Scientific Computing with Python 3

Table of Contents

Scientific Computing with Python 3

Credits

About the Authors

About the Reviewer

www.PacktPub.com

Why subscribe?

Acknowledgement

Preface

What this book covers

What you need for this book

Who this book is for

Python vs Other Languages

Other Python literature

Conventions

Reader feedback

Customer support

Downloading the example code

Downloading the color images of this book

Errata

Piracy

Questions

1. Getting Started

Installation and configuration instructions

Installation

Anaconda

Configuration

Python Shell

Executing scripts

Getting Help

Jupyter – Python notebook

Program and program flow

Comments

Line joining

Basic types

Numbers

Strings

Variables

Lists

Operations on lists

Boolean expressions

Repeating statements with loops

Repeating a task

Break and else

Conditional statements

Encapsulating code with functions

Scripts and modules

Simple modules - collecting functions

Using modules and namespaces

Interpreter

Summary

2. Variables and Basic Types

Variables

Numeric types

Integers

Plain integers

Floating point numbers

Floating point representation

Infinite and not a number

Underflow - Machine Epsilon

Other float types in NumPy

Complex numbers

Complex Numbers in Mathematics

The j notation

Real and imaginary parts

Booleans

Boolean operators

Boolean casting

Automatic Boolean casting

Return values of and and or

Boolean and integer

Strings

Operations on strings and string methods

String formatting

Summary

Exercises

3. Container Types

Lists

Slicing

Strides

Altering lists

Belonging to a list

List methods

In–place operations

Merging lists – zip

List comprehension

Arrays

Tuples

Dictionaries

Creating and altering dictionaries

Looping over dictionaries

Sets

Container conversions

Type checking

Summary

Exercises

4. Linear Algebra – Arrays

Overview of the array type

Vectors and matrices

Indexing and slices

Linear algebra operations

Solving a linear system

Mathematical preliminaries

Arrays as functions

Operations are elementwise

Shape and number of dimensions

The dot operations

The array type

Array properties

Creating arrays from lists

Accessing array entries

Basic array slicing

Altering an array using slices

Functions to construct arrays

Accessing and changing the shape

The shape function

Number of dimensions

Reshape

Transpose

Stacking

Stacking vectors

Functions acting on arrays

Universal functions

Built-in universal functions

Create universal functions

Array functions

Linear algebra methods in SciPy

Solving several linear equation systems with LU

Solving a least square problem with SVD

More methods

Summary

Exercises

5. Advanced Array Concepts

Array views and copies

Array views

Slices as views

Transpose and reshape as views

Array copy

Comparing arrays

Boolean arrays

Checking for equality

Boolean operations on arrays

Array indexing

Indexing with Boolean arrays

Using where

Performance and Vectorization

Vectorization

Broadcasting

Mathematical view

Constant functions

Functions of several variables

General mechanism

Conventions

Broadcasting arrays

The broadcasting problem

Shape mismatch

Typical examples

Rescale rows

Rescale columns

Functions of two variables

Sparse matrices

Sparse matrix formats

Compressed sparse row

Compressed Sparse Column

Row-based linked list format

Altering and slicing matrices in LIL format

Generating sparse matrices

Sparse matrix methods

Summary

6. Plotting

Basic plotting

Formatting

Meshgrid and contours

Images and contours

Matplotlib objects

The axes object

Modifying line properties

Annotations

Filling areas between curves

Ticks and ticklabels

Making 3D plots

Making movies from plots

Summary

Exercises

7. Functions

Basics

Parameters and arguments

Passing arguments - by position and by keyword

Changing arguments

Access to variables defined outside the local namespace

Default arguments

Beware of mutable default arguments

Variable number of arguments

Return values

Recursive functions

Function documentation

Functions are objects

Partial application

Using Closures

Anonymous functions - the  lambda keyword

The lambda construction is always replaceable

Functions as decorators

Summary

Exercises

8. Classes

Introduction to classes

Class syntax

The __init__ method

Attributes and methods

Special methods

Reverse operations

Attributes that depend on each other

The property function

Bound and unbound methods

Class attributes

Class methods

Subclassing and inheritance

Encapsulation

Classes as decorators

Summary

Exercises

9. Iterating

The for statement

Controlling the flow inside the loop

Iterators

Generators

Iterators are disposable

Iterator tools

Generators of recursive sequences

 Arithmetic geometric mean

Convergence acceleration

List filling patterns

List filling with the append method

List from iterators

Storing generated values

When iterators behave as lists

Generator expression

Zipping iterators

Iterator objects

Infinite iterations

The while loop

Recursion

Summary

Exercises

10. Error Handling

What are exceptions?

Basic principles

Raising exceptions

Catching exceptions

User-defined exceptions

Context managers - the with statement

Finding Errors: Debugging

Bugs

The stack

The Python debugger

Overview - debug commands

Debugging in IPython

Summary

11. Namespaces, Scopes, and Modules

Namespace

Scope of a variable

Modules

Introduction

Modules in IPython

The IPython magic command

The variable __name__

Some useful modules

Summary

12. Input and Output

File handling

Interacting with files

Files are iterable

File modes

NumPy methods

savetxt

 loadtxt

Pickling

Shelves

Reading and writing Matlab data files

Reading and writing images

Summary

13. Testing

Manual testing

Automatic testing

Testing the bisection algorithm

Using unittest package

Test setUp and tearDown methods

Parameterizing tests

Assertion tools

Float comparisons

Unit and functional tests

Debugging

Test discovery

Measuring execution time

Timing with a magic function

Timing with the Python module timeit

Timing with a context manager

Summary

Exercises

14. Comprehensive Examples

Polynomials

Theoretical background

Tasks

The polynomial class

Newton polynomial

Spectral clustering

Solving initial value problems

Summary

Exercises

15. Symbolic Computations - SymPy

What are symbolic computations?

Elaborating an example in SymPy

Basic elements of SymPy

Symbols - the basis of all formulas

Numbers

Functions

Undefined functions

Elementary Functions

Lambda - functions

Symbolic Linear Algebra

Symbolic matrices

Examples for Linear Algebra Methods in SymPy

Substitutions

Evaluating symbolic expressions

Example: A study on the convergence order of Newton's Method

Converting a symbolic expression into a numeric function

A study on the parameter dependency of polynomial coefficients

Summary

References

Scientific Computing with Python 3

Scientific Computing with Python 3

Copyright © 2016 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 authors, 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: December 2016

Production reference: 1141216

Published by Packt Publishing Ltd.

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B3 2PB, UK.

ISBN 978-1-78646-351-7

www.packtpub.com

Credits

About the Authors

Claus Führer is a professor of scientific computations at Lund University, Sweden. He has an extensive teaching record that includes intensive programming courses in numerical analysis and engineering mathematics across various levels in many different countries and teaching environments. Claus also develops numerical software in research collaboration with industry and received Lund University’s Faculty of Engineering Best Teacher Award in 2016.

Jan Erik Solem is a Python enthusiast, former associate professor, and currently the CEO of Mapillary, a street imagery computer vision company. He has previously worked as a face recognition expert, founder and CTO of Polar Rose, and computer vision team leader at Apple. Jan is a World Economic Forum technology pioneer and won the Best Nordic Thesis Award 2005-2006 for his dissertation on image analysis and pattern recognition. He is also the author of Programming Computer Vision with Python (O'Reilly 2012).

Olivier Verdier began using Python for scientific computing back in 2007 and received a PhD in mathematics from Lund University in 2009. He has held post-doctoral positions in Cologne, Trondheim, Bergen, and Umeå and is now an associate professor of mathematics at Bergen University College, Norway.

About the Reviewer

Helmut Podhaisky works in the Institute of Mathematics at the Martin Luther University in Halle-Wittenberg, where he teaches mathematics and scientific computing. He has co-authored a book on numerical methods for ordinary differential equations as well as several research papers on numerical methods. For work and fun, he uses Python, Fortran, Octave, Mathematica, and Haskell.

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Acknowledgement

We want to acknowledge the competent and helpful comments and suggestions by Helmut Podhaisky, Halle University, Germany. To have such a partner in the process of writing a book is a big luck and chance for the authors.

We would also like to express our gratitude towards the reviewers of the first edition of this book, [7], Linda Kann, KTH Stockholm, Hans Petter Langtangen, Simula Research Laboratory, and Alf Inge Wang, NTNU Trondheim.

A book has to be tested in teaching. And here we had fantastic partners: the teaching assistants from the course Beräkningsprogramering med Python during the years and the colleagues involved in teaching: Najmeh Abiri, Christian Andersson, Dara Maghdid, Peter Meisrimel, Fatemeh Mohammadi, Azahar Monge, Anna-Maria Persson, Alexandros Sopasakis, Tony Stillfjord, Lund University. Najmeh Abiri also tested most of the Jupyter notebook material which you find on the book's webpage.

A book has not only to be written, it has to be published, and in this process Aishwarya Pandere and Karan Thakkar, PACKT Publishing, were always constructive, friendly and helpful partners bridging different time zones and different text processing tools. Thanks.

Claus Führer, Jan-Erik Solem, Olivier Verdier  Lund, Bergen 2016

Preface

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.

What this book covers

Chapter 1, Getting Started, addresses the main language elements of Python without going into detail. Here we make a brief tour through all. It is a good starting point for those who want to start directly. It is a quick reference for those readers who want in a later chapter understand an example which uses might use constructs like functions before functions were explained in deep .

Chapter 2, Variables and Basic Types, presents the most important and basic types in Python. Float is the more important datatype in scientific computing together with the special numbers nan and inf. Booleans, integers, complex, and strings are other basic datatypes, which will be used throughout this book.

Chapter 3, Container Types, explains how to work with container types, mainly lists. Dictionaries and tuples will be explained as well as indexing and looping, through container objects. Occasionally, one uses even sets as a special container type.

Chapter 4, Linear Algebra, works with the most important objects in linear algebra--vectors and matrices. This book chooses NumPy array as the central tool for describing matrices and even higher order tensors. Arrays have many advanced features and allows also for universal functions acting on matrices or vectors elementwise. The book emphasizes on array indexing, slices, and the dot product as the basic operation in most computing tasks. Some linear algebra examples are worked out to demonstrate the use of SciPy's submodule linalg.

Chapter 5, Advanced Array Concepts, explains some more advanced aspects of arrays. The difference between array copies and views is explained extensively as views make programs using arrays very fast but are often a source for errors, which are hard to debug. The use of Boolean arrays to write effective, compact, and readable code is shown and demonstrated. Finally, the technique of array broadcasting-- a unique feature of NumPy arrays -- is explained by its analogy to operations performed on functions.

Chapter 6, Plotting, shows how to make plots, mainly classical x/yplots but also 3D plots and histograms. Scientific computing requires good tools for visualizing the results. Python's module matplotlib is introduced starting from the handy plotting commands in its submodule pyplot. Finetuning and modifying plots becomes possible by creating graphical objects such as axes. We show how attributes of these objects can be changed and annotations can be made.

Chapter 7, Functions, form the fundamental building block in programming, which is probably nearest to underlying mathematical concepts. Function definition and function calls are explained as the different ways to set function arguments. Anonymous lambda functions are introduced and used in various examples throughout the book.

Chapter 8, Classes, defines objects as instances of classes, which we provide with methods and attributes. In mathematics, class attributes often depend on each other, which requires special programming techniques for setter and getter functions. Basic mathematical operations such as + can be defined for special mathematic datatypes. Inheritance and abstraction are mathematical concepts which are reflected by object oriented programming. We demonstrate the use of inheritance by a simple solver class for ordinary differential equations.

Chapter 9, Iterating, presents iteration using loops and iterators. There is now a chapter in this book without loops and iterations, but here we come to principles of iterators and create own generator objects. In this chapter, you learn why a generator can be exhausted and how infinite loops can be programmed. Python's module itertools is a useful companion for this chapter.

Chapter 10, Error Handling, covers errors and exceptions and how to find and fix them. An error or an exception is an event, which breaks the execution of a program unit. This chapter shows what to do then, that is, how an exception can be handled. You learn to define your own exception classes and how to provide valuable information, which can be used for catching these exceptions. Error handling is more than printing an error message.

Chapter 11, Namespaces, Scopes and Modules, covers Python modules. What are local and global variables? When is a variable known and when is it unknown to a program unit? This is discussed in this chapter. A variable can be passed to a function by a parameter list or tacitly injected by making use of its scope. When should this technique be applied and when not? This chapter tries to give an answer to this central question.

Chapter 12, Input and Output, covers some options for handling data files. Data files are used for storing and providing data for a given problem, often large scale measurements. This chapter describes how this data can be accessed and modified using different formats.

Chapter 13, Testing, focuses on testing for scientific programming. The key tool is unittest, which allows for automatic testing and parametrized tests. By considering the classical bisection algorithm in numerical mathematics, we exemplify different steps to design meaningful tests, which as a side effect also deliver a documentation of the use of a piece of code. Careful testing provides test protocols which can be later helpful when debugging a complex code often written by many different programmers.

Chapter 14, Comprehensive Examples, presents some comprehensive and longer examples together with a brief introduction to the theoretical background and their complete implementation. These examples make use of all constructs shown in the book so far and put them in a larger and more complex context. They are open for extensions by the reader.

Chapter 15, Symbolic Computations - SymPy, speaks about symbolic computations. Scientific computing is mainly numeric computations with inexact data and approximative results. This is contrasted by symbolic computations often formal manipulation, which aims for exact solutions in a closed form expression. In this last chapter of the book, we introduce this technique in Python, which is often used for deriving and verifying theoretically mathematical models and numerical results. We emphasize on high precision floating point evaluation of symbolic expressions.

What you need for this book

You would need Pyhon3.5 or higher, SciPy, NumPy, Matplotlib, IPython shell (we recommend strongly to install Python and its packages through Anaconda). The examples of the book do not have any special hardware requirements on memory and graphics.

Who this book is for

This book is the outcome of a course on Python for scientific computing which is taught at Lund University since 2008. The course expanded over the years, and condensed versions of the material were taught at universities in Cologne, Trondheim, Stavanger, Soran, Lappeenranta and also in computation oriented companies.

Our belief is that Python and its surrounding scientific computing ecosystem — SciPy, NumPY and matplotlib — represent a tremendous progress in scientific computing environment. Python and the aforementioned libraries are free and open source. What’s more, is a modern language featuring all the bells and whistles that this adjective entails: object oriented programming, testing, advanced shell with IPython, etc. When writing this book we had two groups of readers in mind:

The reader who chooses Python as his or her first programming language will use this book in a teacher-led course. The book guides into the different topics and offers background reading and experimenting. A teacher typically selects and orders the material from this book in such a way, that it fits to the specific learning outcomes of an introductory course.

The reader who already has some experience in programming, and some taste for scientific computing or mathematics will use this book as a companion when diving into the world of Scipy and Numpy. Programming in Python can be quite different from programming in MATLAB, say. The book wants to point out the pythonic way of programming, which makes programming a pleasure.

Our goal is to explain the steps to get started with Python in the context of scientific computing. The book may be read either from the first page to the last, or by picking the bits that seem most interesting. Needless to say, as improving one’s programming skills requires considerable practice, it is highly advisable to experiment and play with the examples and the exercises in the book.

We hope that the readers will enjoy programming with Python, SciPy, NumPY and matplotlib as much as we do.

Python vs Other Languages

When it comes to deciding what language to use for a book on scientific computing many factors