Genius Manual
i
Genius Manual
Genius Manual
ii
Copyright © 1997-2014 Jiří (George) Lebl
Copyright © 2004 Kai Willadsen
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(GFDL), Version 1.1 or any later version published by the Free Software Foundation with no Invariant Sections, no Front-Cover
Texts, and no Back-Cover Texts. You can find a copy of the GFDL at this link or in the file COPYING-DOCS distributed with
this manual.
This manual is part of a collection of GNOME manuals distributed under the GFDL. If you want to distribute this manual
separately from the collection, you can do so by adding a copy of the license to the manual, as described in section 6 of the
license.
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trademarks, then the names are in capital letters or initial capital letters.
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FREE DOCUMENTATION LICENSE WITH THE FURTHER UNDERSTANDING THAT:
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Feedback
To report a bug or make a suggestion regarding the Genius Mathematics Tool application or this manual, please visit the Genius
Web page or email me at jirka@5z.com.
Genius Manual
iii
COLLABORATORS
TITLE :
Genius Manual
ACTION
NAME
DATE
SIGNATURE
WRITTEN BY
Jiří Lebl and Kai
Willadsen
March 9, 2015
REVISION HISTORY
NUMBER
DATE
0.2
October 2014
DESCRIPTION
Jiri (George) Lebl jirka@5z.com
NAME
Genius Manual
iv
Contents
1
Introduction
1
2
Getting Started
2
2.1
To Start Genius Mathematics Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2.2
When You Start Genius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
3
4
5
Basic Usage
5
3.1
Using the Work Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
3.2
To Create a New Program
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
3.3
To Open and Run a Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Plotting
7
4.1
Line Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
4.2
Parametric Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
4.3
Slopefield Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.4
Vectorfield Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.5
Surface Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
GEL Basics
5.1
5.2
5.3
14
Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.1.1
Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.1.2
Booleans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.1.3
Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.1.4
Null . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Using Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.2.1
Setting Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2.2
Built-in Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2.3
Previous Result Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Using Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.3.1
Defining Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.3.2
Variable Argument Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Genius Manual
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7
8
9
v
5.3.3
Passing Functions to Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.3.4
Operations on Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.4
Separator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.5
Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.6
Modular Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.7
List of GEL Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Programming with GEL
24
6.1
Conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.2
Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.2.1
While Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.2.2
For Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.2.3
Foreach Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.2.4
Break and Continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.3
Sums and Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.4
Comparison Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.5
Global Variables and Scope of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.6
Parameter variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.7
Returning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.8
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.9
Lvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Advanced Programming with GEL
30
7.1
Error Handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
7.2
Toplevel Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
7.3
Returning Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
7.4
True Local Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7.5
GEL Startup Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7.6
Loading Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Matrices in GEL
34
8.1
Entering Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
8.2
Conjugate Transpose and Transpose Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
8.3
Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Polynomials in GEL
9.1
36
Using Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
10 Set Theory in GEL
37
10.1 Using Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Genius Manual
11 List of GEL functions
vi
38
11.1 Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
11.2 Basic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
11.3 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
11.4 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
11.5 Numeric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
11.6 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
11.7 Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
11.8 Matrix Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
11.9 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
11.10Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
11.11Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
11.12Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
11.13Equation Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
11.14Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
11.15Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
11.16Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
11.17Commutative Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
11.18Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
11.19Symbolic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
11.20Plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
12 Example Programs in GEL
84
13 Settings
86
13.1 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
13.2 Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
13.3 Terminal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
13.4 Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
14 About Genius Mathematics Tool
88
Genius Manual
vii
List of Figures
2.1
Genius Mathematics Tool Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
4.1
Create Plot Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
4.2
Plot Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
4.3
Parametric Plot Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.4
Parametric Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.5
Surface Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Abstract
Manual for the Genius Math Tool.
Genius Manual
1 / 88
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 is some version past 1.0.5, it is mentioned, but
below 1.0.5 you would have to look at the NEWS file.
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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.
Genius Manual
3 / 88
Figure 2.1: Genius Mathematics Tool Window
The Genius Mathematics Tool window contains the following elements:
Menubar. The menus on the menubar contain all of the commands that you need to work with files in Genius Mathematics Tool.
The File menu contains items for loading and saving items and creating new programs. The Load and Run... command
does not open a new window for the program, but just executes the program directly. It is equivalent to the load command.
The Calculator menu controls the calculator engine. It allows you to run the currently selected program or to interrupt the
current calculation. You can also look at the full expression of the last answer (useful if the last answer was too large to fit
onto the console), or you can view a listing of the values of all user defined variables. 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
some either 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.
Genius Manual
4 / 88
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.
Genius Manual
5 / 88
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 expression just looks like mathematics. For example
genius> 30*70 + 67^3.0 + ln(7) * (88.8/100)
or
genius> 62734 + 812634 + 77^4 mod 5
or
genius> | sin(37) - e^7 |
or
genius> sum n=1 to 70 do 1/n
(Last is the harmonic sum from 1 to 70)
To get a list of functions and commands, type:
genius> help
If you wish to get more help on a specific function, type:
genius> help FunctionName
To view this manual, type:
genius> manual
Suppose you have previously saved some GEL commands as a program to a file and you now want to execute them. To load this
program from the file path/to/program.gel, type
genius> load path/to/program.gel
Genius Mathematics Tool keeps track of the current directory. To list files in the current directory type ls, to change directory do
cd directory as in the UNIX command shell.
Genius Manual
3.2
6 / 88
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.
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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.
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Figure 4.1: Create Plot Window
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.
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Figure 4.2: Plot Window
From here you can print out the plot, create encapsulated postscript or a PNG version of the plot or change the zoom. If the
dependent axis was not set correctly you can have Genius fit it by finding out the extrema of the graphed functions.
For plotting using the command line see the documentation of the LinePlot function.
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.
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Figure 4.3: Parametric Plot Tab
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.
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Figure 4.4: Parametric Plot
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.
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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.
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Figure 4.5: Surface Plot
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.
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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 if number is floating point. These are entered in a similar fashion to C notation. You can use E, e or @ as the
exponent delimiter. Note that using the exponent delimiter gives a float even if there is no decimal point in the number. Examples:
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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:
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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).
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.
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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)
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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.
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.
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)
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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.
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Comments
GEL is similar to other scripting languages in that # denotes a comments, 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 pond 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: 2ˆ(5!) *
3ˆ(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, 10ˆ(10ˆ10) % 6 will simply not work (the exponent will be too large), while 10ˆ(10ˆ10) mod
6 is instantaneous. The first expression first tries to compute the integer 10ˆ(10ˆ10) 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.
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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 * eˆ(1i*pi)| returns 3.
See Mathworld for more information.
aˆb Exponentiation, raises a to the bth power.
a.ˆb 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*bˆ-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 a/b.
a.%b Element by element the mod operator. Returns the remainder after element by element integer a./b.
a mod b Modular evaluation operator. The expression a is evaluated modulo b. See 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 (can also be combine 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 (can also
be combine 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
(can also be combine 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.
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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 both a or b are true, else returns false. If given numbers, nonzero numbers are treated as
true.
a xor b Logical xor. Returns true 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 2ˆ-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 imaginary number (multiply a by the imaginary). Note that normally the number i is written as 1i. So the above
is equal to
(a)*1i
`a Quote an identifier so that it doesn’t get evaluated. Or quote a matrix so that it doesn’t get expanded.
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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 -1ˆk is really
(-1)ˆk, but -foo(1)ˆk is really -(foo(1)ˆk). So be careful how you use it and if in doubt, add parentheses.
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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
6.2.1
Loops
While Loops
Syntax:
while <expression1> do
until <expression1> do
do <expression2> while
do <expression2> until
<expression2>
<expression2>
<expression1>
<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.
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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 2ˆ-20 of the step size. That is, even if
we overshoot by 2ˆ-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 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>
)
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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:
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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
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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.
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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.
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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 sequence of
lines as if were entered on the command line. In particular, you do not need to enter the separator at the end of the line (unless it
is of course part of several statements inside parentheses).
The following code will produce an error when entered on the top level of a program, while it will work just fine in a function.
if Something() then
DoSomething()
else
DoSomethingElse()
The problem is that after Genius Mathematics Tool sees the end of line after the second line, it will decide that we have whole
statement and it will execute it. After the execution is done, Genius Mathematics Tool will go on to the next line, it will see
else, and it will produce a parsing error. To fix this, use parentheses. Genius Mathematics Tool will not be satisfied until it has
found that all parentheses are closed.
if Something() then (
DoSomething()
) else (
DoSomethingElse()
)
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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);
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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
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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
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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,
4,
5, 6,
7,
8, 9,
11, 11, 11,
10
10
10
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
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[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.
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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.
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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.
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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, ...)
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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.
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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.
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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:
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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.
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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.
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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).
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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 (-1ˆk)/((2*k+1)ˆ2), 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. eˆx 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.
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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."
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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 an 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 eˆx 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.
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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>
= 3
genius>
=
[0
genius>
=
[2
0
randint(4)
randint(4,2)
1]
randint(4,2,3)
2
0
1
3]
round round (x)
Aliases: Round
Round a number.
sqrt sqrt (x)
Aliases: SquareRoot
The square root. When operating modulo some integer will return either a null or a vector of the square roots. Examples:
genius> sqrt(2)
= 1.41421356237
genius> sqrt(-1)
= 1i
genius> sqrt(4) mod 7
=
[2
5]
genius> 2*2 mod 7
= 4
See Planetmath for more information.
trunc trunc (x)
Aliases: Truncate IntegerPart
Truncate number to an integer (return the integer part).
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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.
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atan2 atan2 (y, x)
Aliases: arctan2
Calculates the arctan2 function. If x>0 then it returns atan(y/x). If x<0 then it returns sign(y) * (pi atan(|y/x|). When x=0 it returns sign(y) * pi/2. atan2(0,0) returns 0 rather then failing.
See Wikipedia or Mathworld for more information.
cos cos (x)
Calculates the cosine function.
See Planetmath for more information.
cosh cosh (x)
Calculates the hyperbolic cosine function.
See Planetmath for more information.
cot cot (x)
The cotangent function.
See Planetmath for more information.
coth coth (x)
The hyperbolic cotangent function.
See Planetmath for more information.
csc csc (x)
The cosecant function.
See Planetmath for more information.
csch csch (x)
The hyperbolic cosecant function.
See Planetmath for more information.
sec sec (x)
The secant function.
See Planetmath for more information.
sech sech (x)
The hyperbolic secant function.
See Planetmath for more information.
sin sin (x)
Calculates the sine function.
See Planetmath for more information.
sinh sinh (x)
Calculates the hyperbolic sine function.
See Planetmath for more information.
tan tan (x)
Calculates the tan function.
See Planetmath for more information.
tanh tanh (x)
The hyperbolic tangent function.
See Planetmath for more information.
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Number Theory
AreRelativelyPrime AreRelativelyPrime (a,b)
Are the real integers a and b relatively prime? Returns true or false.
See Planetmath or Mathworld for more information.
BernoulliNumber BernoulliNumber (n)
Return the nth Bernoulli number.
See Wikipedia or Mathworld for more information.
ChineseRemainder ChineseRemainder (a,m)
Aliases: CRT
Find the x that solves the system given by the vector a and modulo the elements of m, using the Chinese Remainder
Theorem.
See Wikipedia or Planetmath or Mathworld for more information.
CombineFactorizations CombineFactorizations (a,b)
Given two factorizations, give the factorization of the product.
See Factorize.
ConvertFromBase ConvertFromBase (v,b)
Convert a vector of values indicating powers of b to a number.
ConvertToBase ConvertToBase (n,b)
Convert a number to a vector of powers for elements in base b.
DiscreteLog DiscreteLog (n,b,q)
Find discrete log of n base b in Fq , the finite field of order q, where q is a prime, using the Silver-Pohlig-Hellman
algorithm.
See Wikipedia or Planetmath or Mathworld for more information.
Divides Divides (m,n)
Checks divisibility (if m divides n).
EulerPhi EulerPhi (n)
Compute the Euler phi function for n, that is the number of integers between 1 and n relatively prime to n.
See Wikipedia or Planetmath or Mathworld for more information.
ExactDivision ExactDivision (n,d)
Return n/d but only if d divides n. If d does not divide n then this function returns garbage. This is a lot faster for very
large numbers than the operation n/d, but of course only useful if you know that the division is exact.
Factorize Factorize (n)
Return factorization of a number as a matrix. The first row is the primes in the factorization (including 1) and the second
row are the powers. So for example:
genius> Factorize(11*11*13)
=
[1
11
13
1
2
1]
See Wikipedia for more information.
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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 Fq , the finite group of order q. Of course q must be a prime.
FindRandomPrimitiveElementMod FindRandomPrimitiveElementMod (q)
Find a random primitive element in Fq , 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 Fq , 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 Fq , 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 2p -1 is a prime. It does this by looking it up in a table
of known values, which is relatively short. See also MersennePrimeExponents and LucasLehmer.
See Wikipedia, Planetmath, Mathworld or GIMPS for more information.
IsNthPower IsNthPower (m,n)
Tests if a rational number m is a perfect nth power. See also IsPerfectPower and IsPerfectSquare.
IsOdd IsOdd (n)
Tests if an integer is odd.
IsPerfectPower IsPerfectPower (n)
Check an integer is any perfect power, ab .
IsPerfectSquare IsPerfectSquare (n)
Check an integer for being a perfect square of an integer. The number must be a real integer. Negative integers are of
course never perfect squares of real integers.
IsPrime IsPrime (n)
Tests primality of integers, for numbers less than 2.5e10 the answer is deterministic (if Riemann hypothesis is true). For
numbers larger, the probability of a false positive depends on IsPrimeMillerRabinReps. That is the probability of false
positive is 1/4 to the power IsPrimeMillerRabinReps. The default value of 22 yields a probability of about 5.7e-14.
If false is returned, you can be sure that the number is a composite. If you want to be absolutely sure that you have a
prime you can use MillerRabinTestSure but it may take a lot longer.
See Planetmath or Mathworld for more information.
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IsPrimitiveMod IsPrimitiveMod (g,q)
Check if g is primitive in Fq , 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 Fq , 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 bˆ(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 2p -1 is a Mersenne prime using the Lucas-Lehmer test. See also MersennePrimeExponents and IsMersennePrimeExponent.
See Wikipedia or Planetmath or Mathworld for more information.
LucasNumber LucasNumber (n)
Returns the nth Lucas number.
See Wikipedia or Planetmath or Mathworld for more information.
MaximalPrimePowerFactors MaximalPrimePowerFactors (n)
Return all maximal prime power factors of a number.
MersennePrimeExponents MersennePrimeExponents
A vector of known Mersenne prime exponents, that is a list of positive integers p such that 2p -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)ˆreps. 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 or Planetmath or Mathworld for more information.
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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)ˆk 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 Planetmath for more information.
PowerMod PowerMod (a,b,m)
Compute aˆb 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 aˆb mod m is just as fast.
Prime Prime (n)
Aliases: prime
Return the nth prime (up to a limit).
See Planetmath or Mathworld for more information.
PrimeFactors PrimeFactors (n)
Return all prime factors of a number as a vector.
See Mathworld for more information.
PseudoprimeTest PseudoprimeTest (n,b)
Pseudoprime test, returns true if and only if bˆ(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 Fq , the finite group of order q, where q is a prime using the Silver-Pohlig-Hellman
algorithm, given f being the factorization of q-1.
SqrtModPrime SqrtModPrime (n,p)
Find square root of n modulo p (where p is a prime). Null is returned if not a quadratic residue.
See Planetmath or Mathworld for more information.
StrongPseudoprimeTest StrongPseudoprimeTest (n,b)
Run the strong pseudoprime test base b on n.
See Planetmath or Mathworld for more information.
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gcd gcd (a,args...)
Aliases: GCD
Greatest common divisor of integers. You can enter as many integers in the argument list, or you can give a vector or a
matrix of integers. If you give more than one matrix of the same size then GCD is done element by element.
See Planetmath or Mathworld for more information.
lcm lcm (a,args...)
Aliases: LCM
Least common multiplier of integers. You can enter as many integers in the argument list, or you can give a vector or a
matrix of integers. If you give more than one matrix of the same size then LCM is done element by element.
See Planetmath or Mathworld for more information.
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 your column reduce a matrix you can use this to find the
nullity. See cref and Nullity.
DeleteColumn DeleteColumn (M,col)
Delete a column of a matrix.
DeleteRow DeleteRow (M,row)
Delete a row of a matrix.
DiagonalOf DiagonalOf (M)
Gets the diagonal entries of a matrix as a column vector.
See Wikipedia for more information.
DotProduct DotProduct (u,v)
Get the dot product of two vectors. The vectors must be of the same size. No conjugates are taken so this is a bilinear form
even if working over the complex numbers.
See Planetmath for more information.
ExpandMatrix ExpandMatrix (M)
Expands a matrix just like we do on unquoted matrix input. That is we expand any internal matrices as blocks. This is a
way to construct matrices out of smaller ones and this is normally done automatically on input unless the matrix is quoted.
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HermitianProduct HermitianProduct (u,v)
Aliases: InnerProduct
Get the Hermitian product of two vectors. The vectors must be of the same size. This is a sesquilinear form using the
identity matrix.
See Mathworld for more information.
I I (n)
Aliases: eye
Return an identity matrix of a given size, that is n by n. If n is zero, returns null.
See Planetmath for more information.
IndexComplement IndexComplement (vec,msize)
Return the index complement of a vector of indexes. Everything is one based. For example for vector [2,3] and size 5,
we return [1,4,5]. If msize is 0, we always return null.
IsDiagonal IsDiagonal (M)
Is a matrix diagonal.
See Wikipedia or Planetmath for more information.
IsIdentity IsIdentity (x)
Check if a matrix is the identity matrix. Automatically returns false if the matrix is not square. Also works on numbers,
in which case it is equivalent to x==1. When x is null (we could think of that as a 0 by 0 matrix), no error is generated
and false is returned.
IsLowerTriangular IsLowerTriangular (M)
Is a matrix lower triangular. That is, are all the entries above the diagonal zero.
IsMatrixInteger IsMatrixInteger (M)
Check if a matrix is a matrix of an integers (non-complex).
IsMatrixNonnegative IsMatrixNonnegative (M)
Check if a matrix is non-negative, that is if each element is non-negative. Do not confuse positive matrices with positive
semi-definite matrices.
See Wikipedia for more information.
IsMatrixPositive IsMatrixPositive (M)
Check if a matrix is positive, that is if each element is positive (and hence real). In particular, no element is 0. Do not
confuse positive matrices with positive definite matrices.
See Wikipedia for more information.
IsMatrixRational IsMatrixRational (M)
Check if a matrix is a matrix of rational (non-complex) numbers.
IsMatrixReal IsMatrixReal (M)
Check if a matrix is a matrix of real (non-complex) numbers.
IsMatrixSquare IsMatrixSquare (M)
Check if a matrix is square, that is its width is equal to its height.
IsUpperTriangular IsUpperTriangular (M)
Is a matrix upper triangular? That is, a matrix is upper triangular if all 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.
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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).
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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.
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BilinearFormFunction BilinearFormFunction (A)
Return a function that evaluates two vectors with respect to the bilinear form given by A.
CharacteristicPolynomial CharacteristicPolynomial (M)
Aliases: CharPoly
Get the characteristic polynomial as a vector. That is, return the coefficients of the polynomial starting with the constant
term. This is the polynomial defined by det(M-xI). The roots of this polynomial are the eigenvalues of M. See also
CharacteristicPolynomialFunction.
See Planetmath for more information.
CharacteristicPolynomialFunction CharacteristicPolynomialFunction (M)
Get the characteristic polynomial as a function. This is the polynomial defined by det(M-xI). The roots of this polynomial are the eigenvalues of M. See also CharacteristicPolynomial.
See Planetmath for more information.
ColumnSpace ColumnSpace (M)
Get a basis matrix for the columnspace of a matrix. That is, return a matrix whose columns are the basis for the column
space of M. That is the space spanned by the columns of M.
CommutationMatrix CommutationMatrix (m, n)
Return the commutation matrix K(m,n), which is the unique m*n by m*n matrix such that K(m,n) * MakeVector(A)
= MakeVector(A.’) for all m by n matrices A.
CompanionMatrix CompanionMatrix (p)
Companion matrix of a polynomial (as vector).
ConjugateTranspose ConjugateTranspose (M)
Conjugate transpose of a matrix (adjoint). This is the same as the ’ operator.
See Planetmath for more information.
Convolution Convolution (a,b)
Aliases: convol
Calculate convolution of two horizontal vectors.
ConvolutionVector ConvolutionVector (a,b)
Calculate convolution of two horizontal vectors. Return result as a vector and not added together.
CrossProduct CrossProduct (v,w)
CrossProduct of two vectors in R3 as a column vector.
DeterminantalDivisorsInteger DeterminantalDivisorsInteger (M)
Get the determinantal divisors of an integer matrix (not its characteristic).
DirectSum DirectSum (M,N...)
Direct sum of matrices.
DirectSumMatrixVector DirectSumMatrixVector (v)
Direct sum of a vector of matrices.
Eigenvalues Eigenvalues (M)
Aliases: eig
Get the eigenvalues of a square matrix. Currently only works for matrices of size up to 4 by 4, or for triangular matrices
(for which the eigenvalues are on the diagonal).
See Wikipedia or Planetmath or Mathworld for more information.
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Eigenvectors Eigenvectors (M)
Eigenvectors (M, &eigenvalues)
Eigenvectors (M, &eigenvalues, &multiplicities)
Get the eigenvectors of a square matrix. Optionally get also the eigenvalues and their algebraic multiplicities. Currently
only works for matrices of size up to 2 by 2.
See Wikipedia or Planetmath or Mathworld for more information.
GramSchmidt GramSchmidt (v,B...)
Apply the Gram-Schmidt process (to the columns) with respect to inner product given by B. If B is not given then the
standard Hermitian product is used. B can either be a sesquilinear function of two arguments or it can be a matrix giving a
sesquilinear form. The vectors will be made orthonormal with respect to B.
See Planetmath for more information.
HankelMatrix HankelMatrix (c,r)
Hankel matrix.
HilbertMatrix HilbertMatrix (n)
Hilbert matrix of order n.
See Planetmath for more information.
Image Image (T)
Get the image (columnspace) of a linear transform.
InfNorm InfNorm (v)
Get the Inf Norm of a vector, sometimes called the sup norm or the max norm.
InvariantFactorsInteger InvariantFactorsInteger (M)
Get the invariant factors of a square integer matrix (not its characteristic).
InverseHilbertMatrix InverseHilbertMatrix (n)
Inverse Hilbert matrix of order n.
See Planetmath for more information.
IsHermitian IsHermitian (M)
Is a matrix Hermitian. That is, is it equal to its conjugate transpose.
See Planetmath for more information.
IsInSubspace IsInSubspace (v,W)
Test if a vector is in a subspace.
IsInvertible IsInvertible (n)
Is a matrix (or number) invertible (Integer matrix is invertible 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.
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IsPositiveDefinite IsPositiveDefinite (M)
Is M a Hermitian positive definite matrix. That is if HermitianProduct(M*v,v) is always strictly positive for any
vector v. M must be square and Hermitian to be positive definite. The check that is performed is that every principal
submatrix has a non-negative determinant. (See HermitianProduct)
Note that some authors (for example Mathworld) do not require that M be Hermitian, and then the condition is on the real
part of the inner product, but we do not take this view. If you wish to perform this check, just check the Hermitian part of
the matrix M as follows: IsPositiveDefinite(M+M’).
See Planetmath or Mathworld for more information.
IsPositiveSemidefinite IsPositiveSemidefinite (M)
Is M a Hermitian positive semidefinite matrix. That is if HermitianProduct(M*v,v) is always non-negative for any
vector v. M must be square and Hermitian to be positive semidefinite. The check that is performed is that every principal
submatrix has a non-negative determinant. (See HermitianProduct)
Note that some authors do not require that M be Hermitian, and then the condition is on the real part of the inner product,
but we do not take this view. If you wish to perform this check, just check the Hermitian part of the matrix M as follows:
IsPositiveSemidefinite(M+M’).
See Planetmath or Mathworld for more information.
IsSkewHermitian IsSkewHermitian (M)
Is a matrix skew-Hermitian. That is, is the conjugate transpose equal to negative of the matrix.
See Planetmath for more information.
IsUnitary IsUnitary (M)
Is a matrix unitary? That is, does M’*M and M*M’ equal the identity.
See Planetmath or Mathworld for more information.
JordanBlock JordanBlock (n,lambda)
Aliases: J
Get the Jordan block corresponding to the eigenvalue lambda with multiplicity n.
See Planetmath or Mathworld for more information.
Kernel Kernel (T)
Get the kernel (nullspace) of a linear transform.
(See NullSpace)
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.
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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.
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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 R2 .
Rotation3DX Rotation3DX (angle)
Return the matrix corresponding to rotation around origin in R3 about the x-axis.
Rotation3DY Rotation3DY (angle)
Return the matrix corresponding to rotation around origin in R3 about the y-axis.
Rotation3DZ Rotation3DZ (angle)
Return the matrix corresponding to rotation around origin in R3 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 (not its characteristic).
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.
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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.
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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)
DoubleFactorial DoubleFactorial (n)
Double factorial: n(n-2)(n-4)...
See Planetmath for more information.
Factorial Factorial (n)
Factorial: n(n-1)(n-2)...
See Planetmath for more information.
FallingFactorial FallingFactorial (n,k)
Falling factorial: (n)_k = n(n-1)...(n-(k-1))
See Planetmath for more information.
Fibonacci Fibonacci (x)
Aliases: fib
Calculate nth Fibonacci number. That is the number defined recursively by Fibonacci(n) = Fibonacci(n-1) +
Fibonacci(n-2) and Fibonacci(1) = Fibonacci(2) = 1.
See Wikipedia or Planetmath or Mathworld for more information.
FrobeniusNumber FrobeniusNumber (v,arg...)
Calculate the Frobenius number. That is calculate smallest number that cannot be given as a non-negative integer linear
combination of a given vector of non-negative integers. The vector can be given as separate numbers or a single vector.
All the numbers given should have GCD of 1.
See Mathworld for more information.
GaloisMatrix GaloisMatrix (combining_rule)
Galois matrix given a linear combining rule (a_1*x_+...+a_n*x_n=x_(n+1)).
GreedyAlgorithm FrobeniusNumber (n,v)
Find the vector c of non-negative integers such that taking the dot product with v is equal to n. If not possible returns
null. v should be given sorted in increasing order and should consist of non-negative integers.
See Mathworld for more information.
HarmonicNumber HarmonicNumber (n,r)
Aliases: HarmonicH
Harmonic Number, the nth harmonic number of order r.
Hofstadter Hofstadter (n)
Hofstadter’s function q(n) defined by q(1)=1, q(2)=1, q(n)=q(n-q(n-1))+q(n-q(n-2)).
LinearRecursiveSequence LinearRecursiveSequence (seed_values,combining_rule,n)
Compute linear recursive sequence using Galois stepping.
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Multinomial Multinomial (v,arg...)
Calculate multinomial coefficients. Takes a vector of k non-negative integers and computes the multinomial coefficient.
This corresponds to the coefficient in the homogeneous polynomial in k variables with the corresponding powers.
The formula for Multinomial(a,b,c) can be written as:
(a+b+c)! / (a!b!c!)
In other words, if we would have only two elements, then Multinomial(a,b) is the same thing as Binomial(a+b,a)
or Binomial(a+b,b).
See Planetmath, Mathworld, or Wikipedia for more information.
NextCombination NextCombination (v,n)
Get combination that would come after v in call to combinations, first combination should be [1:k]. This function is
useful if you have many combinations to go through and you don’t want to waste memory to store them all.
For example with Combination you would normally write a loop like:
for n in Combinations (4,6) do (
SomeFunction (n)
);
But with NextCombination you would write something like:
n:=[1:4];
do (
SomeFunction (n)
) while not IsNull(n:=NextCombination(n,6));
See also Combinations.
Pascal Pascal (i)
Get the Pascal’s triangle as a matrix. This will return an i+1 by i+1 lower diagonal matrix 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=1}ˆn (-1)ˆk/k!.
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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 rof 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””)*hˆ4*(ba)/180, note that n should be even.
See Planetmath for more information.
CompositeSimpsonsRuleTolerance CompositeSimpsonsRuleTolerance (f,a,b,FourthDerivativeBound,Toler
Integration of f by Composite Simpson’s Rule on the interval [a,b] with the number of steps calculated by the fourth
derivative bound and the desired tolerance.
See Planetmath for more information.
Derivative Derivative (f,x0)
Attempt to calculate derivative by trying first symbolically and then numerically.
EvenPeriodicExtension EvenPeriodicExtension (f,L)
Return a function 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).
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IsContinuous IsContinuous (f,x0)
Try and see if a real-valued function is continuous at x0 by calculating the limit there.
IsDifferentiable IsDifferentiable (f,x0)
Test for differentiability by approximating the left and right limits and comparing.
LeftLimit LeftLimit (f,x0)
Calculate the left limit of a real-valued function at x0.
Limit Limit (f,x0)
Calculate the limit of a real-valued function at x0. Tries to calculate both left and right limits.
MidpointRule MidpointRule (f,a,b,n)
Integration by midpoint rule.
NumericalDerivative NumericalDerivative (f,x0)
Aliases: NDerivative
Attempt to calculate numerical derivative.
NumericalFourierSeriesCoefficients NumericalFourierSeriesCoefficients (f,L,N)
Return a vector of vectors [a,b] where a are the cosine coefficients and b are the sine coefficients of the Fourier series of
f with half-period L (that is defined on [-L,L] and extended periodically) with coefficients up to Nth harmonic computed
numerically. The coefficients are computed by numerical integration using NumericalIntegral.
See Wikipedia or Mathworld for more information.
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.
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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,
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.
Genius Manual
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_0ˆx eˆ(-tˆ2) dt.
See Planetmath for more information.
FejerKernel FejerKernel (n,t)
Fejer kernel of order n evaluated at t
See Planetmath for more information.
GammaFunction GammaFunction (x)
Aliases: Gamma
The Gamma function. Currently only implemented for real values.
See Planetmath or Wikipedia for more information.
KroneckerDelta KroneckerDelta (v)
Returns 1 if and only if all elements are equal.
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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*eˆx. 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*eˆx. 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 Planetmath for more information.
MoebiusMapping MoebiusMapping (z,z2,z3,z4)
Moebius mapping using the cross ratio taking z2,z3,z4 to 1,0, and infinity respectively.
See Planetmath for more information.
MoebiusMappingInftyToInfty MoebiusMappingInftyToInfty (z,z2,z3)
Moebius mapping using the cross ratio taking infinity to infinity and z2,z3 to 1 and 0 respectively.
See Planetmath for more information.
MoebiusMappingInftyToOne MoebiusMappingInftyToOne (z,z3,z4)
Moebius mapping using the cross ratio taking infinity to 1 and z3,z4 to 0 and infinity respectively.
See Planetmath for more information.
MoebiusMappingInftyToZero MoebiusMappingInftyToZero (z,z2,z4)
Moebius mapping using the cross ratio taking infinity to 0 and z2,z4 to 1 and infinity respectively.
See Planetmath for more information.
PoissonKernel PoissonKernel (r,sigma)
Poisson kernel on D(0,1) (not normalized to 1, that is integral of this is 2pi).
PoissonKernelRadius PoissonKernelRadius (r,sigma)
Poisson kernel on D(0,R) (not normalized to 1).
RiemannZeta RiemannZeta (x)
Aliases: zeta
The Riemann zeta function. Currently only implemented for real values.
See Planetmath 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)
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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*xˆ3 + 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 a 2
by n+1 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","Exponential
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>
genius>
genius>
genius>
genius>
genius>
genius>
genius>
LinePlotClear();
lines = EulersMethodFull(‘(x,y)=[y@(2),-y@(1)],0,[1.0,1.0],10.0,500);
lines = ExpandMatrix(lines);
firstline = lines@(,[1,2]);
secondline = lines@(,[1,3]);
LinePlotWindow = [0,10,-2,2];
LinePlotDrawLine(firstline,"color","blue","legend","First");
LinePlotDrawPoints(secondline,"color","red","thickness",3,"legend","Second");
See Mathworld, or Wikipedia for more information.
Version 1.0.10 onwards.
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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 Halleys’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*xˆ3 + 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.
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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*xˆ2 + 2*x + 1 corresponds to the vector [1,2,3]. Returns a column vector of the
two solutions.
See Planetmath or Mathworld for more information.
QuarticFormula QuarticFormula (p)
Compute roots of a quartic (degree 4) polynomial using the quartic formula. The polynomial should be given as a vector
of coefficients. That is 5*xˆ4 + 2*x + 1 corresponds to the vector [1,2,0,0,5]. Returns a column vector of the
four solutions.
See Planetmath, Mathworld, or Wikipedia for more information.
RungeKutta RungeKutta (f,x0,y0,x1,n)
Use classical non-adaptive fourth order Runge-Kutta method to numerically solve y’=f(x,y) for initial x0, y0 going to x1
with n increments, returns y at x1.
Systems can be solved by just having y be a (column) vector everywhere. That is, y0 can be a vector in which case f
should take a number x and a vector of the same size for the second argument and should return a vector of the same size.
See Mathworld, or Wikipedia for more information.
RungeKuttaFull RungeKuttaFull (f,x0,y0,x1,n)
Use classical non-adaptive fourth order Runge-Kutta method to numerically solve y’=f(x,y) for initial x0, y0 going to x1
with n increments, returns a 2 by n+1 matrix with the x and y values. Suitable for plugging into LinePlotDrawLine 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","Exponential
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>
genius>
genius>
genius>
genius>
genius>
genius>
genius>
LinePlotClear();
lines = RungeKuttaFull(‘(x,y)=[y@(2),-y@(1)],0,[1.0,1.0],10.0,100);
lines = ExpandMatrix(lines);
firstline = lines@(,[1,2]);
secondline = lines@(,[1,3]);
LinePlotWindow = [0,10,-2,2];
LinePlotDrawLine(firstline,"color","blue","legend","First");
LinePlotDrawPoints(secondline,"color","red","thickness",3,"legend","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 of an entire matrix.
See Mathworld for more information.
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GaussDistribution GaussDistribution (x,sigma)
Integral of the GaussFunction from 0 to x (area under the normal curve).
See Mathworld for more information.
GaussFunction GaussFunction (x,sigma)
The normalized Gauss distribution function (the normal curve).
See Mathworld for more information.
Median Median (m)
Aliases: median
Calculate median of an entire matrix.
See Mathworld for more information.
PopulationStandardDeviation PopulationStandardDeviation (m)
Aliases: stdevp
Calculate the population standard deviation of a whole matrix.
RowAverage RowAverage (m)
Aliases: RowMean
Calculate average of each row in a matrix.
See Mathworld for more information.
RowMedian RowMedian (m)
Calculate median of each row in a matrix and return a column vector of the medians.
See Mathworld for more information.
RowPopulationStandardDeviation RowPopulationStandardDeviation (m)
Aliases: rowstdevp
Calculate the population standard deviations of rows of a matrix and return a vertical vector.
RowStandardDeviation RowStandardDeviation (m)
Aliases: rowstdev
Calculate the standard deviations of rows of a matrix and return a vertical vector.
StandardDeviation StandardDeviation (m)
Aliases: stdev
Calculate the standard deviation of a whole matrix.
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.
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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).
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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 cˆ<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.
AlphabetToString AlphabetToString (vec,alphabet)
Convert a vector of 0-based alphabet values (positions in the alphabet string) to a string.
StringToASCII StringToASCII (str)
Convert a string to a vector of ASCII values.
StringToAlphabet StringToAlphabet (str,alphabet)
Convert a string to a vector of 0-based alphabet values (positions in the alphabet string), -1’s for unknown letters.
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)))
SymbolicDerivativeTry SymbolicDerivativeTry (f)
Attempt to symbolically differentiate the function f, where f is a function of one variable, returns null if unsuccessful but
is silent. (See SymbolicDerivative)
SymbolicNthDerivative SymbolicNthDerivative (f,n)
Attempt to symbolically differentiate a function n times. (See SymbolicDerivative)
SymbolicNthDerivativeTry SymbolicNthDerivativeTry (f,n)
Attempt to symbolically differentiate a function n times quietly and return null on failure (See SymbolicNthDerivative)
SymbolicTaylorApproximationFunction SymbolicTaylorApproximationFunction (f,x0,n)
Attempt to construct the Taylor approximation function around x0 to the nth degree. (See SymbolicDerivative)
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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.
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.
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.)
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,
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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,0,3,100),"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])
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.
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 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>
genius>
genius>
The
LinePlotDrawPoints(0,0,"color","blue","thickness",3)
LinePlotDrawPoints([0,0;1,-1;-1,-1])
LinePlotDrawPoints(RungeKuttaFull(‘(x,y)=y,0,3,100),"color","blue","legend"," ←Solution")
Available from version 1.0.18 onwards.
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.
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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.
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 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> Slopefield(‘(x,y)=sin(x-y),-5,5,-5,5)
SurfacePlot SurfacePlot (func)
SurfacePlot (func,x1,x2,y1,y2,z1,z2)
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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>
genius>
genius>
genius>
SurfacePlotData([0,0,0;1,0,1;0,1,1;1,1,3])
SurfacePlotData(data,"My data")
SurfacePlotData(data,-1,1,-1,1,0,10)
SurfacePlotData(data,SurfacePlotWindow)
Here’s an example of how to plot in polar coordinates, in particular how to plot the function -rˆ2 * 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])
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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>
genius>
genius>
genius>
SurfacePlotDataGrid([1,2;3,4],[0,1,0,1])
SurfacePlotDataGrid(data,[-1,1,-1,1],"My data")
d:=null; for i=1 to 20 do for j=1 to 10 d@(i,j) = (0.1*i-1)^2-(0.1*j)^2;
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, arrows, 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
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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)
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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
Genius Manual
(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
)
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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.
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 10ˆ-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 10ˆ-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.
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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 typsetting 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.
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Chapter 14
About Genius Mathematics Tool
Genius Mathematics Tool was written by Jiří (George) Lebl (jirka@5z.com). The history of Genius Mathematics Tool goes back
to late 1997. It was the first calculator program for GNOME, but it then grew beyond being just a desktop calculator. To find
more information about Genius Mathematics Tool, please visit the Genius Web page.
To report a bug or make a suggestion regarding this application or this manual, 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.
Jiří 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 WisconsinMadison, and Oklahoma State University. The software has been used for both teaching and research.
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