Genius Manual
Genius Manual
Jiří Lebl
University of Illinois, Urbana-Champaign
[email protected]
Kai Willadsen
University of Queensland, Australia
[email protected]
Genius Manual
by Jiří Lebl and Kai Willadsen
Copyright © 1997-2009 Jiří (George) Lebl
Copyright © 2004 Kai Willadsen
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You can find a copy of the GFDL at this link (ghelp:fdl) or in the file COPYING-DOCS distributed with this manual.
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Feedback
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Feedback Page (ghelp:gnome-feedback).
Table of Contents
1. Introduction............................................................................................................................................1
2. Getting Started.......................................................................................................................................2
2.1. To Start Genius Mathematics Tool..............................................................................................2
2.2. When You Start Genius...............................................................................................................2
3. 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
4. Plotting....................................................................................................................................................7
4.1. Line Plots ....................................................................................................................................7
4.2. Parametric Plots ..........................................................................................................................8
4.3. Slopefield Plots ...........................................................................................................................9
4.4. Vectorfield Plots ........................................................................................................................10
4.5. Surface Plots..............................................................................................................................10
5. GEL Basics ...........................................................................................................................................12
5.1. Values ........................................................................................................................................12
5.1.1. Numbers .......................................................................................................................12
5.1.2. Booleans .......................................................................................................................13
5.1.3. Strings...........................................................................................................................14
5.1.4. Null ...............................................................................................................................14
5.2. Using Variables .........................................................................................................................15
5.2.1. Setting Variables...........................................................................................................15
5.2.2. Built-in Variables..........................................................................................................16
5.2.3. Previous Result Variable...............................................................................................16
5.3. Using Functions ........................................................................................................................16
5.3.1. Defining Functions .......................................................................................................17
5.3.2. Variable Argument Lists...............................................................................................17
5.3.3. Passing Functions to Functions ....................................................................................18
5.3.4. Operations on Functions...............................................................................................18
5.4. Absolute Value / Modulus.........................................................................................................19
5.5. Separator ...................................................................................................................................19
5.6. Modular Evaluation...................................................................................................................19
5.7. List of GEL Operators...............................................................................................................20
6. Programming with GEL .....................................................................................................................25
6.1. Conditionals ..............................................................................................................................25
6.2. Loops.........................................................................................................................................25
6.2.1. While Loops .................................................................................................................25
6.2.2. For Loops......................................................................................................................25
6.2.3. Foreach Loops ..............................................................................................................26
6.2.4. Break and Continue ......................................................................................................26
6.3. Sums and Products ....................................................................................................................27
6.4. Comparison Operators ..............................................................................................................27
6.5. Global Variables and Scope of Variables ..................................................................................27
6.6. Returning...................................................................................................................................29
iii
6.7. References.................................................................................................................................30
6.8. Lvalues ......................................................................................................................................30
7. Advanced Programming with GEL ...................................................................................................32
7.1. Error Handling ..........................................................................................................................32
7.2. Toplevel Syntax.........................................................................................................................32
7.3. Returning Functions..................................................................................................................33
7.4. True Local Variables .................................................................................................................34
7.5. GEL Startup Procedure .............................................................................................................35
7.6. Loading Programs .....................................................................................................................35
8. Matrices in GEL ..................................................................................................................................37
8.1. Entering Matrices ......................................................................................................................37
8.2. Conjugate Transpose and Transpose Operator..........................................................................38
8.3. Linear Algebra ..........................................................................................................................38
9. Polynomials in GEL.............................................................................................................................40
9.1. Using Polynomials ....................................................................................................................40
10. Set Theory in GEL.............................................................................................................................41
10.1. Using Sets ...............................................................................................................................41
11. List of GEL functions ........................................................................................................................42
11.1. Commands ..............................................................................................................................42
11.2. Basic........................................................................................................................................43
11.3. Parameters ...............................................................................................................................49
11.4. Constants .................................................................................................................................54
11.5. Numeric...................................................................................................................................55
11.6. Trigonometry...........................................................................................................................62
11.7. Number Theory .......................................................................................................................68
11.8. Matrix Manipulation ...............................................................................................................79
11.9. Linear Algebra ........................................................................................................................87
11.10. Combinatorics .....................................................................................................................102
11.11. Calculus...............................................................................................................................107
11.12. Functions.............................................................................................................................113
11.13. Equation Solving.................................................................................................................117
11.14. Statistics ..............................................................................................................................119
11.15. Polynomials.........................................................................................................................122
11.16. Set Theory ...........................................................................................................................123
11.17. Miscellaneous......................................................................................................................124
11.18. Symbolic Operations...........................................................................................................125
11.19. Plotting ................................................................................................................................126
12. Example Programs in GEL.............................................................................................................131
13. Settings..............................................................................................................................................133
13.1. Output....................................................................................................................................133
13.2. Precision................................................................................................................................134
13.3. Terminal ................................................................................................................................135
13.4. Memory.................................................................................................................................135
14. About Genius Mathematics Tool ....................................................................................................136
iv
List of Figures
2-1. Genius Mathematics Tool Window ......................................................................................................2
4-1. Create Plot Window..............................................................................................................................7
4-2. Plot Window .........................................................................................................................................7
4-3. Parametric Plot Tab ..............................................................................................................................8
4-4. Parametric Plot .....................................................................................................................................9
4-5. Surface Plot ........................................................................................................................................10
v
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.
This manual describes mostly the graphical version of the calculator, but the language is of course the
same. The command line only version lacks the graphing capabilities and all other capabilities that
require the graphical user interface.
1
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.
2
Chapter 2. Getting Started
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. Finally it allows plotting functions using a user friendly
dialog box.
The other menus have same familiar functions as in other applications.
Toolbar.
The toolbar contains a subset of the commands that you can access from the menubar.
Working area
The working area is the primary method of interacting with the application.
The working area initially has just the Console tab which is the main way of interacting with the
calculator. Here you type expressions and the results are immediately returned after you hit the
Enter key.
Alternatively you can write longer programs and those can appear in separate tabs and can be stored
in files for later retrieval.
3
Chapter 2. Getting Started
4
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 here and it willg et evaluated.
Type your expression into the Console work area and press enter and the expression will be evaluated.
Expressions are written in a language called GEL. The most simple GEL expression just looks like
mathematics. For example
genius> 30*70 + 67^3.0 + ln(7) * (88.8/100)
or
genius> 62734 + 812634 + 77^4 mod 5
or
genius> | sin(37) - e^7 |
or
genius> sum n=1 to 70 do 1/n
(Last is the harmonic sum from 1 to 70)
To get a list of functions and commands, type:
genius> help
If you wish to get more help on a specific function, type:
genius> help FunctionName
To view this manual, type:
genius> manual
Suppose you have previously saved some GEL commands as a program to a file and you now want to
execute them. To load this program from the file path/to/program.gel, type
genius> load path/to/program.gel
5
Chapter 3. Basic Usage
Genius Mathematics Tool keeps track of the current directory. To list files in the current directory type ls,
to change directory do cd directory as in the unix command shell.
3.2. To Create a New Program
To start writing a new program, choose File−→New Program. A new tab will appear in the work area.
You can write a GEL program in this work area. Once you have written your program you can run it by
Calculator−→Run. This will execute your program and will display any output on the Console tab.
Executing a program is equivalent of taking the text of the program and typing it into the console. The
only difference is that this input is done independent of the console and just the output goes onto the
console. Calculator−→Run will always run the currently selected program even if you are on the
Console tab. The currently selected program has its tab in bold type. To select a program, just click on
its tab.
To save the program you’ve just written, choose File−→Save As...
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.
6
Chapter 4. Plotting
Plotting support is only available in the graphical GNOME version. All plotting accessible from the
graphical interface is available from the Create Plot window. You can access this window by either
clicking on the Plot button on the toolbar or selecting Plot from the Calculator menu. You can also
access the plotting functionality by using the plotting functions of the GEL language. See Chapter 5 to
find out how to enter expressions that Genius understands.
4.1. Line Plots
To graph real valued functions of one variable open the Create Plot window. You can also use the
LinePlot function on the command line (see its documentation).
Once you click the Plot button, a window opens up with some notebooks in it. You want to be in the
Function line plot notebook tab, and inside you want to be on the Functions / Expressions notebook
tab. See Figure 4-1.
Figure 4-1. Create Plot Window
Into the text boxes just type in expressions where x is the independent variable. You can also just give
names of functions such as cos rather then having to type cos(x). You can graph up to ten functions. If
you make a mistake and Genius cannot parse the input it will signify this with a warning icon on the right
of the text input box where the error occurred, as well as giving you an error dialog. You can change the
ranges of the dependent and independent variables in the bottom part of the dialog. Pressing the Plot
button produces the graph shown in Figure 4-2.
7
Chapter 4. Plotting
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. See Figure 4-3.
8
Chapter 4. Plotting
Figure 4-3. Parametric Plot Tab
An example of a parametric plot is given in Figure 4-3. Similar operations can be done on such graphs as
can be done on the other line plots. For plotting using the command line see the documentation of the
LinePlotParametric or LinePlotCParametric function.
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.
9
Chapter 4. Plotting
When a slope field is active, there is an extra Solver menu available, through which you can bring up the
solver dialog. Here you can have Genius plot specific solutions for the given initial conditions. You can
either specify initial conditions in the dialog, or you can click on the plot directly to specify the initial
point. While the solver dialog is active, the zooming by clicking and dragging does not work. You have
to close the dialog first if you want to zoom using the mouse.
The solver uses the standard Runge-Kutta method. The plots will stay on the screen until cleared. The
solver will stop whenever it reaches the boundary of the plot window. Zooming does not change the
limits or parameters of the solutions, you will have to clear and redraw them with appropriate
parameters. You can also use the SlopefieldDrawSolution function to draw solutions from the
command line or programs.
4.4. Vectorfield Plots
In the create plot window, you can also choose the Vector field notebook tab to create a two dimensional
vector field plot. Similar operations can be done on such graphs as can be done on the other line plots.
For plotting using the command line see the documentation of the VectorfieldPlot function.
By default the direction and magnitude of the vector field is shown. To only show direction and not the
magnitude, check the appropriate checkbox to normalize the arrow lengths.
When a vector field is active, there is an extra Solver menu available, through which you can bring up
the solver dialog. Here you can have Genius plot specific solutions for the given initial conditions. You
can either specify initial conditions in the dialog, or you can click on the plot directly to specify the
initial point. While the solver dialog is active, the zooming by clicking and dragging does not work. You
have to close the dialog first if you want to zoom using the mouse.
The solver uses the standard Runge-Kutta method. The plots will stay on the screen until cleared.
Zooming does not change the limits or parameters of the solutions, you will have to clear and redraw
them with appropriate parameters. You can also use the VectorfieldDrawSolution function to draw
solutions from the command line or programs.
4.5. Surface Plots
Genius can also plot surfaces. Select the Surface plot tab in the main notebook of the Create Plot
window. Here you can specify a single expression which should use either x and y as real independent
variables or z as a complex variable (where x is the real part of z and y is the imaginary part). For
example to plot the modulus of the cosine function for complex parameters, you could enter |cos(z)|.
This would be equivalent to |cos(x+1i*y)|. See Figure 4-5. For plotting using the command line see
the documentation of the SurfacePlot function.
10
Chapter 4. Plotting
Figure 4-5. Surface Plot
11
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. Genius Mathematics Tool can
therefore be used as a simple calculator, or as a powerful theoretical research tool. The syntax is meant to
have as shallow of a learning curve as possible, especially for use as a calculator.
5.1. Values
Values in GEL can be numbers, Booleans or strings. Values can be used in calculations, assigned to
variables and returned from functions, among other uses.
5.1.1. Numbers
Integers are the first type of number in GEL. Integers are written in the normal way.
1234
Hexidecimal 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:
12
Chapter 5. GEL Basics
1.315
7.887e77
7.887e-77
.3
0.3
77e5
When Genius prints a floating point number it will always append a .0 even if the number is whole. This
is to indicate that floating point numbers are taken as imprecise quantities. When a number is written in
the scientific notation, it is always a floating point number and thus Genius does not print the .0.
The final type of number in gel is the complex numbers. You can enter a complex number as a sum of
real and imaginary parts. The imaginary part ends with an i. Here are examples of entering complex
numbers:
1+2i
8.01i
77*e^(1.3i)
Important: When entering imaginary numbers, a number must be in front of the i. If you use i by
itself, Genius will interpret this as referring to the variable i. If you need to refer to i by itself, use 1i
instead.
In order to use mixed fraction notation with imaginary numbers you must have the mixed fraction in
parentheses. (i.e., (1 2/5)i)
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
13
Chapter 5. GEL Basics
Only addition, subtraction and multiplication are supported. If you mix numbers with Booleans in an
expression then the numbers are converted to Booleans as described above. This means that, for example:
1 == true
always evaluates to true since 1 will be converted to true before being compared to true.
5.1.3. Strings
Like numbers and Booleans, strings in GEL can be stored as values inside variables and passed to
functions. You can also concatenate a string with another value using the plus operator. For example:
a=2+3;"The result is: "+a
will create the string:
The result is: 5
You can also use C-like escape sequences such as \n,\t,\b,\a and \r. To get a \ or " into the string you
can quote it with a \. For example:
"Slash: \\ Quotes: \" Tabs: \t1\t2\t3"
will make a string:
Slash: \ Quotes: " Tabs:
1 2 3
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 contant null or nothing. By nothing we
mean that if you end an expression with a separator ;, it is equivalent to ending it with a separator
followed by a null.
14
Chapter 5. GEL Basics
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.
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:
15
Chapter 5. GEL Basics
x = 3
x := 3
To assign to a variable, use the = or := operators. These operators set the value of the variable and return
the number you set, so you can do things like
a = b = 5
The = and := operators can both be used to set variables. The difference between them is that the :=
operator always acts as an assignment operator, whereas the = operator may be interpreted as testing for
equality when used in a context where a Boolean expression is expected.
For issues regarding the scope of variables, see 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.
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)
16
Chapter 5. GEL Basics
cos(2*pi)
gcd(921,317)
To evaluate a function, enter the name of the function, followed by the arguments (if any) to the function
in parentheses. This will return the result of applying the function to its arguments. The number of
arguments to the function is, of course, different for each function.
There are many built-in functions, such as sin, cos and tan. You can use the help built-in function to
get a list of available functions, or see 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 which Genius can load in order to make available. Example:
function addup(a,b,c) = a+b+c
then addup(1,4,9) yields 14
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
17
Chapter 5. GEL Basics
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)
If you want to pass a function that doesn’t exist yet, you can use an anonymous function (see
Section 5.3.1).
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)
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 does
exp(sin(x)*cos(x)+4)
This can be useful when quickly defining functions. For example to create a function to perform the
above operation, you can just type:
f = exp(sin*cos+4)
This can also be used in plotting. For example, to plot sin squared you can enter:
18
Chapter 5. GEL Basics
LinePlot(sin^2)
Warning
Not all functions can be used in this way. In addition, when you use a binary
operation the functions must take the same arguments.
5.4. Absolute Value / Modulus
You can make an absolute value of something by putting the |’s around it. For example:
|a-b|
In case the expression is a complex number the result will be the modulus (distance from the origin). For
example: |3 * e^(1i*pi)| returns 3.
5.5. Separator
In GEL if you want to type more than one command you have to use the ; operator, which is a way to
separate expressions, such a combined expression will return whatever is the result of the last one, so
suppose you type the following:
3 ; 5
This expression will yield 5.
This will require some parenthesizing to make it unambiguous sometimes, especially if the ; is not the
top most primitive. This slightly differs from other programming languages where the ; is a terminator
of statements, whereas in GEL it’s actually a binary operator. If you are familiar with pascal this should
be second nature. However genius can let you pretend it is a terminator somewhat, if a ; is found at the
end of a parenthesis or a block, genius will itself append a null node to it as if you would have written
;null. This is usefull 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.
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Chapter 5. GEL Basics
5.6. Modular Evaluation
Sometimes when working with large numbers, it might be faster if results are modded after each
calculation. To use it you just add "mod <integer>" after the expression. Example:
2^(5!) * 3^(6!) mod 5
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.
5.7. List of GEL Operators
As everything in gel is really just an expression, it is really just all connected together with operators.
Here is a list of the operators in GEL.
a;b
The separator, just evaluates both a and b, but returns only the result of b.
a=b
The assignment operator. This assigns b to a (a must be a valid lvalue) (note however that this
operator may be translated to == if used in a place where boolean expression is expected)
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Chapter 5. GEL Basics
a:=b
The assignment operator. Assigns b to a (a must be a valid lvalue). This is different from = because
it never gets translated to a ==.
|a|
Absolute value or modulus (if a is a complex number).
See Mathworld (http://mathworld.wolfram.com/AbsoluteValue.html) for more information.
a^b
Exponentiation, raises a to the bth power.
a.^b
Element by element exponentiation. Raise each element of a matrix a to the bth power. Or if b is a
matrix of the same size as a, then do the operation element by element. If a is a number and b is a
matrix then it creates matrix of the same size as b with a raised to all the different powers in b.
a+b
Addition. Adds two numbers, matrices, functions or strings. If you add a string to anything the
result will just be a string.
a-b
Subtraction. Subtract two numbers, matrices or functions.
a*b
Multiplication. This is the normal matrix multiplication.
a.*b
Element by element multiplication if a and b are matrices.
a/b
Division.
a./b
Element by element division.
a\b
Back division. That is this is the same as b/a.
a.\b
Element by element back division.
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Chapter 5. GEL Basics
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 remaineder after element by element 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).
a!=b
Inequality operator, returns true if a does not equal b else returns false.
a<>b
Alternative inequality operator, returns true if a does not equal b else returns false.
a<=b
Less than or equal operator, returns true if a is less than or equal to b else returns false.
a>=b
Greater than or equal operator, returns true if a is greater than or equal to b else returns false.
a<=>b
Comparison operator. If a is equal to b it returns 0, if a is less than b it returns -1 and if a is greater
than b it returns 1.
a and b
Logical and.
a or b
Logical or.
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Chapter 5. GEL Basics
a xor b
Logical xor.
not a
Logical not.
-a
Negation operator.
&a
Variable referencing (to pass a reference to something). See Section 6.7.
*a
Variable dereferencing (to access a referenced varible). See Section 6.7.
a’
Matrix conjugate transpose.
a.’
Matrix transpose, does not conjugate the entries.
[email protected](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.
[email protected](b,)
Get row of a matrix (or rows if b is a vector).
[email protected](b,:)
Same as above.
[email protected](,c)
Get column of a matrix (or columns if c is a vector).
[email protected](:,c)
Same as above.
[email protected](b)
Get an element from a matrix treating it as a vector. This will traverse the matrix row-wise.
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Chapter 5. GEL Basics
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 mamtrix A we could do
[email protected](2:4,)
as 2:4 will return a vector [2,3,4].
a:b:c
Build a vector from a to c with b as a step. That is for example
genius> 1:2:9
=
‘[1, 3, 5, 7, 9]
(a)i
Make a imaginary number (multiply a by the imaginary). Note that normally the number i is
written as 1i. So the above is equal to
(a)*1i
‘a
Quote an identifier so that it doesn’t get evaluated. Or quote a matrix so that it doesn’t get expanded.
Note: The @() operator makes the : operator most useful. With this you can specify regions of a
matrix. So that [email protected](2:4,6) is the rows 2,3,4 of the column 6. Or [email protected](,1:2) will get you the first two
columns of a matrix. You can also assign to the @() operator, as long as the right value is a matrix
that matches the region in size, or if it is any other type of value.
Note: The comparison operators (except for the <=> operator which behaves normally), are not
strictly binary operators, they can in fact be grouped in the normal mathematical way, e.g.:
(1<x<=y<5) is a legal boolean expression and means just what it should, that is (1<x and x≤y and
y<5)
Note: The unitary minus operates in a different fashion depending on where it appears. If it appears
before a number it binds very closely, if it appears in front of an expression it binds less than the
power and factorial operators. So for example -1^k is really (-1)^k, but -foo(1)^k is really
-(foo(1)^k). So be careful how you use it and if in doubt, add parentheses.
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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. Loops
6.2.1. While Loops
Syntax:
while <expression1> do
until <expression1> do
do <expression2> while
do <expression2> until
<expression2>
<expression2>
<expression1>
<expression1>
These are similiar to other languages, however they 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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Chapter 6. Programming with GEL
6.2.2. For Loops
Syntax:
for <identifier> = <from> to <to> do <body>
for <identifier> = <from> to <to> by <increment> do <body>
Loop with identifier being set to all values from <from> to <to>, optionally using an increment other
than 1. These are faster, nicer and more compact than the normal loops such as above, but less flexible.
The identifier must be an identifier and can’t be a dereference. The value of identifier is the last value of
identifier, or <from> if body was never evaluated. The variable is guaranteed to be initialized after a
loop, so you can safely use it. Also the <from>, <to> and <increment> must be non complex values.
The <to> is not guaranteed to be hit, but will never be overshot, for example the following prints out odd
numbers from 1 to 19:
for i = 1 to 20 by 2 do print(i)
6.2.3. Foreach Loops
Syntax:
for <identifier> in <matrix> do <body>
For each element, going row by row from left to right do the body. 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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Chapter 6. Programming with GEL
6.3. Sums and Products
Syntax:
sum <identifier> = <from> to <to> do <body>
sum <identifier> = <from> to <to> by <increment> do <body>
sum <identifier> in <matrix> do <body>
prod <identifier> = <from> to <to> do <body>
prod <identifier> = <from> to <to> by <increment> do <body>
prod <identifier> in <matrix> do <body>
If you substitute for with sum or prod, then you will get a sum or a product instead of a for loop.
Instead of returning the last value, these will return the sum or the product of the values respectively.
If no body is executed (for example sum i=1 to 0 do ...) then sum returns 0 and prod returns 1 as
is the standard convention.
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 expecing 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 arguemnts 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.
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Chapter 6. Programming with GEL
6.5. Global Variables and Scope of Variables
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. This is where GEL
differs from a language such as C. One could describe variables as being semi global in a sense For
example, the following code will print out 5. The function g cannot be called on the top level (outside f
as a will not be defined).
function f() = (a:=5; g());
function g() = print(a);
f();
If you define a variable inside a function it will override any variables defined in calling functions. For
example, we modify the above code and write:
function f() = (a:=5; g());
function g() = print(a);
a:=10;
f();
This code will still print out 5. But if you call g outside of f then you will get a printout of 10. Note that
setting a to 5 inside f does not change the value of a at the top (global) level, so if you now check the
value of a it will still be 10.
Function arguments are exactly like variables defined inside the function, except that they are initialized
with the value that was passed to the function. Other than this point, they are treated just like all other
variables defined inside the function.
Functions are treated exactly like variables. Hence you can locally redefine functions. Normally (on the
top level) you cannot redefine protected variables and functions. But locally you can do this. Consider
the following session:
genius> function f(x) = sin(x)^2
= (‘(x)=(sin(x)^2))
genius> function f(x) = sin(x)^2
= (‘(x)=(sin(x)^2))
genius> function g(x) = ((function sin(x)=x^10);f(x))
= (‘(x)=((sin:=(‘(x)=(x^10)));f(x)))
genius> g(10)
= 1e20
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Chapter 6. Programming with GEL
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 neccessary to set a global variable from inside a function. When this behaviour
is needed, use the set function. Passing a string or a quoted identifier to this function sets the variable
globally (on the top level). For example, to set a to the value 3 you could call:
set(‘a,3)
or:
set("a",3)
The set function always sets the toplevel global. There is no way to set a local variable in some function
from a subroutine. If this is required, must use passing by reference.
So to recap in a more technical language: Genius operates with different numberred contexts. The top
level is the context 0 (zero). Whenever a function is entered, the context is raised, and when the function
returns the context is lowered. A function or a variable is always visible from all higher numbered
contexts. When a variable was defined in a lower numbered context, then setting this variable has the
effect of creating a new local variable in the current context number and this variable will now be visible
from all higher numbered contexts.
There are also true local variables which are not seen from anywhere but the current context. Also when
returning functions by value it may reference variables not visible from higher context and this may be a
problem. See the sections True Local Variables and Returning Functions.
6.6. 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;
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Chapter 6. Programming with GEL
y=y+1;
x=x+1
)
)
6.7. References
It may be neccessary 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 which 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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Chapter 6. Programming with GEL
6.8. 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 varable of name a.
*a
Dereference of an identifier. This will set whatever variable a points to.
[email protected](<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
[email protected](1,1) := 5
[email protected](4:8,3) := [1,2,3,4,5]’
Note that both := and = can be used interchangably. Except if the assignment appears in a condition. It is
thus always safer to just use := when you mean assignment, and == when you mean comparison.
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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 synatax is slightly different if you enter statements on the top level versus when they are inside
parentheses or inside functions. On the top level, enter acts the same as if you press return on the
command line. Therefore think of programs as just sequence of lines as if were entered on the command
line. In particular, you do not need to enter the separator at the end of the line (unless it is of course part
of several statements inside parenteses).
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 parenteses
are closed.
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Chapter 7. Advanced Programming with GEL
if Something() then (
DoSomething()
) else (
DoSomethingElse()
)
7.3. Returning Functions
It is possible to return functions as value. This way you can build functions which 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 which returns a function which 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
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Chapter 7. Advanced Programming with GEL
the function f, and at the top level you define k to be say 5. Then when you run f, the function r will not
put k into the private dictionary because it was global (toplevel) at the time of definition of r.
Sometimes it is better to have more control over how variables are copied into the private dictionary.
Since version 1.0.7, you can specify which variables are copied into the private dictionary by putting
extra square brackets after the arguments with the list of variables to be copied separated by commas. If
you do this, then variables are copied into the private dictionary at time of the function definition, and the
private dictionary is not touched afterwards. For example
function f() = (
k := 5;
function r(x) [k] = (x+k);
k := 10;
r
)
will return a function that when called will add 5 to its argument. The local copy of k was created when
the function was defined.
When you want the function to not have any private dictionary when put empty square brackets after the
argument list. Then no private dictionary will be created at all. Doing this is good to increase efficiency
when a private dictionary is not needed or when you want the function to lookup all variables as it sees
them when called. For example suppose you want the function returned from f to see the value of k from
the toplevel despite there being a local variable of the same name during definition. So the code
function f() = (
k := 5;
function r(x) [] = (x+k);
r
);
k := 10;
g = f();
g(10)
will return 20 and not 15, which would happen if k with a value of 5 was added to the private dictionary.
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)
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Chapter 7. Advanced Programming with GEL
you probably want the function r when passed as g into f to see k as 10 rather than 5, so that the code
returns 11 and not 6. However, as written, the function when executed will see the k that is equal to 5.
There are two ways to solve this. One would be to have r get k in a private dictionary using the square
bracket notation section Returning Functions.
But there is another solution. Since version 1.0.7 there are true local variables. These are variables that
are visible only from the current context and not from any called functions. We could define k as a local
variable in the function f. To do this add a local statement as the first statement in the function (it must
always be the first statement in the function). You can also make any arguments be local variables as
well. That is,
function f(g,x) = (
local g,x,k;
k := 5;
g(x)
);
Then the code will work as expected and prints out 11. Note that the local statement initializes all the
refereced variables (except for function arguments) to a null.
If all variables are to be created as locals you can just pass an asterix instead of a list of variables. In this
case the variables will not be initialized until they are actually set of course. So the following definition
of f will also work:
function f(g,x) = (
local *;
k := 5;
g(x)
);
It is good practice that all functions that take other functions as arguments use local variables. This way
the passed function does not see implementation details and get confused.
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 the library its installed place, you’ll have to first compile it with genius --compile
loader.gel > lib.cgel
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Chapter 7. Advanced Programming with GEL
7.6. Loading Programs
Sometimes you have a larger program that you wrote into a file and want to read in that file. In these
situations, you have two options. You can keep the functions you use most inside the ~/.geniusinit
file. Or if you want to load up a file in a middle of a session (or from within another file), you can type
load <list of filenames> at the prompt. This has to be done on the top level and not inside any function
or whatnot, and it cannot be part of any expression. It also has a slightly different syntax than the rest of
genius, more similiar to a shell. You can enter the file in quotes. If you use the ” quotes, you will get
exactly the string that you typed, if you use the "" quotes, special characters will be unescaped as they are
for strings. Example:
load program1.gel program2.gel
load "Weird File Name With SPACES.gel"
There are also cd, pwd and ls commands built in. cd will take one argument, ls will take an argument
which 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 a sizable library of matrix manipulation and linear
algebra functions.
8.1. Entering Matrices
To enter matrixes, you can use one of the following two syntaxes. You can either enter the matrix on one
line, separating values by commas and rows by semicolons. Or you can enter each row on one line,
separating values by commas. You can also just combine the two methods. So to enter a 3x3 matrix of
numbers 1-9 you could do
[1,2,3;4,5,6;7,8,9]
or
[1, 2, 3
4, 5, 6
7, 8, 9]
Do not use both ’;’ and return at once on the same line though.
You can also use the matrix expansion functionality to enter matricies. For example you can do:
a = [ 1, 2, 3
4, 5, 6
7, 8, 9]
b = [ a, 10
11, 12]
and you should get
[1,
2, 3,
4,
5, 6,
7,
8, 9,
11, 11, 11,
10
10
10
12]
similiarly you can build matricies out of vectors and other stuff like that.
Another thing is that non-specified spots are initialized to 0, so
[1, 2, 3
4, 5
6]
will end up being
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Chapter 8. Matrices in GEL
[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 [email protected](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.
38
Chapter 8. Matrices in GEL
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 which 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.
39
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 which you can assign to something.
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.
40
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.
41
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 interpretor. The file will execute as if it were typed onto the command line.
cd
cd /directory/name
Change working directory to /directory/name.
pwd
pwd
Print the current working directory.
ls
ls
List files in the current directory.
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Chapter 11. List of GEL functions
plugin
plugin plugin_name
Load a plugin. Plugin of that name must be installed on the system in the proper directory.
11.2. Basic
AskString
AskString (query)
AskString (query, default)
Asks a question and lets the user enter a string which it then returns. If the user cancels or closes the
window, then null is returned. The execution of the program is blocked until the user responds. If
default is given, then it is pre-typed in for the user to just press enter on.
Compose
Compose (f,g)
Compose two functions and return a function that is the composition of f and g.
ComposePower
ComposePower (f,n,x)
Compose and execute a function with itself n times, passing x as argument. Returning x if n equals
0. Example:
genius> function f(x) = x^2 ;
genius> ComposePower (f,3,7)
= 5764801
genius> f(f(f(7)))
= 5764801
Evaluate
Evaluate (str)
Parses and evaluates a string.
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Chapter 11. List of GEL functions
GetCurrentModulo
GetCurrentModulo
Get current modulo from the context outside the function. That is, if outside of the function was
executed in modulo (using mod) then this returns what this modulo was. Normally the body of the
function called is not executed in modular arithmetic, and this builtin function makes it possible to
make GEL functions aware of modular arithmetic.
Identity
Identity (x)
Identity function, returns its argument.
IntegerFromBoolean
IntegerFromBoolean (bval)
Make integer (0 for false or 1 for true) from a boolean value. bval can also be a number in
which case a non-zero value will be interpreted as true and zero will be interpretted 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.
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Chapter 11. List of GEL functions
IsFunctionOrIdentifier
IsFunctionOrIdentifier (arg)
Check if argument is a function or an identifier.
IsFunctionRef
IsFunctionRef (arg)
Check if argument is a function reference. This includes variable references.
IsMatrix
IsMatrix (arg)
Check if argument is a matrix. Even though null is sometimes considered an empty matrix, the
function IsMatrix does not consider null a matrix.
IsNull
IsNull (arg)
Check if argument is a null.
IsString
IsString (arg)
Check if argument is a text string.
IsValue
IsValue (arg)
Check if argument is a number.
Parse
Parse (str)
Parses but does not evaluate a string. Note that certain precomputation is done during the parsing
stage.
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Chapter 11. List of GEL functions
SetFunctionFlags
SetFunctionFlags (id,flags...)
Set flags for a function, currently "PropagateMod" and "NoModuloArguments". If
"PropagateMod" is set, then the body of the function is evaluated in modular arithmetic when the
function is called inside a block that was evaluated using modular arithmetic (using mod). If
"NoModuloArguments", then the arguments of the function are never evaluated using modular
arithmetic.
SetHelp
SetHelp (id,category,desc)
Set the category and help description line for a function.
SetHelpAlias
SetHelpAlias (id,alias)
Sets up a help alias.
chdir
chdir (dir)
Changes current directory, same as the cd.
display
display (str,expr)
Display a string and an expression with a colon to separate them.
error
error (str)
Prints a string to the error stream (onto the console).
exit
exit
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Chapter 11. List of GEL functions
Aliases: quit
Exits the program.
false
false
Aliases: False FALSE
The false boolean value.
manual
manual
Displays the user manual.
print
print (str)
Prints an expression and then print a newline. The argument str can be any expression. It is made
into a string before being printed.
printn
printn (str)
Prints an expression without a trailing newline. The argument str can be any expression. It is made
into a string before being printed.
protect
protect (id)
Protect a variable from being modified. This is used on the internal GEL functions to avoid them
being accidentally overridden.
ProtectAll
ProtectAll ()
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Chapter 11. List of GEL functions
Protect all currently defined variables, parameters and functions from being modified. This is used
on the internal GEL functions to avoid them being accidentally overridden. Normally Genius
Mathematics Tool considers unprotected variables as user defined.
set
set (id,val)
Set a global variable. The id can be either a string or a quoted identifier as follows. For example:
set(‘x,1)
will set the global variable x to the value 1.
string
string (s)
Make a string. This will make a string out of any argument.
true
true
Aliases: True TRUE
The true boolean value.
undefine
undefine (id)
Alias: Undefine
Undefine a variable. This includes locals and globals, every value on all context levels is wiped.
This function should really not be used on local variables. A vector of identifiers can also be passed
to undefine several variables.
UndefineAll
UndefineAll ()
Undefine all unprotected global variables (including functions and parameters). Normally Genius
Mathematics Tool considers protected variables as system defined functions and variables. Note that
48
Chapter 11. List of GEL functions
UndefineAll only removes the global definition of symbols not local ones, so that it may be run
from inside other functions safely.
unprotect
unprotect (id)
Unprotect a variable from being modified.
UserVariables
UserVariables ()
Return a vector of identifiers of user defined (unprotected) global variables.
wait
wait (secs)
Waits a specified number of seconds. secs must be nonnegative. Zero is accepted and nothing
happens in this case, except possibly user interface events are processed.
version
version
Returns the version of Genius as a horizontal 3-vector with major version first, then minor version
and finally patchlevel.
warranty
warranty
Gives the warranty information.
11.3. Parameters
ChopTolerance
ChopTolerance = number
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Chapter 11. List of GEL functions
Tolerance of the Chop function.
ContinuousNumberOfTries
ContinuousNumberOfTries = number
How many iterations to try to find the limit for continuity and limits.
ContinuousSFS
ContinuousSFS = number
How many successive steps to be within tolerance for calculation of continuity.
ContinuousTolerance
ContinuousTolerance = number
Tolerance for continuity of functions and for calculating the limit.
DerivativeNumberOfTries
DerivativeNumberOfTries = number
How many iterations to try to find the limit for derivative.
DerivativeSFS
DerivativeSFS = number
How many successive steps to be within tolerance for calculation of derivative.
DerivativeTolerance
DerivativeTolerance = number
Tolerance for calculating the derivatives of functions.
ErrorFunctionTolerance
ErrorFunctionTolerance = number
Tolerance of the ErrorFunction.
50
Chapter 11. List of GEL functions
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.
LinePlotWindow
LinePlotWindow = [x1,x2,y1,y2]
Sets the limits for line plotting functions such as LinePlot.
LinePlotDrawLegends
LinePlotDrawLegends = true
Tells genius to draw the legends for line plotting functions such as LinePlot.
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Chapter 11. List of GEL functions
MaxDigits
MaxDigits = number
Maximum digits to display.
MaxErrors
MaxErrors = number
Maximum errors to display.
MixedFractions
MixedFractions = boolean
If true, mixed fractions are printed.
NumericalIntegralFunction
NumericalIntegralFunction = function
The function used for numerical integration in NumericalIntegral.
NumericalIntegralSteps
NumericalIntegralSteps = number
Steps to perform in NumericalIntegral.
OutputChopExponent
OutputChopExponent = number
When another number in the object being printed (a matrix or a value) is greater than
10-OutputChopWhenExponent, and the number being printed is less than 10-OutputChopExponent, then display 0.0
instead of the number.
Output is never chopped if OutputChopExponent is zero. It must be a nonnegative integer.
If you want output always chopped according to OutputChopExponent, then set
OutputChopWhenExponent, to something greater than or equal to OutputChopExponent.
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Chapter 11. List of GEL functions
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.
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.
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Chapter 11. List of GEL functions
SumProductTolerance
SumProductTolerance = number
Tolerance for InfiniteSum and InfiniteProduct.
SurfacePlotWindow
SurfacePlotWindow = [x1,x2,y1,y2,z1,z2]
Sets the limits for surface plotting (See SurfacePlot).
VectorfieldNormalized
VectorfieldNormalized = true
Should the vectorfield plotting have normalized arrow length. If true, vector fields will only show
direction and not magnitude. (See VectorfieldPlot).
11.4. Constants
CatalanConstant
CatalanConstant
Catalan’s Constant, approximately 0.915... It is defined to be the series where terms are
(-1^k)/((2*k+1)^2), where k ranges from 0 to infinity.
See Mathworld (http://mathworld.wolfram.com/CatalansConstant.html) for more information.
EulerConstant
EulerConstant
Aliases: gamma
Euler’s Constant gamma. Sometimes called the Euler-Mascheroni constant.
See Wikipedia (http://en.wikipedia.org/wiki/Euler-Mascheroni_constant) or Planetmath
(http://planetmath.org/encyclopedia/MascheroniConstant.html) or Mathworld
(http://mathworld.wolfram.com/Euler-MascheroniConstant.html) for more information.
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Chapter 11. List of GEL functions
GoldenRatio
GoldenRatio
The Golden Ratio.
See Wikipedia (http://en.wikipedia.org/wiki/Golden_ratio) or Planetmath
(http://planetmath.org/encyclopedia/GoldenRatio.html) or Mathworld
(http://mathworld.wolfram.com/GoldenRatio.html) for more information.
Gravity
Gravity
Free fall acceleration at sea level.
See Wikipedia (http://en.wikipedia.org/wiki/Standard_gravity) for more information.
e
e
The base of the natural logarithm. e^x is the exponential function exp. This is the number
approximately 2.71828182846...
See Wikipedia (http://en.wikipedia.org/wiki/E_(mathematical_constant)) or Planetmath
(http://planetmath.org/encyclopedia/E.html) or Mathworld (http://mathworld.wolfram.com/e.html)
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 (http://en.wikipedia.org/wiki/Pi) or Planetmath
(http://planetmath.org/encyclopedia/Pi.html) or Mathworld (http://mathworld.wolfram.com/Pi.html)
for more information.
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Chapter 11. List of GEL functions
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.
See Wikipedia (http://en.wikipedia.org/wiki/Absolute_value), Planetmath (absolute value)
(http://planetmath.org/encyclopedia/AbsoluteValue.html), Planetmath (modulus)
(http://planetmath.org/encyclopedia/ModulusOfComplexNumber.html), Mathworld (absolute value)
(http://mathworld.wolfram.com/AbsoluteValue.html) or Mathworld (complex modulus)
(http://mathworld.wolfram.com/ComplexModulus.html) for more information.
Chop
Chop (x)
Replace very small number with zero.
ComplexConjugate
ComplexConjugate (z)
Aliases: conj Conj
Calculates the complex conjugate of the complex number z. If z is a vector or matrix, all its
elements are conjugated.
See Wikipedia (http://en.wikipedia.org/wiki/Complex_conjugate) for more information.
Denominator
Denominator (x)
Get the denominator of a rational number.
See Wikipedia (http://en.wikipedia.org/wiki/Denominator) for more information.
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Chapter 11. List of GEL functions
FractionalPart
FractionalPart (x)
Return the fractional part of a number.
See Wikipedia (http://en.wikipedia.org/wiki/Fractional_part) for more information.
Im
Im (z)
Aliases: ImaginaryPart
Get the imaginary part of a complex number.
See Wikipedia (http://en.wikipedia.org/wiki/Imaginary_part) for more information.
IntegerQuotient
IntegerQuotient (m,n)
Division without remainder.
IsComplex
IsComplex (num)
Check if argument is a complex (non-real) number.
IsComplexRational
IsComplexRational (num)
Check if argument is a possibly complex rational number.
IsFloat
IsFloat (num)
Check if argument is a floating point number (non-complex).
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Chapter 11. List of GEL functions
IsGaussInteger
IsGaussInteger (num)
Aliases: IsComplexInteger
Check if argument is a possibly complex integer.
IsInteger
IsInteger (num)
Check if argument is an integer (non-complex).
IsNonNegativeInteger
IsNonNegativeInteger (num)
Check if argument is a non-negative real integer.
IsPositiveInteger
IsPositiveInteger (num)
Aliases: IsNaturalNumber
Check if argument is a positive real integer. Note that we accept the convention that 0 is not a
natural number.
IsRational
IsRational (num)
Check if argument is a rational number (non-complex).
IsReal
IsReal (num)
Check if argument is a real number.
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Chapter 11. List of GEL functions
Numerator
Numerator (x)
Get the numerator of a rational number.
See Wikipedia (http://en.wikipedia.org/wiki/Numerator) for more information.
Re
Re (z)
Aliases: RealPart
Get the real part of a complex number.
See Wikipedia (http://en.wikipedia.org/wiki/Real_part) for more information.
Sign
Sign (x)
Aliases: sign
Return the sign of a number. That is returns -1 if value is negative, 0 if value is zero and 1 if value
is positive. If x is a complex value then Sign returns the direction or 0.
ceil
ceil (x)
Aliases: Ceiling
Get the lowest integer more than or equal to n.
exp
exp (x)
The exponential function. This is the function e^x where e is the base of the natural logarithm.
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Chapter 11. List of GEL functions
See Wikipedia (http://en.wikipedia.org/wiki/Exponential_function) or Planetmath
(http://planetmath.org/encyclopedia/LogarithmFunction.html) or Mathworld
(http://mathworld.wolfram.com/ExponentialFunction.html) for more information.
float
float (x)
Make number a floating point value. That is returns the floating point representation of the number
x.
floor
floor (x)
Aliases: Floor
Get the highest integer less than or equal to n.
ln
ln (x)
The natural logarithm, the logarithm to base e.
log
log (x)
log (x,b)
Logarithm of x base b (calls DiscreteLog if in modulo mode), if base is not given, e is used.
log10
log10 (x)
Logarithm of x base 10.
log2
log2 (x)
Aliases: lg
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Chapter 11. List of GEL functions
Logarithm of x base 2.
max
max (a,args...)
Aliases: Max Maximum
Returns the maximum of arguments or matrix.
min
min (a,args...)
Aliases: Min Minimum
Returns the minimum of arguments or matrix.
rand
rand (size...)
Generate random float in the range [0,1). If size is given then a matrix (if two numbers are
specified) or vector (if one number is specified) of the given size returned.
randint
randint (max,size...)
Generate random integer in the range [0,max). If size is given then a matrix (if two numbers are
specified) or vector (if one number is specified) of the given size returned. For example
genius> randint(4)
= 3
genius> randint(4,2)
=
[0
1]
genius> randint(4,2,3)
=
[2
0
2
0
1
3]
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Chapter 11. List of GEL functions
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 (http://planetmath.org/encyclopedia/SquareRoot.html) for more information.
trunc
trunc (x)
Aliases: Truncate IntegerPart
Truncate number to an integer (return the integer part).
11.6. Trigonometry
acos
acos (x)
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Chapter 11. List of GEL functions
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
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Chapter 11. List of GEL functions
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.
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Chapter 11. List of GEL functions
See Wikipedia (http://en.wikipedia.org/wiki/Arctangent) or Mathworld
(http://mathworld.wolfram.com/InverseTangent.html) for more information.
atanh
atanh (x)
Aliases: arctanh
The arctanh (inverse tanh) function.
atan2
atan2 (y, x)
Aliases: arctan2
Calculates the arctan2 function. If x>0 then it returns atan(y/x). If x<0 then it returns sign(y)
* (pi - atan(|y/x|). When x=0 it returns sign(y) * pi/2. atan2(0,0) returns 0 rather
then failing.
See Wikipedia (http://en.wikipedia.org/wiki/Atan2) or Mathworld
(http://mathworld.wolfram.com/InverseTangent.html) for more information.
cos
cos (x)
Calculates the cosine function.
See Planetmath (http://planetmath.org/encyclopedia/DefinitionsInTrigonometry.html) for more
information.
cosh
cosh (x)
Calculates the hyperbolic cosine function.
See Planetmath (http://planetmath.org/encyclopedia/HyperbolicFunctions.html) for more
information.
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Chapter 11. List of GEL functions
cot
cot (x)
The cotangent function.
See Planetmath (http://planetmath.org/encyclopedia/DefinitionsInTrigonometry.html) for more
information.
coth
coth (x)
The hyperbolic cotangent function.
See Planetmath (http://planetmath.org/encyclopedia/HyperbolicFunctions.html) for more
information.
csc
csc (x)
The cosecant function.
See Planetmath (http://planetmath.org/encyclopedia/DefinitionsInTrigonometry.html) for more
information.
csch
csch (x)
The hyperbolic cosecant function.
See Planetmath (http://planetmath.org/encyclopedia/HyperbolicFunctions.html) for more
information.
sec
sec (x)
The secant function.
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Chapter 11. List of GEL functions
See Planetmath (http://planetmath.org/encyclopedia/DefinitionsInTrigonometry.html) for more
information.
sech
sech (x)
The hyperbolic secant function.
See Planetmath (http://planetmath.org/encyclopedia/HyperbolicFunctions.html) for more
information.
sin
sin (x)
Calculates the sine function.
See Planetmath (http://planetmath.org/encyclopedia/DefinitionsInTrigonometry.html) for more
information.
sinh
sinh (x)
Calculates the hyperbolic sine function.
See Planetmath (http://planetmath.org/encyclopedia/HyperbolicFunctions.html) for more
information.
tan
tan (x)
Calculates the tan function.
See Planetmath (http://planetmath.org/encyclopedia/DefinitionsInTrigonometry.html) for more
information.
tanh
tanh (x)
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Chapter 11. List of GEL functions
The hyperbolic tangent function.
See Planetmath (http://planetmath.org/encyclopedia/HyperbolicFunctions.html) for more
information.
11.7. Number Theory
AreRelativelyPrime
AreRelativelyPrime (a,b)
Are the real integers a and b relatively prime? Returns true or false.
See Planetmath (http://planetmath.org/encyclopedia/RelativelyPrime.html) or Mathworld
(http://mathworld.wolfram.com/RelativelyPrime.html) for more information.
BernoulliNumber
BernoulliNumber (n)
Return the nth Bernoulli number.
See Wikipedia (http://en.wikipedia.org/wiki/Bernoulli_number) or Mathworld
(http://mathworld.wolfram.com/BernoulliNumber.html) 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 (http://en.wikipedia.org/wiki/Chinese_remainder_theorem) or Planetmath
(http://planetmath.org/encyclopedia/ChineseRemainderTheorem.html) or Mathworld
(http://mathworld.wolfram.com/ChineseRemainderTheorem.html) for more information.
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Chapter 11. List of GEL functions
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 algoritm.
See Wikipedia (http://en.wikipedia.org/wiki/Discrete_logarithm) or Planetmath
(http://planetmath.org/encyclopedia/DiscreteLogarithm.html) or Mathworld
(http://mathworld.wolfram.com/DiscreteLogarithm.html) 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.
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Chapter 11. List of GEL functions
See Wikipedia (http://en.wikipedia.org/wiki/Euler_phi) or Planetmath
(http://planetmath.org/encyclopedia/EulerPhifunction.html) or Mathworld
(http://mathworld.wolfram.com/TotientFunction.html) 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
1
11
2
13
1]
See Wikipedia (http://en.wikipedia.org/wiki/Factorization) for more information.
Factors
Factors (n)
Return all factors of n in a vector. This includes all the non-prime factors as well. It includes 1 and
the number itself. So for example to print all the perfect numbers (those that are sums of their
factors) up to the number 1000 you could do (this is of course very inefficent)
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.
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Chapter 11. List of GEL functions
This is a fairly good factorization if your number is the product of two factors that are very close to
each other.
See Wikipedia (http://en.wikipedia.org/wiki/Fermat_factorization) 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)
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Chapter 11. List of GEL functions
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 (http://en.wikipedia.org/wiki/Mersenne_prime), Planetmath
(http://planetmath.org/encyclopedia/MersenneNumbers.html), Mathworld
(http://mathworld.wolfram.com/MersennePrime.html) or GIMPS (http://www.mersenne.org/) 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.
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Chapter 11. List of GEL functions
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 (http://planetmath.org/encyclopedia/PrimeNumber.html) or Mathworld
(http://mathworld.wolfram.com/PrimeNumber.html) for more information.
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 b^(n-1) == 1 mod n. This calles the
PseudoprimeTest
IsStrongPseudoprime
IsStrongPseudoprime (n,b)
Test if n is a strong pseudoprime to base b but not a prime.
Jacobi
Jacobi (a,b)
Aliases: JacobiSymbol
Calculate the Jacobi symbol (a/b) (b should be odd).
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Chapter 11. List of GEL functions
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 (http://planetmath.org/encyclopedia/LegendreSymbol.html) or Mathworld
(http://mathworld.wolfram.com/LegendreSymbol.html) 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 (http://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test) or
Planetmath (http://planetmath.org/encyclopedia/LucasLhemer.html) or Mathworld
(http://mathworld.wolfram.com/Lucas-LehmerTest.html) for more information.
LucasNumber
LucasNumber (n)
Returns the nth Lucas number.
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Chapter 11. List of GEL functions
See Wikipedia (http://en.wikipedia.org/wiki/Lucas_number) or Planetmath
(http://planetmath.org/encyclopedia/LucasNumbers.html) or Mathworld
(http://mathworld.wolfram.com/LucasNumber.html) 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 (http://en.wikipedia.org/wiki/Mersenne_prime), Planetmath
(http://planetmath.org/encyclopedia/MersenneNumbers.html), Mathworld
(http://mathworld.wolfram.com/MersennePrime.html) or GIMPS (http://www.mersenne.org/) for
more information.
MillerRabinTest
MillerRabinTest (n,reps)
Use the Miller-Rabin primality test on n, reps number of times. The probability of false positive is
(1/4)^reps. It is probably usually better to use IsPrime since that is faster and better on smaller
integers.
See Wikipedia (http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test) or
Planetmath (http://planetmath.org/encyclopedia/MillerRabinPrimeTest.html) or Mathworld
(http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html) for more information.
MillerRabinTestSure
MillerRabinTestSure (n)
Use the Miller-Rabin primality test on n with enough bases that assuming the Generalized Reimann
Hypothesis the result is deterministic.
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Chapter 11. List of GEL functions
See Wikipedia (http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test) or
Planetmath (http://planetmath.org/encyclopedia/MillerRabinPrimeTest.html) or Mathworld
(http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html) for more information.
ModInvert
ModInvert (n,m)
Returns inverse of n mod m.
See Mathworld (http://mathworld.wolfram.com/ModularInverse.html) for more information.
MoebiusMu
MoebiusMu (n)
Return the Moebius mu function evaluated in n. That is, it returns 0 if n is not a product of distinct
primes and (-1)^k if it is a product of k distinct primes.
See Planetmath (http://planetmath.org/encyclopedia/MoebiusFunction.html) or Mathworld
(http://mathworld.wolfram.com/MoebiusFunction.html) for more information.
NextPrime
NextPrime (n)
Returns the least prime greater than n. Negatives of primes are considered prime and so to get the
previous prime you can use -NextPrime(-n).
This function uses the GMP’s mpz_nextprime which in turn uses the probabilistic Miller-Rabin
test (See also MillerRabinTest). The probability of false positive is not tunable, but is low enough
for all practical purposes.
See Planetmath (http://planetmath.org/encyclopedia/PrimeNumber.html) or Mathworld
(http://mathworld.wolfram.com/PrimeNumber.html) for more information.
PadicValuation
PadicValuation (n,p)
Returns the padic valuation (number of trailing zeros in base p).
76
Chapter 11. List of GEL functions
See Planetmath (http://planetmath.org/encyclopedia/PAdicValuation.html) for more information.
PowerMod
PowerMod (a,b,m)
Compute a^b mod m. The b’s power of a modulo m. It is not neccessary to use this function as it is
automatically used in modulo mode. Hence a^b mod m is just as fast.
Prime
Prime (n)
Aliases: prime
Return the nth prime (up to a limit).
See Planetmath (http://planetmath.org/encyclopedia/PrimeNumber.html) or Mathworld
(http://mathworld.wolfram.com/PrimeNumber.html) for more information.
PrimeFactors
PrimeFactors (n)
Return all prime factors of a number as a vector.
See Mathworld (http://mathworld.wolfram.com/PrimeFactor.html) for more information.
PseudoprimeTest
PseudoprimeTest (n,b)
Pseudoprime test, returns true if and only if b^(n-1) == 1 mod n
See Planetmath (http://planetmath.org/encyclopedia/Pseudoprime.html) or Mathworld
(http://mathworld.wolfram.com/Pseudoprime.html) for more information.
RemoveFactor
RemoveFactor (n,m)
77
Chapter 11. List of GEL functions
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 (http://planetmath.org/encyclopedia/Divisibility.html) or Mathworld
(http://mathworld.wolfram.com/Factor.html) 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 algoritm, 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 (http://planetmath.org/encyclopedia/QuadraticResidue.html) or Mathworld
(http://mathworld.wolfram.com/QuadraticResidue.html) for more information.
StrongPseudoprimeTest
StrongPseudoprimeTest (n,b)
Run the strong pseudoprime test base b on n.
See Planetmath (http://planetmath.org/encyclopedia/StrongPseudoprime.html) or Mathworld
(http://mathworld.wolfram.com/StrongPseudoprime.html) for more information.
gcd
gcd (a,args...)
Aliases: GCD
Greatest common divisor of integers. You can enter as many integers in the argument list, or you
can give a vector or a matrix of integers. If you give more than one matrix of the same size then
GCD is done element by element.
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Chapter 11. List of GEL functions
See Planetmath (http://planetmath.org/encyclopedia/GreatestCommonDivisor.html) or Mathworld
(http://mathworld.wolfram.com/GreatestCommonDivisor.html) 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 (http://planetmath.org/encyclopedia/LeastCommonMultiple.html) or Mathworld
(http://mathworld.wolfram.com/LeastCommonMultiple.html) 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)
79
Chapter 11. List of GEL functions
Remove column(s) and row(s) from a matrix.
CompoundMatrix
CompoundMatrix (k,A)
Calculate the kth compund matrix of A.
CountZeroColumns
CountZeroColumns (M)
Count the number of zero columns in a matrix. For example Once you column reduce a matrix you
can use this to find the nullity. See cref and Nullity.
DeleteColumn
DeleteColumn (M,col)
Delete a column of a matrix.
DeleteRow
DeleteRow (M,row)
Delete a row of a matrix.
DiagonalOf
DiagonalOf (M)
Gets the diagonal entries of a matrix as a column vector.
See Wikipedia (http://en.wikipedia.org/wiki/Diagonal_of_a_matrix#Matrices) 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.
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Chapter 11. List of GEL functions
See Planetmath (http://planetmath.org/encyclopedia/DotProduct.html) for more information.
ExpandMatrix
ExpandMatrix (M)
Expands a matrix just like we do on unquoted matrix input. That is we expand any internal matrices
as blocks. This is a way to construct matrices out of smaller ones and this is normally done
automatically on input unless the matrix is quoted.
HermitianProduct
HermitianProduct (u,v)
Aliases: InnerProduct
Get the hermitian product of two vectors. The vectors must be of the same size. This is a
sesquilinear form using the identity matrix.
See Mathworld (http://mathworld.wolfram.com/HermitianInnerProduct.html) 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 (http://planetmath.org/encyclopedia/IdentityMatrix.html) 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)
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Chapter 11. List of GEL functions
Is a matrix diagonal.
See Wikipedia (http://en.wikipedia.org/wiki/Diagonal_matrix) or Planetmath
(http://planetmath.org/encyclopedia/DiagonalMatrix.html) 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 below 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 nonnegative, that is if each element is nonnegative. Do not confuse positive
matrices with positive semi-definite matrices.
See Wikipedia (http://en.wikipedia.org/wiki/Positive_matrix) 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 (http://en.wikipedia.org/wiki/Positive_matrix) for more information.
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Chapter 11. List of GEL functions
IsMatrixRational
IsMatrixRational (M)
Check if a matrix is a matrix of rational (non-complex) numbers.
IsMatrixReal
IsMatrixReal (M)
Check if a matrix is a matrix of real (non-complex) numbers.
IsMatrixSquare
IsMatrixSquare (M)
Check if a matrix is square, that is its width is equal to its height.
IsUpperTriangular
IsUpperTriangular (M)
Is a matrix upper triangular? That is, a matrix is upper triangular if all all the entries below the
diagonal are zero.
IsValueOnly
IsValueOnly (M)
Check if a matrix is a matrix of numbers only. Many internal functions make this check. Values can
be any number including complex numbers.
IsVector
IsVector (v)
Is argument a horizontal or a vertical vector. Genius does not distinguish between a matrix and a
vector and a vector is just a 1 by n or n by 1 matrix.
IsZero
IsZero (x)
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Chapter 11. List of GEL functions
Check if a matrix is composed of all zeros. Also works on numbers, in which case it is equivalent to
x==0. When x is null (we could think of that as a 0 by 0 matrix), no error is generated and true is
returned as the condition is vacuous.
LowerTriangular
LowerTriangular (M)
Returns a copy of the matrix M with all the entries above the diagonal set to zero.
MakeDiagonal
MakeDiagonal (v,arg...)
Aliases: diag
Make diagonal matrix from a vector.
See Wikipedia (http://en.wikipedia.org/wiki/Diagonal_matrix) or Planetmath
(http://planetmath.org/encyclopedia/DiagonalMatrix.html) 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 vecgtor. That is we add all the elements and return a
number that is the sum of all the elements.
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Chapter 11. List of GEL functions
MatrixSumSquares
MatrixSumSquares (A)
Calculate the sum of squares of all elements in a matrix or vector.
OuterProduct
OuterProduct (u,v)
Get the outer product of two vectors. That is, suppose that u and v are vertical vectors, then the
outer product is v * u.’.
ReverseVector
ReverseVector (v)
Reverse elements in a vector.
RowSum
RowSum (m)
Calculate sum of each row in a matrix and return a vertical vector with the result.
RowSumSquares
RowSumSquares (m)
Calculate sum of squares of each row in a matrix.
RowsOf
RowsOf (M)
Gets the rows of a matrix as a vertical vector. Each element of the vector is a horizontal vector
which is the corresponding row of M. This function is useful if you wish to loop over the rows of a
matrix. For example, as for r in RowsOf(M) do something(r).
SetMatrixSize
SetMatrixSize (M,rows,columns)
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Chapter 11. List of GEL functions
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.
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 [email protected](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.
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Chapter 11. List of GEL functions
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
AuxilliaryUnitMatrix
AuxilliaryUnitMatrix (n)
Get the auxilliary unit matrix of size n. This is a square matrix matrix with that is all zero except the
superdiagonal being all ones. It is the Jordan block matrix of one zero eigenvalue.
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Chapter 11. List of GEL functions
See Planetmath (http://planetmath.org/encyclopedia/JordanCanonicalFormTheorem.html) or
Mathworld (http://mathworld.wolfram.com/JordanBlock.html) for more information on Jordan
Cannonical Form.
BilinearForm
BilinearForm (v,A,w)
Evaluate (v,w) with respect to the bilinear form given by the matrix A.
BilinearFormFunction
BilinearFormFunction (A)
Return a function that evaluates two vectors with respect to the bilinear form given by A.
CharacteristicPolynomial
CharacteristicPolynomial (M)
Aliases: CharPoly
Get the characteristic polynomial as a vector. That is, return the coefficients of the polynomial
starting with the constant term. This is the polynomial defined by det(M-xI). The roots of this
polynomial are the eigenvalues of M. See also CharacteristicPolynomialFunction.
See Planetmath (http://planetmath.org/encyclopedia/CharacteristicEquation.html) 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 (http://planetmath.org/encyclopedia/CharacteristicEquation.html) for more
information.
ColumnSpace
ColumnSpace (M)
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Chapter 11. List of GEL functions
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 (http://planetmath.org/encyclopedia/ConjugateTranspose.html) 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)
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Chapter 11. List of GEL functions
CrossProduct of two vectors in R3.
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 (http://en.wikipedia.org/wiki/Eigenvalue) or Planetmath
(http://planetmath.org/encyclopedia/Eigenvalue.html) or Mathworld
(http://mathworld.wolfram.com/Eigenvalue.html) for more information.
Eigenvectors
Eigenvectors (M)
Eigenvectors (M, &eigenvalues)
Eigenvectors (M, &eigenvalues, &multiplicities)
Get the eigenvectors of a square matrix. Optionally get also the eigenvalues and their algebraic
multiplicities. Currently only works for matrices of size up to 2 by 2.
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Chapter 11. List of GEL functions
See Wikipedia (http://en.wikipedia.org/wiki/Eigenvector) or Planetmath
(http://planetmath.org/encyclopedia/Eigenvector.html) or Mathworld
(http://mathworld.wolfram.com/Eigenvector.html) 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 (http://planetmath.org/encyclopedia/GramSchmidtOrthogonalization.html) for more
information.
HankelMatrix
HankelMatrix (c,r)
Hankel matrix.
HilbertMatrix
HilbertMatrix (n)
Hilbert matrix of order n.
See Planetmath (http://planetmath.org/encyclopedia/HilbertMatrix.html) 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.
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Chapter 11. List of GEL functions
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 (http://planetmath.org/encyclopedia/HilbertMatrix.html) for more information.
IsHermitian
IsHermitian (M)
Is a matrix hermitian. That is, is it equal to its conjugate transpose.
See Planetmath (http://planetmath.org/encyclopedia/HermitianMatrix.html) for more information.
IsInSubspace
IsInSubspace (v,W)
Test if a vector is in a subspace.
IsInvertible
IsInvertible (n)
Is a matrix (or number) invertible (Integer matrix is invertible iff it’s invertible over the integers).
IsInvertibleField
IsInvertibleField (n)
Is a matrix (or number) invertible over a field.
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Chapter 11. List of GEL functions
IsNormal
IsNormal (M)
Is M a normal matrix. That is, does M*M’ == M’*M.
See Planetmath (http://planetmath.org/encyclopedia/NormalMatrix.html) or Mathworld
(http://mathworld.wolfram.com/NormalMatrix.html) for more information.
IsPositiveDefinite
IsPositiveDefinite (M)
Is M a hermitian positive definite matrix. That is if HermitianProduct(M*v,v) is always strictly
positive for any vector v. M must be square and hermitian to be positive definite. The check that is
performed is that every principal submatrix has a nonnegative 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 (http://planetmath.org/encyclopedia/PositiveDefinite.html) or Mathworld
(http://mathworld.wolfram.com/PositiveDefiniteMatrix.html) for more information.
IsPositiveSemidefinite
IsPositiveSemidefinite (M)
Is M a hermitian positive semidefinite matrix. That is if HermitianProduct(M*v,v) is always
nonnegative 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 nonnegative 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 (http://planetmath.org/encyclopedia/PositiveSemidefinite.html) or Mathworld
(http://mathworld.wolfram.com/PositiveSemidefiniteMatrix.html) for more information.
IsSkewHermitian
IsSkewHermitian (M)
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Chapter 11. List of GEL functions
Is a matrix skew-hermitian. That is, is the conjugate transpose equal to negative of the matrix.
See Planetmath (http://planetmath.org/encyclopedia/SkewHermitianMatrix.html) for more
information.
IsUnitary
IsUnitary (M)
Is a matrix unitary? That is, does M’*M and M*M’ equal the identity.
See Planetmath (http://planetmath.org/encyclopedia/UnitaryTransformation.html) or Mathworld
(http://mathworld.wolfram.com/UnitaryMatrix.html) for more information.
JordanBlock
JordanBlock (n,lambda)
Aliases: J
Get the Jordan block corresponding to the eigenvalue lambda with multiplicity n.
See Planetmath (http://planetmath.org/encyclopedia/JordanCanonicalFormTheorem.html) or
Mathworld (http://mathworld.wolfram.com/JordanBlock.html) for more information.
Kernel
Kernel (T)
Get the kernel (nullspace) of a linear transform.
(See NullSpace)
LUDecomposition
LUDecomposition (A, L, U)
Get the LU decomposition of A and store the result in the L and U which should be references. It
returns true if successful. For example suppose that A is a square matrix, then after running:
genius> LUDecomposition(A,&L,&U)
You will have the lower matrix stored in a variable called L and the upper matrix in a variable called
U.
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Chapter 11. List of GEL functions
This is the LU decomposition of a matrix aka Crout and/or Cholesky reduction. (ISBN
0-201-11577-8 pp.99-103) The upper triangular matrix features a diagonal of values 1 (one). This is
not Doolittle’s Method which features the 1’s diagonal on the lower matrix.
Not all matrices have LU decompositions, for example [0,1;1,0] does not and this function
returns false in this case and sets L and U to null.
See Planetmath (http://planetmath.org/encyclopedia/LUDecomposition.html) or Mathworld
(http://mathworld.wolfram.com/LUDecomposition.html) for more information.
Minor
Minor (M,i,j)
Get the i-j minor of a matrix.
See Planetmath (http://planetmath.org/encyclopedia/Minor.html) 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 (http://planetmath.org/encyclopedia/Nullspace.html) for more information.
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Chapter 11. List of GEL functions
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 (http://planetmath.org/encyclopedia/Nullity.html) for more information.
OrthogonalComplement
OrthogonalComplement (M)
Get the orthogonal complement of the columnspace.
PivotColumns
PivotColumns (M)
Return pivot columns of a matrix, that is columns which have a leading 1 in row reduced form. Also
returns the row where they occur.
Projection
Projection (v,W,B...)
Projection of vector v onto subspace W with respect to inner product given by B. If B is not given
then the standard hermitian product is used. B can either be a sesquilinear function of two arguments
or it can be a matrix giving a sesquilinear form.
QRDecomposition
QRDecomposition (A, Q)
Get the QR decomposition of a square matrix A, returns the upper triangular matrix R and sets Q to
the orthogonal (unitary) matrix. Q should be a reference or null if you don’t want any return. For
example:
genius> R = QRDecomposition(A,&Q)
You will have the upper triangular matrix stored in a variable called R and the orthogonal (unitary)
matrix stored in Q.
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Chapter 11. List of GEL functions
See Planetmath (http://planetmath.org/encyclopedia/QRDecomposition.html) or Mathworld
(http://mathworld.wolfram.com/QRDecomposition.html) 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 (http://planetmath.org/encyclopedia/RayleighQuotient.html) 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 (http://planetmath.org/encyclopedia/RayleighQuotient.html) for more information
on Rayleigh quotient.
Rank
Rank (M)
Aliases: rank
Get the rank of a matrix.
See Planetmath (http://planetmath.org/encyclopedia/SylvestersLaw.html) for more information.
RosserMatrix
RosserMatrix ()
Rosser matrix, a classic symmetric eigenvalue test problem.
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Chapter 11. List of GEL functions
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.
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Chapter 11. List of GEL functions
SmithNormalFormField
SmithNormalFormField (A)
Smith Normal Form for fields (will end up with 1’s on the diagonal).
SmithNormalFormInteger
SmithNormalFormInteger (M)
Smith Normal Form for square integer matrices (not its characteristic).
SolveLinearSystem
SolveLinearSystem (M,V,args...)
Solve linear system Mx=V, return solution V if there is a unique solution, null otherwise. Extra two
reference parameters can optionally be used to get the reduced M and V.
ToeplitzMatrix
ToeplitzMatrix (c, r...)
Return the Toeplitz matrix constructed given the first column c and (optionally) the first row r. If
only the column c is given then it is conjugated and the nonconjugated version is used for the first
row to give a Hermitian matrix (if the first element is real of course).
See Planetmath (http://planetmath.org/encyclopedia/ToeplitzMatrix.html) for more information.
Trace
Trace (M)
Aliases: trace
Calculate the trace of a matrix. That is the sum of the diagonal elements.
See Planetmath (http://planetmath.org/encyclopedia/Trace.html) for more information.
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Chapter 11. List of GEL functions
Transpose
Transpose (M)
Transpose of a matrix. This is the same as the .’ operator.
See Planetmath (http://planetmath.org/encyclopedia/Transpose.html) for more information.
VandermondeMatrix
VandermondeMatrix (v)
Aliases: vander
Return the Vandermonde matrix.
VectorAngle
VectorAngle (v,w,B...)
The angle of two vectors with respect to inner product given by B. If B is not given then the standard
hermitian product is used. B can either be a sesquilinear function of two arguments or it can be a
matrix giving a sesquilinear form.
VectorSpaceDirectSum
VectorSpaceDirectSum (M,N)
The direct sum of the vector spaces M and N.
VectorSubspaceIntersection
VectorSubspaceIntersection (M,N)
Intersection of the subspaces given by M and N.
VectorSubspaceSum
VectorSubspaceSum (M,N)
The sum of the vector spaces M and N, that is {w | w=m+n, m in M, n in N}.
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Chapter 11. List of GEL functions
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 (http://en.wikipedia.org/wiki/Determinant) or Planetmath
(http://planetmath.org/encyclopedia/Determinant2.html) 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 (http://en.wikipedia.org/wiki/Row_echelon_form) or Planetmath
(http://planetmath.org/encyclopedia/RowEchelonForm.html) for more information.
rref
rref (M)
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Chapter 11. List of GEL functions
Aliases: RREF ReducedRowEchelonForm
Get the reduced row echelon form of a matrix. That is, apply gaussian elimination together with
backaddition to M.
See Wikipedia (http://en.wikipedia.org/wiki/Reduced_row_echelon_form) or Planetmath
(http://planetmath.org/encyclopedia/ReducedRowEchelonForm.html) for more information.
11.10. Combinatorics
Catalan
Catalan (n)
Get n’th catalan number.
See Planetmath (http://planetmath.org/encyclopedia/CatalanNumbers.html) 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 (http://planetmath.org/encyclopedia/DoubleFactorial.html) for more information.
Factorial
Factorial (n)
Factorial: n(n-1)(n-2)...
See Planetmath (http://planetmath.org/encyclopedia/Factorial.html) for more information.
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Chapter 11. List of GEL functions
FallingFactorial
FallingFactorial (n,k)
Falling factorial: (n)_k = n(n-1)...(n-(k-1))
See Planetmath (http://planetmath.org/encyclopedia/FallingFactorial.html) 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 (http://en.wikipedia.org/wiki/Fibonacci_number) or Planetmath
(http://planetmath.org/encyclopedia/FibonacciSequence.html) or Mathworld
(http://mathworld.wolfram.com/FibonacciNumber.html) for more information.
FrobeniusNumber
FrobeniusNumber (v,arg...)
Calculate the Frobenius number. That is calculate smallest number that cannot be given as a
nonnegative integer linear combination of a given vector of nonnegative integers. The vector can be
given as separate numbers or a single vector. All the numbers given should have GCD of 1.
See Mathworld (http://mathworld.wolfram.com/FrobeniusNumber.html) 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)
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Chapter 11. List of GEL functions
Find the vector c of nonnegative 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 nonnegative
integers.
See Mathworld (http://mathworld.wolfram.com/GreedyAlgorithm.html) for more information.
HarmonicNumber
HarmonicNumber (n,r)
Aliases: HarmonicH
Harmonic Number, the n’th harmonic number of order r.
Hofstadter
Hofstadter (n)
Hofstadter’s function q(n) defined by q(1)=1, q(2)=1, q(n)=q(n-q(n-1))+q(n-q(n-2)).
LinearRecursiveSequence
LinearRecursiveSequence (seed_values,combining_rule,n)
Compute linear recursive sequence using galois stepping.
Multinomial
Multinomial (v,arg...)
Calculate multinomial coefficients. Takes a vector of k nonnegative 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 (http://planetmath.org/encyclopedia/MultinomialTheorem.html), Mathworld
(http://mathworld.wolfram.com/MultinomialCoefficient.html), or Wikipedia
(http://en.wikipedia.org/wiki/Multinomial_theorem) for more information.
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Chapter 11. List of GEL functions
NextCombination
NextCombination (v,n)
Get combination that would come after v in call to combinations, first combination should be
[1:k]. This function is useful if you have many combinations to go through and you don’t want to
waste memory to store them all.
For example with Combination you would normally write a loop like:
for n in Combinations (4,6) do (
SomeFunction (n)
);
But with NextCombination you would write something like:
n:=[1:4];
do (
SomeFunction (n)
) while not IsNull(n:=NextCombination(n,6));
See also Combinations.
Pascal
Pascal (i)
Get the Pascal’s triangle as a matrix. This will return an i+1 by i+1 lower diagonal matrix which is
the Pascal’s triangle after i iterations.
See Planetmath (http://planetmath.org/encyclopedia/PascalsTriangle.html) for more information.
Permutations
Permutations (k,n)
Get all permutations of k numbers from 1 to n as a vector of vectors.
See Mathworld (http://mathworld.wolfram.com/Permutation.html) or Wikipedia
(http://en.wikipedia.org/wiki/Permutation) for more information.
RisingFactorial
RisingFactorial (n,k)
Aliases: Pochhammer
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Chapter 11. List of GEL functions
(Pochhammer) Rising factorial: (n)_k = n(n+1)...(n+(k-1)).
See Planetmath (http://planetmath.org/encyclopedia/RisingFactorial.html) for more information.
StirlingNumberFirst
StirlingNumberFirst (n,m)
Aliases: StirlingS1
Stirling number of the first kind.
See Planetmath (http://planetmath.org/encyclopedia/StirlingNumbersOfTheFirstKind.html) or
Mathworld (http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html) for more
information.
StirlingNumberSecond
StirlingNumberSecond (n,m)
Aliases: StirlingS2
Stirling number of the second kind.
See Planetmath (http://planetmath.org/encyclopedia/StirlingNumbersSecondKind.html) or
Mathworld (http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html) for more
information.
Subfactorial
Subfactorial (n)
Subfactorial: n! times sum_{k=1}^n (-1)^k/k!.
Triangular
Triangular (nth)
Calculate the n’th triangular number.
See Planetmath (http://planetmath.org/encyclopedia/TriangularNumbers.html) for more
information.
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Chapter 11. List of GEL functions
nCr
nCr (n,r)
Aliases: Binomial
Calculate combinations, that is, the binomial coefficient. n can be any real number.
See Planetmath (http://planetmath.org/encyclopedia/Choose.html) for more information.
nPr
nPr (n,r)
Calculate the number of permutations of size rof numbers from 1 to n.
See Mathworld (http://mathworld.wolfram.com/Permutation.html) or Wikipedia
(http://en.wikipedia.org/wiki/Permutation) for more information.
11.11. Calculus
CompositeSimpsonsRule
CompositeSimpsonsRule (f,a,b,n)
Integration of f by Composite Simpson’s Rule on the interval [a,b] with n subintervals with error of
max(f””)*h^4*(b-a)/180, note that n should be even.
See Planetmath (http://planetmath.org/encyclopedia/SimpsonsRule.html) for more information.
CompositeSimpsonsRuleTolerance
CompositeSimpsonsRuleTolerance (f,a,b,FourthDerivativeBound,Tolerance)
Integration of f by Composite Simpson’s Rule on the interval [a,b] with the number of steps
calculated by the fourth derivative bound and the desired tolerance.
See Planetmath (http://planetmath.org/encyclopedia/SimpsonsRule.html) for more information.
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Chapter 11. List of GEL functions
Derivative
Derivative (f,x0)
Attempt to calculate derivative by trying first symbolically and then numerically.
EvenPeriodicExtension
EvenPeriodicExtension (f,L)
Return a function which is even periodic extension of f with half period L. That is a function
defined on the interval [0,L] extended to be even on [-L,L] and then extended to be periodic with
period 2*L.
See also OddPeriodicExtension and PeriodicExtension.
FourierSeriesFunction
FourierSeriesFunction (a,b,L)
Return a function which is a Fourier series with the coefficients given by the vectors a (sines) and b
(cosines). Note that [email protected](1) is the constant coefficient! That is, [email protected](n) refers to the term
cos(x*(n-1)*pi/L), while [email protected](n) refers to the term sin(x*n*pi/L). Either a or b can be
null.
See Wikipedia (http://en.wikipedia.org/wiki/Fourier_series) or Mathworld
(http://mathworld.wolfram.com/FourierSeries.html) 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)
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Chapter 11. List of GEL functions
Try to calculate an infinite sum for a single parameter function.
InfiniteSum2
InfiniteSum2 (func,arg,start,inc)
Try to calculate an infinite sum for a double parameter function with func(arg,n).
IsContinuous
IsContinuous (f,x0)
Try and see if a real-valued function is continuous at x0 by calculating the limit there.
IsDifferentiable
IsDifferentiable (f,x0)
Test for differentiability by approximating the left and right limits and comparing.
LeftLimit
LeftLimit (f,x0)
Calculate the left limit of a real-valued function at x0.
Limit
Limit (f,x0)
Calculate the limit of a real-valued function at x0. Tries to calculate both left and right limits.
MidpointRule
MidpointRule (f,a,b,n)
Integration by midpoint rule.
NumericalDerivative
NumericalDerivative (f,x0)
Aliases: NDerivative
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Chapter 11. List of GEL functions
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 (http://en.wikipedia.org/wiki/Fourier_series) or Mathworld
(http://mathworld.wolfram.com/FourierSeries.html) for more information.
NumericalFourierSeriesFunction
NumericalFourierSeriesFunction (f,L,N)
Return a function which is the Fourier series of f with half-period L (that is defined on [-L,L] and
extended periodically) with coefficients up to Nth harmonic computed numerically. This is the
trigonometric real series composed of sines and cosines. The coefficients are computed by
numerical integration using NumericalIntegral.
See Wikipedia (http://en.wikipedia.org/wiki/Fourier_series) or Mathworld
(http://mathworld.wolfram.com/FourierSeries.html) for more information.
NumericalFourierCosineSeriesCoefficients
NumericalFourierCosineSeriesCoefficients (f,L,N)
Return a vector of coefficients of the the cosine Fourier series of f with half-period L. That is, we
take f defined on [0,L] take the even periodic extension and compute the Fourier series, which
only has sine terms. The series is computed up to the Nth harmonic. The coefficients are computed
by numerical integration using NumericalIntegral. Note that [email protected](1) is the constant coefficient!
That is, [email protected](n) refers to the term cos(x*(n-1)*pi/L).
See Wikipedia (http://en.wikipedia.org/wiki/Fourier_series) or Mathworld
(http://mathworld.wolfram.com/FourierCosineSeries.html) for more information.
NumericalFourierCosineSeriesFunction
NumericalFourierCosineSeriesFunction (f,L,N)
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Chapter 11. List of GEL functions
Return a function which is the cosine Fourier series of f with half-period L. That is, we take f
defined on [0,L] take the even periodic extension and compute the Fourier series, which only has
cosine terms. The series is computed up to the Nth harmonic. The coefficients are computed by
numerical integration using NumericalIntegral.
See Wikipedia (http://en.wikipedia.org/wiki/Fourier_series) or Mathworld
(http://mathworld.wolfram.com/FourierCosineSeries.html) for more information.
NumericalFourierSineSeriesCoefficients
NumericalFourierSineSeriesCoefficients (f,L,N)
Return a vector of coefficients of the the sine Fourier series of f with half-period L. That is, we take
f defined on [0,L] take the odd periodic extension and compute the Fourier series, which only has
sine terms. The series is computed up to the Nth harmonic. The coefficients are computed by
numerical integration using NumericalIntegral.
See Wikipedia (http://en.wikipedia.org/wiki/Fourier_series) or Mathworld
(http://mathworld.wolfram.com/FourierSineSeries.html) for more information.
NumericalFourierSineSeriesFunction
NumericalFourierSineSeriesFunction (f,L,N)
Return a function which is the sine Fourier series of f with half-period L. That is, we take f defined
on [0,L] take the odd periodic extension and compute the Fourier series, which only has sine
terms. The series is computed up to the Nth harmonic. The coefficients are computed by numerical
integration using NumericalIntegral.
See Wikipedia (http://en.wikipedia.org/wiki/Fourier_series) or Mathworld
(http://mathworld.wolfram.com/FourierSineSeries.html) for more information.
NumericalIntegral
NumericalIntegral (f,a,b)
Integration by rule set in NumericalIntegralFunction of f from a to b using NumericalIntegralSteps
steps.
NumericalLeftDerivative
NumericalLeftDerivative (f,x0)
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Chapter 11. List of GEL functions
Attempt to calculate numerical left derivative.
NumericalLimitAtInfinity
NumericalLimitAtInfinity (_f,step_fun,tolerance,successive_for_success,N)
Attempt to calculate the limit of f(step_fun(i)) as i goes from 1 to N.
NumericalRightDerivative
NumericalRightDerivative (f,x0)
Attempt to calculate numerical right derivative.
OddPeriodicExtension
OddPeriodicExtension (f,L)
Return a function which is odd periodic extension of f with half period L. That is a function defined
on the interval [0,L] extended to be odd on [-L,L] and then extended to be periodic with period
2*L.
See also EvenPeriodicExtension and PeriodicExtension.
OneSidedFivePointFormula
OneSidedFivePointFormula (f,x0,h)
Compute one-sided derivative using five point formula.
OneSidedThreePointFormula
OneSidedThreePointFormula (f,x0,h)
Compute one-sided derivative using three-point formula.
PeriodicExtension
PeriodicExtension (f,a,b)
Return a function which is the periodic extension of f defined on the interval [a,b] and has period
b-a.
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Chapter 11. List of GEL functions
See also OddPeriodicExtension and EvenPeriodicExtension.
RightLimit
RightLimit (f,x0)
Calculate the right limit of a real-valued function at x0.
TwoSidedFivePointFormula
TwoSidedFivePointFormula (f,x0,h)
Compute two-sided derivative using five-point formula.
TwoSidedThreePointFormula
TwoSidedThreePointFormula (f,x0,h)
Compute two-sided derivative using three-point formula.
11.12. Functions
Argument
Argument (z)
Aliases: Arg arg
argument (angle) of complex number.
DirichletKernel
DirichletKernel (n,t)
Dirichlet kernel of order n.
DiscreteDelta
DiscreteDelta (v)
Returns 1 iff all elements are zero.
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Chapter 11. List of GEL functions
ErrorFunction
ErrorFunction (x)
Aliases: erf
The error function, 2/sqrt(pi) * int_0^x e^(-t^2) dt.
See Planetmath (http://planetmath.org/encyclopedia/ErrorFunction.html) for more information.
FejerKernel
FejerKernel (n,t)
Fejer kernel of order n evaluated at t
See Planetmath (http://planetmath.org/encyclopedia/FejerKernel.html) for more information.
GammaFunction
GammaFunction (x)
Aliases: Gamma
The Gamma function. Currently only implemented for real values.
See Planetmath (http://planetmath.org/encyclopedia/GammaFunction.html) for more information.
KroneckerDelta
KroneckerDelta (v)
Returns 1 iff all elements are equal.
MinimizeFunction
MinimizeFunction (func,x,incr)
Find the first value where f(x)=0.
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Chapter 11. List of GEL functions
MoebiusDiskMapping
MoebiusDiskMapping (a,z)
Moebius mapping of the disk to itself mapping a to 0.
See Planetmath (http://planetmath.org/encyclopedia/MobiusTransformation.html) 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 (http://planetmath.org/encyclopedia/MobiusTransformation.html) 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 (http://planetmath.org/encyclopedia/MobiusTransformation.html) 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 (http://planetmath.org/encyclopedia/MobiusTransformation.html) 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.
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Chapter 11. List of GEL functions
See Planetmath (http://planetmath.org/encyclopedia/MobiusTransformation.html) 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 (http://planetmath.org/encyclopedia/RiemannZetaFunction.html) 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 (http://en.wikipedia.org/wiki/Unit_step) for more information.
cis
cis (x)
The cis function, that is the same as cos(x)+1i*sin(x)
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Chapter 11. List of GEL functions
deg2rad
deg2rad (x)
Convert degrees to radians.
rad2deg
rad2deg (x)
Convert radians to degrees.
11.13. Equation Solving
CubicFormula
CubicFormula (p)
Compute roots of a cubic (degree 3) polynomial using the cubic formula. The polynomial should be
given as a vector of coefficients. That is 4*x^3 + 2*x + 1 corresponds to the vector [1,2,0,4].
Returns a column vector of the three solutions. The first solution is always the real one as a cubic
always has one real solution.
See Planetmath (http://planetmath.org/encyclopedia/CubicFormula.html), Mathworld
(http://mathworld.wolfram.com/CubicFormula.html), or Wikipedia
(http://en.wikipedia.org/wiki/Cubic_equation) for more information.
EulersMethod
EulersMethod (f,x0,y0,x1,n)
Use classical Euler’s method to numerically solve y’=f(x,y) for initial x0, y0 going to x1 with n
increments, returns y at x1.
Systems can be solved by just having y be a (column) vector everywhere. That is, y0 can be a
vector in which case f should take a number x and a vector of the same size for the second
argument and should return a vector of the same size.
See Mathworld (http://mathworld.wolfram.com/EulerForwardMethod.html), or Wikipedia
(http://en.wikipedia.org/wiki/Eulers_method) for more information.
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Chapter 11. List of GEL functions
FindRootBisection
FindRootBisection (f,a,b,TOL,N)
Find root of a function using the bisection method. TOL is the desired tolerance and N is the limit on
the number of iterations to run, 0 means no limit. The function returns a vector
[success,value,iteration], where success is a boolean indicating success, value is the
last value computed, and iteration is the number of iterations done.
FindRootFalsePosition
FindRootFalsePosition (f,a,b,TOL,N)
Find root of a function using the method of false position. TOL is the desired tolerance and N is the
limit on the number of iterations to run, 0 means no limit. The function returns a vector
[success,value,iteration], where success is a boolean indicating success, value is the
last value computed, and iteration is the number of iterations done.
FindRootMullersMethod
FindRootMullersMethod (f,x1,x2,x3,TOL,N)
Find root of a function using the Muller’s method. TOL is the desired tolerance and N is the limit on
the number of iterations to run, 0 means no limit. The function returns a vector
[success,value,iteration], where success is a boolean indicating success, value is the
last value computed, and iteration is the number of iterations done.
FindRootSecant
FindRootSecant (f,a,b,TOL,N)
Find root of a function using the secant method. TOL is the desired tolerance and N is the limit on
the number of iterations to run, 0 means no limit. The function returns a vector
[success,value,iteration], where success is a boolean indicating success, value is the
last value computed, and iteration is the number of iterations done.
PolynomialRoots
PolynomialRoots (p)
Compute roots of a polynomial (degrees 1 through 4) using one of the formulas for such
polynomials. The polynomial should be given as a vector of coefficients. That is 4*x^3 + 2*x +
1 corresponds to the vector [1,2,0,4]. Returns a column vector of the solutions.
The function calls QuadraticFormula, CubicFormula, and QuarticFormula.
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Chapter 11. List of GEL functions
QuadraticFormula
QuadraticFormula (p)
Compute roots of a quadratic (degree 2) polynomial using the quadratic formula. The polynomial
should be given as a vector of coefficients. That is 3*x^2 + 2*x + 1 corresponds to the vector
[1,2,3]. Returns a column vector of the two solutions.
See Planetmath (http://planetmath.org/encyclopedia/QuadraticFormula.html) or Mathworld
(http://mathworld.wolfram.com/QuadraticFormula.html) for more information.
QuarticFormula
QuarticFormula (p)
Compute roots of a quartic (degree 4) polynomial using the quartic formula. The polynomial should
be given as a vector of coefficients. That is 5*x^4 + 2*x + 1 corresponds to the vector
[1,2,0,0,5]. Returns a column vector of the four solutions.
See Planetmath (http://planetmath.org/encyclopedia/QuarticFormula.html), Mathworld
(http://mathworld.wolfram.com/QuarticEquation.html), or Wikipedia
(http://en.wikipedia.org/wiki/Quartic_equation) 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 (http://mathworld.wolfram.com/Runge-KuttaMethod.html), or Wikipedia
(http://en.wikipedia.org/wiki/Runge-Kutta_methods) for more information.
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Chapter 11. List of GEL functions
11.14. Statistics
Average
Average (m)
Aliases: average Mean mean
Calculate average of an entire matrix.
See Mathworld (http://mathworld.wolfram.com/ArithmeticMean.html) for more information.
GaussDistribution
GaussDistribution (x,sigma)
Integral of the GaussFunction from 0 to x (area under the normal curve).
See Mathworld (http://mathworld.wolfram.com/NormalDistribution.html) for more information.
GaussFunction
GaussFunction (x,sigma)
The normalized Gauss distribution function (the normal curve).
See Mathworld (http://mathworld.wolfram.com/NormalDistribution.html) for more information.
Median
Median (m)
Aliases: median
Calculate median of an entire matrix.
See Mathworld (http://mathworld.wolfram.com/StatisticalMedian.html) for more information.
PopulationStandardDeviation
PopulationStandardDeviation (m)
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Chapter 11. List of GEL functions
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 (http://mathworld.wolfram.com/ArithmeticMean.html) for more information.
RowMedian
RowMedian (m)
Calculate median of each row in a matrix and return a column vector of the medians.
See Mathworld (http://mathworld.wolfram.com/StatisticalMedian.html) 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)
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Chapter 11. List of GEL functions
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 (http://planetmath.org/encyclopedia/PolynomialLongDivision.html) for more
information.
IsPoly
IsPoly (p)
Check if a vector is usable as a polynomial.
MultiplyPoly
MultiplyPoly (p1,p2)
Multiply two polynomials (as vectors).
NewtonsMethodPoly
NewtonsMethodPoly (poly,guess,epsilon,maxn)
Run newton’s method on a polynomial to attempt to find a root, returns after two successive values
are within epsilon or after maxn tries (then returns null).
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Chapter 11. List of GEL functions
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).
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Chapter 11. List of GEL functions
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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Chapter 11. List of GEL functions
11.17. 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.
11.18. 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)))
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Chapter 11. List of GEL functions
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)
11.19. Plotting
LinePlot
LinePlot (func1,func2,func3,...)
LinePlot (func1,func2,func3,x1,x2,y1,y2)
Plot a function (or several functions) with a line. First up to 10 arguments are functions, then
optionally you can specify the limits of the plotting window as x1, x2, y1, y2. If limits are not
specified, then the currently set limits apply (See LinePlotWindow)
The parameter LinePlotDrawLegends controls the drawing of the legend.
Examples:
126
Chapter 11. List of GEL functions
genius> LinePlot(sin,cos)
genius> LinePlot(‘(x)=x^2,-1,1,0,1)
LinePlotClear
LinePlotClear ()
Show the line plot window and clear out functions and any other lines that were drawn.
LinePlotDrawLine
LinePlotDrawLine (x1,y1,x2,y2,...)
LinePlotDrawLine (v,...)
Draw a line from x1,y1 to x2,y2. x1,y1, x2,y2 can be replaced by an n by 2 matrix for a longer
line.
Extra parameters can be added to specify line color, thickness, arrows, and the plotting window. You
can do this by adding a string "color", "thickness", "window", or "arrow", and after it either
the color string, the thicknes as an integer, the window as 4-vector, and for arrow either "origin",
"end", "both", or "none". For "window" we can specify "fit" rather than a vector in which
case, the x range will be set precisely and the y range will be set with five percent borders around
the line.
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")
LinePlotParametric
LinePlotParametric (xfunc,yfunc,...)
LinePlotParametric (xfunc,yfunc,t1,t2,tinc)
LinePlotParametric (xfunc,yfunc,t1,t2,tinc,x1,x2,y1,y2)
Plot a parametric function with a line. First come the functions for x and y then optionally the t
limits as t1,t2,tinc, then optionally the limits as x1,x2,y1,y2.
If limits are not specified, then the currently set limits apply (See LinePlotWindow).
The parameter LinePlotDrawLegends controls the drawing of the legend.
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Chapter 11. List of GEL functions
LinePlotCParametric
LinePlotCParametric (func,...)
LinePlotCParametric (func,t1,t2,tinc)
LinePlotCParametric (func,t1,t2,tinc,x1,x2,y1,y2)
Plot a parametric complex valued function with a line. First comes the function that returns x+iy,
then optionally the t limits as t1,t2,tinc, then optionally the limits as x1,x2,y1,y2.
If limits are not specified, then the currently set limits apply (See LinePlotWindow).
The parameter LinePlotDrawLegends controls the drawing of the legend.
SlopefieldClearSolutions
SlopefieldClearSolutions ()
Clears the solutions drawn by the SlopefieldDrawSolution function.
SlopefieldDrawSolution
SlopefieldDrawSolution (x, y, dx)
When a slope field plot is active, draw a solution with the specified initial condition. The standard
Runge-Kutta method is used with increment dx. Solutions stay on the graph until a different plot is
shown or until you call SlopefieldClearSolutions. You can also use the graphical interface to
draw solutions and specify initial conditions with the mouse.
SlopefieldPlot
SlopefieldPlot (func)
SlopefieldPlot (func,x1,x2,y1,y2)
Plot a slope field. The function func should take two real numbers x and y, or a single complex
number. Optionally you can specify the limits of the plotting window as x1, x2, y1, y2. If limits are
not specified, then the currently set limits apply (See LinePlotWindow).
The parameter LinePlotDrawLegends controls the drawing of the legend.
Examples:
genius> Slopefield(‘(x,y)=sin(x-y),-5,5,-5,5)
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Chapter 11. List of GEL functions
SurfacePlot
SurfacePlot (func)
SurfacePlot (func,x1,x2,y1,y2,z1,z2)
Plot a surface function which takes either two arguments or a complex number. First comes the
function then optionally limits as x1, x2, y1, y2, z1, z2. If limits are not specified, then the
currently set limits apply (See SurfacePlotWindow). Genius can only plot a single surface
function at this time.
Examples:
genius> SurfacePlot(|sin|,-1,1,-1,1,0,1.5)
genius> SurfacePlot(‘(x,y)=x^2+y,-1,1,-1,1,-2,2)
genius> SurfacePlot(‘(z)=|z|^2,-1,1,-1,1,0,2)
VectorfieldClearSolutions
VectorfieldClearSolutions ()
Clears the solutions drawn by the VectorfieldDrawSolution function.
VectorfieldDrawSolution
VectorfieldDrawSolution (x, y, dt, tlen)
When a vector field plot is active, draw a solution with the specified initial condition. The standard
Runge-Kutta method is used with increment dt for an interval of length tlen. Solutions stay on the
graph until a different plot is shown or until you call VectorfieldClearSolutions. You can
also use the graphical interface to draw solutions and specify initial conditions with the mouse.
VectorfieldPlot
VectorfieldPlot (funcx, funcy)
VectorfieldPlot (funcx, funcy, x1, x2, y1, y2)
Plot a two dimensional vector field. The function funcx should be the dx/dt of the vectorfield and
the function funcy should be the dy/dt of the vectorfield. The functions should take two real
numbers x and y, or a single complex number. When the parameter VectorfieldNormalized is
true, then the magnitude of the vectors is normalized. That is, only the direction and not the
magnitude is shown.
Optionally you can specify the limits of the plotting window as x1, x2, y1, y2. If limits are not
specified, then the currently set limits apply (See LinePlotWindow).
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Chapter 11. List of GEL functions
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)
130
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("ref: argument not a value only matrix");bailout);
s := min(rows(m), columns(m));
i := 1;
d := 1;
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Chapter 12. Example Programs in GEL
while d <= s and i <= columns(m) do (
# This just makes the anchor element non-zero if at
# all possible
if [email protected](d,i) == 0 then (
j := d+1;
while j <= rows(m) do (
if [email protected](j,i) == 0 then
(j=j+1;continue);
a := [email protected](j,);
[email protected](j,) := [email protected](d,);
[email protected](d,) := a;
j := j+1;
break
)
);
if [email protected](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 [email protected](j,i) != 0 then (
[email protected](j,) := [email protected](j,)-([email protected](j,i)/[email protected](d,i))*[email protected](d,)
);
j := j+1
);
[email protected](d,) := [email protected](d,) * (1/[email protected](d,i));
d := d+1;
i := i+1
);
m
)
132
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 10^-x (0=never chop)
How to chop output. But only when other numbers nearby are large. See the documentation of the
paramter OutputChopExponent.
Only chop numbers when another number is greater than 10^-x
When to chop output. This is set by the paramter OutputChopWhenExponent. See the
documentation of the paramter OutputChopExponent.
133
Chapter 13. Settings
Remember output settings across sessions
Should the output settings in the Number/Expression output options frame be remembered for
next session. Does not apply to the Error/Info output options frame.
If unchecked, either the default or any previously saved settings are used each time Genius starts up.
Note that settings are saved at the end of the session, so if you wish to change the defaults check this
box, restart Genius Mathematics Tool and then uncheck it again.
Display errors in a dialog
If set the errors will be displayed in a seprate dialog, if unset the errors will be printed on the
console.
Display information messages in a dialog
If set the information messages will be displayed in a seprate dialog, if unset the information
messages will be printed on the console.
Maximum errors to display
The maximum number of errors to return on one evaluation (MaxErrors). If you set this to 0 then
all errors are always returned. Usually if some loop causes many errors, then it is unlikely that you
will be able to make sense out of more than a few of these, so seeing a long list of errors is usually
not helpful.
In addition to these preferences, there are some preferences that can only be changed by setting them in
the workspace console. For others that may affect the output see 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 effect how matrices (and
perhaps other stuff) is printed, useful for pasting into documents. Normal style is the default human
readable printing style of Genius Mathematics Tool. The other styles are for typsetting in LaTeX,
MathML (XML), or in Troff.
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.
134
Chapter 13. Settings
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 avoids the problem of running out of memory if you do
something by mistake which uses too much memory, such as a recursion without end. This could
slow your computer and make it hard to even interrupt the program.
Once the limit is reached, Genius Mathematics Tool asks if you wish to interrupt the computation or
if you wish to continue. If you continue, no limit is applied and it will be possible to run your
computer out of memory. The limit will be applied again next time you execute a program or an
expression on the Console regardless of how you answered the question.
Setting the limit to zero means there is no limit to the amount of memory that genius uses.
135
Chapter 14. About Genius Mathematics Tool
Genius Mathematics Tool was written by Jiří (George) Lebl (<[email protected]>). 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 (http://www.jirka.org/genius.html).
To report a bug or make a suggestion regarding this application or this manual, follow the directions in
this document (ghelp:gnome-feedback).
This program is distributed under the terms of the GNU General Public license as published by the Free
Software Foundation; either version 2 of the License, or (at your option) any later version. A copy of this
license can be found at this link (ghelp:gpl), or in the file COPYING included with the source code of
this program.
Jiří Lebl was during various parts of the development partially supported for the work by NSF grant
DMS 0900885 and the University of Illinois at Urbana-Champaign. The software has been used for both
teaching and research.
136
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