FriCAS User Guide

Contents:

• 0.1 Introduction to FriCAS
  • 0.1.1 Symbolic Computation
  • 0.1.2 Numeric Computation
  • 0.1.3 Graphics
  • 0.1.4 HyperDoc
  • 0.1.5 Interactive Programming
  • 0.1.6 Data Structures
  • 0.1.7 Mathematical Structures
  • 0.1.8 Pattern Matching
  • 0.1.9 Polymorphic Algorithms
  • 0.1.10 Extensibility
• 0.2 A Technical Introduction
  • 0.2.1 Types are Defined by Abstract Datatype Programs
  • 0.2.2 The Type of Basic Objects is a Domain or Subdomain
  • 0.2.3 Domains Have Types Called Categories
  • 0.2.4 Operations Can Refer To Abstract Types
  • 0.2.5 Categories Form Hierarchies
  • 0.2.6 Domains Belong to Categories by Assertion
  • 0.2.7 Packages Are Clusters of Polymorphic Operations
  • 0.2.8 The Interpreter Builds Domains Dynamically
  • 0.2.9 FriCAS Code is Compiled
  • 0.2.10 FriCAS is Extensible
• 0.3 Using FriCAS as a Pocket Calculator
  • 0.3.1 Basic Arithmetic
  • 0.3.2 Type Conversion
  • 0.3.3 Useful Functions
• 0.4 Using FriCAS as a Symbolic Calculator
  • 0.4.1 Expressions Involving Symbols
  • 0.4.2 Complex Numbers
  • 0.4.3 Number Representations
  • 0.4.4 Modular Arithmetic
• 0.5 General Points about FriCAS
  • 0.5.1 Computation Without Output
  • 0.5.2 Accessing Earlier Results
  • 0.5.3 Splitting Expressions Over Several Lines
  • 0.5.4 Comments and Descriptions
  • 0.5.5 Control of Result Types
• 0.6 Data Structures in FriCAS
  • 0.6.1 Lists
  • 0.6.2 Segmented Lists
  • 0.6.3 Streams
  • 0.6.4 Arrays, Vectors, Strings, and Bits
  • 0.6.5 Flexible Arrays
• 0.7 Functions, Choices, and Loops
  • 0.7.1 Reading Code from a File
  • 0.7.2 Blocks
  • 0.7.3 Functions
  • 0.7.4 Choices
  • 0.7.5 Loops
  • 0.7.5.1 The repeat loop
  • 0.7.5.2 The while loop
  • 0.7.5.3 The for loop
• 1.0 An Overview of FriCAS
• 1.1 Starting Up and Winding Down
  • 1.1.1 Clef
• 1.2 Typographic Conventions
• 1.3 The FriCAS Language
  • 1.3.1 Arithmetic Expressions
  • 1.3.2 Previous Results
  • 1.3.3 Some Types
  • 1.3.4 Symbols, Variables, Assignments, and Declarations
  • 1.3.5 Conversion
  • 1.3.6 Calling Functions
  • 1.3.7 Some Predefined Macros
  • 1.3.8 Long Lines
  • 1.3.9 Comments
• 1.4 Numbers
• 1.5 Data Structures
• 1.6 Expanding to Higher Dimensions
• 1.7 Writing Your Own Functions
• 1.8 Polynomials
• 1.9 Limits
• 1.10 Series
• 1.11 Derivatives
• 1.12 Integration
• 1.13 Differential Equations
• 1.14 Solution of Equations
• 1.15 System Commands
  • 1.15.1 Undo
• 1.16 Graphics
• 2.0 Using Types and Modes
• 2.1 The Basic Idea
  • 2.1.1 Domain Constructors
• 2.2 Writing Types and Modes
  • 2.2.1 Types with No Arguments
  • 2.2.2 Types with One Argument
  • 2.2.3 Types with More Than One Argument
  • 2.2.4 Modes
  • 2.2.5 Abbreviations
• 2.3 Declarations
• 2.4 Records
• 2.5 Unions
  • 2.5.1 Unions Without Selectors
  • 2.5.2 Unions With Selectors
• 2.6 The Any Domain
• 2.7 Conversion
• 2.8 Subdomains Again
• 2.9 Package Calling and Target Types
• 2.10 Resolving Types
• 2.11 Exposing Domains and Packages
• 2.12 Commands for Snooping
• 3.0 Using HyperDoc
• 3.1 Headings
• 3.2 Key Definitions
• 3.3 Scroll Bars
• 3.4 Input Areas
• 3.5 Radio Buttons and Toggles
• 3.6 Search Strings
  • 3.6.1 Logical Searches
• 3.7 Example Pages
• 3.8 X Window Resources for HyperDoc
• 4.0 Input Files and Output Styles
• 4.1 Input Files
• 4.2 The .axiom.input File
• 4.3 Common Features of Using Output Formats
• 4.4 Monospace Two-Dimensional Mathematical Format
• 4.5 TeX Format
• 4.6 IBM Script Formula Format
• 4.7 FORTRAN Format
• 5.0 Interactive Language
• 5.1 Immediate and Delayed Assignments
• 5.2 Blocks
• 5.3 if-then-else
• 5.4 Loops
  • 5.4.1 Compiling vs. Interpreting Loops
  • 5.4.2 return in Loops
  • 5.4.3 break in Loops
  • 5.4.4 break vs. => in Loop Bodies
  • 5.4.5 More Examples of break
  • 5.4.6 iterate in Loops
  • 5.4.7 while Loops
  • 5.4.8 for Loops
  • 5.4.9 for i in n..m repeat
  • 5.4.10 for i in n..m by s repeat
  • 5.4.11 for i in n.. repeat
  • 5.4.12 for x in l repeat
  • 5.4.13 Such that Predicates
  • 5.4.14 Parallel Iteration
  • 5.4.15 Mixing Loop Modifiers
• 5.5 Creating Lists and Streams with Iterators
• 5.6 An Example: Streams of Primes
• 6.0 Functions, Macros and Rules
• 6.1 Functions vs. Macros
• 6.2 Macros
• 6.3 Introduction to Functions
• 6.4 Declaring the Type of Functions
• 6.5 One-Line Functions
• 6.6 Declared vs. Undeclared Functions
• 6.7 Functions vs. Operations
• 6.8 Delayed Assignments vs. Functions with No Arguments
• 6.9 How FriCAS Determines What Function to Use
• 6.10 Compiling vs. Interpreting
• 6.11 Piece-Wise Function Definitions
  • 6.11.1 A Basic Example
  • 6.11.2 Picking Up the Pieces
  • 6.11.3 Predicates
• 6.12 Caching Previously Computed Results
• 6.13 Recurrence Relations
• 6.14 Making Functions from Objects
• 6.15 Functions Defined with Blocks
• 6.16 Free and Local Variables
• 6.17 Anonymous Functions
  • 6.17.1 Some Examples
  • 6.17.2 Declaring Anonymous Functions
• 6.18 Example: A Database
• 6.19 Example: A Famous Triangle
• 6.20 Example: Testing for Palindromes
• 6.21 Rules and Pattern Matching
• 7.0 Graphics
• 7.1 Two-Dimensional Graphics
  • 7.1.1 Plotting Two-Dimensional Functions of One Variable
  • 7.1.2 Plotting Two-Dimensional Parametric Plane Curves
  • 7.1.3 Plotting Plane Algebraic Curves
  • 7.1.4 Two-Dimensional Options
  • 7.1.5 Color
  • 7.1.6 Palette
  • 7.1.7 Two-Dimensional Control-Panel
  • 7.1.7.1 Transformations
  • 7.1.7.2 Messages
  • 7.1.7.3 Multiple Graphs
  • 7.1.7.4 Buttons
  • 7.1.8 Operations for Two-Dimensional Graphics
  • 7.1.9 Addendum: Building Two-Dimensional Graphs
  • 7.1.9.1 Creating a Two-Dimensional Viewport from a List of Points
  • 7.1.9.2 Creating a Two-Dimensional Viewport of a List of Points from a File
  • 7.1.10 Addendum: Appending a Graph to a Viewport Window Containing a Graph
• 7.2 Three-Dimensional Graphics
  • 7.2.1 Plotting Three-Dimensional Functions of Two Variables
  • 7.2.2 Plotting Three-Dimensional Parametric Space Curves
  • 7.2.3 Plotting Three-Dimensional Parametric Surfaces
  • 7.2.4 Three-Dimensional Options
  • 7.2.5 The makeObject Command
  • 7.2.6 Building Three-Dimensional Objects From Primitives
  • 7.2.6.1 Cube Example
  • 7.2.7 Coordinate System Transformations
  • 7.2.8 Three-Dimensional Clipping
  • 7.2.9 Three-Dimensional Control-Panel
  • 7.2.9.1 Transformations
  • 7.2.9.2 Messages
  • 7.2.9.3 Colormap
  • 7.2.9.4 Buttons
  • 7.2.9.5 Light
  • 7.2.9.6 View Volume
  • 7.2.10 Operations for Three-Dimensional Graphics
  • 7.2.11 Customization using .Xdefaults
• 8.0 Advanced Problem Solving
• 8.1 Numeric Functions
• 8.2 Polynomial Factorization
  • 8.2.1 Integer and Rational Number Coefficients
  • 8.2.2 Finite Field Coefficients
  • 8.2.3 Simple Algebraic Extension Field Coefficients
  • 8.2.4 Factoring Rational Functions
• 8.3 Manipulating Symbolic Roots of a Polynomial
  • 8.3.1 Using a Single Root of a Polynomial
  • 8.3.2 Using All Roots of a Polynomial
• 8.4 Computation of Eigenvalues and Eigenvectors
• 8.5 Solution of Linear and Polynomial Equations
  • 8.5.1 Solution of Systems of Linear Equations
  • 8.5.2 Solution of a Single Polynomial Equation
  • 8.5.3 Solution of Systems of Polynomial Equations
• 8.6 Limits
• 8.7 Laplace Transforms
• 8.8 Integration
• 8.9 Working with Power Series
  • 8.9.1 Creation of Power Series
  • 8.9.2 Coefficients of Power Series
  • 8.9.3 Power Series Arithmetic
  • 8.9.4 Functions on Power Series
  • 8.9.5 Converting to Power Series
  • 8.9.6 Power Series from Formulas
  • 8.9.7 Substituting Numerical Values in Power Series
  • 8.9.8 Example: Bernoulli Polynomials and Sums of Powers
• 8.10 Solution of Differential Equations
  • 8.10.1 Closed-Form Solutions of Linear Differential Equations
  • 8.10.2 Closed-Form Solutions of Non-Linear Differential Equations
  • 8.10.3 Power Series Solutions of Differential Equations
• 8.11 Finite Fields
  • 8.11.1 Modular Arithmetic and Prime Fields
  • 8.11.2 Extensions of Finite Fields
  • 8.11.3 Irreducible Modulus Polynomial Representations
  • 8.11.4 Cyclic Group Representations
  • 8.11.5 Normal Basis Representations
  • 8.11.6 Conversion Operations for Finite Fields
  • 8.11.7 Utility Operations for Finite Fields
• 8.12 Primary Decomposition of Ideals
• 8.13 Computation of Galois Groups
• 8.14 Non-Associative Algebras and Modelling Genetic Laws
• 9.1 AssociationList
• 9.2 BalancedBinaryTree
• 9.3 BasicOperator
• 9.4 BinaryExpansion
• 9.5 BinarySearchTree
• 9.6 CardinalNumber
• 9.7 CartesianTensor
  • 7.1 Forming tensors
  • 7.2 Multiplication
  • 7.3 Selecting Components
  • 7.4 Contraction
  • 7.5 Transpositions
  • 7.6 Arithmetic
  • 7.7 Specific Tensors
  • 7.8 Properties of the CartesianTensor domain
  • 7.9 Tensor Calculus
• 9.8 Character
• 9.9 CharacterClass
• 9.10 CliffordAlgebra
  • 9.10.1 The Complex Numbers as a Clifford Algebra
  • 9.10.2 The Quaternion Numbers as a Clifford Algebra
  • 9.10.3 The Exterior Algebra on a Three Space
  • 9.10.4 The Dirac Spin Algebra
• 9.11 Complex
• 9.12 ContinuedFraction
• 9.13 CycleIndicators
• 9.14 DeRhamComplex
• 9.15 DecimalExpansion
• 9.16 DistributedMultivariatePolynomial
• 9.17 DoubleFloat
• 9.18 EqTable
• 9.19 Equation
• 9.20 Exit
• 9.21 Expression
• 9.22 Factored
  • 9.22.1 Decomposing Factored Objects
  • 9.22.2 Expanding Factored Objects
  • 9.22.3 Arithmetic with Factored Objects
  • 9.22.4 Creating New Factored Objects
  • 9.22.5 Factored Objects with Variables
• 9.23 FactoredFunctions2
• 9.24 File
• 9.25 FileName
• 9.26 FlexibleArray
• 9.27 Float
  • 9.27.1 Introduction to Float
  • 9.27.2 Conversion Functions
  • 9.27.3 Output Functions
  • 9.27.4 An Example: Determinant of a Hilbert Matrix
• 9.28 Fraction
• 9.29 FullPartialFractionExpansion
• 9.30 GeneralSparseTable
• 9.31 GroebnerFactorizationPackage
• 9.32 Heap
• 9.33 HexadecimalExpansion
• 9.34 Integer
  • 9.34.1 Basic Functions
  • 9.34.2 Primes and Factorization
  • 9.34.3 Some Number Theoretic Functions
• 9.35 IntegerLinearDependence
• 9.36 IntegerNumberTheoryFunctions
• 9.37 Kernel
• 9.38 KeyedAccessFile
• 9.39 LexTriangularPackage
• 9.40 LazardSetSolvingPackage
• 9.41 Library
• 9.42 LieExponentials
• 9.43 LiePolynomial
• 9.44 LinearOrdinaryDifferentialOperator
  • 9.44.1 Differential Operators with Series Coefficients
• 9.45 LinearOrdinaryDifferentialOperator1
  • 9.45.1 Differential Operators with Rational Function Coefficients
• 9.46 LinearOrdinaryDifferentialOperator2
  • 9.46.1 Differential Operators with Constant Coefficients
  • 9.46.2 Differential Operators with Matrix Coefficients Operating on Vectors
• 9.47 List
  • 9.47.1 Creating Lists
  • 9.47.2 Accessing List Elements
  • 9.47.3 Changing List Elements
  • 9.47.4 Other Functions
  • 9.47.5 Dot, Dot
• 9.48 LyndonWord
• 9.49 Magma
• 9.50 MakeFunction
• 9.51 MappingPackage1
• 9.52 Matrix
  • 9.52.1 Creating Matrices
  • 9.52.2 Operations on Matrices
• 9.53 MultiSet
• 9.54 MultivariatePolynomial
• 9.55 None
• 9.56 Octonion
• 9.57 OneDimensionalArray
• 9.58 Operator
• 9.59 OrderedVariableList
• 9.60 OrderlyDifferentialPolynomial
• 9.61 PartialFraction
• 9.62 Permanent
• 9.63 Polynomial
• 9.64 Quaternion
• 9.65 RadixExpansion
• 9.66 RealClosure
• 9.67 RegularTriangularSet
• 9.68 RomanNumeral
• 9.69 Segment
• 9.70 SegmentBinding
• 9.71 Set
• 9.72 SingleInteger
• 9.73 SparseTable
• 9.74 SquareMatrix
• 9.75 SquareFreeRegularTriangularSet
• 9.76 Stream
• 9.77 String
• 9.78 StringTable
• 9.79 Symbol
• 9.80 Table
• 9.81 TextFile
• 9.82 TwoDimensionalArray
• 9.83 UnivariatePolynomial
• 9.84 UniversalSegment
• 9.85 Vector
• 9.86 Void
• 9.87 WuWenTsunTriangularSet
• 9.88 XPBWPolynomial
• 9.89 XPolynomial
• 9.90 XPolynomialRing
• 9.91 ZeroDimensionalSolvePackage
• 10.0 Interactive Programming
• 10.1 Drawing Ribbons Interactively
• 10.2 A Ribbon Program
• 10.3 Coloring and Positioning Ribbons
• 10.4 Points, Lines, and Curves
• 10.5 A Bouquet of Arrows
• 10.6 Diversion: When Things Go Wrong
• 10.7 Drawing Complex Vector Fields
• 10.8 Drawing Complex Functions
• 10.9 Functions Producing Functions
• 10.10 Automatic Newton Iteration Formulas
• 11.0 Packages
• 11.1 Names, Abbreviations, and File Structure
• 11.2 Syntax
• 11.3 Abstract Datatypes
• 11.4 Capsules
• 11.5 Input Files vs. Packages
• 11.6 Compiling Packages
• 11.7 Parameters
• 11.8 Conditionals
• 11.9 Testing
• 11.10 How Packages Work
• 12.0 Categories
• 12.1 Definitions
• 12.2 Exports
• 12.3 Documentation
• 12.4 Hierarchies
• 12.5 Membership
• 12.6 Defaults
• 12.7 FriCASs
• 12.8 Correctness
• 12.9 Attributes
• 12.10 Parameters
• 12.11 Conditionals
• 12.12 Anonymous Categories
• 13.0 Domains
• 13.1 Domains vs. Packages
• 13.2 Definitions
• 13.3 Category Assertions
• 13.4 A Demo
• 13.5 Browse
• 13.6 Representation
• 13.7 Multiple Representations
• 13.8 Add Domain
• 13.9 Defaults
• 13.10 Origins
• 13.11 Short Forms
• 13.12 Example 1: Clifford Algebra
• 13.13 Example 2: Building A Query Facility
  • 13.13.1 A Little Query Language
  • 13.13.2 The Database Constructor
  • 13.13.3 Query Equations
  • 13.13.4 DataLists
  • 13.13.5 Index Cards
  • 13.13.6 Creating a Database
  • 13.13.7 Putting It All Together
  • 13.13.8 Example Queries
• 14.0 Browse
• 14.1 The Front Page: Searching the Library
  • 14.1.1 Constructors
  • 14.1.2 Operations
  • 14.1.3 Attributes
  • 14.1.4 General
  • 14.1.5 Documentation
  • 14.1.6 Complete
• 14.2 The Constructor Page
  • 14.2.1 Constructor Page Buttons
  • 14.2.1.1 Description
  • 14.2.1.2 Operations
  • 14.2.1.3 Attributes
  • 14.2.1.4 Examples
  • 14.2.1.5 Exports
  • 14.2.1.6 Related Operations
  • 14.2.2 Cross Reference
  • 14.2.2.1 Parents
  • 14.2.2.2 Ancestors
  • 14.2.2.3 Relatives
  • 14.2.2.4 Dependents
  • 14.2.2.5 Lineage
  • 14.2.2.6 Clients
  • 14.2.2.7 Benefactors
  • 14.2.2.8 Children
  • 14.2.2.9 Descendants
  • 14.2.2.10 Domains
  • 14.2.3 Views Of Constructors
  • 14.2.3.1 names
  • 14.2.3.2 abbrs
  • 14.2.3.3 kinds
  • 14.2.3.4 files
  • 14.2.3.5 parameters
  • 14.2.3.6 filter
  • 14.2.3.7 documentation
  • 14.2.3.8 conditions
  • 14.2.4 Giving Parameters to Constructors
• 14.3 Miscellaneous Features of Browse
  • 14.3.1 The Description Page for Operations
  • 14.3.1.1 Arguments
  • 14.3.1.2 Returns
  • 14.3.1.3 Origin
  • 14.3.1.4 Conditions
  • 14.3.1.5 Description
  • 14.3.1.6 Where
  • 14.3.2 Views of Operations
  • 14.3.2.1 names
  • 14.3.2.2 filter
  • 14.3.2.3 documentation
  • 14.3.2.4 signatures
  • 14.3.2.5 parameters
  • 14.3.2.6 origins
  • 14.3.2.7 conditions
  • 14.3.2.8 usage
  • 14.3.2.9 implementation
  • 14.3.2.10 generalize
  • 14.3.2.11 all domains
  • 14.3.3 Capitalization Convention
• 15.0 FriCAS Packages
• 15.1 Important Things to Read First
• 15.2 The New FriCAS Library Compiler
• 15.3 The NAG Library Link
  • 15.3.1 Interpreting NAG Documentation
  • 15.3.1.1 Correspondence Between Fortran and FriCAS types
  • 15.3.1.2 Classification of NAG parameters
  • 15.3.1.3 IFAIL
  • 15.3.2 Using the Link
  • 15.3.3 Providing values for Argument Subprograms
  • 15.3.3.1 Providing ASPs via FortranExpression
  • 15.3.3.2 Providing ASPs via FortranCode
  • 15.3.3.3 Providing ASPs via FileName
  • 15.3.4 General Fortran-generation utilities in FriCAS
  • 15.3.4.1 Template Manipulation
  • 15.3.4.2 Manipulating the Fortran Output Stream
  • 15.3.4.3 Fortran Types
  • 15.3.4.4 FortranScalarType
  • 15.3.4.5 FortranType
  • 15.3.4.6 SymbolTable
  • 15.3.4.7 TheSymbolTable
  • 15.3.4.8 Advanced Fortran Code Generation
  • 15.3.4.9 Switch
  • 15.3.4.10 FortranCode
  • 15.3.4.11 FortranProgram
  • 15.3.5 Some technical information
• 15.4 Interactive Front-end and Language
• 15.5 Library
• 15.6 HyperTex
• 15.7 Documentation
• 16.0 FriCAS System Commands
• 16.1 Introduction
• 16.2 )abbreviation
• 16.3 )boot
• 16.4 )cd
• 16.5 )close
• 16.6 )clear
• 16.7 )compile
• 16.8 )display
• 16.9 )edit
• 16.10 )fin
• 16.11 )frame
• 16.12 )help
• 16.13 )history
• 16.14 )library
• 16.15 )lisp
• 16.16 )load
• 16.17 )trace
• 16.18 )pquit
• 16.19 )quit
• 16.20 )read
• 16.21 )set
• 16.22 )show
• 16.23 )spool
• 16.24 )synonym
• 16.25 )system
• 16.26 )trace
• 16.27 )undo
• 16.28 )what
• 17.1 FriCAS Categories
• 18.1 FriCAS Domains
• 19.1 FriCAS Packages
• 21.0 Programs for AXIOM Images
• 21.1 images1.input
• 21.2 images2.input
• 21.3 images3.input
• 21.4 images5.input
• 21.5 images6.input
• 21.6 images7.input
• 21.7 images8.input
• 21.8 conformal.input
• 21.9 tknot.input
• 21.10 ntube.input
• 21.11 dhtri.input
• 21.12 tetra.input
• 21.13 antoine.input
• 21.14 scherk.input
• Index

Indices and tables

• Index
• Module Index
• Search Page