Solving Problems Numerically | Sage Beginner's Guide

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Sage Beginner's Guide

By : Craig Finch

3.9 (11)

Sage Beginner's Guide

3.9 (11)

By: Craig Finch

Overview of this book

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

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

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

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

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

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

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

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

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

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

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

Chapter 8. Solving Problems Numerically

The previous chapter described how to use Sage to solve many difficult problems in symbolic mathematics. While this capability is very useful, many real-world problems do not lend themselves to symbolic computation. Some differential equations don't have closed-form solutions, and not every integral can be computed in terms of elementary functions. In other cases, a function value may have to be computed from a look-up table that was derived from experimental results, which precludes symbolic computation. In this chapter, we will demonstrate some of the tools in Sage that allow us to solve problems numerically.

We will learn how to:

Find the roots of an equation
Compute integrals and derivatives numerically
Find minima and maxima of functions
Compute discrete Fourier transforms, and apply window functions
Numerically solve an ordinary differential equation (ODE), and systems of ODEs
Use optimization techniques to fit curves and find minima
Explore the probability...

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