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Sage Beginner's Guide

By : Craig Finch
1 (1)
Book Image

Sage Beginner's Guide

1 (1)
By: Craig Finch

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Sage Beginner's Guide
Craig Finch
May, 2011 | 364 pages
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Table of Contents (17 chapters)
Sage Beginner's Guide
Sage Beginner's Guide
Credits
Credits
About the Author
About the Author
About the Reviewers
About the Reviewers
www.PacktPub.com
www.PacktPub.com
Preface
Preface
Free Chapter
1
What Can You Do with Sage?
What Can You Do with Sage?
Getting started
Using Sage as a powerful calculator
More advanced graphics
A practical example: analysing experimental data
Time for action – fitting the standard curve
Time for action – plotting experimental data
Time for action – fitting a growth model
Summary
2
Installing Sage
Installing Sage
Before you begin
Installing a binary version of Sage on Windows
Installing a binary version of Sage on OS X
Installing a binary version of Sage on GNU/Linux
Building Sage from source
Summary
3
Getting Started with Sage
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
Using the notebook interface
Time for action – doing calculations with the notebook interface
Displaying results of calculations
Operators and variables
Time for action – using strings
Callable symbolic expressions
Time for action – defining callable symbolic expressions
Functions
Time for action – calling functions
Time for action – defining and using your own functions
Time for action – defining a function with keyword arguments
Objects
Time for action – working with objects
Summary
4
Introducing Python and Sage
Introducing Python and Sage
Python 2 and Python 3
Writing code for Sage
Sequence types: lists, tuples, and strings
Time for action – creating lists
Time for action – accessing items in a list
Time for action – returning multiple values from a function
Time for action – working with strings
For loops
Time for action – iterating over lists
Time for action – computing a solution to the diffusion equation
Time for action – using a list comprehension
While loops and text file I/O
Time for action – saving data in a text file
Time for action – reading data from a text file
If statements and conditional expressions
Storing data in a dictionary
Time for action – defining and accessing dictionaries
Lambda forms
Time for action – using lambda to create an anonymous function
Summary
5
Vectors, Matrices, and Linear Algebra
Vectors, Matrices, and Linear Algebra
Vectors and vector spaces
Time for action – working with vectors
Time for action – manipulating elements of vectors
Matrices and matrix spaces
Time for action – solving a system of linear equations
Time for action – accessing elements and parts of a matrix
Time for action – manipulating matrices
Time for action – matrix algebra
Time for action – trying other matrix methods
Time for action – computing eigenvalues and eigenvectors
Time for action – computing the QR factorization
Time for action – computing the singular value decomposition
An introduction to NumPy
Time for action – creating NumPy arrays
Time for action – working with NumPy arrays
Time for action – creating matrices in NumPy
Summary
6
Plotting with Sage
Plotting with Sage
Confusion alert: Sage plots and matplotlib
Plotting in two dimensions
Time for action – plotting symbolic expressions
Time for action – plotting a function with a pole
Time for action – plotting a parametric function
Time for action – making a polar plot
Time for action – plotting a vector field
Time for action – making a scatter plot
Time for action – plotting a list
Time for action – plotting with graphics primitives
Using matplotlib
Time for action – plotting functions with matplotlib
Time for action – getting the matplotlib figure object
Time for action – improving polar plots
Time for action – making a bar chart
Time for action – making a pie chart
Time for action – plotting a histogram
Plotting in three dimensions
Time for action – make an interactive 3D plot
Time for action – parametric plots in 3D
Time for action – making some contour plots
Summary
7
Making Symbolic Mathematics Easy
Making Symbolic Mathematics Easy
Using the notebook interface
Defining symbolic expressions
Time for action – defining callable symbolic expressions
Time for action – defining relational expressions
Time for action – relational expressions with assumptions
Manipulating expressions
Time for action – manipulating expressions
Time for action – working with rational functions
Time for action – substituting symbols in expressions
Time for action – expanding and factoring polynomials
Time for action – manipulating trigonometric expressions
Time for action – simplifying expressions
Solving equations and finding roots
Time for action – solving equations
Time for action – finding roots
Differential and integral calculus
Time for action – calculating limits
Time for action – calculating derivatives
Time for action – calculating integrals
Series and summations
Time for action – computing sums of series
Time for action – finding Taylor series
Laplace transforms
Time for action – computing Laplace transforms
Solving ordinary differential equations
Time for action – solving an ordinary differential equation
Summary
8
Solving Problems Numerically
Solving Problems Numerically
Sage and NumPy
Solving equations and finding roots numerically
Time for action – finding roots of a polynomial
Finding minima and maxima of functions
Time for action – minimizing a function of one variable
Time for action – minimizing a function of several variables
Numerical approximation of derivatives
Time for action – approximating derivatives with differences
Time for action – computing gradients
Numerical integration
Time for action – numerical integration
Time for action – numerical integration with NumPy
Discrete Fourier transforms
Time for action – computing discrete Fourier transforms
Time for action – plotting window functions
Solving ordinary differential equations
Time for action – solving a first-order ODE
Time for action – solving a higher-order ODE
Time for action – alternative method of solving a system of ODEs
Numerical optimization
Time for action – linear programming
Time for action – least squares fitting
Time for action – a constrained optimization problem
Probability
Time for action – accessing probability distribution functions
Summary
9
Learning Advanced Python Programming
Learning Advanced Python Programming
How to write good software
Object-oriented programming
Time for action – defining a class that represents a tank
Time for action – making the tanks move
Time for action – creating your first module
Time for action – creating a vehicle base class
Time for action – creating a combat simulation package
Potential pitfalls when working with classes and instances
Time for action – using class and instance attributes
Time for action – more about class and instance attributes
Time for action – creating empty classes and functions
Handling errors gracefully
Time for action – raising and handling exceptions
Time for action – creating custom exception types
Unit testing
Time for action – creating unit tests for the Tank class
Summary
10
Where to go from here
Where to go from here
Typesetting equations with LaTeX
Time for action – PDF output from the notebook interface
Time for action – working with LaTeX markup in the notebook interface
Time for action – putting it all together
Speeding up execution
Time for action – detecting collisions between spheres
Time for action – detecting collisions: command-line version
Time for action – faster collision detection
Time for action – using NumPy
Time for action – optimizing collision detection with Cython
Calling Sage from Python
Time for action – calling Sage from a Python script
Introducing Python decorators
Time for action – introducing the Python decorator
Making interactive graphics
Time for action – making interactive controls
Using interactive controls
Summary
Index
Index

Numerical approximation of derivatives

In the previous chapter, we learned how to use Sage to compute derivatives of symbolic functions. Now, we will learn how to approximate derivatives numerically.

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