Sage Beginner's Guide

358 Pages·2011·24.292 MB·English

by  Finch C.

Sage
Beginner's Guide
Copyright © 2011 Packt Publishing
All rights reserved. No part of this book may be reproduced, stored in a retrieval system,
or transmitted in any form or by any means, without the prior written permission of the
publisher, except in the case of brief quotations embedded in critical articles or reviews.
Every effort has been made in the preparation of this book to ensure the accuracy of the
information presented. However, the information contained in this book is sold without
warranty, either express or implied. Neither the author, nor Packt Publishing, and its dealers
and distributors will be held liable for any damages caused or alleged to be caused directly or
indirectly by this book.
Packt Publishing has endeavored to provide trademark information about all of the
companies and products mentioned in this book by the appropriate use of capitals.
However, Packt Publishing cannot guarantee the accuracy of this information.
First published: May 2011
Production Reference: 1250411
Published by Packt Publishing Ltd.
32 Lincoln Road
Olton
Birmingham, B27 6PA, UK.
ISBN 978-1-849514-46-0
www.packtpub.com
Cover Image by Ed Maclean ([email protected])
About the Author
Craig Finch is a Ph. D. Candidate in the Modeling and Simulation program at the University
of Central Florida (UCF). He earned a Bachelor of Science degree from the University
of Illinois at Urbana-Champaign and a Master of Science degree from UCF, both in
electrical engineering. Craig worked as a design engineer for TriQuint Semiconductor, and
currently works as a research assistant in the Hybrid Systems Lab at the UCF NanoScience
Technology Center. Craig's professional goal is to develop tools for computational science
and engineering and use them to solve difficult problems. In particular, he is interested
in developing tools to help biologists study living systems. Craig is committed to using,
developing, and promoting open-source software. He provides documentation and "how-to"
examples on his blog at http://www.shocksolution.com.
I would like to thank my advisers, Dr. J. Hickman and Dr. Tom Clarke, for
giving me the opportunity to pursue my doctorate. I would also like to
thank my parents for buying the Apple IIGS computer that started it all.
About the Reviewers
Dr. David Kirkby is a chartered engineer living in Essex, England. David has a B.Sc. in
Electrical and Electronic Engineering, an M.Sc. in Microwaves and OptoElectronics, and a
Ph.D. in Medical Physics. Despite David's Ph.D. being in Medical Physics, it was primarily an
engineering project, measuring the optical properties of human tissue, with a mixture of
Monte Carlo modeling, radio frequency design, and laser optics. David was awarded his Ph.D.
in 1999 from University College London.
Although not a mathematician, Dr. Kirkby has made extensive use of mathematical software.
Most of his experience has been with MathematicaTM from Wolfram Research, although he
has used both MATLABTM and SimulinkTM too.
David is the author of a number of open-source projects, including software for modeling
transmission lines using finite difference (http://atlc.sourceforge.net/), design of
Yagi-Uda antennas (http://www.g8wrb.org/yagi/) which can use a genetic algorithm
for optimization, as well as software for data collection and analysis from electronic test
equipment. David once wrote a web-based interface to MathematicaTM (http://witm.
sourceforge.net/) which allows MathematicaTM to be used from a personal computer,
PDA or smartphone.
Soon after the Sage project was started by Professor William Stein, Dr. Kirkby joined the
development of Sage. He primarily worked on the successful port of Sage to the Solaris and
OpenSolaris operating systems and encourages other developers to write portable code,
conforming to POSIX standard, avoiding GNUisms.
Professionally, David's skill sets include computer modeling, radio frequency design,
analogue circuit design, electromagnetic compatibility and optics—both free space
and integrated. David has also been a Solaris system administrator for the University of
Washington where the Sage project is based.
When not working on writing software, David enjoys playing chess, gardening, and
spending time with his wife Lin and dog Smudge.
Readers wishing to contact Dr. Kirkby can do so via his website http://www.drkirkby.
co.uk/ where details of his consulting services may be found.
Minh Nguyen has been a contributor to the Sage project since December 2007. Over the
years, he has worked on various aspects of Sage ranging from the standard documentation and
modules such as cryptography, number theory, and graph theory to the Sage build system. He
regularly maintains the Sage website and works on book projects that aim to provide in-depth
documentation on using Sage to study cryptography and mathematics. More of his rantings
can be found at http://mvngu.wordpress.com.
Table of Contents
Preface 1
Chapter 1: What Can You Do with Sage? 9
Getting started 9
Using Sage as a powerful calculator 12
Symbolic mathematics 14
Linear algebra 17
Solving an ordinary differential equation 18
More advanced graphics 19
Visualising a three-dimensional surface 20
Typesetting mathematical expressions 21
A practical example: analysing experimental data 22
Time for action – fitting the standard curve 22
Time for action – plotting experimental data 24
Time for action – fitting a growth model 25
Summary 26
Chapter 2: Installing Sage 29
Before you begin 29
Installing a binary version of Sage on Windows 30
Downloading VMware Player 30
Installing VMWare Player 30
Downloading and extracting Sage 30
Launching the virtual machine 31
Start Sage 32
Installing a binary version of Sage on OS X 33
Downloading Sage 34
Installing Sage 34
Starting Sage 34
Table of Contents
Installing a binary version of Sage on GNU/Linux 35
Downloading and decompressing Sage 35
Running Sage from your user account 36
Installing for multiple users 37
Building Sage from source 38
Prerequisites 38
Downloading and decompressing source tarball 39
Building Sage 39
Installation 39
Summary 39
Chapter 3: Getting Started with Sage 41
How to get help with Sage 41
Starting Sage from the command line 42
Using the interactive shell 43
Time for action – doing calculations on the command line 43
Getting help 45
Command history 46
Tab completion 47
Interactively tracing execution 48
Using the notebook interface 48
Starting the notebook interface 49
Time for action – doing calculations with the notebook interface 52
Getting help in the notebook interface 54
Working with cells 54
Working with code 55
Closing the notebook interface 55
Displaying results of calculations 56
Operators and variables 56
Arithmetic operators 57
Numerical types 58
Integers and rational numbers 58
Real numbers 59
Complex numbers 60
Symbolic expressions 60
Defining variables on rings 61
Combining types in expressions 62
Strings 62
Time for action – using strings 62
Callable symbolic expressions 63
Time for action – defining callable symbolic expressions 64
Automatically typesetting expressions 65
[ ii ]
Table of Contents
Functions 66
Time for action – calling functions 66
Built-in functions 68
Numerical approximations 68
The reset and restore functions 69
Defining your own functions 70
Time for action – defining and using your own functions 70
Functions with keyword arguments 72
Time for action – defining a function with keyword arguments 72
Objects 73
Time for action – working with objects 74
Getting help with objects 75
Summary 77
Chapter 4: Introducing Python and Sage 79
Python 2 and Python 3 79
Writing code for Sage 80
Long lines of code 81
Running scripts 81
Sequence types: lists, tuples, and strings 82
Time for action – creating lists 82
Getting and setting items in lists 85
Time for action – accessing items in a list 85
List functions and methods 87
Tuples: read-only lists 87
Time for action – returning multiple values from a function 87
Strings 89
Time for action – working with strings 90
Other sequence types 92
For loops 92
Time for action – iterating over lists 92
Time for action – computing a solution to the diffusion equation 94
List comprehensions 99
Time for action – using a list comprehension 99
While loops and text file I/O 101
Time for action – saving data in a text file 101
Time for action – reading data from a text file 103
While loops 105
Parsing strings and extracting data 105
Alternative approach to reading from a text file 106
If statements and conditional expressions 107
Storing data in a dictionary 108
[ iii ]
Table of Contents
Time for action – defining and accessing dictionaries 108
Lambda forms 110
Time for action – using lambda to create an anonymous function 110
Summary 111
Chapter 5: Vectors, Matrices, and Linear Algebra 113
Vectors and vector spaces 113
Time for action – working with vectors 114
Creating a vector space 115
Creating and manipulating vectors 116
Time for action – manipulating elements of vectors 116
Vector operators and methods 117
Matrices and matrix spaces 118
Time for action – solving a system of linear equations 118
Creating matrices and matrix spaces 120
Accessing and manipulating matrices 120
Time for action – accessing elements and parts of a matrix 120
Manipulating matrices 122
Time for action – manipulating matrices 122
Matrix algebra 124
Time for action – matrix algebra 124
Other matrix methods 125
Time for action – trying other matrix methods 126
Eigenvalues and eigenvectors 127
Time for action – computing eigenvalues and eigenvectors 127
Decomposing matrices 129
Time for action – computing the QR factorization 129
Time for action – computing the singular value decomposition 131
An introduction to NumPy 133
Time for action – creating NumPy arrays 133
Creating NumPy arrays 134
NumPy types 135
Indexing and selection with NumPy arrays 136
Time for action – working with NumPy arrays 136
NumPy matrices 137
Time for action – creating matrices in NumPy 137
Learning more about NumPy 139
Summary 139
Chapter 6: Plotting with Sage 141
Confusion alert: Sage plots and matplotlib 141
Plotting in two dimensions 141
[ iv ]