Sage Beginners Guide电子书

Sage

Table of Contents

Sage

Credits

About the Author

About the Reviewers

www.PacktPub.com

Support files, eBooks, discount offers and more

Why Subscribe?

Free Access for Packt account holders

Preface

What this book covers

What you need for this book

Who this book is for

Conventions

Time for action - heading

What just happened?

Pop quiz - heading

Have a go hero - heading

Reader feedback

Customer support

Downloading the example code

Errata

Piracy

Questions

1. What Can You Do with Sage?

Getting started

Using Sage as a powerful calculator

Symbolic mathematics

Have a go hero -

Linear algebra

Solving an ordinary differential equation

More advanced graphics

Visualising a three-dimensional surface

Typesetting mathematical expressions

A practical example: analysing experimental data

Time for action - fitting the standard curve

What just happened?

Time for action - plotting experimental data

What just happened?

Time for action - fitting a growth model

What just happened?

Summary

2. Installing Sage

Before you begin

Installing a binary version of Sage on Windows

Downloading VMware Player

Installing VMWare Player

Downloading and extracting Sage

Launching the virtual machine

Start Sage

Installing a binary version of Sage on OS X

Downloading Sage

Installing Sage

Starting Sage

Installing a binary version of Sage on GNU/Linux

Downloading and decompressing Sage

Running Sage from your user account

Installing for multiple users

Building Sage from source

Prerequisites

Downloading and decompressing source tarball

Building Sage

Installation

Summary

3. Getting Started with Sage

How to get help with Sage

Starting Sage from the command line

Using the interactive shell

Time for action - doing calculations on the command line

What just happened?

Getting help

Command history

Tab completion

Interactively tracing execution

Using the notebook interface

Starting the notebook interface

Time for action - doing calculations with the notebook interface

What just happened?

Getting help in the notebook interface

Working with cells

Working with code

Closing the notebook interface

Have a go hero - using the notebook interface

Displaying results of calculations

Operators and variables

Arithmetic operators

Pop quiz - working with operators

Numerical types

Integers and rational numbers

Real numbers

Complex numbers

Symbolic expressions

Defining variables on rings

Combining types in expressions

Pop quiz - understanding types

Strings

Time for action - using strings

What just happened?

Callable symbolic expressions

Time for action - defining callable symbolic expressions

What just happened?

Automatically typesetting expressions

Functions

Time for action - calling functions

What just happened?

Have a go hero - make some more plots

Built-in functions

Numerical approximations

The reset and restore functions

Defining your own functions

Time for action - defining and using your own functions

What just happened?

Functions with keyword arguments

Time for action - defining a function with keyword arguments

What just happened?

Objects

Time for action - working with objects

What just happened?

Getting help with objects

Summary

4. Introducing Python and Sage

Python 2 and Python 3

Writing code for Sage

Long lines of code

Running scripts

Sequence types: lists, tuples, and strings

Time for action - creating lists

What just happened?

Getting and setting items in lists

Time for action - accessing items in a list

What just happened?

Pop quiz - lists and indexing

List functions and methods

Tuples: read-only lists

Time for action - returning multiple values from a function

What just happened?

Strings

Time for action - working with strings

What just happened?

Other sequence types

For loops

Time for action - iterating over lists

What just happened?

Time for action - computing a solution to the diffusion equation

What just happened?

Pop quiz - lists and for loops

Have a go hero - adding another for loop

List comprehensions

Time for action - using a list comprehension

What just happened?

While loops and text file I/O

Time for action - saving data in a text file

What just happened?

Time for action - reading data from a text file

What just happened?

While loops

Parsing strings and extracting data

Alternative approach to reading from a text file

Have a go hero - define a function for reading the text file

Have a go hero - replace a for loop with a while loop

If statements and conditional expressions

Storing data in a dictionary

Time for action - defining and accessing dictionaries

What just happened?

Lambda forms

Time for action - using lambda to create an anonymous function

What just happened?

Summary

5. Vectors, Matrices, and Linear Algebra

Vectors and vector spaces

Time for action - working with vectors

What just happened?

Creating a vector space

Creating and manipulating vectors

Time for action - manipulating elements of vectors

What just happened?

Vector operators and methods

Matrices and matrix spaces

Time for action - solving a system of linear equations

What just happened?

Creating matrices and matrix spaces

Accessing and manipulating matrices

Time for action - accessing elements and parts of a matrix

What just happened?

Manipulating matrices

Time for action - manipulating matrices

What just happened?

Pop quiz - manipulating matrices

Matrix algebra

Time for action - matrix algebra

What just happened?

Other matrix methods

Time for action - trying other matrix methods

What just happened?

Eigenvalues and eigenvectors

Time for action - computing eigenvalues and eigenvectors

What just happened?

Have a go hero - verifying the eigenvalues and eigenvectors

Decomposing matrices

Time for action - computing the QR factorization

What just happened?

Time for action - computing the singular value decomposition

What just happened?

An introduction to NumPy

Time for action - creating NumPy arrays

What just happened?

Creating NumPy arrays

NumPy types

Indexing and selection with NumPy arrays

Time for action - working with NumPy arrays

What just happened?

Have a go hero - replacing lists with NumPy arrays

NumPy matrices

Time for action - creating matrices in NumPy

What just happened?

Learning more about NumPy

Summary

6. Plotting with Sage

Confusion alert: Sage plots and matplotlib

Plotting in two dimensions

Plotting symbolic expressions with Sage

Time for action - plotting symbolic expressions

What just happened?

Time for action - plotting a function with a pole

What just happened?

Time for action - plotting a parametric function

What just happened?

Time for action - making a polar plot

What just happened?

Time for action - plotting a vector field

What just happened?

Plotting data in Sage

Time for action - making a scatter plot

What just happened?

Time for action - plotting a list

What just happened?

Using graphics primitives

Time for action - plotting with graphics primitives

What just happened?

Using matplotlib

Time for action - plotting functions with matplotlib

What just happened?

Using matplotlib to "tweak" a Sage plot

Time for action - getting the matplotlib figure object

What just happened?

Time for action - improving polar plots

What just happened?

Plotting data with matplotlib

Time for action - making a bar chart

What just happened?

Time for action - making a pie chart

What just happened?

Time for action - plotting a histogram

What just happened?

Plotting in three dimensions

Time for action - make an interactive 3D plot

What just happened?

Higher quality output

Parametric 3D plotting

Time for action - parametric plots in 3D

What just happened?

Contour plots

Time for action - making some contour plots

What just happened?

Summary

7. Making Symbolic Mathematics Easy

Using the notebook interface

Defining symbolic expressions

Time for action - defining callable symbolic expressions

What just happened?

Relational expressions

Time for action - defining relational expressions

What just happened?

Time for action - relational expressions with assumptions

What just happened?

Manipulating expressions

Time for action - manipulating expressions

What just happened?

Manipulating rational functions

Time for action - working with rational functions

What just happened?

Substitutions

Time for action - substituting symbols in expressions

What just happened?

Expanding and factoring polynomials

Time for action - expanding and factoring polynomials

What just happened?

Manipulating trigonometric expressions

Time for action - manipulating trigonometric expressions

What just happened?

Logarithms, rational functions, and radicals

Time for action - simplifying expressions

What just happened?

Solving equations and finding roots

Time for action - solving equations

What just happened?

Finding roots

Time for action - finding roots

What just happened?

Differential and integral calculus

Time for action - calculating limits

What just happened?

Derivatives

Time for action - calculating derivatives

What just happened?

Integrals

Time for action - calculating integrals

What just happened?

Series and summations

Time for action - computing sums of series

What just happened?

Taylor series

Time for action - finding Taylor series

What just happened?

Have a go hero - Taylor series

Laplace transforms

Time for action - computing Laplace transforms

What just happened?

Solving ordinary differential equations

Time for action - solving an ordinary differential equation

What just happened?

Summary

8. Solving Problems Numerically

Sage and NumPy

Solving equations and finding roots numerically

Time for action - finding roots of a polynomial

What just happened?

Finding minima and maxima of functions

Time for action - minimizing a function of one variable

What just happened?

Functions of more than one variable

Time for action - minimizing a function of several variables

What just happened?

Numerical approximation of derivatives

Time for action - approximating derivatives with differences

What just happened?

Computing gradients

Time for action - computing gradients

What just happened?

Numerical integration

Time for action - numerical integration

What just happened?

Numerical integration with NumPy

Time for action - numerical integration with NumPy

What just happened?

Discrete Fourier transforms

Time for action - computing discrete Fourier transforms

What just happened?

Window functions

Time for action - plotting window functions

What just happened?

Have a go hero - using window functions

Solving ordinary differential equations

Time for action - solving a first-order ODE

What just happened?

Solving a system of ODEs

Time for action - solving a higher-order ODE

What just happened?

Solving the system using the GNU Scientific Library

Time for action - alternative method of solving a system of ODEs

What just happened?

Numerical optimization

Time for action - linear programming

What just happened?

Fitting a function to a noisy data set

Time for action - least squares fitting

What just happened?

Constrained optimization

Time for action - a constrained optimization problem

What just happened?

Probability

Time for action - accessing probability distribution functions

What just happened?

Summary

9. Learning Advanced Python Programming

How to write good software

Object-oriented programming

Time for action - defining a class that represents a tank

What just happened?

Making our tanks move

Time for action - making the tanks move

What just happened?

Have a go hero - checking the values passed to __init__

Creating a module for our classes

Time for action - creating your first module

What just happened?

Expanding our simulation to other kinds of vehicles

Time for action - creating a vehicle base class

What just happened?

Creating a package for our simulation

Time for action - creating a combat simulation package

What just happened?

Have a go hero - adding another derived class

Potential pitfalls when working with classes and instances

Time for action - using class and instance attributes

What just happened?

Time for action - more about class and instance attributes

What just happened?

Creating empty classes and functions

Time for action - creating empty classes and functions

What just happened?

Handling errors gracefully

Time for action - raising and handling exceptions

What just happened?

Exception types

Have a go hero - raising exceptions in the __init__ method of Tank

Creating your own exception types

Time for action - creating custom exception types

What just happened?

Unit testing

Time for action - creating unit tests for the Tank class

What just happened?

Strategies for unit testing

Have a go hero - creating some unit tests

Summary

10. Where to go from here

Typesetting equations with LaTeX

Installing LaTeX

Time for action - PDF output from the notebook interface

What just happened?

The view function in the interactive shell

LaTeX mark-up in the notebook interface

Time for action - working with LaTeX markup in the notebook interface

What just happened?

Time for action - putting it all together

What just happened?

Have a go hero - Bessel functions of the second kind

Speeding up execution

Time for action - detecting collisions between spheres

What just happened?

Time for action - detecting collisions: command-line version

What just happened?

Tips for measuring runtimes

Optimizing our algorithm

Time for action - faster collision detection

What just happened?

Optimizing with NumPy

Time for action - using NumPy

What just happened?

More about NumPy

Optimizing with Cython

Time for action - optimizing collision detection with Cython

What just happened?

Have a go hero - further optimization with Cython

Calling Sage from Python

Time for action - calling Sage from a Python script

What just happened?

Have a go hero - solving an ODE symbolically from Python

Introducing Python decorators

Time for action - introducing the Python decorator

What just happened?

Pop quiz - understanding function decorators

Have a go hero - improving the decorator

Have a go hero - the memoize decorator

Making interactive graphics

Time for action - making interactive controls

What just happened?

Using interactive controls

Time for action - an interactive example

What just happened?

Have a go hero - Taylor series

Summary

Index