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 Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License (GFDL), Version 1.1 or any later version published by the Free Software Foundation with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. You can find a copy of the GFDL at this link (ghelp:fdl) or in the file COPYING-DOCS distributed with this manual. This manual is part of a collection of GNOME manuals distributed under the GFDL. If you want to distribute this manual separately from the collection, you can do so by adding a copy of the license to the manual, as described in section 6 of the license. 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SHOULD ANY DOCUMENT OR MODIFIED VERSION PROVE DEFECTIVE IN ANY RESPECT, YOU (NOT THE INITIAL WRITER, AUTHOR OR ANY CONTRIBUTOR) ASSUME THE COST OF ANY NECESSARY SERVICING, REPAIR OR CORRECTION. THIS DISCLAIMER OF WARRANTY CONSTITUTES AN ESSENTIAL PART OF THIS LICENSE. NO USE OF ANY DOCUMENT OR MODIFIED VERSION OF THE DOCUMENT IS AUTHORIZED HEREUNDER EXCEPT UNDER THIS DISCLAIMER; AND 2. UNDER NO CIRCUMSTANCES AND UNDER NO LEGAL THEORY, WHETHER IN TORT (INCLUDING NEGLIGENCE), CONTRACT, OR OTHERWISE, SHALL THE AUTHOR, INITIAL WRITER, ANY CONTRIBUTOR, OR ANY DISTRIBUTOR OF THE DOCUMENT OR MODIFIED VERSION OF THE DOCUMENT, OR ANY SUPPLIER OF ANY OF SUCH PARTIES, BE LIABLE TO ANY PERSON FOR ANY DIRECT, INDIRECT, SPECIAL, INCIDENTAL, OR CONSEQUENTIAL DAMAGES OF ANY CHARACTER INCLUDING, WITHOUT LIMITATION, DAMAGES FOR LOSS OF GOODWILL, WORK STOPPAGE, COMPUTER FAILURE OR MALFUNCTION, OR ANY AND ALL OTHER DAMAGES OR LOSSES ARISING OUT OF OR RELATING TO USE OF THE DOCUMENT AND MODIFIED VERSIONS OF THE DOCUMENT, EVEN IF SUCH PARTY SHALL HAVE BEEN INFORMED OF THE POSSIBILITY OF SUCH DAMAGES. Feedback To report a bug or make a suggestion regarding the Genius Mathematics Tool application or this manual, follow the directions in the GNOME 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. 19 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) 20 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. 21 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. 22 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. 23 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. 24 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. 25 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> ) 26 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. 27 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 28 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; 29 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. 30 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. 31 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. 32 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 33 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) 34 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 35 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 36 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 37 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. 42 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. 43 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. 44 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. 45 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 46 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 () 47 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 49 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. 51 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. 52 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. 53 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. 54 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. 55 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. 56 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). 57 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. 58 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. 59 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 60 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] 61 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) 62 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 63 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. 64 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. 65 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. 66 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) 67 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. 68 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. 69 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. 70 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) 71 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. 72 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). 73 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. 74 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. 75 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. 78 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. 80 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) 81 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. 82 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) 83 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. 84 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) 85 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. 86 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. 87 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) 88 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) 89 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. 90 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. 91 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. 92 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) 93 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. 94 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. 95 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. 96 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. 97 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. 98 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. 99 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}. 100 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) 101 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. 102 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) 103 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. 104 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 105 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. 106 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. 107 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) 108 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 109 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) 110 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) 111 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. 112 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. 113 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. 114 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. 115 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) 116 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. 117 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. 118 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. 119 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) 120 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) 121 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). 122 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). 123 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). 124 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))) 125 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. 127 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) 128 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). 129 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; 131 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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