Sage
Reference Manual
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Front Matter
Sage
Reference Manual
William Stein
wstein@gmail.com
Date:
July 11, 2008
Front Matter
Contents
1. Introduction
2. The Sage Command Line
2.1 Attach a file to a running instance of Sage
2.2 Interactively tracing execution of a command
2.3 Sage: Command Line Arguments
3. The Sage Notebook
3.1 The Sage Notebook object
3.2 A Cell
3.3 A Worksheet
3.4 The Sage Notebook Twisted Web Server
3.5 Javascript (AJAX) Component of
Sage
Notebook
3.6 Customization of the Notebook
3.7 Sage Notebook CSS
3.8 Support for the Notebook (introspection and setup)
3.9 Sage Notebook: Introspection
3.10 Wiki Interactive Web Page
4. Symbolic Calculus
4.1 Symbolic Computation
4.2 Symbolic Equations and Inequalities
4.3 Functional notation support for common calculus methods
4.4 A Sample Session using Sympy
4.5 Calculus Tests and Examples
4.6 Further examples from Wester's paper
5. 2D Graphics
5.1 2D Plotting
5.2 Animated plots
6. 3D Graphics
6.1 Introduction
6.2 Parametric Plots
6.3 List Plots
6.4 Plotting Functions
6.5 Platonic Solids.
6.6 Lines, Frames, Spheres, Points, Dots, and Text
6.7 Base classes for 3D Graphics objects and plotting
6.8 The Tachyon 3D Ray Tracer
7. Games
7.1 Sudoku Solver
8. Graph Theory
8.1 Graph Theory
8.1.1 Graph Format
8.1.2 Generators
8.1.3 Labels
8.1.4 Database
8.1.5 Visualization
8.2 A collection of constructors of common graphs
8.3 N.I.C.E. - Nice (as in open source) Isomorphism Check Engine
8.4 Graph Database Module
8.5 A module for dealing with lists of graphs
9. Constants
9.1 Mathematical constants
10. Functions
10.1 SAGE Functions Class
10.2 Transcendental Functions
10.3 Piecewise-defined Functions
10.4 Orthogonal Polynomials
10.5 Special Functions
11. Basic Structure
11.1 Abstract base class for SAGE objects
11.2 Base class for parent objects with generators
11.3 Formal sums
11.4 Factorizations
11.5 Elements
11.5.1 The Abstract Element Class Hierarchy
11.5.2 How to Define a New Element Class
11.6 Mutability Pyrex Implementation
11.7 Sequences
11.8 Sets
11.9 The set of prime numbers
12. Miscellaneous
12.1 Miscellaneous functions
12.2 SAGE package management commands
12.3 Get resource usage of process
12.4 Multidimensional enumeration
12.5 Installing shortcut scripts
12.6 SAGE Interface to the HG/Mercurial Revision Control System
12.7 Functional notation
12.8 Latex printing support
12.9 Logging of SAGE sessions
12.10 Object persistence
12.11 Support for persistent functions in .sage files
12.12 Evaluating a String in SAGE
12.13 Miscellaneous arithmetic functions
13. Databases
13.1 Cremona's tables of elliptic curves
13.2 The Stein-Watkins table of elliptic curves
13.3 John Jones's tables of number fields
13.4 Linear codes
13.5 Interface to Sloane On-Line Encyclopedia of Integer Sequences
13.6 Frank Luebeck's tables of Conway polynomials over finite fields
13.7 Tables of zeros of the Riemann-Zeta function
14. Interpreter Interfaces
14.1 Common Interface Functionality
14.2 Interface to Axiom
14.3 Interface to GAP
14.3.1 First Examples
14.3.2 GAP and Singular
14.3.3 Saving and loading objects
14.3.4 Long Input
14.3.5 Changing which GAP is used
14.4 Interface to GP/Pari
14.5 Interface to the Gnuplot interpreter
14.6 Interface to KASH
14.6.1 Issues
14.6.2 Tutorial
14.6.3 Long Input
14.7 Interface to Magma
14.7.1 Parameters
14.7.2 Multiple Return Values
14.7.3 Long Input
14.7.4 Other Examples
14.8 Interface to Maple
14.8.1 Tutorial
14.9 Interface to MATLAB
14.9.1 Tutorial
14.10 Interface to Maxima
14.10.1 Tutorial
14.10.2 Examples involving matrices
14.10.3 Laplace Transforms
14.10.4 Continued Fractions
14.10.5 Special examples
14.10.6 Miscellaneous
14.10.7 Interactivity
14.10.8 Latex Output
14.10.9 Long Input
14.11 Interface to Mathematica
14.11.1 Tutorial
14.11.2 Long Input
14.11.3 Loading and saving
14.12 Interface to mwrank
14.13 Interface to Octave
14.13.1 Computation of Special Functions
14.13.2 Tutorial
14.14 Interface to SAGE
14.15 Interface to Singular
14.15.1 Introduction
14.15.2 Tutorial
14.15.3 Computing the Genus
14.15.4 An Important Concept
14.15.5 Long Input
14.16 The Tachyon Ray Tracer
15. C/C++ Library Interfaces
15.1 PARI C-library interface
15.2 Victor Shoup's NTL C++ Library
15.3 Cremona's mwrank C++ library
16. Networking and Grid Computing
16.1 Wiki Interactive Web Page
16.2 Distributed Sage
17. Cryptography
17.1 Cryptosystems
17.2 Ciphers
17.3 Classical Cryptosystems
17.4 Classical Ciphers
17.5 Stream Cryptosystems
17.6 Stream Ciphers
17.7 Linear feedback shift register (LFSR) sequence commands
17.8 Small Scale Variants of the AES (SR) Polynomial System Generator
17.9 Multivariate Polynomial Systems
18. Combinatorics
18.1 Combinatorial Functions
18.2 Functions that compute some of the sequences in Sloane's tables
18.3 Compute Bell and Uppuluri-Carpenter numbers
18.4 Alternating Sign Matrices
18.5 Cartesian Products
18.6 Combinations
18.7 Combinatorial Algebras
18.8 Signed Compositions
18.9 Compositions
18.10 Exact Cover Problem via Dancing Links
18.11 Dyck Words
18.12 Finite combinatorial classes
18.13 Paths in Directed Acyclic Graphs
18.14 Kostka-Foulkes Polynomials
18.15 Lyndon words
18.16 Miscellaneous
18.17 Necklaces
18.18 Partition/Diagram Algebras
18.19 Partitions
18.20 Permutations
18.21 q-Analogues
18.22 Ribbons
18.23 Schubert Polynomials
18.24 Ordered Set Partitions
18.25 Set Partitions
18.26 Skew Partitions
18.27 Skew Tableaux
18.28 Subsets
18.29 Subwords
18.30 Symmetric Functions
18.31 Hall-Littlewood Polynomials
18.32 Jack Polynomials
18.33 Macdonald Polynomials - under development
18.34 Symmetric Group Algebra
18.35 Tableaux
18.36 Tools
18.37 Tuples
18.38 Words
19. Probability
19.1 Random variables and probability spaces
20. Category Theory
20.1 Categories
20.2 Homsets
20.3 Morphisms
20.4 Functors
21. Monoids
21.1 Free Monoids
21.2 Monoid Elements
21.3 Free abelian monoids
21.4 Abelian monoid elements
22. Groups
22.1 Base class for groups
22.2 Multiplicative Abelian Groups
22.3 Abelian group elements
22.4 Homomorphisms of abelian groups
22.5 Basic functionality for dual groups of finite multiplicative Abelian groups
22.6 Permutation groups
22.7 Permutation group elements
22.8 Permutation group homomorphisms
22.9 Rubik's cube group functions
22.10 Matrix Groups
22.11 Matrix Group Elements
22.12 Homomorphisms Between Matrix Groups
22.13 Matrix Group Homsets
22.14 Linear Groups
22.15 General Linear Groups
22.16 Special Linear Groups
22.17 Orthogonal Linear Groups
22.18 Symplectic Linear Groups
22.19 Unitary Groups
and
23. General Rings, Ideals, and Morphisms
23.1 Ideals
23.2 Monoid of Ring Ideals
23.3 Homomorphisms of rings
23.4 Space of homomorphisms between two rings
23.5 Infinity Rings
23.6 Fraction Field of Integral Domains
23.7 Fraction Field Elements
23.8 Quotient Rings
23.9 Quotient Ring Elements
24. Standard Commutative Rings
24.1 Ring
of Integers
24.2 Elements of the ring
of integers
24.3 Ring
of integers modulo
24.4 Elements of
24.5 Field
of Rational Numbers
24.6 Rational Numbers
24.7 Finite Fields
24.8 Elements of Finite Fields
25. p-adic Rings
25.1 Introduction to the
-adics
25.2 Terminology and types of
-adics
25.2.1 Fixed Modulus Rings
25.2.2 Capped Absolute Rings
25.2.3 Capped Relative Rings and Fields
25.2.4 Lazy Rings and Fields
25.2.5 Unramified Extensions
26. Fixed and Arbitrary Precision Numerical Fields
26.1 Double Precision Real Numbers
26.2 Double Precision Complex Numbers
26.3 Field of Arbitrary Precision Real Numbers
26.4 Field of Arbitrary Precision Complex Numbers
26.5 Arbitrary Precision Complex Numbers
26.6 Field of Arbitrary Precision Real Intervals
27. Number Fields
27.1 Number Fields
27.2 Number Field Elements
28. Polynomial Rings
28.1 Univariate Polynomial Rings
28.2 Univariate Polynomial Base Class
28.3 Quotients of Univariate Polynomial Rings
28.4 Elements of Quotients of Univariate Polynomial Rings
28.5 Term Orderings
28.6 Multivariate Polynomial Rings
28.7 Multivariate Polynomials
28.8 Ideals in multivariate polynomial rings
28.9 Boolean Polynomials
28.9.1 Implementation specific notes
28.9.2 Access to the original P
OLY
B
O
R
I
interface
28.10 Generic Convolution
29. Power Series Rings
29.1 Univariate Power Series Rings
29.2 Power Series
29.3 Laurent Series Rings
29.4 Laurent Series
30. Algebras
30.1 Free algebras
30.2 Free algebra elements
30.3 Free algebra quotients
30.4 Free algebra quotient elements
31. Quaternion Algebras
31.1 Quaternion algebras
31.2 Quaternion algebra elements
31.3 Quaternion orders
31.4 Quaternion order elements
31.5 Quaternion ideal
31.6 Quaternion ideal elements
32. Matrices and Spaces of Matrices
Introduction
32.1 Matrix Spaces
32.2 Matrix Constructor
32.3 Matrices over an arbitrary ring
32.3.1 Implementation and Design
32.4 Abstract base class for matrices
32.5 Base class for matrices, part 0
32.6 Base class for matrices, part 1
32.7 Base class for matrices, part 2
32.8 Generic Asymptotically Fast Strassen Algorithms
32.9 Minimal Polynomials of Linear Recurrence Sequences
32.10 Base class for dense matrices
32.11 Base class for sparse matrices
32.12 Dense Matrices over a general ring
32.13 Sparse Matrices over a general ring
32.14 Dense matrices over
for
small
32.15 Sparse matrices over
for
small
32.16 Dense matrices over the integer ring
32.17 Dense matrices over the rational field
32.18 Dense matrices over the real double field
32.19 Dense matrices over the Complex Double Field
33. Modules
33.1 Abstract base class for modules
33.2 Free modules
33.3 Elements of free modules
33.4 Complex double vectors
33.5 Real double vectors
33.6 Homspaces between free modules
33.7 Morphisms of free modules
33.8 Morphisms defined by a matrix
34. Combinatorial Geometry
34.1 Lattice and reflexive polytopes
34.2 Groebner Fans
34.3 Polytopes
35.
L
-functions
35.1 Rubinstein's
-function Calculator
35.2 Watkins Symmetric Power
-function Calculator
35.3 Dokchitser's L-functions Calculator
36. Schemes
36.1 Scheme implementation overview
36.1.1 TODO List
36.2 Schemes
36.3 Spec of a ring
36.4 Scheme obtained by glueing two other schemes
36.5 Points on schemes
36.6 Ambient Spaces
36.7 Affine
space over a ring
36.8 Projective
space over a ring
36.9 Algebraic schemes
36.10 Set of homomorphisms between two schemes
36.11 Scheme morphism
36.12 Divisors on schemes
37. Elliptic and Plane Curves
37.1 Plane curve constructors
37.2 Affine plane curves over a general ring
37.3 Plane curves over a general ring
37.4 Elliptic curve constructor
37.5 Elliptic curves over a general ring
37.6 Elliptic curves over a general field
37.7 Elliptic curves over the rational numbers
37.8 Elliptic curves over finite fields
37.9 Formal groups of elliptic curves
37.10 Computation of Frobenius matrix on Monsky-Washnitzer cohomology
38. Hyperelliptic Curves
38.1 Hyperelliptic curve constructor
38.2 Hyperelliptic curves over a finite field
38.3 Hyperelliptic curves over a general ring
38.4 Constructor for Jacobian of a hyperelliptic curve
38.5 Jacobian of a Hyperelliptic curve of Genus 2
38.6 Jacobian of a General Hyperelliptic Curve
38.7 Rational point sets on a Jacobian
38.8 Jacobian `morphism' as a class in the Picard group
38.9 Conductor and Reduction Types for Genus 2 Curves
39. Coding Theory
39.1 Linear Codes
39.2 AUTHOR:
39.3 This module implements functions useful for studying binary self-dual codes
39.4 Bounds for Parameters of Codes
40. Modular Forms: General Hecke Algebras and Hecke Modules
40.1 Congruence subgroups of SL2(Z)
40.2 Dirichlet characters
40.3 The set
of cusps
40.4 Dimensions of spaces of modular forms
40.5 Conjectural Slopes of Hecke Polynomial
40.6 Hecke modules
40.7 Submodule of a Hecke module
40.8 Ambient Hecke modules
40.9 Elements of Hecke modules
40.10 Hom spaces between objects of the category of hecke modules over a given base ring
40.11 Morphism of Hecke modules
40.12 Hecke algebras and modules
40.13 Hecke operators
41. Modular Symbols
41.1 Creation of modular symbols spaces
41.2 Space of modular symbols (base class)
41.3 Ambient spaces of modular symbols
41.4 Subspace of ambient spaces of modular symbols
41.5 A single element of an ambient space of modular symbols
41.6 Manin symbols
41.7 Space of boundary modular symbols
41.8 Heilbronn matrix computation
41.9 List of Elements of
41.10 Relation matrices for ambient modular symbols spaces
42. Modular Forms
42.1 Creating Spaces of Modular Forms
42.2 Generic spaces of modular forms
42.3 Ambient Spaces of Modular Forms
42.4 Modular Forms with Character
42.5 Modular Forms for
over
42.6 Modular Forms for
over
42.7 Modular Forms over a Non-minimal Base Ring
42.8 Submodules of spaces of modular forms
42.9 The Cuspidal Subspace
42.10 The Eisenstein Subspace
42.11 Eisenstein Series
42.12 Elements of modular forms spaces
42.13 Hecke Operators on
-expansions
42.14 Numerical computation of newforms
42.15 The Victor Miller Basis
42.16 Ambient Spaces of Modular Forms
42.17 Compute spaces of half-integral weight modular forms
43. Modular Abelian Varieties
43.1 Constructors for certain modular abelian varieties
43.2 Base class for modular abelian varieties
43.3 Ambient Jacobian Abelian Variety
43.4 Finite subgroups of modular abelian varieties
43.5 Torsion points on modular abelan varieties.
43.6 Torsion subgroups of modular abelian varieties
43.7 Cuspidal subgroups of modular abelian varieties
43.8 Hecke operators on modular abelian varieties
43.9 Homology of modular abelian varieties
43.10 Spaces of homomorphisms between modular abelian varieties
43.11 Morphisms between modular abelian varieties, including Hecke operators
A. History and License
A.1 License
A.1.1 The GNU General Public License
Module Index
Index
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Release 2008.07.11, documentation updated on July 11, 2008.
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