string "fit" 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, legend should be a string that can be used as
the legend in the graph. That is, if legends are being
printed.
Examples:
genius> SurfacePlotDrawLine(0,0,0,1,1,1,"color","blue","thickness",3)
genius> SurfacePlotDrawLine([0,0,0;1,-1,2;-1,-1,-3])
Available from version 1.0.19 onwards.
SurfacePlotDrawPoints
SurfacePlotDrawPoints (x,y,z,...)
SurfacePlotDrawPoints (v,...)
Draw a point at x,y,z. The input can be an n by 3 matrix
for n different points. This function has essentially
the same input as SurfacePlotDrawLine.
Extra parameters can be added to specify line color,
thickness, the plotting window, or legend. You can do
this by adding an argument string "color", "thickness",
"window", or "legend", and after it specify the color,
the thickness, the window as 6-vector, or the legend.
The color should be either a string indicating the
common English word for the color that GTK will
recognize such as "red", "blue", "yellow", etc...
Alternatively the color can be specified in RGB format
as "#rgb", "#rrggbb", or "#rrrrggggbbbb", where the r,
g, or b are hex digits of the red, green, and blue
components of the color. Finally the color can also be
specified as a real vector specifying the red green and
blue components where the components are between 0 and
1.
The window should be given as usual as
[x1,x2,y1,y2,z1,z2], or alternatively can be given as a
string "fit" 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, legend should be a string that can be used as
the legend in the graph. That is, if legends are being
printed.
Examples:
genius> SurfacePlotDrawPoints(0,0,0,"color","blue","thickness",3)
genius> SurfacePlotDrawPoints([0,0,0;1,-1,2;-1,-1,1])
Available from version 1.0.19 onwards.
VectorfieldClearSolutions
VectorfieldClearSolutions ()
Clears the solutions drawn by the
VectorfieldDrawSolution function.
Version 1.0.6 onwards.
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.
Version 1.0.6 onwards.
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)
__________________________________________________________
Chapter 12. 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("MyOwnREF: 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
)
__________________________________________________________
Chapter 13. 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 Section 5.2 about evaluating and assigning to
variables, and Section 11.3 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
__________________________________________________________
13.1. 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 parameter
OutputChopExponent.
Only chop numbers when another number is greater than 10^-x
When to chop output. This is set by the parameter
OutputChopWhenExponent. See the documentation of the
parameter 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 separate
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
separate 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 Section
11.3.
IntegerOutputBase
The base that will be used to output integers
OutputStyle
A string, can be "normal", "latex", "mathml" or "troff"
and it will affect 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 typesetting in LaTeX, MathML (XML), or in Troff.
__________________________________________________________
13.2. 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.
__________________________________________________________
13.3. 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.
__________________________________________________________
13.4. 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 limit avoids the problem
of running out of memory if you do something by mistake
that 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.
__________________________________________________________
Chapter 14. About Genius Mathematics Tool
Genius Mathematics Tool was written by Jiř (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, send email to me (the author) or post to the
mailing list (see the web page).
This program is distributed under the terms of the GNU General
Public license as published by the Free Software Foundation;
either version 3 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.
Jiř Lebl was during various parts of the development partially
supported for the work by NSF grants DMS 0900885, DMS 1362337,
the University of Illinois at Urbana-Champaign, the University
of California at San Diego, the University of
Wisconsin-Madison, and Oklahoma State University. The software
has been used for both teaching and research.
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