4. Linear Algebra – Arrays | Scientific Computing with Python 3

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Scientific Computing with Python 3

By : Claus Führer, Claus Fuhrer, Jan Erik Solem, Olivier Verdier

4 (2)

Scientific Computing with Python 3

4 (2)

By: Claus Führer, Claus Fuhrer, Jan Erik Solem, Olivier Verdier

Overview of this book

Python can be used for more than just general-purpose programming. It is a free, open source language and environment that has tremendous potential for use within the domain of scientific computing. This book presents Python in tight connection with mathematical applications and demonstrates how to use various concepts in Python for computing purposes, including examples with the latest version of Python 3. Python is an effective tool to use when coupling scientific computing and mathematics and this book will teach you how to use it for linear algebra, arrays, plotting, iterating, functions, polynomials, and much more.

Preface

Preface

What this book covers

What you need for this book

Who this book is for

Other Python literature

Conventions

Reader feedback

Customer support

Free Chapter

1. Getting Started

1. Getting Started

Installation and configuration instructions

Program and program flow

Basic types

Repeating statements with loops

Conditional statements

Encapsulating code with functions

Scripts and modules

Interpreter

Summary

2. Variables and Basic Types

2. Variables and Basic Types

Variables

Numeric types

Booleans

Strings

Summary

Exercises

3. Container Types

3. Container Types

Lists

Arrays

Tuples

Dictionaries

Sets

Container conversions

Type checking

Summary

Exercises

4. Linear Algebra – Arrays

4. Linear Algebra – Arrays

Overview of the array type

Mathematical preliminaries

The array type

Accessing array entries

Functions to construct arrays

Accessing and changing the shape

Stacking

Functions acting on arrays

Linear algebra methods in SciPy

Summary

Exercises

5. Advanced Array Concepts

5. Advanced Array Concepts

Array views and copies

Comparing arrays

Boolean operations on arrays

Array indexing

Performance and Vectorization

Broadcasting

Sparse matrices

Summary

6. Plotting

6. Plotting

Basic plotting

Formatting

Meshgrid and contours

Images and contours

Matplotlib objects

Making 3D plots

Making movies from plots

Summary

Exercises

7. Functions

7. Functions

Basics

Parameters and arguments

Variable number of arguments

Return values

Recursive functions

Function documentation

Functions are objects

Anonymous functions - the lambda keyword

Functions as decorators

Summary

Exercises

8. Classes

8. Classes

Introduction to classes

Attributes and methods

Attributes that depend on each other

Bound and unbound methods

Class attributes

Class methods

Subclassing and inheritance

Encapsulation

Classes as decorators

Summary

Exercises

9. Iterating

9. Iterating

The for statement

Controlling the flow inside the loop

Iterators

Convergence acceleration

List filling patterns

When iterators behave as lists

Iterator objects

Infinite iterations

Summary

Exercises

10. Error Handling

10. Error Handling

What are exceptions?

Finding Errors: Debugging

Summary

11. Namespaces, Scopes, and Modules

11. Namespaces, Scopes, and Modules

Namespace

Scope of a variable

Modules

Summary

12. Input and Output

12. Input and Output

File handling

NumPy methods

Pickling

Shelves

Reading and writing Matlab data files

Reading and writing images

Summary

13. Testing

13. Testing

Manual testing

Automatic testing

Using unittest package

Parameterizing tests

Assertion tools

Float comparisons

Unit and functional tests

Debugging

Test discovery

Measuring execution time

Summary

Exercises

14. Comprehensive Examples

14. Comprehensive Examples

Polynomials

The polynomial class

Newton polynomial

Spectral clustering

Solving initial value problems

Summary

Exercises

15. Symbolic Computations - SymPy

15. Symbolic Computations - SymPy

What are symbolic computations?

Basic elements of SymPy

Elementary Functions

Symbolic Linear Algebra

Examples for Linear Algebra Methods in SymPy

Substitutions

Evaluating symbolic expressions

Converting a symbolic expression into a numeric function

Summary

References

References

Linear algebra methods in SciPy

SciPy offers a large range of methods from numerical linear algebra in its scipy.linalg module. Many of these methods are Python wrapping programs from LAPACK, a collection of well-approved FORTRAN subroutines used to solve linear equation systems and eigenvalue problems. Linear algebra methods are the core of any method in scientific computing, and the fact that SciPy uses wrappers instead of pure Python code makes these central methods extremely fast. We present in detail here how two linear algebra problems are solved with SciPy to give you a flavour of this module.

Solving several linear equation systems with LU

Let A be an n × n matrix and b₁ , b₂ , ..., b_k be a sequence of n-vectors. We consider the problem to find n vectors x_i such that:

We assume that the vectors b_i are not known simultaneously. In particular, it is quite a common situation that the i^th problem has to be solved before b_i+1 becomes available.

LU factorization is a way to organize...

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