Symbolic Algebra and Mathematics with Xcas

# Symbolic Algebra and Mathematics with Xcas ### Renée De Graeve, Bernard Parisse1, Jay Belanger2 Sections written by Luka Marohnić3

© 2002, 2007 Renée De Graeve, Bernard Parisse
renee.degraeve@wanadoo.fr
bernard.parisse@ujf-grenoble.fr

# Contents

# Chapter 1 Introduction

## 1.1 Notations used in this manual

User input in Xcas typically takes one of the following three forms:

Commands that you enter on the command line.
  For example, to compute sinπ/4, you can type

  sin(pi/4)

  The output will be typeset in blue and the messages in green.
Keyboard shortcuts.
  These are indicated by separating the prefix key and the standard key with a plus +. For example, to exit an Xcas session, you can type the control key along with the q key, which will be denoted by Ctrl+Q.
Menu commands.
  When denoting menu items (which are typeset in sans serif font), the submenus are connected by ▸ . For example, from within Xcas you can choose the File menu, then choose the Open submenu, and then choose the File item. This will be indicated by File ▸ Open ▸ File.

When describing command syntax, specific values that you enter for arguments are in typewriter font, while argument placeholders that should be replaced by actual values are in italics. Optional arguments will be enclosed by angular brackets. For example, you can find the derivative of an expression with the diff command (see Section 13.2.1), which takes the form diff(expr ⟨,x ⟩) where expr is an expression and x is (optionally) a variable or a list of variables.

In Xcas there exist many synonymous commands (i.e. having different names but the same effect). When a group of synonyms is described, the syntax details are provided only for the first command name and it is implied that the instructions apply to the synonyms as well.

In the index, the entries which are part of Xcas language are written in typewriter font (command names in black and programming keywords in blue) or italics (option names and specific values like color names).

## 1.2 Interfaces for the giac library

The giac library is a C++ mathematics library. It comes with two interfaces use directly; a graphical interface and a command-line interface. All interfaces can do symbolic and numeric calculations, use giac’s programming language, and have a built in help function.

The graphical interface is called Xcas, and is the most full-featured interface. Xcas has additional help features to make it easy to use, plus it has a built-in spreadsheet, it can do dynamic geometry and it can do turtle graphics. The output given by this interface is typeset; for example:

sqrt(2)

| √ |
2

The command-line interface can be run inside a terminal, and in a graphical environment can also draw graphs. The output given by this interface is in text form; for example:

sqrt(2)

sqrt(2)

There is also a web version, which can be run through a javascript-enabled browser (it works best with Firefox), either over the Internet or from local files. Other programs (for example, TeXmacs) have interfaces for the command-line version. Some of these interfaces, such as the two mentioned here, typeset their output.

- Xcas interface
- Command-line interface
- Firefox interface
- TeXmacs interface
- Checking the version of giac that you are using

# Chapter 2 Xcas interface

## 2.1 Entry levels

The Xcas interface can run several independent calculation sessions, each session will be contained in a separate tab. Before you understand the Xcas interface, it would help to be familiar with the components of a session.

Each session can have any number of input levels. Each input level will have a number to the left of it; the number is used to identify the level. Each level can have one of the following:

Command line entry
  — This is the default; you can open a new command line with Alt+N.

  You can enter a giac command (or a series of commands separated by semicolons) on a command line and send it to be evaluated by hitting enter. The result will then be displayed, and another command line will appear. You can also scroll through the command history with Ctrl+Up and Ctrl+Down.

  If the output is a number or an expression, then it will appear in blue text in a small area below the input region; this area will be an expression editor (see Section 2.9). There will be a scrollbar and a small button M to the right of this area, which is a menu providing various options.

  If the output is a graphic, then it will appear in a graphing area below the input region. To the right of the graphic will be a control panel which use to manipulate the graphic (see Section 18.1.1).

Expression editor
  — See Section 2.9. You can open an expression editor with Alt+E.
Two-dimensional geometry screen
  — See Section 18.1.1. You can open a two-dimensional geometry screen with Alt+G. This level will have a screen, as well as a control panel, menus and a command line to control the screen.
Three-dimensional geometry screen
  — See Section 18.1.1. You can open a three-dimensional geometry screen with Alt+H. This level will have a screen, as well as a control panel, menus and a command line to control the screen.
Turtle graphics screen
  — You can open a turtle graphics screen with Alt+D. This level will have a screen, as well as a program editor and command line.
Spreadsheet
  — See Section 2.10. You can open a spreadsheet with Alt+T. A spreadsheet can open a graphic screen.
Program editor
  — See Section 24.1.1. You can open a program editor with Alt+P.
Comment line
  — See Section 2.8.2. You can open a comment line with Alt+C.

Levels can be moved up and down in a session, or even moved to a different session.

The level containing the cursor is the current level. The current level can be evaluated or re-evaluated by typing enter.

You can select a level (for later operations) by clicking on the number in the white box to the left of the level. Once selected, the box containing the number turns black. You can select a range of levels by clicking on the number for the beginning level, and then holding the shift key while you click on the number for the ending level.

You can copy the instructions in a range of levels by selecting the range, and then clicking the middle mouse button on the number of the target level.

## 2.2 Starting window

> Figure 2.1: The initial Xcas window

When you first start Xcas, you get a largely blank window, as shown in Figure 2.1. The first row will consist of the main menus; you can save and load Xcas sessions, configure Xcas and its interface and run various commands with entries from these menus.

The second row will contains tabs; one tab for each session that you are running in Xcas. Each tab will have the name of its session, or Unnamed if the session has no name. The first time you start Xcas, there will be only one session, which will be unnamed.

The third row will contain various buttons.

- The button ? opens the help index (the same as the Help ▸ Index menu entry; see Section 2.3.3). If there is a command on the command line, the help index will open at this command.
- The button Save saves the session to a file. The first time you click on it you will be prompted for a file name ending in .xws in which to save the session. The button will be pink if the session is not saved or if it has changed since the last change, it will be green once the session is saved. The name in the title will be the name of the file used to save the session.
- The wide button, which in the picture above says Config: exact real RAD 12 xcas 12.6728M, is a status line indicating the current Xcas configuration (see Section 2.5). If the session is unsaved, it will begin with Config:; if the session is saved in a file filename.xws, this button will begin with Config filename.xws:. Other information on this status line:
  1. exact or approx
     This tells you whether Xcas will give you exact values, such as √2 when possible, or gives you decimal approximations, such as 1.4142135. (See Section 2.5.4.)
  2. real, cplx or CPLX.

     When this shows real (for example), then Xcas will by default only find real solutions of equations. When this shows cplx, then Xcas will find complex solutions of equations. When this shows CPLX, then Xcas will regard variables as complex; for example, it won’t simplify re(z) (the real part of the variable z) to z. (See sections 2.5.5 and 2.5.6.)

  3. RAD or DEG.

     This tells you whether angles, as in trigonometric arguments, are measured in radians or degrees. (See Section 2.5.3.)

  4. An integer.

     This tells you how many significant digits will be used in floating point calculations. (See Section 2.5.1.)

  5. Xcas, python ^=**, python ^=xor, Maple, MuPAD, or TI89.

     This tells you what syntax Xcas will use. Xcas can be set to emulate the languages of Python, Maple, MuPAD, or the TI89 series of calculators. (See Section 2.5.2.)

  6. The last item tells you how much memory Xcas is using.
  Clicking on this status line button opens a window where you can configure the settings shown on this line as well as some other settings; you can also open the window with the menu item Cfg ▸ CAS Configuration (see Section 2.5.7).
- The button STOP (in red) is used to halt a computation which is running on too long.
- The button Kbd toggles an on-screen scientific keyboard at the bottom of the window, which is shown in Figure 2.2.

  > Figure 2.2: On-screen scientific keyboard in Xcas

  Along the right hand side of the keyboard are some keys that can be used to change the keyboard.
  - The X key hides the keyboard, just like pressing the Kbd button again.
  - The cmds key toggles a menu bar at the bottom of the screen which can be used as an alternate menu or persistent submenu. This bar will contain buttons home, <<, some menu titles, >>, var, cust and X.

    The << and >> buttons scroll through menu items. Clicking on one of the menu buttons will perform the appropriate action or replace the menu items by its submenu items. When submenu items appear, there will also be a BACK button to return to the previous menu. Clicking on the home button returns the menu buttons to the main menu.

    After the menu buttons is a var button. This replaces the menu buttons by buttons representing the variables that you have defined. After that is a cust button, which displays commands that you store in a list variable CST (see section 3.3.10).

    The last button, X, closes the menu bar.

  - The msg key brings up a message window at the bottom of the window which will give you helpful messages; for example, if you save a graphic, it will tell you the name of the file it is saved in and how to include it in a LATEX file.
  - The abc key toggles the keyboard between the scientific keyboard and an alphabetic keyboard.
- The button X closes the current session.

## 2.3 Getting help

Xcas is an extensive program, but using it is simplified with several different ways of getting help. The help menu (see section 2.4.4) has several submenus for various forms of help, some of which are mentioned below.

- Tooltips
- HTML help
- Help index
- Getting help in a commandline

## 2.4 Menus

The menus provide different ways to work with Xcas and its sessions, as well as ways of inserting functions and constants into the current session. Selecting a menu item corresponding to a function or constant brings up the help index (see section 2.3.3) with the chosen function or constant selected.

- File menu
- Edit menu
- Cfg menu
- Help menu
- Toolbox menu
- Expression menu
- Cmds menu
- Prg menu
- Graphic menu
- Geo menu
- Spreadsheet menu
- Phys menu
- Highschool menu
- Turtle menu

## 2.5 Configuring Xcas

- Number of significant digits
- Language mode
- Units for angles
- Exact or approximate values
- Complex numbers
- Complex variables
- Configuring the computations
- Configuring the graphics
- More configuration
- Configuration file

## 2.6 Printing and saving

- Saving a session
- Saving a spreadsheet
- Saving a program
- Printing a session

## 2.7 Translating to other computer languages

Xcas can export a session, or parts of a session, to typesetting languages such as LATEX and MathML.

- Translating an expression to LATEX
  - Example
- Translating the entire session to LATEX
- Translating graphical output to LATEX
  - Example
- Translating an expression to MathML
  - Example
- Translating a spreadsheet to MathML
- Indent an XML string
  - Example
- Export to presentation or content MathML
  - Examples
- Configuring markup export
  - Examples
- Translating a Maple file to Xcas

## 2.8 Entry in Xcas

If you enter a command into Xcas and press Enter, the result will appear in the output box below the input, colored in blue. Any messages generated during the computation will be shown in the box between the input and output, colored in green (the box is otherwise hidden). For example:

integrate(a/(x+a)^2,x=0..inf)

No checks were made for singular points of antiderivative -1/(x+a) for definite integration in [0,+infinity]

1 |  |  |  |  |  |  |  |  |  |

Note that line breaks in a command entry can be entered by pressing Shift+Enter.

- Suppressing output
  - Example
- Entering comments
- Previous outputs
  - Example

## 2.9 Editing expressions

You can enter expressions on the command line, but Xcas also has a built-in expression editor that use to enter expressions in two dimensions, the way they normally look when typeset. When you have an expression in the editor, you can also manipulate subexpressions apart from the entire expression.

- Entering expressions in the editor: an example
- Subexpressions
- Manipulating subexpressions

## 2.10 Spreadsheets

- Opening a spreadsheet
- Spreadsheet window

# Chapter 3 CAS building blocks

## 3.1 Constants

- Numbers
- Symbolic constants
- Testing for undefined and infinity symbols
  - Examples

## 3.2 Sequences, sets and lists

- Sequences
- Sets
- Lists
- Accessing elements
  - Examples

## 3.3 Variables

- Variable names
- Assigning values to variables
- Assignment by reference
- Copying lists
- Incrementing variables
- Storing and recalling variables and their values
  - Example
- Copying variables
  - Example
- Assumptions on variables
  - Examples
- Unassigning variables
  - Examples
- CST variable

## 3.4 Functions

- Defining functions

# Chapter 4 Files and directories

## 4.1 Files

- Writing variable values to a file
  - Example
  - Example
- Writing output to a file
  - Example
- Reading files
  - Examples
- Reading CSV data
  - Examples
  - Application: loading and sorting real-world data

## 4.2 Directories

- Working directories
  - Examples
- Internal directories

# Chapter 5 Booleans and strings

## 5.1 Booleans

- Boolean values
- Comparison operators
- Defining functions with boolean tests
  - Example
  - Example
  - Examples
- Logical operators
  - Examples
- Transforming a boolean expression to a list
  - Examples
- Transforming a list into a boolean expression
  - Examples
- Evaluating booleans
  - Examples

## 5.2 Strings

- Characters and strings
  - Examples
  - Example
- Newline character
  - Example
- Length of a string
  - Example
- Extracting portions of a string
  - Examples
- Finding and removing leading/trailing whitespace
  - Examples
- Splitting strings into lists of tokens
  - Examples
- Concatenation of a list of strings
  - Examples
- ASCII code of a character
  - Example
- ASCII code of a string
  - Examples
- String defined by the ASCII codes of its characters
  - Example
- Finding a character in a string
  - Examples
- Concatenating objects into a string
  - Examples
- Adding an object to a string
  - Examples
- Transforming a real number into a string
  - Example
  - Example
- Transforming a string into a number
  - Examples
- Levensthein distance
  - Example
- Hamming distance
  - Example

## 5.3 Bitwise operators

- Basic operators
  - Examples
- Bitwise Hamming distance
  - Example

## 5.4 Writing an integer in a different base

- Writing an integer in base 2, 8 or 16
  - Examples
- Writing an integer in an arbitrary base
  - Examples
  - Examples

# Chapter 6 Sequences, lists, and sets

## 6.1 Sequences and lists

- Defining a sequence or a list
  - Examples
  - Examples
- Making a sequence or a list
  - Examples
  - Examples of sequences being used
- Length of a sequence or list
  - Examples
- Getting the first element of a sequence or list
  - Examples
- Getting a sequence or list without the first element
  - Example
- Getting an element of a sequence or a list
  - Examples
  - Example
- Finding a subsequence or a sublist
  - Examples
- Concatenating sequences
  - Example
- The + operator applied on sequences and lists
  - Examples
- Transforming sequences into lists and lists into sequences
  - Examples

## 6.2 Values of a sequence un

- Array of values of a sequence
  - Example
- Solving a recurrence relation or a system
  - Examples
- Table of values and graph of a recurrent sequence
  - Examples

## 6.3 Operations on lists

- Sizes of lists within a list
  - Example
- Creating a list by using a function
  - Examples
- Creating a list with zeros
  - Example
- Creating a list of integers
  - Examples
- Selecting elements of a list
  - Example
- Obtaining left and right portions of a list
  - Examples
- Modifying the elements of a list
  - Examples
- Removing elements from a list
  - Examples
- Inserting an element into a list or a string
  - Examples
- Appending an element at the end of a list
  - Examples
- Prepending an element at the beginning of a list
  - Examples
- Concatenating two lists or a list and an element
  - Examples
- Flattening a list
  - Example
- Reversing order in a list
  - Examples
- Rotating a list
  - Examples
- Shifting the elements of a list
  - Examples
- Sorting
  - Examples
- Sorting a list by increasing order
  - Examples
- Sorting a list by decreasing order
  - Examples
- Sorting by a permutation
  - Examples
- Counting elements equal to a given value
  - Example
- Counting elements smaller than a given value
  - Example
- Counting elements greater than a given value
  - Example
- Sum of elements of a list
  - Example
- Sum of list (or matrix) elements transformed by a function
  - Examples
- Cumulated sum of the elements of a list
  - Examples
- Products
  - Examples
  - Examples
- Applying a function of one variable to the elements of a list
  - Examples
- Applying a bivariate function to the elements of two lists
  - Examples
- Folding sequences
  - Examples
- List of differences of consecutive terms
  - Example
- Creating a matrix from a list
  - Examples
- Creating a list from a matrix
  - Example

## 6.4 Operations on sets and lists

- Defining sets
  - Examples
- Testing if a value is in a list or a set
  - Examples
- Union of two sets or of two lists
  - Examples
- Intersection of two sets or of two lists
  - Examples
- Difference of two sets or of two lists
  - Examples
- Cartesian products

## 6.5 Ranges of values

- Definition of a range of values
  - Examples
- Center of a range of values
  - Examples
- Ranges of values defined by their center
  - Examples

## 6.6 Intervals

- Defining intervals
  - Examples
- Obtaining endpoints of an interval
  - Examples
- Interval arithmetic
- Midpoint of an interval
  - Example
- Union of intervals
  - Examples
- Intersection of intervals
  - Example
- Testing if an object is in an interval
  - Examples
- Converting a number into an interval
  - Examples
- Converting box constraints from matrix to interval form
  - Example

# Chapter 7 Numbers

## 7.1 Integers (and Gaussian Integers)

Xcas can manage integers with unlimited precision, such as the following (see Section 12.1.1):

factorial(100)

| 9332621544394415268169923885626670049071596826438162 |  |  |  |  |  |  |  |  |
| 1468592963895217599993229915608941463976156518286253 |  |  |  |  |  |  |  |  |
| 697920827223758251185210916864000000000000000000000000 |  |  |  |  |  |  |  |  |

Gaussian integers are numbers of the form a+ib, where a and b are in ℤ. For most functions in this section, use Gaussian integers in place of integers.

- GCD
  - Examples
- GCD of a list of integers
  - Example
- Least common multiple
  - Examples
- Decomposition into prime factors
  - Examples
- List of prime factors
  - Examples
- Matrix of factors
  - Examples
- Divisors of a number
  - Examples
- Integer Euclidean quotient
  - Examples
- Integer Euclidean remainder
  - Examples
  - Example
  - Example
- Euclidean quotient and Euclidean remainder of two integers
  - Examples
- Testing evenness
  - Examples
- Testing oddness
  - Examples
- Testing pseudo-primality
  - Examples
- Testing primality
  - Examples
  - Examples
- Smallest pseudo-prime greater than n
  - Example
- Greatest pseudo-prime less than n
  - Example
- The nth pseudo-prime number
  - Examples
- Counting pseudo-primes less than or equal to n
  - Examples
- Bézout’s identity
  - Example
- Chinese remainders
  - Examples
  - Example
- Solving au+bv=c in ℤ
  - Example
- Solving a2+b2=p in ℤ
  - Example
- Solving Diophantine equations
  - Examples
- Euler indicatrix
  - Example
  - Example
- Legendre symbol
  - Examples
- Jacobi symbol
  - Examples
- Listing all compositions of an integer into k parts
  - Examples
- Day of the week
  - Examples

## 7.2 Rational numbers

- Transform a floating point number into a rational
  - Examples
- Integral and fractional part of a rational number
  - Examples
- Numerator of a fraction after simplification
  - Examples
- Denominator of a fraction after simplification
  - Example
- Numerator and denominator of a fraction
  - Example
- Simplifying a pair of integers
  - Examples
- Continued fraction representation of a real
  - Examples
- Transforming a continued fraction representation into a real
  - Examples
- Bernoulli numbers
  - Example
- Access to PARI/GP commands

## 7.3 Real numbers

- Evaluating a real at a given precision
  - Examples
- Standard arithmetic operators for real numbers
  - Examples
  - Example
- Prefixed division on reals
  - Examples
- nth root
  - Examples
- Exponential integral function
  - Examples
  - Examples
- Logarithmic integral function
  - Example
- Cosine integral function
  - Examples
- Sine integral function
  - Example
- Heaviside step function
  - Examples
- Dirac distribution
  - Examples
- Error function
  - Examples
- Complementary error function
  - Examples
- Gamma function
  - Examples
- Upper incomplete γ function
  - Examples
- Lower incomplete γ function
  - Examples
- Beta function
  - Examples
- Derivatives of the DiGamma function
  - Examples
- ζ function
  - Examples
- Airy functions
  - Examples

## 7.4 Complex numbers

Note that complex numbers are also used to represent points in the plane (see Section 25.5.2). Some functions and operators which work on complex numbers also work on points.

- Usual arithmetic operators for complex numbers
- Real and imaginary parts of a complex number
  - Examples
- Writing a complex number z in rectangular form
  - Example
- Modulus and argument of a complex number
  - Examples
- Normalized complex number
  - Example
- Conjugate of a complex number
  - Example
- Multiplication by the complex conjugate
  - Examples
- Barycenter of complex numbers
  - Example

## 7.5 Algebraic numbers

- Definition
- Minimum polynomial of an algebraic number
  - Examples

# Chapter 8 Operators and functions

## 8.1 Operators or infixed functions

An operator is an infixed function. For example, the arithmetic functions +, -, *, /, and ^ are operators. (See Section 7.3.2 and Section 7.4.1.)

- Special Xcas operators
- Defining custom operators
  - Examples

## 8.2 Functions and expressions with symbolic variables

- Difference between a function and an expression
- Transforming an expression into a function
  - Examples
  - Examples
  - Example
- Top and leaves of an expression
  - Examples
  - Examples
  - Examples

## 8.3 Functions

- Context-dependent functions.
  - Examples
  - Examples of - and ’-’.
  - Examples of * and ’*’.
  - Examples of / and ’/’.
- Standard functions
  - Example
  - Example
  - Examples
  - Examples
  - Examples
  - Examples
  - Examples
  - Examples
  - Examples
  - Example
  - Examples
  - Examples
  - Example
  - Example
  - Example
  - Example
  - Example
  - Example
  - Examples
  - Examples
  - Examples
  - Examples
- Defining algebraic functions
  - Example
  - Example
  - Examples
- Composing functions
- Defining a function with history
  - Example
  - Example

## 8.4 Getting information about univariate real functions

- Domain of a function
  - Examples
- Table of variations of a function
  - Examples

# Chapter 9 Algebraic expressions

## 9.1 Evaluation and substitution

- Expression evaluation
  - Examples
- Changing the evaluation level
  - Example
- Algebraic expression evaluation
- Preventing evaluation
  - Example
- Forcing evaluation
- Distribution of multiplication over addition
  - Example
- Canonical form
  - Examples
- Multiplication by the conjugate quantity
  - Examples
- Separation of variables
  - Examples
- Factoring
  - Examples
- Zeros of an expression
  - Examples
- Substituting a variable by a value
  - Examples
- Substituting a variable by a value
  - Examples
  - Examples
- Substituting a variable by a value
  - Examples
- Substituting a variable by a value (Maple and MuPAD compatibility)
  - Examples
  - Examples
- Substituting a subexpression by another expression
  - Examples
- Eliminating one or more variables from a list of equations
  - Examples
- Primitive evaluation at boundaries
  - Example
- Extracting subexpressions
  - Examples

## 9.2 Periodic functions

- Defining periodic expressions
  - Example
- Finding a period of an expression
  - Examples

## 9.3 Equations

- Defining an equation
  - Example
- Transforming an equation into a difference
  - Example
- Transforming an equation into a list
  - Example
- Left side of an equation
  - Example
- Right side of an equation
  - Example
- Solving equation(s)
  - Examples
  - Examples

## 9.4 Utility functions

- Replacing small values by zero
  - Examples
- Finding symbolic variables in an expression
  - Examples
- List of variables and of expressions
  - Examples
- List of variables of an algebraic expressions
  - Examples
- Testing if a variable is in an expression
  - Examples
- Numeric evaluation
  - Examples
- Rational approximation
  - Examples

# Chapter 10 Rewriting algebraic expressions

## 10.1 General rewriting and simplification routines

- Regrouping expressions
  - Example
- Normal form
  - Examples
- Simplifying
  - Examples
- Automatic simplification
  - Examples
- Normal form for rational functions
  - Examples
- Simplification of expressions involving Dirac delta distribution
  - Examples
- Replacing signum with Heaviside step function and vice versa
- Rewriting with Heaviside function
- Examples
- Rewriting with absolute values
  - Examples
- Rewriting an expression with different options
  - Examples

## 10.2 Trigonometry

Xcas can evaluate the trigonometric functions in either radians or degrees (see Section 8.3.2). It can also manipulate them algebraically.

- Expanding a trigonometric expression
- Linearizing a trigonometric expression
  - Examples
- Increasing the phase by π/2 in a trigonometric expression
  - Examples
- Putting together sine and cosine of the same angle
  - Examples
- Simplifying
  - Example
- Simplifying trigonometric expressions
  - Examples
- Transforming arccos into arcsin
  - Example
- Transforming arccos into arctan
  - Example
- Transforming arcsin into arccos
  - Example
- Transforming arcsin into arctan
  - Example
- Transforming arctan into arcsin
  - Example
- Transforming arctan into arccos
  - Example
- Transforming complex exponentials into sin and cos
  - Examples
- Transforming tan(x) into sin(x)/cos(x)
  - Example
- Transforming sin(x) into cos(x)tan(x)
  - Example
- Transforming cos(x) into sin(x)/tan(x)
  - Example
- Rewriting tan(x) in terms of sin(2x) and cos(2x)
  - Example
- Rewriting tan(x) in terms of cos(2x) and sin(2x)
  - Example
- Rewriting sin, cos, tan in terms of half tangent
  - Examples
- Rewriting trigonometric/hyperbolic functions in terms of half tangent/exponentials
  - Examples
- Transforming trigonometric functions into complex exponentials
  - Examples
- Transforming inverse trigonometric functions into logarithms
  - Example
- Simplifying and expressing preferentially with sines
  - Example
- Simplifying and expressing preferentially with cosines
  - Example
- Simplifying and expressing preferentially with tangents
  - Example

## 10.3 Exponentials and logarithms

- Rewriting hyperbolic functions as exponentials
  - Example
- Expanding exponentials
  - Example
- Expanding logarithms
  - Example
- Linearizing exponentials
  - Examples
- Collecting logarithms
  - Examples
- Expanding powers
  - Example
- Rewriting a power as an exponential
  - Example
- Rewriting exp(nln(x)) as a power
  - Example
- Simplifying complex exponentials
  - Example

## 10.4 Rewriting transcendental expressions

- Expanding transcendental expressions
  - Examples
- Combining terms of the same type
  - Examples

# Chapter 11 Polynomials

## 11.1 Basic functions for polynomials

- Polynomials of a single variable
- Polynomials of several variables
- Apply a function to the internal sparse format of a polynomial
  - Example
- Converting to a symbolic polynomial
  - Examples
- Converting from a symbolic polynomial
  - Examples
- Transforming a polynomial in internal format into a list and back
  - Examples
- Coefficients of a polynomial
  - Examples
- Polynomial degree
  - Examples
- Polynomial valuation
  - Examples
- Leading coefficient of a polynomial
  - Examples
- Trailing coefficient degree of a polynomial
  - Examples
- Polynomial evaluation
  - Examples
- Polynomial evaluation with Horner algorithm
  - Example
- Rewriting in terms of the powers of (x−a)
  - Examples
- Factoring xn in a polynomial
  - Example
- GCD of the coefficients of a polynomial
  - Example
- Primitive part of a polynomial
  - Example
- Factoring
  - Examples
- Square-free factorization
  - Examples
- List of factors
  - Examples
- Computing with the exact root of a polynomial
  - Example
- Exact roots of a polynomial
  - Examples
- Coefficients of a polynomial defined by its roots
  - Example
- Truncating to order n
  - Examples
- Converting a series expansion into a polynomial
  - Examples
- Changing the order of variables
  - Example
- Random polynomials
  - Examples
- Random lists
  - Example

## 11.2 Polynomial interpolation and fitting

- Lagrange polynomial
  - Examples
- Natural splines
- Natural interpolation
  - Examples
- Numerical spline interpolation
  - Supported spline types
  - Example
- Rational interpolation
  - Examples
- Rational interpolation without poles
  - Examples
- Trigonometric interpolation
  - Example
- Least-squares polynomial approximation
  - Example
  - Examples
- Minimax polynomial approximation
  - Example

## 11.3 Arithmetic and polynomials

Polynomials are represented by expressions or by lists of coefficients in decreasing power order. In the first case, for instructions requiring a main variable (like extended gcd computations), the variable used by default is x if not specified. For coefficients in ℤ/nℤ, use % n for each coefficient of the list or apply it to the entire expression defining the polynomial.

- Divisors of a polynomial
  - Examples
- Euclidean quotient
  - Examples
- Euclidean remainder
  - Examples
- Quotient and remainder
  - Examples
- GCD of two polynomials with the Euclidean algorithm
  - Examples
- Choosing the GCD algorithm of two polynomials
  - Examples
- LCM of two polynomials
  - Examples
- Bézout’s identity
  - Examples
- Solving au+bv=c over polynomials
  - Examples
- Chinese remainders
  - Examples
- Cyclotomic polynomial
  - Examples
- Sturm sequences and number of sign changes of P on (a,b]
  - Examples
- Sylvester matrix of two polynomials and resultant
  - Example
  - Examples using the resultant.

## 11.4 Exact bounds for roots of a polynomial

- Exact bounds for real roots of a polynomial
  - Examples
- Exact bounds for positive real roots of a polynomial
  - Examples
- An upper bound for the positive real roots of a polynomial
  - Examples
- A lower bound for the positive real roots of a polynomial
  - Examples
- Exact values of rational roots of a polynomial
  - Examples
- Exact bounds for complex roots of a polynomial
  - Examples
- Exact values of the complex rational roots of a polynomial
  - Example

## 11.5 Orthogonal polynomials

- Legendre polynomials
  - Examples
- Hermite polynomial
  - Examples
- Laguerre polynomials
  - Examples
- Chebyshev polynomials of the first kind
  - Examples
- Chebyshev polynomial of the second kind
  - Examples

## 11.6 Gröbner basis and Gröbner reduction

- Gröbner basis
  - Example
  - Examples
- Gröbner reduction
  - Examples
- Testing if a (list of) polynomial(s) belongs to an ideal given by a Gröbner basis
  - Examples
- Building a polynomial from its evaluation
  - Examples

## 11.7 Rational functions

- Numerator
  - Examples
- Numerator after simplification
  - Examples
- Denominator
  - Examples
- Denominator after simplification
  - Examples
- Numerator and denominator
  - Examples
- Simplifying
  - Example
- Common denominator
  - Example
- Polynomial and fractional part
  - Example
- Partial fraction expansion
  - Example

## 11.8 Exact roots and poles

- Roots and poles of a rational function
  - Examples
  - Example
- Rational function given by roots and poles
  - Example

## 11.9 Computing in ℤ/pℤ or in ℤ/pℤ[x]

The way to compute over ℤ/pℤ or over ℤ/pℤ[x] depends on the syntax mode:

- In Xcas mode, an object n over ℤ/pℤ is written as n%p.

  The representation is the symmetric representation: e.g. 11%13 returns -2%13.

- In Maple mode, integers modulo p are represented like usual integers instead of using specific modular integers. To avoid confusion with standard commands, modular commands are written with a capital letter (inert form) and followed by the mod command.

  The Maple commands are discussed in Section 11.10.

- - Examples
- Expanding and reducing
  - Example
- Addition in ℤ/pℤ or in ℤ/pℤ[x]
  - Examples
- Subtraction in ℤ/pℤ or in ℤ/pℤ[x]
  - Examples
- Multiplication in ℤ/pℤ or in ℤ/pℤ[x]
  - Examples
- Euclidean quotient
  - Example
- Euclidean remainder
  - Example
- Euclidean quotient and euclidean remainder
  - Example
- Division in ℤ/pℤ or in ℤ/pℤ[x]
- Power in ℤ/pℤ and in ℤ/pℤ[x]
  - Examples
- Computing an (mod p )
  - Examples
- Inverse in ℤ/pℤ
  - Example
- Rebuilding a fraction from its value modulo p
  - Examples
- GCD in ℤ/pℤ[x]
  - Example
- Factoring over ℤ/pℤ[x]
  - Example
- Determinant of a matrix in ℤ/pℤ
  - Example
- Inverse of a matrix with coefficients in ℤ/pℤ
  - Example
- Row reduction to echelon form in ℤ/pℤ
  - Example
  - Example
- Construction of a Galois field
  - Examples
- Factoring a polynomial with coefficients in a Galois field
  - Example

## 11.10 Computing in ℤ/pℤ[x] using Maple syntax

You can set Xcas to work in Maple mode rather than native Xcas mode (see Section 2.5.2).

- Euclidean quotient
  - Examples
- Euclidean remainder
  - Examples
- GCD in ℤ/pℤ[x]
  - Examples
- Factoring in ℤ/pℤ[x]
  - Example
- Determinant of a matrix with coefficients in ℤ/pℤ
  - Example
- Inverse of a matrix in ℤ/pℤ
  - Example
- Row reduction to echelon form in ℤ/pℤ
  - Example

# Chapter 12 Combinatorics

## 12.1 Combinatorial analysis

- Factorials
  - Examples
- Binomial coefficients
  - Example
- Counting permutations
  - Example
- Wilf-Zeilberger pairs
  - Example

## 12.2 Permutations

A permutation p of size n is a bijection from {0,…,n−1} to itself and is represented by the list:

[p(0),p(1),p(2),…,p(n−1)].

For example, the permutation p represented by [1,3,2,0] is the function from [0,1,2,3] to [0,1,2,3] defined by p(0)=1, p(1)=3, p(2)=2, p(3)=0.

A cycle c of size p, represented by the list [a0,…,ap−1] (0≤ ak≤ n−1), is the permutation such that

c(ai)=ai+1 for 0≤ i≤ p−2, c(ap−1)=a0, c(k)=k otherwise.

For example, the cycle c represented by the list [3,2,1] is the permutation c defined by c(3)=2, c(2)=1, c(1)=3, c(0)=0 (i.e. the permutation represented by the list [0,3,1,2]).

- Random permutations
  - Example
- Previous and next permutation
  - Examples
- Decomposing a permuation into a product of disjoint cycles
  - Examples
- Product of cycles to permutation
  - Examples
- Transforming a cycle into a permutation
  - Example
- Transforming a permutation into a matrix
  - Example
- Checking for a permutation
  - Examples
- Checking for a cycle
  - Examples
- Product of two permutations
  - Examples
- Signature of a permutation
  - Example
- Inverse of a permutation
  - Example
- Inverse of a cycle
  - Example
- Order of a permutation
  - Examples
- Group generated by two permutations
  - Example

# Chapter 13 Calculus

## 13.1 Limits

- Univariate function limits
  - Examples

## 13.2 Derivative

- Derivatives and partial derivatives
  - Examples
- Functional derivative
  - Examples
- Implicit differentiation
  - Examples
- Approximating derivatives of discrete functions
  - Examples

## 13.3 Integration

- Antiderivatives and definite integrals
  - Examples
- Primitives and definite integrals
  - Examples
- Discrete summation
  - Examples
- Riemann sums
  - Examples
- Integration by parts
  - Example
- Change of variables
- Integrals and limits
  - Examples
- Length of an arc
  - Examples

## 13.4 Differential equations

This section is limited to symbolic (or exact) solutions of differential equations. For numeric solutions of differential equations, see odesolve (Section 22.4.1). For graphic representation of solutions of differential equations, see plotfield (Section 18.9.1), plotode (Section 18.8.3) and interactive_plotode (Section 18.9.2).

- Solving differential equations
  - Examples
  - Examples of first order linear equations
  - Examples of first differential equations with separable variables
  - Examples of Bernoulli differential equations a(x)y′+b(x)y=c(x)yn where n is a real constant
  - Examples of first order homogeneous differential equations y′=F(y/x)
  - Examples of first order differential equations with an integrating factor
  - Example of first order differential equations without x
  - Examples of first order Clairaut differential equations y=x y′+f(y′)
- Laplace transform
  - Examples
  - Applications
- Solving linear homogeneous second-order ODE with rational coefficients
  - Examples

## 13.5 Taylor series and asymptotic expansions

- Dividing by increasing power order
  - Example
- Series expansion
  - Example
  - Examples
- Inverse of a series
  - Examples
- Residue of an expression at a point
  - Example
- Padé expansion
  - Examples

## 13.6 Z-transform

- Z-transform of a sequence
  - Examples
- Inverse Z-transform of a rational function
  - Examples

## 13.7 Multivariate calculus

- Gradient
  - Example
- Laplacian
  - Examples
- Hessian matrix
  - Examples
- Divergence
  - Example
- Rotational
  - Example
- Potential
  - Example
- Conservative flux field
  - Example
- Determining where a function is convex
  - Examples

## 13.8 Calculus of variations

- Motivation: the Brachistochrone
- Euler-Lagrange equations
  - Examples
- Solving the Brachistochrone Problem
- Jacobi equation
  - Example
- Finding conjugate points
  - Example
- An example: finding the surface of revolution with minimal area

# Chapter 14 Vectors, matrices, and tables

## 14.1 Functions for vectors

- Norms of a vector
- Normalizing a vector
  - Example
- Term by term sum of two lists
  - Examples
- Term by term difference of two lists
  - Example
- Term by term product of two lists
  - Example
- Term by term quotient of two lists
  - Example
- Scalar product
  - Example
- Cross product
  - Example

## 14.2 Matrices

- Entering matrices
  - Examples
- Special matrices
  - Examples
  - Example
  - Examples
  - Example
  - Examples
  - Example
- Combining matrices
  - Examples
  - Examples
  - Examples
  - Examples
- Creating a matrix with a formula or function
  - Examples
- Getting the parts of a matrix
  - Examples
  - Examples
- Modifying matrices
  - Examples
  - Examples
  - Example
  - Examples
  - Examples
  - Examples
  - Example

## 14.3 Functions for matrices

- Matrix evaluation
- Addition and subtraction of two matrices
  - Examples
- Multiplication of two matrices
  - Example
- Addition of elements of a column of a matrix
  - Example
- Cumulated sum of elements of each column of a matrix
  - Example
- Product of elements of each column of a matrix
  - Example
- Power of a matrix
  - Example
- Hadamard product
  - Examples
- Hadamard division
  - Example
- Hadamard power
  - Example
- Elementary row operations
  - Example
  - Examples
  - Examples
  - Example
  - Example
- Counting elements of a matrix which satisfy a given property
  - Examples
- Counting elements equal to a given value
  - Example
- Counting elements smaller than a given value
  - Example
- Counting elements greater than a given value
  - Example
- Dimension of a matrix
  - Example
- Number of rows
  - Example
- Number of columns
  - Example

## 14.4 Sparse matrices

- Tables
  - Example
- Defining sparse matrices
  - Example
- Operations on sparse matrices
  - Example

## 14.5 Statistics on lists and matrices

- Lists
  - Examples
- Matrices
  - Examples

# Chapter 15 Linear algebra

## 15.1 Basic matrix operations

- Transpose of a matrix
  - Example
- Inverse of a matrix
  - Example
- Trace of a matrix
  - Example
- Determinant of a matrix
  - Examples
  - Examples
- Rank of a matrix
  - Examples
- Transconjugate of a matrix
  - Example
- Equivalent matrix
  - Examples
- Basis of a linear subspace
  - Example
- Basis of the intersection of two subspaces
  - Example
- Image of a linear function
  - Example
- Kernel of a linear function
  - Example
  - Examples
- Subspace generated by the columns of a matrix
  - Examples
- Subspace generated by the rows of a matrix
  - Examples
- Testing positive definiteness of a symmetric matrix
  - Example

## 15.2 Matrix reduction

- Eigenvalues
  - Examples
- Jordan normal form
  - Examples
- Eigenvectors
  - Examples
- Rational Jordan matrix
  - Examples
- Jordan form of a matrix
  - Examples
- Powers of a square matrix
  - Example
- Characteristic polynomial
  - Examples
- Characteristic polynomial using Hessenberg algorithm
  - Examples
- Minimal polynomial
  - Examples
- Adjoint matrix
  - Examples
- Companion matrix of a polynomial
  - Examples
- Hessenberg matrix reduction
  - Example
- Hermite normal form
  - Example
- Smith normal form in ℤ
  - Example
- Smith normal form
  - Examples

## 15.3 Matrix factorizations

Note that most matrix factorization algorithms are implemented numerically, only a few of them will work symbolically.

- Cholesky decomposition
  - Examples
  - Example
- QR decomposition
  - Examples
- QR decomposition (for TI compatibility)
  - Example
- LQ decomposition (HP compatible)
  - Examples
- LU decomposition
  - Example
- LU decomposition (for TI compatibility)
  - Example
- Singular values (HP compatible)
  - Examples
- Singular value decomposition
  - Examples
- LDL decomposition
  - Example
- Computing inertia of a symmetric matrix
  - Examples
- Short basis of a lattice
  - Examples

## 15.4 Matrix norms

See Section 14.1.1 for different norms on vectors.

- Frobenius norm
  - Example
- ℓ2 matrix norm
  - Example
- ℓ∞ matrix norm
  - Example
- Matrix row norm
  - Example
- Matrix column norm
  - Example
- Operator norm of a matrix
  - Examples

## 15.5 Isometries

An isometry of ℝn is a distance-preserving map. In ℝ2, the isometries are made up of:

- reflection across a line.
- rotation about a point.
- translation.

In ℝ3, the isometries are made up of:

- reflection across a plane.
- rotation about a line.
- translation.

An isometry is direct if it preserves orientation (it does not involve a reflection), otherwise it is indirect.

An n× n matrix A determines an isometry of the function f:ℝn→ℝn by f(x)=Ax) is an isometry. Such isometries fix the origin, so they cannot involve translation, can only rotate about the origin, and the line or plane of reflection will pass through the origin.

An isometry in ℝ2 can be characterized by:

- The angle of rotation (for a direct isometry) or a normal to the line of reflection (for an indirect isometry).
- 1 or -1 to indicate whether it is direct or indirect; 1 for direct and -1 for indirect.

An isometry in ℝ3 can be characterized by:

- The direction of an axis of rotation.
- The angle of rotation (for a direct isometry) or a normal to the plane of reflection (for an indirect isometry).
- 1 or -1 to indicate whether it is direct or indirect; 1 for direct and -1 for indirect.

- Recognizing an isometry
  - Examples
- Finding the matrix of an isometry
  - Examples

## 15.6 Quadratic forms

- Matrix of a quadratic form
  - Example
- Transforming a matrix into a quadratic form
  - Examples
- Reducing a quadratic form
  - Example
- Conjugate gradient algorithm
  - Examples
- Gram-Schmidt orthonormalization
  - Examples
- Graph of a conic
  - Example
- Conic reduction
  - Example
  - Example
- Graph of a quadric
  - Example
- Quadric reduction
  - Example
  - Example

## 15.7 Linear systems

The augmented matrix of the system A X=b is either the matrix obtained by gluing the column vector b to the right of the matrix A (as with border(A,tran(b))), representing A X=b, or the matrix obtained by gluing the column vector −b to the right of the matrix A, representing A x−b=0.

- Matrix of a system
  - Examples
- Gauss reduction of a matrix
  - Example
- Gauss-Jordan reduction
  - Examples
- Solving AX=b
  - Examples
- Step by step Gauss-Jordan reduction of a matrix
  - Examples
- Solving linear systems
  - Examples
  - Example
  - Example
- Solving a linear system using the Jacobi method
  - Examples
- Solving a linear system using the Gauss-Seidel method
  - Examples
- Least squares solution of a linear system
  - Examples
- Finding linear recurrences
  - Examples

# Chapter 16 Optimization

## 16.1 Linear Programming

Linear programming problems involve maximizing a linear functionals under linear equality or inequality constraints. The simplest case can be solved directly by the so-called simplex algorithm. Most cases require you to solve an auxiliary linear programming problem to find an initial vertex for the simplex algorithm.

- Simplex algorithm
  - Example
- Solving (mixed integer) linear programming problems
  - Solving an LP problem in symbolic form
  - Examples
  - Solving an LP problem in matrix form
  - Examples
  - Solving MIP (Mixed Integer Programming) problems
  - Examples
  - Example
  - Example
  - Solving problems in floating-point arithmetic
  - Example
  - Loading problems from files
- Transportation problem
  - Examples

## 16.2 Analytical methods for nonlinear optimization

- Constrained global optimization
  - Examples
- Local extrema
  - Examples

## 16.3 Numerical methods for nonlinear optimization

- Univariate global minimization on a segment
  - Examples
- Minimization on bounded convex polyhedra
  - Examples
- Derivative-free constrained optimization
  - Examples
- Solving general nonlinear programming problems
  - Examples
- Simulated annealing minimization
  - Examples

# Chapter 17 Metric properties of curves

## 17.1 The center of curvature

Let Γ be a curve in space parameterized by a continuously differentiable function, and M0 be a point on the curve. The curve will have an arclength parameterization; namely, it can be parameterized by a function M(s), where M(0)=M0 and |s| is the length of the curve from M0 to M(s), in the direction of the curve if s>0 and the opposite direction if s<0.

For such a Γ, the vector T(s)=M′(s) will be the unit tangent to the curve at M(s), and N(s)=T′(s) will be perpendicular to the tangent. The circle through M(s) with center at M(s)+N(s) is called the osculating circle to Γ at M(s). Informally, the osculating circle is the circle through M(s) which most closely approximates Γ. The set of all centers of curvature is another curve, called the evolute of Γ.

The radius of the osculating circle is |N(s)| and is called the radius of curvature of Γ at M(s). The reciprocal of this is called the curvature of Γ at M(s).

## 17.2 Computing the curvature and related values

A curve can be described in Xcas with a parametrization or with a curve object. Various curve objects are described in chapters 25 and 26. The commands in this section can work with curves described either way. You can get the equation of a curve object with the equation command (see Section 25.12.7).

- Curvature of a curve
  - Examples
- Osculating circle of a curve
  - Examples
- Evolute of a curve
  - Examples

# Chapter 18 Graphs

## 18.1 Generalities

Most graph instructions take expressions as arguments. A few exceptions (mostly Maple-compatibility instructions) also accept functions. Some optional arguments, like color, thickness, can be used as optional attributes in all graphic instructions. They are described below.

- The graphic screen
- Graph and geometric objects attributes
  - Individual attributes
  - Examples
  - Global attributes
  - Examples
  - Formatting textual annotations
  - Example
- Colors and color functions
  - RGB and HSV colors
  - Examples
  - Colormaps
  - Examples
  - Color interpolation and RGB to XYZ conversion
  - Examples

## 18.2 Graph of a function

- 2D graph
  - Examples
- 3D graph
  - Functions of two variables
  - Examples
  - 3D graph with rainbow colors
  - “4D” graph
- 2D graph for Maple compatibility
  - Example
- 3D surfaces for Maple compatibility
  - Examples
- A note on graphing expressions

## 18.3 Graph of a line and tangent to a graph

- Drawing a line
  - Examples
- Drawing a 2D horizontal line
  - Example
- Drawing a 2D vertical line
  - Example
- Tangent to a 2D graph
  - Example
- Tangent to a 2D graph
  - Example
- Plotting a line with a point and the slope
  - Example
- Intersection of a 2D graph with the axis

## 18.4 Area graphs

- Graphing inequalities with two variables
  - Examples
- Computing the area under a curve
  - Examples
- Graphing the area under a curve
  - Examples

## 18.5 Contour and density graphs for surfaces

- Contour lines
  - Examples
- 2D graph of a surface with colors
  - Example

## 18.6 Implicit graphs

The plotimplicit or implicitplot command draws curves or surfaces defined by an implicit expression or equation. If the option unfactored is given as the last argument, the original expression is taken unmodified. Otherwise, the expression is normalized, then replaced by the factorization of the numerator of its normalization.

Each factor of the expression corresponds to a component of the implicit curve or surface. For each factor, Xcas tests if it is of total degree less or equal to 2, in which case conic or quadric is called. Otherwise the numeric implicit solver is called.

Optional step and ranges arguments may be passed to the numeric implicit solver, note that they are dismissed for each component that is a conic or a quadric.

- 2D implicit curve
  - Example
- 3D implicit surface
  - Examples

## 18.7 Parametric curves and surfaces

The plotparam or paramplot or DrawParm command draws parametric curves and surfaces.

- 2D parametric curve
  - Examples
- 3D parametric surface
  - Examples

## 18.8 Plotting curves and sequences

- Bezier curves
  - Examples
  - Examples
- Curves in polar coordinates
  - Examples
- Plotting solutions of differential equations
  - Examples
- Graphing recurrent sequences
  - Example

## 18.9 Plotting fields

- Tangent field
  - Example
- Interactive plotting of solutions of a differential equation
  - Example

## 18.10 Animated graphs

Xcas can display animated 2D, 3D or “4D” graphs. This is done first by computing a sequence of graphic objects, then after completion, by displaying the sequence in a loop. To stop or start the animation, click on the button ▸ (to the left of Menu).

- Animation of a 2D graph
  - Examples
- Animation of a 3D graph
  - Example
- Animation of a sequence of graphic objects
  - Examples

# Chapter 19 Statistics

## 19.1 One variable statistics

Xcas has several functions to perform statistics. Data is typically given as a list of numbers, such as A:=[0,1,2,3,4,5,6,7,8,9,10,11]. This particular list will be used in several examples. Statistics on matrices are discussed in Section 14.5.2.

- Mean
  - Examples
- Variance
  - Examples
- Standard deviation
  - Examples
- Population standard deviation
  - Examples
- Median
  - Examples
- Quartiles
  - Example
- Quantiles
  - Examples
- Whisker boxes
  - Example
- Classes
  - Examples
- Histograms
  - Example
  - Example
- Accumulating terms
  - Example
- Frequencies
  - Example
- Cumulative frequencies
  - Examples
- Bar plots
  - Examples
- Pie charts
  - Examples

## 19.2 Two variable statistics

- Covariance and correlation
  - Examples
  - Example
  - Example
- Scatterplots
  - Examples
- Polygonal paths
  - Examples
  - Examples
  - Example
- Linear regression
  - Example
  - Example
- Exponential regression
  - Example
  - Example
- Logarithmic regression
  - Example
  - Example
- Power regression
  - Example
  - Example
- Polynomial regression
  - Example
  - Example
- Logistic regression
  - Example
  - Example

## 19.3 Random numbers

- Initializing the random number generator
- Generating uniformly distributed random numbers
  - Examples
  - Example
  - Example
  - Examples
  - Example
- Sampling from probability distributions
  - Example
  - Examples
  - Example
  - Example
  - Example
  - Example
  - Example
  - Example
  - Example
  - Example
  - Example
- Random variables
  - Examples
- Generating random vectors and lists
  - Examples
- Generating random matrices
  - Examples

## 19.4 Density and distribution functions

- Distributions and inverse distributions
- Uniform distribution
  - Example
  - Examples
  - Example
- Binomial distribution
  - Examples
  - Examples
  - Example
- Negative binomial distribution
  - Example
  - Examples
  - Example
- Multinomial distribution
  - Example
- Poisson distribution
  - Example
  - Examples
  - Example
- Normal distributions
  - Examples
  - Examples
  - Examples
  - Examples
- Student’s distribution
  - Example
  - Examples
  - Example
  - Example
- χ2 distribution
  - Example
  - Examples
  - Example
  - Example
- Fisher-Snédécor distribution
  - Example
  - Examples
  - Example
  - Example
- Gamma distribution
  - Example
  - Examples
  - Example
- Beta distribution
  - Example
  - Examples
  - Example
- Geometric distribution
  - Example
  - Examples
  - Example
- Cauchy distribution
  - Examples
  - Examples
  - Example
- Exponential distribution
  - Example
  - Examples
  - Example
- Weibull distribution
  - Example
  - Examples
  - Example
- Kolmogorov-Smirnov distribution
  - Example
- Wilcoxon or Mann-Whitney distribution
  - Examples
  - Examples
  - Examples
- Moment generating functions for probability distributions
  - Examples
- Cumulative distribution functions
  - Examples
- Inverse distribution functions
  - Example
- Kernel density estimation
  - Examples
- Distribution fitting by maximum likelihood
  - Examples
- Markov chains
  - Example
- Generating a random walks
  - Examples

## 19.5 Hypothesis testing

- General
- Testing the mean with the Z test
  - Examples
- Testing the mean with the T test
  - Examples
- Testing a distribution with the χ2 distribution
  - Examples
- Testing a distribution with the Kolmogorov-Smirnov distribution
  - Examples

# Chapter 20 Signal Processing

## 20.1 Basic functions

- Boxcar function
  - Examples
- Rectangle function
  - Example
- Triangle function
  - Example
- Cardinal sine function
  - Examples

## 20.2 Common operations on signals

- Root mean square
  - Examples
- Finding and removing leading/trailing zeros
  - Example
- Cross-correlation of two signals
  - Examples
- Auto-correlation of a signal
  - Example
- Convolution of two signals or functions
  - Examples
  - Application

## 20.3 Filters

- Low-pass filtering
  - Example
- High-pass filtering
  - Example
- Moving-average filter
  - Example
- Performing thresholding operations on an array
  - Examples

## 20.4 Transforms

- Fourier coefficients
  - Examples
- Continuous Fourier Transform
  - Examples
  - Examples
- Defining symbolic transform pairs
  - Examples
- Discrete Fourier Transform and Fast Fourier Transform
  - Examples
  - Applications
- Short-time Fourier transform
  - Example
- Hilbert transform
  - Examples
- Analytic representation of a real signal
  - Examples
  - Instantaneous amplitude, frequency and phase
  - Examples
- Empirical mode decomposition
- Example
- Hilbert-Huang transform
  - Examples
- Discrete wavelet transform
  - Examples

## 20.5 Window functions

- Bartlett-Hann window
  - Example
- Blackman-Harris window
  - Example
- Blackman window
  - Example
- Bohman window
  - Example
- Cosine window
  - Example
- Gaussian window
  - Example
- Hamming window
  - Example
- Hann-Poisson window
  - Example
- Hann window
  - Example
- Parzen window
  - Example
- Poisson window
  - Example
- Riemann window
  - Example
- Triangular window
  - Example
- Tukey window
  - Example
- Welch window
  - Example

# Chapter 21 Data analysis and machine learning

## 21.1 Clustering

Clustering is a form of unsupervised machine learning. Xcas can perform hierarchical clustering and k-means clustering.

- Hierarchical clustering
  - Examples
- k-means clustering
  - Example

## 21.2 Artificial neural networks

This section explains how to create and train feed-forward neural networks in Xcas.

A feed-forward network, more precisely a multilayer perceptron, is comprised of a sequence of at least three neuron layers. The first layer is called the input layer and the last layer is called the output layer. Layers are essentially vectors of numerical data. Each component of such vector corresponds to an activated neuron. The network processes the given input by applying the formula lk+1=fk(WkTlk), where lk is the kth layer of length nk, lk+1 is the (k+1)-th layer of length nk+1, Wk is a nk× nk+1 weight matrix and fk is the activation function at the k+1-th layer (which is applied element-wise). Here k=1,2,…,L−1, where L is the number of layers. Each layer except the final one contains a bias neuron with linear activation. Bias neurons are auxiliary neurons created during construction of the network.

Multilayer perceptrons are trained by using backpropagation to compute the approximate gradient (with respect to the weight parameters in the matrices Wk) of the error function which compares the output to the expected output (or ground truth). Precisely, after training data is fed to the network, the obtained output is compared to the expected output and finally the error is backpropagated to find the weight adjustment enabling the network to predict the expected output more accurately.

The examples in Section 21.2.2 demonstrate how to prepare training data, construct neural networks, train them and test their accuracy.

##### Remark.

The commands in this section require that giac is linked to GSL.

- Creating neural networks
  - Examples
- Training a neural network
  - Examples

# Chapter 22 Numerical computations

Real numbers may have an exact representation (e.g. rationals, symbolic expressions involving square roots or constants like π, …) or approximate representation, which means that internally the real is represented by a rational (with a denominator that is a power of the basis of the representation) close to the real. Inside Xcas, the standard scientific notation is used for approximate representation; that is a mantissa (with a point as decimal separator) optionally followed by the letter e and an integer exponent.

Note that, for example, the real number 10−4 is an exact number but 1e–4 is an approximate representation of this number.

## 22.1 Floating point representation

This section discusses how real numbers are represented.

- Digits
- Representation by hardware floats
  - Examples of representations of normalized floats
- Approximate evaluation
  - Examples

## 22.2 Computing derivatives and integrals

- Approximating derivatives
  - Examples
- Approximating definite integrals
  - Example

## 22.3 Solving equations

- Newton method
  - Examples
- Finding approximate solutions of equations involving one variable
  - Examples
- Finding approximate solutions to systems of equations
  - Examples
  - Example
- Numeric roots of a polynomial
  - Examples

## 22.4 Solving differential equations

- Approximating solutions of y′=f(t,y)
  - Examples
- Approximating solutions of the system v′=f(t,v)
  - Examples
- Approximating solutions of y′′=f(t,y,y′) with boundary conditions
  - Examples

## 22.5 Numerical factorization of a matrix

Numeric Cholesky, QR, LU, and SVD factorizations of a matrix can be carried out, as described in Section 15.3.

# Chapter 23 Unit objects and physical constants

## 23.1 Unit objects

- Notation of unit objects
  - Examples
- Computing with units
  - Examples
- Converting units into MKSA units
  - Example
- Converting units
  - Examples
- Converting between Celsius and Fahrenheit
  - Examples
- Factoring a unit
  - Examples
- Simplifying units
  - Example

## 23.2 Constants

- Notation of physical constants
  - Example

# Chapter 24 Programming

Xcas has a programming language that allows you to write complex programs in several different syntax flavours. This section explains the details of the language, how to enter programs in Xcas (most notably new functions) and how to debug them. In the end, it is briefly explained how to extend and use giac from C++.

## 24.1 Functions, programs and scripts

- Program editor
- Functions
- Local variables
- Default values of the parameters
- Programs
- Scripts
- Code blocks

## 24.2 Basic instructions

- Comments
- Input
  - Examples
  - Examples
- Reading a single keystroke
- Checking required conditions
  - Example
- Checking the type of the argument
  - Examples
  - Subtypes
  - Example
  - Object comparison
  - Examples
- Printing
  - Examples
  - Example
- Displaying exponents
  - Examples
- Infixed assignments
- Assignment by copying
  - Example
- Difference between operators := and =<
  - Example
  - Examples

## 24.3 Control structures

- Conditional statements
  - Examples
  - Examples
- Switch statement
  - Example
- For loop
  - Example
  - Example
  - Example
- Repeat loop
  - Example
- While loop
  - Example
- Breaking out of loop
  - Example
- Skipping to the next iteration of a loop
  - Example
- Changing the order of execution
  - Example

## 24.4 Errors

- Handling errors
  - Examples
- Throwing exceptions
  - Example

## 24.5 Other useful instructions

- Defining a function with a variable number of arguments
  - Examples
- Assignments in a program
- Converting strings to giac expressions
  - Examples
  - Examples
- Creating symbols from strings
  - Examples
  - Examples
- Converting giac expressions to strings
  - Examples
- Converting real numbers to strings
  - Examples
- Working with the graphics screen
- Pausing a program
  - Example

## 24.6 Debugging

- Starting the debugger
  - Example

## 24.7 Linking to and extending the giac library

- Using giac inside a C++ program
- Defining new giac functions

# Chapter 25 Two-dimensional graphics

## 25.1 Introduction

- Points, vectors and complex numbers
- Clearing the DispG screen
- Toggling the axes

## 25.2 Basic commands

- Drawing unit vectors in the plane
  - Example
- Drawing dotted paper
  - Example
- Drawing lined paper
  - Example
- Drawing grid paper
  - Example
- Drawing triangular paper
  - Example

## 25.3 Display features of graphics

- Graphic features
- Parameters for changing features
  - Parameters for local features
  - Examples (of the filled option)
  - Parameters for global features
  - Example
- Commands for global display features
  - Example
  - Examples
- Defining geometric objects without drawing them
  - Examples

## 25.4 Geometric demonstrations with sliding parameters

Variables should be unspecified to demonstrate a general geometric result, but need to have specific values when drawing. There are a couple of different approaches to deal with this.

One approach is to use the assume command (see Section 3.3.8). If a variable is assumed to have a value, then that value will be used in graphics but the variable will still be unspecified for calculations. For example:

assume(a=2.1):; A:=point(a+i)

but the variable a will still be treated as a variable in calculations:

distance(0,A)

| √ |
⎛ ⎝ | −a | ⎞ ⎠ | 2+1

Another approach would be to use the point or pointer mode in a geometry screen. If no geometry screen is shown, the command Alt+G or the Geo ▸ New figure 2d menu will open a screen. Clicking on the Mode button right above the graphic screen and choosing pointer or point will put the screen in pointer or point mode. If a point is defined and displayed, such as with A:=point(2.1+i), then clicking on the name of the point (A in this case) with the right mouse button will bring up a configuration screen. As long as there is a point defined with non-symbolic values, there will be a symb box on the configuration screen. Selecting the symb box and choosing OK will be equivalent to the commands:

assume(Ax=[2.1,-8.16901408451,8.16901408451]); assume(Ay=[1,-5.0,5.0]

This brings up two lines beneath the arrows to the right of the screen which can be used to change the assumed values of Ax and Ay. Also, the point A will be redefined as point(Ax,Ay) (see Figure 25.1).

> Figure 25.1: A 2D-geometry screen in Xcas

## 25.5 Points in the plane

- Points and complex numbers
- Point in the plane
  - Example
  - Example
- Difference and sum of two points in the plane
- Defining random points in the plane
  - Examples
- Points in polar coordinates
  - Example
- Finding a point of intersection of two objects in the plane
  - Example
- Finding the points of intersection of two geometric objects in the plane
  - Examples
- Finding the orthocenter of a triangle in the plane
  - Example
- Finding the midpoint of a segment in the plane
- Barycenter in the plane
  - Example
- Isobarycenter of n points in the plane
- Center of a circle in the plane
  - Example
- Vertices of a polygon in the plane
  - Examples
- Vertices of a polygon in the plane, closed
- A point on a geometric object in the plane

## 25.6 Lines in plane geometry

- Lines and directed lines in the plane
  - Example
- Half-lines in the plane
  - Example
- Line segments in the plane
  - Example
- Vectors in the plane
  - Example
  - Examples
- Parallel lines in the plane
  - Example
- Perpendicular lines in the plane
  - Example
- Tangents to curves in the plane
  - Examples
- Median of a triangle in the plane
  - Example
- Altitude of a triangle
  - Example
- Perpendicular bisector of a segment in the plane
  - Example
- Angle bisector
- Exterior angle bisector

## 25.7 Triangles in the plane

See Section 26.6 for triangles in space.

- Arbitrary triangles in the plane
  - Example
- Isosceles triangles in the plane
  - Examples
- Right triangles in the plane
  - Examples
- Equilateral triangles in the plane
  - Examples

## 25.8 Quadrilaterals in the plane

See Section 26.7 for quadrilaterals in space.

- Squares in the plane
  - Examples
- Rhombuses in the plane
  - Examples
- Rectangles in the plane
  - Examples
  - Example
- Parallelograms in the plane
  - Examples
- Arbitrary quadrilaterals in the plane
  - Example

## 25.9 Other polygons in the plane

See Section 26.8 for polygons in space.

- Regular hexagons in the plane
  - Examples
- Regular polygons in the plane
  - Examples
- General polygons in the plane
  - Examples
- Polygonal lines in the plane
  - Examples
- Convex hulls
  - Example

## 25.10 Circles

- Circles and arcs in the plane
- Circular arcs
  - Examples
- Circles (TI compatibility)
  - Example
- Inscribed circles
  - Example
- Circumscribed circles
  - Example
- Excircles
  - Example
- Power of a point relative to a circle
  - Example
- Radical axis of two circles
  - Example

## 25.11 Other conic sections

- Ellipse in the plane
  - Examples
- Hyperbola in the plane
  - Examples
- Parabola in the plane
  - Examples

## 25.12 Coordinates in the plane

- Affix of a point or vector
  - Examples
- Abscissa of a point or vector in the plane
  - Examples
- Ordinate of a point or vector in the plane
  - Examples
- Coordinates of a point, vector or line in the plane
  - Examples
- Rectangular coordinates of a point
  - Example
- Polar coordinates of a point
  - Example
- Cartesian equation of a geometric object in the plane
  - Example
- Parametric equation of a geometric object in the plane
  - Examples

## 25.13 Measurements

- Measurement and display
  - Examples
  - Example
- Distance between objects in the plane
  - Examples
  - Example
- Squared length of a segment in the plane
  - Example
- Measure of an angle in the plane
  - Examples
  - Converting between radians and degrees
- Graphical representation of the area of a polygon
  - Examples
- Area of a polygon
  - Examples
- Perimeter of a polygon
  - Examples
- Slope of a line
  - Examples
- Radius of a circle
  - Example
- Length of a vector
  - Example
- Angle of a vector
  - Example
- Normalize a complex number
  - Example

## 25.14 Transformations

- General remarks
- Translations in the plane
  - Examples
- Reflections in the plane
  - Examples
- Rotation in the plane
  - Examples
- Homothety in the plane
  - Example
- Similarity in the plane
  - Example
- Inversion in the plane
  - Example
- Orthogonal projection in the plane
  - Examples

## 25.15 Properties

- Checking whether a point is on an object in the plane
  - Examples
- Checking whether three points are collinear in the plane
  - Examples
- Checking whether four points are concyclic in the plane
  - Examples
- Checking whether a point is in a polygon or circle
  - Examples
- Checking whether an object is an equilateral triangle in the plane
  - Examples
- Checking whether an object in the plane is an isosceles triangle
  - Examples
- Checking whether an object in the plane is a right triangle or a rectangle
  - Examples
- Checking whether an object in the plane is a square
  - Examples
- Checking whether an object in the plane is a rhombus
  - Examples
- Checking whether an object in the plane is a parallelogram
  - Examples
- Checking whether two lines in the plane are parallel
  - Examples
- Checking whether two lines in the plane are perpendicular
  - Examples
- Checking whether two circles in the plane are orthogonal
  - Examples
- Checking whether elements are conjugates
  - Examples
- Checking whether four points form a harmonic division
  - Examples
- Checking whether lines form a bundle
  - Example
- Checking whether circles form a bundle
  - Example

## 25.16 Harmonic division

- Finding a point dividing a segment in the harminic ratio k
  - Examples
- Cross ratio of four collinear points
  - Examples
- Harmonic division
  - Examples
- Harmonic conjugate
  - Examples
- Pole and polar
  - Example
  - Examples
- Polar reciprocal
  - Example

## 25.17 Loci and envelopes

- Loci
  - Example
  - Examples
- Envelopes
  - Example
- Trace of a geometric object
  - Example

# Chapter 26 Three-dimensional graphics

## 26.1 Introduction

- 3D geometry graphics screen
- Changing the view

## 26.2 The axes

- Drawing unit vectors
  - Example
  - Example
  - Example

## 26.3 Points in space

- Defining a point in three-dimensions
  - Example
- Defining a random point in three-dimensions
  - Example
- Finding an intersection point of two objects in space
  - Examples
- Finding the intersection points of two objects in space
  - Examples
- Finding the midpoint of a segment in space
  - Example
- Finding the barycenter of a set of points in space
  - Examples
- Finding the isobarycenter of a set of points in space
  - Example

## 26.4 Lines in space

- Lines and directed lines in space
  - Examples
- Half lines in space
  - Example
- Segments in space
  - Example
- Vectors in space
  - Examples
- Parallel lines and planes in space
  - Examples
- Perpendicular lines and planes in space
  - Examples
- Planes orthogonal to lines and lines orthogonal to planes in space
  - Examples
- Common perpendiculars to lines in space
  - Example

## 26.5 Planes in space

See also sections 26.4.6 and 26.4.7 for planes perpendicular and orthogonal to lines and planes.

- Planes in space
  - Example
- Bisector plane in space
  - Example
- Tangent planes in space
  - Examples

## 26.6 Triangles in space

- Drawing triangles in space
  - Example
- Isosceles triangles in space
  - Examples
- Right triangles in space
  - Examples
- Equilateral triangles in space
  - Examples

## 26.7 Quadrilaterals in space

See Section 25.8 for quadrilaterals in the plane.

- Squares in space
  - Examples
- Rhombuses in space
  - Examples
- Rectangles in space
  - Examples
- Parallelograms in space
  - Examples
- Arbitrary quadrilaterals in space
  - Example

## 26.8 Polygons in space

See Section 25.9 for polygons in the plane.

- Hexagons in space
  - Examples
- Regular polygons in space
  - Examples
- General polygons in space
  - Example
- Polygonal lines in space
  - Example

## 26.9 Circles and conics in space

- Circles in space
  - Examples
- Ellipses in space
  - Example
- Hyperbolas in space
  - Example
- Parabolas in space
  - Example

## 26.10 Three-dimensional coordinates

- Abscissa of a three-dimensional point
  - Example
- Ordinate of a three-dimensional point
  - Example
- Cote of a three-dimensional point
  - Example
- Coordinates of a point, vector or line in space
  - Examples
- Cartesian equation of an object in space
  - Examples
- Parametric equation of an object in space
  - Examples
- Length of a segment in space
  - Example
- Squared length of a segment in space
  - Example
- Measure of an angle in space
  - Examples

## 26.11 Properties

- Checking whether object in space is in another object
  - Examples
- Checking whether points and/or lines in space are coplanar
  - Examples
- Checking whether lines and/or planes in space are parallel
  - Examples
- Checking whether lines and/or planes in space are perpendicular
  - Examples
- Checking whether two lines or two spheres in space are orthogonal
  - Examples
- Checking whether points in space are collinear
  - Examples
- Checking whether points in space are concyclic
  - Examples
- Checking whether points in space are cospherical
  - Examples
- Checking whether an object in space is an equilateral triangle
  - Examples
- Checking whether an object in space is an isosceles triangle
  - Examples
- Checking whether an object in space is a right triangle or a rectangle
  - Examples
- Checking whether an object in space is a square
  - Examples
- Checking whether an object in space is a rhombus
  - Examples
- Checking whether an object in space is a parallelogram
  - Examples

## 26.12 Transformations in space

- General remarks
- Translation in space
  - Examples
- Reflection in space with respect to a plane, line or point
  - Examples
- Rotation in space
  - Examples
- Homothety in space
  - Examples
- Similarity in space
  - Examples
- Inversion in space
  - Examples
- Orthogonal projection in space
  - Examples

## 26.13 Surfaces

- Cones
  - Example
- Half-cones
  - Example
- Cylinders
  - Example
- Spheres
  - Examples
- Graph of a function of two variables
  - Example
- Graph of parametric equations in space
  - Example

## 26.14 Solids

- Cubes
  - Examples
- Tetrahedrons
  - Examples
- Parallelepipeds
  - Examples
- Prisms
  - Example
- Polyhedra
- Vertices
- Faces
- Edges

## 26.15 Platonic solids

To specify a Platonic solid, Xcas works with the center, a vertex and a third point to specify a plane of symmetry. To speed up calculations, it may be useful to use approximate calculations, which can be ensured with the evalf command. For example, instead of:

centered_cube([0,0,0],[3,2,1],[1,1,0])

it would typically be better to use:

centered_cube(evalf([0,0,0],[3,2,1],[1,1,0]))

since it is faster.

- Centered tetrahedra
  - Example
- Centered cubes
  - Examples
- Octahedra
  - Example
- Dodecahedra
  - Examples
- Icosahedra
  - Examples

# Chapter 27 Multimedia

## 27.1 Images

- Image structure in Xcas
- Creating and loading images
  - Examples
- Loading images to the legacy format
  - Example
- Viewing images
  - Example
  - Examples
- Exporting images
  - Examples
  - Examples
- New images from existing ones
  - Examples
- Modifying images
- Extracting data from images
- Trimming image margins
  - Example
- Blending images
  - Example

## 27.2 Audio

Xcas has support for audio objects in wav (Waveform Audio File) format. The following commands are provided for working with audio objects.

- For creating and playing audio objects, there are:
  - createwav, for creating audio objects (see Section 27.2.1).
  - playsnd, for playing audio objects (see Section 27.2.14).
- For reading and writing wave files, there are:
  - readwav, for reading audio files from disk to audio objects (see Section 27.2.3).
  - writewav, for writing audio files to disk from audio objects (see Section 27.2.4).
- For manipulating audio objects, there are:
  - set_channel_data, for modifying the contents of an audio clip (see Section 27.2.9).
  - normalize, for scaling the amplitude of an audio clip (see Section 27.2.15).
  - stereo2mono, for averaging channels to a single channel (see Section 27.2.5).
  - resample, for changing of the sample rate (see Section 27.2.11).
  - reverse, for reversing audio in time (see Section 27.2.2).
  - inv or - (unary operator), for inverting channel data (see Section 27.2.2).
  - + (infixed operator), for concatenation of audio clips (see Section 27.2.2).
  - splice, for splicing audio clips together with a crossfade (see Section 27.2.17).
  - mixdown, for mixing several clips down to one clip (see Section 27.2.18).
- For getting information from an audio object, there are:
  - about, for printing out the properties of an audio clip (see Section 27.2.6).
  - length or size, for obtaining the audio clip length in samples (see Section 27.2.6).
  - channels, for finding the number of channels (see Section 27.2.6).
  - bit_depth, for finding the number of bits in each sample value (see Section 27.2.6).
  - samplerate, for finding the number of samples per second (see Section 27.2.6).
  - duration, for finding the duration of the audio in seconds (see Section 27.2.6).
  - channel_data, for extracting channel data from an audio clip (see Section 27.2.8).
  - rms, for computing the RMS (root mean square) of an audio clip (see Section 27.2.16).
  - plotwav, for visualizing a waveform (see Section 27.2.12).
  - plotspectrum, for visualizing power spectra (see Section 27.2.13).

- Creating audio clips
  - Examples
- New audio clips from existing ones
- Reading wav files from disk
  - Example
- Writing wav files to disk
  - Example
- Averaging channel data
  - Example
- Audio clip properties
- Preparing digital sound data
  - Examples
- Extracting samples from audio clips
  - Example
- Setting samples in audio clips
  - Example
- Trimming silence
  - Example
- Changing the sampling rate of an audio clip
  - Example
- Visualizing waveforms
  - Examples
- Visualizing power spectra
  - Examples
- Listening to a digital sound
- Normalizing audio
  - Example
- Loudness estimation
  - Example
- Joining audio clips together
  - Example
- Mixing audio clips together
  - Example

# Index

’, 9.1.4 ’*’, 14.1.7 ’+’, 5.2.13, 14.1.3 ’-’, 14.1.4 (), 3.2.1, 6.1.1, 9.1.14, 27.1.6, 27.2.2 *, 6.4.6, 6.6.3, 7.3.2, 7.4.1, 8.3.1, 11.9.4, 14.1.7, 14.3.3 *=, 3.3.5 +, 5.2.13, 5.2.14, 6.1.9, 6.6.3, 7.3.2, 7.4.1, 8.3.1, 11.9.2, 14.1.3, 14.3.2, 25.5.3, 27.2.2 +=, 3.3.5 +infinity, 3.1.2 ,, 6.1.8 -, 6.6.3, 7.3.2, 7.4.1, 8.3.1, 11.9.3, 14.1.4, 14.3.2, 25.5.3, 27.1.7, 27.2.2 -=, 3.3.5 ->, 3.4.1, 8.2.1 -inf, 3.1.2 -infinity, 3.1.2 .*, 14.1.5, 14.3.8 .+, 14.1.3, 14.3.2 .-, 14.1.4, 14.3.2 .., 6.5.1 ./, 14.1.6, 14.3.9 .^, 14.3.10 .xcasrc, 2.5.10 /, 6.6.3, 7.3.2, 7.4.1, 8.3.1, 11.9.8, 15.1.2 //, 2.8.2, 24.2.1 /=, 3.3.5 :;, 2.8.1 :=, 3.3.2, 3.4.1, 8.2.1, 14.2.6, 24.2.8 <, 5.1.2 =, 3.3.2 =<, 3.3.3, 5.1.2, 14.2.6, 24.2.8 ==, 5.1.2 =>, 3.3.2, 3.4.1, 10.1.10, 23.1.4, 24.2.8 >, 5.1.2 >=, 5.1.2 ?:, 5.1.3 [], 3.2.3, 3.2.4, 6.1.6, 6.1.10, 6.4.1, 14.2.5, 27.1.8, 27.2.8 8.1.1, 8.3.4 8.1.1, 8.3.4 $, 6.1.2, 8.1.1 %, 7.1.9, 8.1.1, 11.9 %%%{ %%%}, 11.1.2 %{ %}, 3.2.2, 6.4.1 %e, 3.1.2 %i, 3.1.2 %pi, 3.1.2 &*, 14.3.3 &&, 5.1.4 &^, 14.3.7 ^, 7.3.2, 7.4.1, 11.9.9, 14.3.7, 15.1.2 _, 23.1.1, 23.2.1 {}, 24.1.2 ||, 5.1.4 !, 12.1.1 ", 5.2.1 \n, 5.2.2 Airy_Ai, 7.3.19 Airy_Bi, 7.3.19 a2q, 15.6.2 abcuv, 11.3.9 about, 3.3.8, 20.4.1, 27.2.6 abs, 7.4.4, 8.3.2, 25.13.10 abscissa, 25.12.2, 26.10.1 accumulate_head_tail, 19.1.11 acos, 8.3.2 acos2asin, 10.2.7 acos2atan, 10.2.8 acosh, 8.3.2 acot, 8.3.2 acsc, 8.3.2 add, 6.3.24 additionally, 3.3.8, 3.3.8 addtable, 20.4.3, 20.4.3 adjoint_matrix, 15.2.10 affix, 25.12.1 algsubs, 9.1.16 algvar, 9.4.4 altitude, 25.6.9 and, 3.3.8, 5.1.4 angle, 25.13.4, 26.10.9 angle_radian, 2.5.3 angleat, 25.13.1 angleatraw, 25.13.1 animate, 18.10.1 animate3d, 18.10.2 animation, 18.10.3 ans, 2.8.3 append, 6.3.10 apply, 14.2.6 approx, 22.1.3 approx_mode, 2.5.4 arc, 25.10.2 arccos, 8.3.2 arccosh, 8.3.2 archive, 3.3.6 arcLen, 13.3.8 arcsin, 8.3.2 arcsinh, 8.3.2 arctan, 8.3.2 arctanh, 8.3.2 area, 18.4.2, 25.13.6 areaat, 25.13.1 areaatraw, 25.13.1 areaplot, 18.4.3, 25.13.5 arg, 7.4.4, 25.13.11 args, 24.5.1 array, 10.1.10 as_function_of, 8.3.5 asc, 5.2.9 asec, 8.3.2 asin, 8.3.2 asin2acos, 10.2.9 asin2atan, 10.2.10 asinh, 8.3.2 assert, 24.2.4 assign, 3.3.2 assume, 3.3.8, 16.1.2, 20.4.1, 24.1.3, 25.4 at, 6.1.6, 6.1.7, 14.2.5 atan, 8.3.2 atan2acos, 10.2.12 atan2asin, 10.2.11 atanh, 8.3.2 atrig2ln, 10.2.22 augment, 6.3.12, 14.2.3 auto_correlation, 20.2.4 autosimplify, 10.1.4 axes, 18.1.2, 25.3.2 Beta, 7.3.16 Binary, 8.1.2 Bisection method, 22.3.2 BlockDiagonal, 14.2.2 Brent method, 22.3.2 bandwidth, 19.4.22 bar_plot, 19.1.14 bareiss, 15.1.4 barplot, 19.1.14 bartlett_hann_window, 20.5.1 barycenter, 7.4.8, 25.5.10, 26.3.6 base, 5.4.2, 10.1.10 basis, 15.1.8 batons, 19.2.2 begin, 24.1.7 bernoulli, 7.2.9 betad, 19.4.12 betad_cdf, 19.4.12 betad_icdf, 19.4.12 bezier, 18.8.1 binary, 5.4.1 binomial, 12.1.2, 19.4.3 binomial_cdf, 19.4.3 binomial_icdf, 19.4.3 bins, 19.4.22 bisection_solver, 22.3.2 bisector, 25.6.11 bit_depth, 27.2.6 bitand, 5.3.1 bitor, 5.3.1 bitxor, 5.3.1 black, 18.1.2 blackman_harris_window, 20.5.2 blackman_window, 20.5.3 block_size, 21.2.1 blockmatrix, 14.2.3 blue, 18.1.2 bohman_window, 20.5.4 boolean, 5.1.1 border, 14.2.3 box_constraints, 6.6.9 boxcar, 20.1.1 boxplot, 19.1.8 boxwhisker, 14.5.1, 19.1.8 break, 24.3.6 breakpoint, 24.6.1 brent_solver, 22.3.2 brown, 18.1.2 bvpsolve, 22.4.3 Celsius2Fahrenheit, 23.1.5 Ci, 7.3.7 Circle, 25.10.3 CLI interface, 1.2.2 ClrDraw, 24.5.7 ClrGraph, 24.5.7 ClrIO, 24.2.6 Colors, 18.1.3 CopyVar, 3.3.7 CST, 3.3.10 c1oc2, 12.2.9 c1op2, 12.2.9 camembert, 19.1.15 canonical_form, 9.1.7 cap_flat_line, 18.1.2 cap_round_line, 18.1.2 cap_square_line, 18.1.2 cas_setup, 2.5.10 case, 24.3.2 cat, 5.2.12, 5.2.14 catch, 24.4.1 cauchy, 19.4.14 cauchy_cdf, 19.4.14 cauchy_icdf, 19.4.14 cauchyd, 19.4.14 cauchyd_cdf, 19.4.14 cauchyd_icdf, 19.4.14 cd, 4.2.1 cdf, 19.4.20 ceil, 8.3.2 ceiling, 8.3.2 center, 25.5.12 center2interval, 6.5.3 centered_cube, 26.15.2 centered_tetrahedron, 26.15.1 cFactor, 2.5.5 cfactor, 2.5.5, 9.1.10 cfsolve, 22.3.2, 22.3.3, 22.3.3 changebase, 15.1.7 channel_data, 27.1.8, 27.2.8 channels, 27.2.6 char, 5.2.10 charpoly, 15.2.7 chinrem, 11.3.10 chisquare, 19.4.9 chisquare_cdf, 19.4.9 chisquare_icdf, 19.4.9 chisquaret, 19.5.4 cholesky, 15.3.1 chrem, 7.1.20 circle, 25.10.1, 26.9.1 circumcircle, 25.10.5 classes, 19.1.9 cluster, 21.1.1 coeff, 11.1.7 coeffs, 11.1.7 col, 14.2.5 colDim, 14.3.18 coldim, 14.3.18 collect, 11.1.18 colNorm, 15.4.5 colnorm, 15.4.5 color, 25.3.3 colormap, 18.1.3 colSwap, 14.3.11 colspace, 15.1.12 colswap, 14.3.11 comb, 12.1.2 combine, 10.4.2 comDenom, 11.7.7 comment, 2.8.2 comments, 2.8.2, 24.2.1 common_perpendicular, 26.4.8 companion, 15.2.11 compare, 24.2.5 complex, 24.2.5 complex_mode, 2.5.5 complex_variables, 2.5.6 complexroot, 11.4.6 concat, 5.2.7, 6.3.12, 14.2.3 cone, 26.13.1 confrac, 7.2.7 conic, 15.6.6 conj, 7.4.6 conjugate_equation, 13.8.5 conjugate_gradient, 15.6.4 cont, 24.6.1 contains, 6.4.2, 6.6.7 content, 11.1.16 contfrac, 10.1.10 continue, 24.3.7 contourplot, 18.5.1 convert, 5.4.2, 6.6.8, 7.2.7, 10.1.10, 11.1.6, 11.1.25, 23.1.4 convertir, 5.4.2, 10.1.10, 11.1.25 convex, 13.7.8 convexhull, 25.9.5 convolution, 20.2.5 coordinates, 25.12.4, 26.10.4 copy, 3.3.4, 24.2.9 correlation, 19.2.1 cos, 8.3.2, 10.1.10 cos2sintan, 10.2.16 cosh, 8.3.2 cosine_window, 20.5.5 cot, 8.3.2 cote, 26.10.3 count, 6.3.25, 14.3.12 count_eq, 6.3.21, 14.3.13 count_inf, 6.3.22, 14.3.14 count_sup, 6.3.23, 14.3.15 courbe_polaire, 18.8.2 covariance, 19.2.1 covariance_correlation, 19.2.1 cpartfrac, 11.7.9 crationalroot, 11.4.7 createwav, 27.2.1 cross, 14.1.8 cross_correlation, 20.2.3 cross_entropy, 21.2.1 cross_point, 18.1.2 cross_ratio, 25.16.2 crossP, 14.1.8 crossproduct, 14.1.8 cSolve, 9.3.6 csc, 8.3.2 cube, 26.14.1 cumSum, 5.2.7, 6.3.26, 14.3.5 cumulated_frequencies, 19.1.13 curl, 13.7.5 curvature, 17.2.1 curve, 22.4.1 cyan, 18.1.2 cycle2perm, 12.2.5 cycleinv, 12.2.12 cycles2permu, 12.2.4 cyclotomic, 11.3.11 cylinder, 26.13.3 Delete, 8.1.2 DelFold, 4.2.2 DelVar, 3.3.9 Det, 11.10.5 DIGITS, 2.5.1 Digits, 2.5.1, 7.3.1, 22.1.3 Dirac, 7.3.10 Discrete Hilbert transform, 20.4.7 Disp, 24.2.6 DispG, 2.7.3, 24.5.7 DispHome, 24.5.7 Distance function Bitwise Hamming, 5.3.2 Euclidean, 25.13.2, 26.10.7 Hamming, 5.2.17 Levensthein, 5.2.16 Squared Euclidean, 25.13.3, 26.10.8 Taxicab, 21.1.1 DOM_COMPLEX, 24.2.5 DOM_FLOAT, 24.2.5 DOM_FUNC, 24.2.5 DOM_IDENT, 24.2.5 DOM_INT, 24.2.5 DOM_LIST, 24.2.5 DOM_RAT, 24.2.5 DOM_STRING, 24.2.5 DOM_SYMBOLIC, 24.2.5 DrawParm, 18.7 DrawPol, 18.8.2 DrawSlp, 18.3.6 DrwCtour, 18.5.1 dash_line, 18.1.2 dashdot_line, 18.1.2 dashdotdot_line, 18.1.2 dayofweek, 7.1.28 debug, 24.6.1 debugger, 24.1.7, 24.6.1 default, 24.3.2 degree, 11.1.8 degrees, 25.13.4 del, 3.3.9 delcols, 14.2.6 delrows, 14.2.6 deltalist, 6.3.31 denom, 7.2.4, 11.7.4 densityplot, 18.5.2 derive, 13.2.1, 13.7.1 deriver, 13.2.1, 13.7.1 deSolve, 13.4.1 desolve, 13.4.1 det, 11.9.15, 15.1.4 det_minor, 15.1.4 dfc, 7.2.7 dfc2f, 7.2.8 diag, 14.2.2 diff, 13.2.1, 13.7.1 dim, 14.3.16 directories, 4.2 display, 18.1.2, 25.3.3, 27.1.4 distance, 25.13.2, 26.10.7 distance2, 25.13.3, 26.10.8 distanceat, 25.13.1 distanceatraw, 25.13.1 div, 7.1.8 divergence, 13.7.4 divide, 11.3.4 divis, 11.3.1 division_point, 25.16.1 divisors, 7.1.7 divpc, 13.5.1 dnewton_solver, 22.3.3 do, 24.3.3 dodecahedron, 26.15.4 domain, 8.4.1 dot, 14.1.7 dot_paper, 25.2.2 dotP, 14.1.7 dotprod, 14.1.7 double, 24.2.5 droit, 9.3.5 dsolve, 13.4.1 duration, 27.2.6 dwt, 20.4.10 Ei, 7.3.5 ERROR, 24.4.2 EXPR, 24.2.5 e, 3.1.2 e2r, 11.1.5 egcd, 11.3.8 egv, 15.2.3 egvl, 15.2.2 eigenvals, 15.2.1 eigenvalues, 15.2.2 eigenvectors, 15.2.3 eigenvects, 15.2.3 eigVc, 15.2.3 eigVl, 15.2.2 element, 25.5.15 elif, 24.3.1 eliminate, 9.1.17 ellipse, 25.11.1, 26.9.2 else, 24.3.1 emd, 20.4.8 end, 24.1.2, 24.1.7, 24.3.1 end_for, 24.3.3 endfunction, 24.1.2 envelope, 25.17.2 epsilon, 9.4.1 epsilon2zero, 9.4.1 equal, 9.3.1 equal2diff, 9.3.2 equal2list, 9.3.3 equation, 25.12.7, 26.10.5 equilateral_triangle, 25.7.4, 26.6.4 erase, 25.1.2 erf, 7.3.11 erfc, 7.3.12 error, 24.4.2 euler, 7.1.24 euler_gamma, 3.1.2 euler_lagrange, 13.8.2 eval, 9.1.1 eval_level, 9.1.2 evala, 9.1.3 evalb, 5.1.7 evalc, 7.4.3 evalf, 3.3.8, 7.3.1, 9.4.6, 22.1.3 evalm, 14.3.1 even, 7.1.11 evolute, 17.2.3 exact, 7.2.1, 9.4.7 exbisector, 25.6.12 excircle, 25.10.6 exp, 8.3.2, 10.1.10 exp2list, 5.1.5 exp2pow, 10.3.8 exp2trig, 10.2.13 expand, 9.1.6 expexpand, 10.3.2, 10.3.6 expln, 10.1.10 exponential, 19.4.15 exponential_cdf, 19.4.15 exponential_icdf, 19.4.15 exponential_regression, 19.2.5 exponential_regression_plot, 19.2.5 exponentiald, 19.4.15 exponentiald_cdf, 19.4.15 exponentiald_icdf, 19.4.15 export_mathml, 2.7.7 expr, 5.2.15, 24.5.3 expression, 24.2.5 expression editor, 2.9 expression tree, 2.9.2 extract_measure, 25.13.1 extrema, 16.2.2 ezgcd, 11.3.6 FALSE, 5.1.1 Factor, 11.10.4 Fahrenheit2Celsius, 23.1.5 False positions method, 22.3.2 Firefox interface, 1.2.3 Formatting text, 18.1.2 Fourier, 20.4.2 FUNC, 24.2.5 f2nd, 7.2.5, 11.7.5 faces, 26.14.7 factor, 9.1.10, 11.9.14, 11.9.19 factor_xn, 11.1.15 factorial, 12.1.1 factoriser, 11.9.14 factors, 11.1.20 false, 5.1.1 falsepos_solver, 22.3.2 fclose, 4.1.2 fcoeff, 11.8.2 fdistrib, 9.1.6 feuille, 8.2.3 fft, 20.4.4 fi, 24.3.1 fieldplot, 18.9.1 filled, 18.1.2 find_minimum, 16.3.1 findhelp, 2.3.4 fisher, 19.4.10 fisher_cdf, 19.4.10 fisher_icdf, 19.4.10 fisherd, 19.4.10 fitdistr, 19.4.23 fitpoly, 11.2.8 flatten, 6.3.13, 27.1.8 float, 16.1.2 float2rational, 7.2.1, 9.4.7 floor, 8.3.2 fMax, 16.2.1, 16.3.3 fMin, 16.2.1, 16.3.3 foldl, 6.3.30 foldr, 6.3.30 fopen, 4.1.2 for, 24.3.3, 24.3.3, 24.3.3 format, 24.5.6 fourier, 20.4.2 fourier_an, 20.4.1 fourier_bn, 20.4.1 fourier_cn, 20.4.1 fPart, 8.3.2 fprint, 4.1.2 frac, 8.3.2 fracmod, 11.9.12 frame_2d, 25.2.1 frame_3d, 26.2.1 frames, 18.10.1, 18.10.2, 25.3.2 frank_wolfe, 16.3.2 frequences, 19.1.12 frequences_cumulees, 19.1.13 frequencies, 19.1.12 frobenius_norm, 15.4.1 from, 24.3.3 froot, 11.8.1 fsolve, 22.3.2, 22.3.3 fullparfrac, 10.1.10 func, 21.2.1, 24.2.5 funcplot, 18.2.1, 26.13.5 function, 24.1.2 function_diff, 13.2.2 fxnd, 7.2.5, 11.7.5 Gamma, 7.3.13 Gcd, 7.1.1, 11.3.5, 11.10.3 GetFold, 4.2.2 GF, 11.9.18 gammad, 19.4.11 gammad_cdf, 19.4.11 gammad_icdf, 19.4.11 gauche, 9.3.4 gauss, 15.6.3 gauss15, 18.4.2 gauss_seidel_linsolve, 15.7.8 gaussian_window, 20.5.6 gaussquad, 22.2.2 gbasis, 11.6.1 gcd, 7.1.1, 11.3.5, 11.9.13 gcdex, 11.3.8 genpoly, 11.6.4 geometric, 19.4.13 geometric_cdf, 19.4.13 geometric_icdf, 19.4.13 getDenom, 7.2.4, 11.7.3 getKey, 24.2.3 getNum, 7.2.3, 11.7.1 getType, 24.2.5 giac, 1.2.2, 1.2.5 gl_material, 18.1.2, 18.1.2 gl_ortho, 25.3.2 gl_quaternion, 18.1.2, 25.3.2 gl_rotation, 18.1.2, 25.3.2 gl_shownames, 18.1.2, 25.3.2 gl_texture, 18.1.2, 18.1.2, 25.3.2, 25.3.2, 27.1.4 gl_x, 18.1.2, 25.3.2 gl_x_axis_color, 25.3.2 gl_x_axis_name, 18.1.2, 25.3.2 gl_x_axis_unit, 18.1.2, 25.3.2 gl_x_tick, 18.1.2 gl_xtick, 25.3.2 gl_y, 18.1.2, 25.3.2 gl_y_axis_color, 25.3.2 gl_y_axis_name, 18.1.2, 25.3.2 gl_y_axis_unit, 18.1.2, 25.3.2 gl_y_tick, 18.1.2 gl_ytick, 25.3.2 gl_z, 18.1.2, 25.3.2 gl_z_axis_color, 25.3.2 gl_z_axis_name, 18.1.2, 25.3.2 gl_z_axis_unit, 18.1.2, 25.3.2 gl_z_tick, 18.1.2 gl_ztick, 25.3.2 gold, 18.1.2 goto, 24.3.8 grad, 13.7.1 gramschmidt, 15.6.5 graph2tex, 2.7.3 graph3d2tex, 2.7.3 graphe_suite, 18.8.4 greduce, 11.6.2 green, 18.1.2 grey, 18.1.2 grid_paper, 25.2.4 groupermu, 12.2.14 Heaviside, 7.3.9 Heaviside2sign, 10.1.7 hadamard, 14.3.8 half_cone, 26.13.2 half_line, 25.6.2, 26.4.2 halftan, 10.2.19 halftan_hyp2exp, 10.2.20 halt, 24.6.1 hamdist, 5.2.17, 5.3.2 hamming_window, 20.5.7 hann_poisson_window, 20.5.8 hann_window, 20.5.9 harmonic_conjugate, 25.16.4 harmonic_division, 25.16.3 has, 9.4.5 hasard, 19.3.2 head, 5.2.4, 6.1.4 hermite, 11.5.2 hessenberg, 15.2.12 hessian, 13.7.3 heugcd, 11.3.6 hexadecimal, 5.4.1 hexagon, 25.9.1, 26.8.1 hht, 20.4.9 hidden_name, 18.1.2 highpass, 20.3.2 hilbert, 14.2.2, 20.4.6, 20.4.7 histogram, 19.1.10 histogramme, 19.1.10 hold, 9.1.4 homothety, 25.14.5, 26.12.5 horner, 11.1.13 hsv, 18.1.3 hsv2rgb, 18.1.3 hybrid_solver, 22.3.3 hybridj_solver, 22.3.3 hybrids_solver, 22.3.3 hybridsj_solver, 22.3.3 hyp2exp, 10.3.1 hyperbola, 25.11.2, 26.9.3 IFTE, 5.1.3 Input, 24.2.2 InputStr, 24.2.2 Int, 13.3.1 Inverse, 11.10.6 i, 3.1.2 i[], 6.6.1 iabcuv, 7.1.21 ibasis, 15.1.9 ibpdv, 13.3.5 ibpu, 13.3.5 icas, 1.2.2 icdf, 19.4.21 ichinrem, 7.1.20 ichrem, 7.1.20 icomp, 7.1.27 icosahedron, 26.15.5 id, 8.3.2 identifier, 24.2.5 identity, 14.2.2 idivis, 7.1.7 idn, 14.2.2 idwt, 20.4.10 iegcd, 7.1.19 if, 24.3.1 ifactor, 7.1.4 ifactors, 7.1.5 ifft, 20.4.4 ifourier, 20.4.2 ifte, 5.1.3 igamma, 7.3.15 igcdex, 7.1.19 ihermite, 15.2.13 ilaplace, 13.4.2 im, 7.4.2 imag, 7.4.2 image, 15.1.10, 27.1.2 imfplot, 20.4.8 implicitdiff, 13.2.3 implicitplot, 18.6 in, 24.3.3 in_ideal, 11.6.3 incircle, 25.10.4 indets, 9.4.2 inequationplot, 18.4.1 inertia, 15.3.10 inf, 3.1.2 infinity, 3.1.2 input, 24.2.2 inString, 5.2.11 insert, 6.3.9 instfreq, 20.4.7 instphase, 20.4.7 int, 13.3.1 intDiv, 7.1.8 integer, 16.1.2, 16.3.4, 24.2.5 integer, 3.3.8 integrate, 13.3.1 inter, 25.5.7, 26.3.4 interactive_odeplot, 18.9.2 interactive_plotode, 18.9.2 internal directories, 4.2.2 interp, 11.2.4, 18.1.3, 27.1.10 intersect, 6.4.4, 6.6.6 interval, 10.1.10 interval2center, 6.5.2 inv, 11.9.11, 11.9.16, 15.1.2, 27.1.7, 27.2.2 inverse, 11.9.11, 11.9.16 inversion, 25.14.7, 26.12.7 invisible_point, 18.1.2 invlaplace, 13.4.2 invztrans, 13.6.2 iPart, 8.3.2 iquo, 7.1.8 iquorem, 7.1.10 iratrecon, 11.9.12 irem, 7.1.9 is_collinear, 25.15.2, 26.11.6 is_concyclic, 25.15.3, 26.11.7 is_conjugate, 25.15.14 is_coplanar, 26.11.2 is_cospherical, 26.11.8 is_cycle, 12.2.8 is_element, 25.15.1, 26.11.1 is_equilateral, 25.15.5, 26.11.9 is_harmonic, 25.15.15 is_harmonic_circle_bundle, 25.15.17 is_harmonic_line_bundle, 25.15.16 is_inside, 25.15.4 is_isosceles, 25.15.6, 26.11.10 is_orthogonal, 25.15.13, 26.11.5 is_parallel, 25.15.11, 26.11.3 is_parallelogram, 25.15.10, 26.11.14 is_permu, 12.2.7 is_perpendicular, 25.15.12, 26.11.4 is_prime, 7.1.14 is_pseudoprime, 7.1.13 is_rectangle, 25.15.7, 26.11.11 is_rhombus, 25.15.9, 26.11.13 is_square, 25.15.8, 26.11.12 isinf, 3.1.3 ismith, 15.2.14 isnan, 3.1.3 isobarycenter, 25.5.11, 26.3.7 isolve, 7.1.23 isom, 15.5.1 isopolygon, 25.9.2, 26.8.2 isosceles_triangle, 25.7.2, 26.6.2 isPrime, 7.1.14 isposdef, 15.1.14 isprime, 7.1.14 istft, 20.4.5 ithprime, 7.1.17 JordanBlock, 14.2.2 jacobi_equation, 13.8.4 jacobi_linsolve, 15.7.7 jacobi_symbol, 7.1.26 join, 5.2.7 jordan, 15.2.5 jusqua, 24.3.4 kde, 19.4.22 keep_pivot, 15.7.3 ker, 15.1.11 kernel, 15.1.11 kernel_density, 19.4.22 kill, 24.6.1 kmeans, 21.1.2 kolmogorovd, 19.4.17 kolmogorovt, 19.5.5 kovacicsols, 13.4.3 LaTeX, 2.7.1, 2.7.2 LateX, 2.7.3 LIST, 24.2.5 | Li, 7.3.6 Line, 25.6.3 LineHorz, 18.3.2 LineTan, 18.3.4 LineVert, 18.3.3 LQ, 15.3.4 LSQ, 15.7.9 LU, 15.3.6 l1norm, 14.1.1, 15.4.5, 15.4.6 l2norm, 14.1.1, 15.4.2, 15.4.6 label, 24.3.8 labels, 18.1.2 lagrange, 11.2.1, 15.1.4 laguerre, 11.5.3 laplace, 13.4.2, 20.4.3 laplacian, 13.7.2 latex, 2.7.1 lcm, 7.1.3, 11.3.7 lcoeff, 11.1.10 ldegree, 11.1.9 ldl, 15.3.9 learning_rate, 21.2.1 left, 5.2.4, 6.3.6, 6.6.2, 8.2.3, 9.3.4 left_rectangle, 18.4.2 legend, 18.1.2 legend, 25.3.3 legendre, 11.5.1 legendre_symbol, 7.1.25 length, 5.2.3, 6.1.3 levenshtein, 5.2.16 lgcd, 7.1.2 lhs, 9.3.4 ligne_polygonale, 19.2.3 limit, 13.1.1 lin, 10.3.4 linabs, 10.1.9 line, 18.3.1, 25.6.1, 26.4.1 line_inter, 25.5.6, 26.3.3 line_paper, 25.2.3 line_segments, 26.14.8 line_width_1, 18.1.2 line_width_2, 18.1.2 line_width_3, 18.1.2 line_width_4, 18.1.2 line_width_5, 18.1.2 line_width_6, 18.1.2 line_width_7, 18.1.2 linear_interpolate, 19.2.3 linear_regression, 19.2.4 linear_regression_plot, 19.2.4 linfnorm, 15.4.4, 15.4.6 linsolve, 15.7.6 linstep, 10.1.8 list, 10.1.10 list2exp, 5.1.6 list2mat, 6.3.32 listplot, 19.2.3 lists, 6.3 lll, 15.3.11 ln, 8.3.2, 10.1.10 lname, 9.4.2 lncollect, 10.3.5 lnexpand, 10.3.3 local, 24.1.2 locus, 25.17.1 log, 8.3.2 log10, 8.3.2 log2, 8.3.2 log_loss, 21.2.1 logarithmic_regression, 19.2.6 logarithmic_regression_plot, 19.2.6 logb, 8.3.2 logistic_regression, 19.2.9 logistic_regression_plot, 19.2.9 loi_normal, 19.4.7 lowpass, 20.3.1 lp_assume, 16.1.2 lp_bestlocalbound, 16.1.2 lp_bestprojection, 16.1.2 lp_binary, 16.1.2 lp_binaryvariables, 16.1.2 lp_breadthfirst, 16.1.2 lp_depthfirst, 16.1.2 lp_depthlimit, 16.1.2 lp_firstfractional, 16.1.2 lp_gaptolerance, 16.1.2 lp_heuristic, 16.1.2 lp_hybrid, 16.1.2 lp_integer, 16.1.2 lp_integervariables, 16.1.2 lp_interiorpoint, 16.1.2 lp_iterationlimit, 16.1.2 lp_lastfractional, 16.1.2 lp_maxcuts, 16.1.2 lp_maximize, 16.1.2 lp_method, 16.1.2 lp_mostfractional, 16.1.2 lp_nodelimit, 16.1.2 lp_nodeselect, 16.1.2 lp_nonnegative, 16.1.2 lp_nonnegint, 16.1.2 lp_presolve, 16.1.2 lp_pseudocost, 16.1.2 lp_simplex, 16.1.2 lp_timelimit, 16.1.2 lp_varselect, 16.1.2 lp_verbose, 16.1.2 lpsolve, 16.1.2 lsq, 15.7.9 lu, 15.3.5 lvar, 9.4.3 MAT, 24.2.5 MathML, 2.7.4, 2.7.5, 2.7.7 MSE, 21.2.1 magenta, 18.1.2 make_symbol, 24.5.4 makelist, 6.3.2, 24.5.2 makemat, 14.2.4 makesuite, 6.1.10 map, 6.3.28, 6.3.28, 11.1.3 maple2xcas, 2.7.9 maple_ifactors, 7.1.6 markov, 19.4.24 mat2list, 6.3.33 mathml, 2.7.4 matpow, 15.2.6 matrix, 10.1.10, 14.2.4 matrix_norm, 15.4.6 max, 8.3.2 maximize, 16.1.2, 16.2.1 maxnorm, 14.1.1, 15.4.3 mean, 14.5.1, 14.5.2, 19.1.1 mean, 14.5.1, 19.1.1 median, 14.5.1, 19.1.5 median, 14.5.1, 19.1.5, 19.1.6 median_line, 25.6.8 member, 6.4.2 mgf, 19.4.19 mid, 5.2.4, 6.1.7 middle_point, 18.4.2, 18.4.3 midpoint, 6.6.4, 25.5.9, 26.3.5 min, 8.3.2, 19.1.6 minimax, 11.2.9 minimize, 16.2.1 minor_det, 15.1.4 minus, 6.4.5 mixdown, 27.2.18 mkisom, 15.5.2 mksa, 23.1.3 mod, 7.1.9, 8.1.1 modgcd, 11.3.6 mods, 7.1.9 momentum, 21.2.1 moving_average, 20.3.3 mRow, 14.3.11 mRowAdd, 14.3.11 mul, 6.3.27, 6.3.27 mult_c_conjugate, 7.4.7 mult_conjugate, 9.1.8 multinomial, 19.4.5 mustache, 19.1.8 NewFold, 4.2.2 Newton method, 22.3.2 NONE, 24.2.5 NUM, 24.2.5 Nullspace, 15.1.11 navy, 18.1.2 nCr, 12.1.2 ncols, 14.3.18 nDeriv, 22.2.1 negbinomial, 19.4.4 negbinomial_cdf, 19.4.4 negbinomial_icdf, 19.4.4 neural_network, 21.2.1 newList, 6.3.3 newMat, 14.2.2 newton, 22.3.1 newton_solver, 22.3.2 newtonj_solver, 22.3.3 nextperm, 12.2.2 nextprime, 7.1.15 nInt, 22.2.2 nlp_binary, 16.3.4 nlp_binaryvariables, 16.3.4 nlp_initialpoint, 16.3.4 nlp_integer, 16.3.4 nlp_integervariables, 16.3.4 nlp_iterationlimit, 16.3.4 nlp_maximize, 16.3.4 nlp_method, 16.3.4 nlp_nonnegative, 16.3.4 nlp_nonnegint, 16.3.4 nlp_precision, 16.3.4 nlp_presolve, 16.3.4 nlp_tolerance, 16.3.4 nlp_verbose, 16.3.4 nlpsolve, 16.3.4 nodisp, 2.8.1, 25.3.4 nonnegint, 16.1.2, 16.3.4 nop, 6.1.10 nops, 6.1.3 norm, 14.1.1, 15.4.2, 15.4.6 normal, 10.1.2, 11.9.1, 11.9.2, 11.9.3, 11.9.4 normal_cdf, 19.4.7 normal_icdf, 19.4.7 normald, 19.4.7 normald_cdf, 19.4.7 normald_icdf, 19.4.7 normalize, 7.4.5, 14.1.2, 25.13.12, 27.2.15 normalt, 19.5.2 not, 5.1.4 nPr, 12.1.3 nprimes, 7.1.18 nrows, 14.3.17 nSolve, 22.3.2 nstep, 18.2.1, 18.2.2, 18.10.1, 18.10.2, 25.3.2 nuage_points, 19.2.2 nullspace, 15.1.11 numdiff, 13.2.4 numer, 7.2.3, 11.7.2 Output, 24.2.2 Ox_2d_unit_vector, 25.2.1 Ox_3d_unit_vector, 26.2.1 Oy_2d_unit_vector, 25.2.1 Oy_3d_unit_vector, 26.2.1 Oz_3d_unit_vector, 26.2.1 octahedron, 26.15.3 octal, 5.4.1 od, 24.3.3 odd, 7.1.12 odeplot, 18.8.3 odesolve, 22.4.1, 22.4.2 olive, 18.1.2 op, 6.1.10, 8.2.3 open_polygon, 25.9.4, 26.8.4 or, 3.3.8, 5.1.4 orange, 18.1.2 ord, 5.2.8 order_size, 13.5.2 ordinate, 25.12.3, 26.10.2 orthocenter, 25.5.8 orthogonal, 26.4.7 osculating_circle, 17.2.2 output, 24.2.2 Pause, 24.5.8 PIC, 24.2.5 Piecewise defined functions, 5.1.3 Postfix, 8.1.2 Prefix, 8.1.2 Psi, 7.3.17 p1oc2, 12.2.9 p1op2, 12.2.9 pa2b2, 7.1.22 pade, 13.5.5 parabola, 25.11.3, 26.9.4 parallel, 25.6.5, 26.4.5 parallelepiped, 26.14.3 parallelogram, 25.8.4, 26.7.4 parameq, 25.12.8, 26.10.6 paramplot, 18.7, 26.13.6 parfrac, 10.1.10 pari, 7.2.10 part, 9.1.19 partfrac, 10.1.10, 11.7.9 parzen_window, 20.5.10 pcar, 15.2.7 pcar_hessenberg, 15.2.8 pcoef, 11.1.23 pcoeff, 11.1.23 perimeter, 25.13.7 perimeterat, 25.13.1 perimeteratraw, 25.13.1 period, 9.2.2 periodic, 9.2.1 perm, 12.1.3 perminv, 12.2.11 permu2cycles, 12.2.3 permu2mat, 12.2.6 permuorder, 12.2.13 perpen_bisector, 25.6.10, 26.5.2 perpendicular, 25.6.6, 26.4.6 peval, 11.1.12 pi, 3.1.2 piecewise, 5.1.3 pink, 18.1.2 pivot, 15.7.5 plane, 26.5.1 playsnd, 27.2.14 plex, 11.6.1, 11.6.3 plot, 18.2.3 plot3d, 18.2.4 plotarea, 18.4.3, 25.13.5 plotcontour, 18.5.1 plotdensity, 18.5.2 plotfield, 18.9.1 plotfunc, 18.2.1, 18.2.2, 26.13.5 plotimf, 20.4.8 plotimplicit, 18.6 plotinequation, 18.4.1 plotlist, 19.2.3 plotode, 18.8.3 plotparam, 18.7, 18.7.2 plotpolar, 18.8.2 plotseq, 18.8.4 plotspectrum, 27.2.13 plotwav, 27.2.12 plus_point, 18.1.2 pmin, 7.5.2, 15.2.9 point, 25.5.2, 26.3.1 point2d, 25.5.4 point3d, 26.3.2 point_milieu, 18.4.3 point_point, 18.1.2 point_polar, 25.5.5 point_width_1, 18.1.2 point_width_2, 18.1.2 point_width_3, 18.1.2 point_width_4, 18.1.2 point_width_5, 18.1.2 point_width_6, 18.1.2 point_width_7, 18.1.2 poisson, 19.4.6 poisson_cdf, 19.4.6 poisson_icdf, 19.4.6 poisson_window, 20.5.11 polar, 25.16.5 polar_coordinates, 25.12.6 polar_point, 25.5.5 polarplot, 18.8.2 pole, 25.16.5 poly1, 11.1.1 poly2symb, 11.1.4 polyEval, 11.1.12 polygon, 25.9.3, 26.8.3 polygonplot, 19.2.3 polyhedron, 26.14.5 polynom, 10.1.10, 11.1.25 polynomial_regression, 19.2.8 polynomial_regression_plot, 19.2.8 poslbdLMQ, 11.4.4 posubLMQ, 11.4.3 potential, 13.7.6 pow2exp, 10.3.7 power_regression, 19.2.7 power_regression_plot, 19.2.7 powermod, 11.9.10 powerpc, 25.10.7 powexpand, 10.3.6 powmod, 11.9.10 prepend, 6.3.11 preval, 9.1.18 prevperm, 12.2.2 prevprime, 7.1.16 primpart, 11.1.17 print, 24.2.6 printpow, 24.2.7 prism, 26.14.4 proc, 24.1.2 product, 6.3.27, 6.3.27, 6.3.27, 14.3.6, 14.3.8 program, 24.1.5 projection, 25.14.8, 26.12.8 proot, 22.3.4 propFrac, 7.2.2 propfrac, 7.2.2, 11.7.8 psrgcd, 11.3.6 ptayl, 11.1.14 purge, 3.3.8, 3.3.9, 20.4.1, 24.1.3 purple, 18.1.2 pwd, 4.2.1 pyramid, 26.14.2 QR, 15.3.3 Quo, 11.3.2, 11.10.1 q2a, 15.6.1 qr, 15.3.2 quadric, 15.6.8 quadrilateral, 25.8.5, 26.7.5 quandrant1, 18.1.2 quandrant2, 18.1.2 quandrant3, 18.1.2 quandrant4, 18.1.2 quantile, 19.1.7 quantile, 14.5.1, 19.1.7 quantiles, 14.5.1, 14.5.2 quartile1, 19.1.6 quartiles, 14.5.1, 14.5.2, 19.1.6 quartiles, 14.5.1, 19.1.6 quest, 2.8.3 quo, 11.3.2, 11.9.5 quorem, 11.3.4, 11.9.7 quote, 5.2.1, 9.1.4 RandSeed, 19.3.1 REDIM, 14.2.6 REPLACE, 14.2.6 ReLU, 21.2.1 Rem, 11.3.3, 11.10.2 Rref, 11.10.7 r2e, 11.1.4 radians, 25.13.4 radical_axis, 25.10.8 radius, 25.13.9 rand, 19.3.2 randbetad, 19.3.3 randbinomial, 19.3.3 randchisquare, 19.3.3 randexp, 19.3.3 randfisher, 19.3.3 randgammad, 19.3.3 randgeometric, 19.3.3 randint, 19.3.2 randMat, 19.3.6 randmarkov, 19.4.25 randmatrix, 19.3.6 randmultinomial, 19.3.3 randNorm, 19.3.3 randnorm, 19.3.3 random, 19.3.2 random_variable, 19.3.4 randPoly, 11.1.27 randperm, 12.2.1 randpoisson, 19.3.3 randpoly, 11.1.27 randseed, 19.3.1 randstudent, 19.3.3 randvar, 19.3.4 randvector, 19.3.5 range, 6.3.4 rank, 15.1.5 ranm, 11.1.28, 19.3.6 rat_jordan, 15.2.4 ratinterp, 11.2.6 rational, 24.2.5 rational_det, 15.1.4 rationalroot, 11.4.5 ratnormal, 10.1.5 rdiv, 7.3.3 re, 7.4.2 read, 4.1.3 readrgb, 27.1.3 readwav, 27.2.3 real, 7.4.2, 24.2.5 reciprocation, 25.16.6 rect, 20.1.2 rectangle, 25.8.3, 26.7.3 rectangle_droit, 18.4.3 rectangle_gauche, 18.4.3 rectangle_left, 18.4.3 rectangle_right, 18.4.3 rectangular_coordinates, 25.12.5 red, 18.1.2 redim, 14.2.6 reduced_conic, 15.6.7 reduced_quadric, 15.6.9 ref, 15.7.2 reflection, 25.14.3, 26.12.3 regroup, 10.1.1 rem, 11.3.3, 11.9.6 remain, 7.1.9 remove, 6.3.8 reorder, 11.1.26 repeat, 24.3.4 repeter, 24.3.4 replace, 14.2.6 resample, 27.2.11 residue, 13.5.4 resoudre, 9.1.10, 18.3.7 restart, 3.3.9 resultant, 11.3.13 return, 24.1.2 reverse, 27.2.2 reverse_rsolve, 15.7.10 revert, 13.5.3 revlex, 11.6.1 revlist, 6.3.14 rgb, 18.1.3 rgb2hsv, 18.1.3 rgb2xyz, 18.1.3 rhombus, 25.8.2, 26.7.2 rhombus_point, 18.1.2 rhs, 9.3.5 riemann_window, 20.5.12 right, 5.2.4, 6.3.6, 6.6.2, 8.2.3, 9.3.5 right_rectangle, 18.4.2 right_triangle, 25.7.3, 26.6.3 risch, 13.3.2 rm_a_z, 3.3.9 rm_all_vars, 3.3.9 rmbreakpoint, 24.6.1 rms, 20.2.1, 27.2.16 rmwatch, 24.6.1 romberg, 22.2.2 rombergm, 18.4.2 rombergt, 18.4.2 root, 7.3.4 rootof, 11.1.21 roots, 11.1.22 rotate, 6.3.15 rotation, 25.14.4, 26.12.4 round, 8.3.2 row, 14.2.5 rowAdd, 14.3.11 rowDim, 14.3.17 rowdim, 14.3.17 rowNorm, 15.4.4 rownorm, 15.4.4, 15.4.6 rowSwap, 14.3.11 rowspace, 15.1.13 rowswap, 14.3.11 rref, 11.9.17, 15.7.3 rsolve, 6.2.2 SCALE, 14.3.11 SCALEADD, 14.3.11 SCHUR, 15.2.12 Secant method, 22.3.2 SetFold, 4.2.2 Si, 7.3.8 SortA, 6.3.18 SortD, 6.3.19 STR, 24.2.5 Steffenson method, 22.3.2 Store, 3.3.2 SVL, 15.3.7 sample, 19.3.2 samplerate, 27.2.6 scalar_product, 14.1.7 scalarProduct, 14.1.7 scale, 14.3.11 scaleadd, 14.3.11 scatterplot, 19.2.2 sec, 8.3.2 secant_solver, 22.3.2 segment, 25.6.3, 25.6.4, 26.4.3 select, 6.3.5 semi_augment, 14.2.3 seq, 6.1.2 seq[], 3.2.1, 6.1.1 seqplot, 18.8.4 seqsolve, 6.2.2 series, 13.5.2 set[], 3.2.2, 6.4.1 set_channel_data, 27.1.7, 27.2.9 shift, 6.3.16 shift_phase, 10.2.3 shuffle, 12.2.1 sign, 8.3.2 sign2Heaviside, 10.1.7 signature, 12.2.10 similarity, 25.14.6, 26.12.6 simp2, 7.2.6, 11.7.6 simplex algorithm, 16.1.1 simplex_reduce, 16.1.1 simplify, 10.1.3, 10.2.5 simplifyDirac, 10.1.6 simpson, 18.4.2 simulated_annealing, 16.3.5 simult, 15.7.4 sin, 8.3.2, 10.1.10 sin2costan, 10.2.15 sinc, 20.1.4 sincos, 10.1.10 sincos, 10.2.13 single_inter, 25.5.6, 26.3.3 sinh, 8.3.2 size, 5.2.3, 6.1.3, 27.1.8 sizes, 6.3.1 slope, 25.13.8 slopeat, 25.13.1 slopeatraw, 25.13.1 smith, 15.2.15 smod, 7.1.9 snedecor, 19.4.10 snedecor_cdf, 19.4.10 snedecor_icdf, 19.4.10 solid_line, 18.1.2 solve, 9.1.10, 9.3.6, 18.3.7 sommet, 8.2.3 sort, 6.3.17 sortperm, 6.3.20 soundsec, 27.2.7 sphere, 26.13.4 splice, 27.2.17 spline, 11.2.3 split, 5.2.6, 9.1.9 spreadsheet, 2.10 sq, 8.3.2 sqrfree, 11.1.19 sqrt, 8.3.2 square, 25.8.1, 26.7.1 square_point, 18.1.2 srand, 19.3.1 sst, 24.6.1 sst_in, 24.6.1 standard deviation, 14.5.1, 14.5.2, 19.1.3 of a sample, 14.5.1, 14.5.2, 19.1.3 of the population, 14.5.1, 14.5.2, 19.1.4 star_point, 18.1.2 stdDev, 19.1.4 stddev, 14.5.1 stddevp, 14.5.1, 19.1.4 stdev, 19.1.3 steffenson_solver, 22.3.2 step, 24.3.3 stereo2mono, 27.2.5 stft, 20.4.5 sto, 3.3.2 string, 10.1.10, 24.2.5, 24.5.5 strip, 5.2.5 student, 19.4.8 student_cdf, 19.4.8 student_icdf, 19.4.8 studentd, 19.4.8 studentt, 19.5.3 sturm, 11.3.12 sturmab, 11.3.12 sturmseq, 11.3.12 subexpression, 2.9.2 subexpressions, 2.9.2, 2.9.3 subMat, 14.2.5 subs, 9.1.15 subsop, 6.3.7, 14.2.6 subst, 9.1.13 subtype, 24.2.5 sum, 5.2.7, 6.3.24, 13.3.3, 14.3.4 sum_riemann, 13.3.4 supposons, 3.3.8 suppress, 6.3.8 surd, 8.3.2 svd, 15.3.8 svl, 15.3.7 swapcol, 14.3.11 swaprow, 14.3.11 switch, 24.3.2 switch_axes, 25.1.3 sylvester, 11.3.13 symb2poly, 11.1.5 symbol, 24.2.5 symbol, 24.1.3 symbol_array, 24.5.4 syst2mat, 15.7.1 TeXmacs interface, 1.2.4 TRUE, 5.1.1 table, 14.4.1 tablefunc, 6.2.1 tableseq, 6.2.3 tabvar, 8.4.2 tail, 5.2.4, 6.1.5 tan, 8.3.2, 10.1.10 tan2cossin2, 10.2.18 tan2sincos, 10.2.14 tan2sincos2, 10.2.17 tangent, 18.3.5, 25.6.7, 26.5.3 tanh, 8.3.2 taylor, 13.5.2 tCollect, 10.2.4 tchebyshev1, 11.5.4 tchebyshev2, 11.5.5 tcoeff, 11.1.11 tcollect, 10.2.4 tdeg, 11.6.1, 11.6.3 tExpand, 10.4.1 teal, 18.1.2 tetrahedron, 26.14.2 texpand, 10.4.1 textinput, 24.2.2 then, 24.3.1 thickness, 25.3.2 thiele, 11.2.5 threshold, 20.3.4 throw, 24.4.2 title, 18.1.2, 21.2.1, 25.3.2 tlin, 10.2.2 to, 24.3.3 topology, 21.2.1 tpsolve, 16.1.3 trace, 15.1.3, 25.17.3 train, 21.2.2 tran, 15.1.1 translation, 25.14.2, 26.12.2 transpose, 15.1.1 trapeze, 18.4.3 trapezoid, 18.4.2, 18.4.3 tri, 20.1.3 triangle, 25.7.1, 26.6.1 triangle_paper, 25.2.5 triangle_point, 18.1.2 triangle_window, 20.5.13 trig2exp, 10.2.21 trigcos, 10.2.24 trigexpand, 10.2.1 triginterp, 11.2.7 trigsimplify, 10.2.6 trigsin, 10.2.23 trigtan, 10.2.25 trim, 5.2.5, 20.2.2, 27.1.9, 27.2.10 trn, 15.1.6 true, 5.1.1 trunc, 8.3.2 truncate, 11.1.24 try, 24.4.1 tsimplify, 10.3.9 tstep, 18.7.1, 18.8.2, 22.4.1, 25.3.2, 25.17.1, 25.17.1 tukey_window, 20.5.14 tuple, 6.4.6 type, 24.2.5 Unary, 8.1.2 Unquoted, 4.1.2 UTPC, 19.4.9 UTPF, 19.4.10 UTPN, 19.4.7 UTPT, 19.4.8 ufactor, 23.1.6 ugamma, 7.3.14 unapply, 8.2.2 unarchive, 3.3.6 unfactored, 18.6.1 uniform, 19.4.2 uniform_cdf, 19.4.2 uniform_icdf, 19.4.2 uniformd, 19.4.2 uniformd_cdf, 19.4.2 uniformd_icdf, 19.4.2 union, 6.4.3, 6.6.5 unitV, 7.4.5, 14.1.2 unquote, 9.1.5 until, 24.3.4 user_operator, 8.1.2 usimplify, 23.1.7 ustep, 25.3.2 VAR, 24.2.5 VARS, 3.3.9, 4.2.2 VAS, 11.4.1 VAS_positive, 11.4.2 valuation, 11.1.9 vandermonde, 14.2.2 variable, 3.3.1 variance, 14.5.1, 14.5.2, 19.1.2 variance, 14.5.1, 19.1.2 vector, 24.2.5 vector, 25.6.4, 26.4.4 vectors, 6.3 version, 1.2.5 vertices, 25.5.13, 26.14.6 vertices_abc, 25.5.13 vertices_abca, 25.5.14 violet, 18.1.2 vpotential, 13.7.7 vstep, 25.3.2 WAIT, 24.5.8 watch, 24.6.1 weibull, 19.4.16 weibull_cdf, 19.4.16 weibull_icdf, 19.4.16 weibulld, 19.4.16 weibulld_cdf, 19.4.16 weibulld_icdf, 19.4.16 weight_decay, 21.2.1 weights, 21.2.1 welch_window, 20.5.15 when, 5.1.3 while, 24.3.5 whisker box, 14.5.1, 14.5.2, 19.1.8 white, 18.1.2 widget_size, 2.5.10 wilcoxonp, 19.4.18 wilcoxons, 19.4.18 wilcoxont, 19.4.18 write, 4.1.1 writergb, 27.1.5 writewav, 27.2.4 wz_certificate, 12.1.4 Xcas interface, 1.2.1, 2.1 xcas.rc, 2.5.10 xcas_mode, 2.5.2, 2.5.10 xml_print, 2.7.6 xor, 5.1.4 xstep, 18.2.1, 18.10.1, 18.10.2, 25.3.2 xyz2rgb, 18.1.3 xyztrange, 2.5.10 yellow, 18.1.2 ystep, 18.2.2, 18.10.2, 25.3.2 Zeta, 7.3.18 zeros, 9.1.11 zip, 6.3.29 zstep, 18.6.2, 25.3.2 ztrans, 13.6.1

1
  Université de Grenoble, initial translation of parts of the French user manual
2
  Full translation and improvements
3
  Optimization, signal processing. The graph theory is in a separate manual.

> This document was translated from LATEX by HEVEA.