Tutorial - SageMath
Tutorial
Release 9.7 The Sage Development Team
Sep 20, 2022
CONTENTS
1 Introduction
3
1.1 Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Ways to Use Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Longterm Goals for Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 A Guided Tour
7
2.1 Assignment, Equality, and Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Getting Help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Functions, Indentation, and Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Basic Algebra and Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.6 Some Common Issues with Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.7 Basic Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.8 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.9 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.10 Parents, Conversion and Coercion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.11 Finite Groups, Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.12 Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.13 Some More Advanced Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 The Interactive Shell
55
3.1 Your Sage Session . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Logging Input and Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3 Paste Ignores Prompts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4 Timing Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5 Other IPython tricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.6 Errors and Exceptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.7 Reverse Search and Tab Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.8 Integrated Help System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.9 Saving and Loading Individual Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.10 Saving and Loading Complete Sessions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 Interfaces
69
4.1 GP/PARI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 GAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3 Singular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.4 Maxima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5 Sage, LaTeX and Friends
75
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
i
5.2 Basic Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.3 Customizing LaTeX Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.4 Customizing LaTeX Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.5 An Example: Combinatorial Graphs with tkz-graph . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.6 A Fully Capable TeX Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.7 External Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6 Programming
83
6.1 Loading and Attaching Sage files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.2 Creating Compiled Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.3 Standalone Python/Sage Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.4 Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.5 Lists, Tuples, and Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.6 Dictionaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.7 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.8 Iterators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.9 Loops, Functions, Control Statements, and Comparisons . . . . . . . . . . . . . . . . . . . . . . . . 91
6.10 Profiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7 Using SageTeX
95
7.1 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.2 Make SageTeX known to TeX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.3 SageTeX documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.4 SageTeX and TeXLive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
8 Afterword
99
8.1 Why Python? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.2 I would like to contribute somehow. How can I? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
8.3 How do I reference Sage? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
9 Appendix
103
9.1 Arithmetical binary operator precedence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
10 Bibliography
105
11 Indices and tables
107
Bibliography
109
Index
111
ii
Tutorial, Release 9.7
Sage is free, open-source math software that supports research and teaching in algebra, geometry, number theory, cryptography, numerical computation, and related areas. Both the Sage development model and the technology in Sage itself are distinguished by an extremely strong emphasis on openness, community, cooperation, and collaboration: we are building the car, not reinventing the wheel. The overall goal of Sage is to create a viable, free, open-source alternative to Maple, Mathematica, Magma, and MATLAB.
This tutorial is the best way to become familiar with Sage in only a few hours. You can read it in HTML or PDF versions, or from the Sage notebook (click Help, then click Tutorial to interactively work through the tutorial from within Sage).
This work is licensed under a Creative Commons Attribution-Share Alike 3.0 License.
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