Python for Mathematics. The Python Series

Preface xv
Section I Overview
Chapter 1 ■ Introduction....3
1.1 WHO IS THIS BOOK FOR?....3
1.1.1 How is this book different from similar books?....4
1.2 WHAT IS IN THIS BOOK?....6
1.2.1 How is this book organised?....6
1.2.2 How to use this book?....7
1.2.3 How is coded is played in this book?....7
1.3 WHAT IS NOT THIS BOOK?....9
Section II Tools for Mathematics
Chapter 2 ■ Using Notebooks....13
2.1 INTRODUCTION....13
2.2 TUTORIAL....13
2.2.1 Installation....13
2.2.2 Starting a Jupyter notebook server....14
2.2.3 Creating a new notebook....14
2.2.4 Organising your files....14
2.2.5 Writing some basic Python code....17
2.2.6 Writing markdown....20
2.2.7 Saving your notebook to a different format....20
2.3 HOWTO....21
2.3.1 Install Anaconda....21
2.3.2 Start a Jupyter notebook server....21
2.3.3 Create a new notebook....21
2.3.4 Find/open a notebook....21
2.3.5 Run Python code....21
2.3.6 Carry out basic arithmetic operations....22
2.3.7 Write markdown....22
2.3.8 Write basic LaTeX....22
2.3.9 Save the output in a different format....23
2.4 EXERCISES....24
2.5 FURTHER INFORMATION....24
2.5.1 Why use anaconda?....24
2.5.2 Why use Jupyter?....24
2.5.3 Why can I not double click on a Jupyter notebook file?....25
2.5.4 Where can I find keyboard shortcuts for using Jupyter....25
2.5.5 What is markdown?....25
2.5.6 What is LaTeX?....25
2.5.7 Can I use \( and \) instead of $ for LATEX?....26
2.5.8 What is a mark up language?....26
Chapter 3 ■ Algebra....27
3.1 INTRODUCTION....27
3.2 TUTORIAL....27
3.3 HOWTO....33
3.3.1 Createasymbolicnumericvalue....33
3.3.2 Getthenumericalvalueofasymbolicexpression....34
3.3.3 Factoranexpression....35
3.3.4 Expandanexpression....35
3.3.5 Simplifyanexpression....36
3.3.6 Solveanequation....36
3.3.7 Substituteavalueintoanexpression....38
3.4 EXERCISES....38
3.5 FURTHERINFORMATION....39
3.5.1 Whyissomecodeinseparatelibraries?....39
3.5.2 WhydoIneedtousesympy?....39
3.5.3 WhydoIsometimessee from sympyimport*?....40
3.5.4 Howtoextractasolutionfromtheoutputof sympy.solveset?....42
3.5.5 WhydoIsometimessee import sympyassym?....44
Chapter 4 ■ Calculus....45
4.1 TUTORIAL....45
4.2 HOWTO....50
4.2.1 Calculatethederivativeofanexpression....50
4.2.2 Calculatetheindefiniteintegralofanexpression....51
4.2.3 Calculatethedefiniteintegralofanexpression....51
4.2.4 Use ∞....52
4.2.5 Calculatelimitsofanexpression....52
4.3 EXERCISES....53
4.4 FURTHERINFORMATION....53
4.4.1 Howcanyouplotafunction....53
Chapter 5 ■ Matrices....56
5.1 TUTORIAL....56
5.2 HOWTO....59
5.2.1 Createamatrix....59
5.2.2 Calculatethedeterminantofamatrix....60
5.2.3 Calculatetheinverseofamatrix....60
5.2.4 Multiplymatricesbyascalar....61
5.2.5 Addmatricestogether....61
5.2.6 Multiplymatricestogether....62
5.2.7 Createavector....62
5.2.8 Solvealinearsystem....62
5.3 EXERCISES....63
5.4 FURTHERINFORMATION....64
5.4.1 Whydoesthisbooknotdiscusscommentingofcode?....64
5.4.2 Whyuse @ for matrixmultiplicationandnot *?....64
5.4.3 IsNumpyalibrarythatcanbeusedforlinearalgebra?....65
Chapter 6 ■ Combinatorics....66
6.1 TUTORIAL....66
6.2 HOWTO....70
6.2.1 Createatuple....70
6.2.2 Howtoaccessparticularelementsinatuple....70
6.2.3 Createbooleanvariables....70
6.2.4 Createaniterablewithagivennumberofitems....73
6.2.5 Createpermutationsofagivensetofelements....73
6.2.6 Createcombinationsofagivensetofelements....74
6.2.7 Summingitemsinaniterable....75
6.2.8 Directlycompute n!....76
6.2.9 Directlycompute
n
i
76
6.2.10 Directlycompute nPi....77
6.3 EXERCISES....77
6.4 FURTHERINFORMATION....78
6.4.1 Arethereotherwaystoaccesselementsintuples?....78
6.4.2 Whydo range, itertools.permutations, and
itertools.combinations not directlygivetheelements?....78
6.4.3 Howdoesthesummationnotation correspondtothecode?78
x ■ Contents
Chapter 7 ■ Probability....80
7.1 TUTORIAL....80
7.2 HOWTO....87
7.2.1 Createalist....87
7.2.2 Defineafunction....89
7.2.3 Callafunction....89
7.2.4 Runcodebasedonacondition....90
7.2.5 Createalistusingalistcomprehension....92
7.2.6 Summingitemsinalist....93
7.2.7 Samplefromaniterable....94
7.2.8 Samplearandomnumber....95
7.2.9 Reproducerandomevents....95
7.3 EXERCISES....96
7.4 FURTHERINFORMATION....97
7.4.1 WhatisthedifferencebetweenaPythonlistandaPythontuple?97
7.4.2 Whydoesthesumofbooleanscountthe Trues?....97
7.4.3 Whatisthedifferencebetween print and return?....98
7.4.4 HowdoesPythonsamplerandomness?....99
7.4.5 Whatisthedifferencebetweenadocstringandacomment....99
Chapter 8 ■ Sequences....102
8.1 TUTORIAL....102
8.2 HOWTO....104
8.2.1 Defineafunctionusingrecursion....104
8.3 EXERCISES....105
8.4 FURTHERINFORMATION....106
8.4.1 Whatarethedifferencesbetweenrecursionanditeration?....106
8.4.2 Whatiscaching?....108
Chapter 9 ■ Statistics....110
9.1 TUTORIAL....110
9.2 HOWTO....118
9.2.1 Calculatemeasuresofspreadandtendency....118
9.2.2 Calculatethesamplecovariance....122
9.2.3 CalculatethePearsoncorrelationcoefficient....123
9.2.4 Fitalineofbestfit....123
9.2.5 Createaninstanceofthenormaldistribution....124
9.2.6 Usethecumulativedistributionfunctionofanormaldistribution125
9.2.7 Usetheinversecumulativedistributionfunctionofanormal
distribution....125
9.3 EXERCISES....126
Contents ■ xi
9.4 FURTHERINFORMATION....129
9.4.1 Whatisthedifferencebetweenthesampleandthepopulation
varianceandstandarddeviation?....129
9.4.2 Howtoplotalineofbestfit?....130
9.4.3 WhatotherstatisticstoolsexistinPython?....131
9.4.4 Whatisthedifferencebetweenmachinelearningandstatistics?131
Chapter 10 ■ DifferentialEquations....132
10.1 TUTORIAL....132
10.2 HOWTO....134
10.2.1 Createasymbolicfunction....134
10.2.2 Createadifferentialequation....135
10.2.3 Obtainthegeneralsolutionofadifferentialequation....136
10.2.4 Obtaintheparticularsolutionofadifferentialequation....137
10.3 EXERCISES....137
10.4 FURTHERINFORMATION....138
10.4.1 Howtosolveasystemofdifferentialequations?....138
10.4.2 Howtosolvedifferentialequationsnumerically?....139
Section III Building Tools
Chapter 11 ■ Variables,ConditionalsandLoops....145
11.1 TUTORIAL....145
11.2 HOWTO....148
11.2.1 Defineanintegervariable....148
11.2.2 Defineafloatvariable....149
11.2.3 Defineastringvariable....149
11.2.4 Defineabooleanvariable....150
11.2.5 Checkthetypeofavariable....150
11.2.6 Manipulatenumericvariables....151
11.2.7 Includevariablesinstrings....152
11.2.8 Combinecollectionsofbooleanvariables....153
11.2.9 Runcode if a conditionholds....153
11.2.10 Repeatcode for a givensetofvariables....154
11.2.11 Repeatcode while a givenconditionholds....155
11.2.12 Iterateoverpairsofitemsfromtwoiterables....156
11.2.13 Iterateoverandindexitemsfromaniterable....156
11.3 EXERCISES....157
11.4 FURTHERINFORMATION....158
11.4.1 WhycanInotonlyusea while loop?....158
11.4.2 WhyshouldInotcheckifabooleanisequalto True or False....159
xii ■ Contents
Chapter 12 ■ Functions andDataStructures....161
12.1 TUTORIAL....161
12.2 HOWTO....164
12.2.1 Defineafunction....164
12.2.2 Writeadocstring....164
12.2.3 Createatuple....165
12.2.4 Createalist....165
12.2.5 Createalistusingalistcomprehension....165
12.2.6 Combinelists....165
12.2.7 Appendanelementtoalist....166
12.2.8 Removeanelementfromalist....166
12.2.9 Sortalist....167
12.2.10 Createasortedlistfromaniterable....167
12.2.11 Accessanelementofaniterable....168
12.2.12 Findtheindexofanelementinaniterable....168
12.2.13 Accessanelementofaniterableusingnegativeindexing....168
12.2.14 Sliceaniterable....169
12.2.15 Findthenumberofelementsinaniterable....169
12.2.16 Createaset....170
12.2.17 Dosetoperations....170
12.2.18 Createhashtables....171
12.2.19 Accesselementinahashtable....172
12.2.20 Iterateoverkeysinahashtable....173
12.2.21 Iterateovervaluesinahashtable....173
12.2.22 Iterateoverpairsofkeysandvalueinahashtable....173
12.3 EXERCISES....174
12.4 FURTHERINFORMATION....175
12.4.1 Whatformatscanbeusedtowriteadocstring?....175
12.4.2 Aretheretoolsavailabletoassistwithwritingdocstrings?175
12.4.3 Apartfromremovingduplicatesandsetoperationswhatarethe
advantagesofusing set?....175
Chapter 13 ■ Object-OrientedProgramming....176
13.1 TUTORIAL....176
13.2 HOWTO....181
13.2.1 Defineaclass....181
13.2.2 Createaninstanceoftheclass....181
13.2.3 Createanattribute....182
13.2.4 Createandcallamethod....183
13.2.5 Howtocreateandcallmagicmethods....184
13.2.6 Useinheritance....186
Contents ■ xiii
13.3 EXERCISES....187
13.4 FURTHERINFORMATION....188
13.4.1 Howtopronouncethedoubleunderscore?....188
13.4.2 Whatisthe self variablefor?....188
13.4.3 Whyuse CamelCase for classesbut snake case for functions?190
13.4.4 Whatisthedifferencebetweenamethodandafunction?....190
Chapter 14 ■ Using aCommandLineandanEditor....191
14.1 TUTORIAL....191
14.2 HOWTO....196
14.2.1 Navigatedirectoriesusingthecommandline....196
14.2.2 Createanewdirectoryusingthecommandline....197
14.2.3 Seethecontentsofadirectoryinthecommandline....197
14.2.4 RunPythoncodeinafile....198
14.2.5 RunPythoncodewithoutusingafileorJupyter....198
14.2.6 InstallVScodeplugins....198
14.3 EXERCISES....199
14.4 FURTHERINFORMATION....199
14.4.1 Whydoyouneedtousethe print function withaneditor?199
14.4.2 CanyouuseaPythonplugintorunmycodefrominsidemyeditor?199
14.4.3 CanIopenaJupyternotebookinsideVScode?....200
14.4.4 WhatisthedifferencebetweenanIntegratedDevelopment
Environmentandaneditor?....200
14.4.5 CanIuse \( and \) instead of $ for LATEX?....200
Chapter 15 ■ Modularisation....201
15.1 TUTORIAL....201
15.2 HOWTO....207
15.2.1 ImportcodefromPythonfiles....207
15.2.2 Breakupcodeintomodularcomponents....208
15.3 EXERCISES....210
15.4 FURTHERINFORMATION....212
15.4.1 Whymodularise?....212
15.4.2 WhydoIgetanimporterror?....213
15.4.3 HowdoImakemyfileimportablefromotherdirectories?....213
Chapter 16 ■ Documentation....214
16.1 TUTORIAL....214
16.1.1 Writingatutorial....214
16.1.2 Writingthehow-toguides....216
16.1.3 Writingtheexplanationssection....218
xiv ■ Contents
16.1.4 Writingthereferencesection....221
16.2 HOWTO....222
16.2.1 Writedocumentation....222
16.2.2 Writemarkdown....223
16.3 EXERCISES....225
16.4 FURTHERINFORMATION....225
16.4.1 Whatisdocumentation?....225
16.4.2 Whatisthepurposeofthefourseparatesectionsindocumentation?226
16.4.3 Whatalternativesaretheretowritingdocumentationinthe
README.md file?....226
Chapter 17 ■ Testing....227
17.1 TUTORIAL....227
17.1.1 Writingtestsforcode....227
17.1.2 Testingdocumentation....231
17.1.3 Documentingthetests....236
17.2 HOWTO....237
17.2.1 Writean assert statement....237
17.2.2 Write assert statementsforcodethatactsrandomly....238
17.2.3 Writeatestfile....239
17.2.4 Formatdoctests....241
17.2.5 Rundoctests....242
17.3 EXERCISES....242
17.4 FURTHERINFORMATION....242
17.4.1 Whyaretestswrittenasfunctions?....242
17.4.2 Isthereamoreefficientwaytoruntests?....242
17.4.3 Whatshouldbetested?....243
17.4.4 Whydoyouneeddoctests?....243
17.4.5 Whatistestdrivendevelopment?....243
17.4.6 Howaremodularisation,documentationandtestingrelated?244
Section IV AboutThisBook
Chapter 18 ■ HowThisBookIsWritten....247
Bibliography....249
Index....251

Python for Mathematics introduces readers to effective methods for doing mathematics using the Python programming language. Most programming texts introduce readers to the building blocks of programming and build up to using more sophisticated tools for a specific purpose, like doing particular mathematical tasks. This is akin to teaching someone how to forge metal so as to make a nail, and then slowly working up to using sophisticated power tools so as to actually build something. This book does things in a different way, by first getting readers to begin using and understanding the tools that are going to be helpful to them as mathematicians, and only then moving onto the granular details. In this way, the practical application of the tools can aid in the understanding of the theory.

Features

• Complete with engaging, practical exercises
• Many useful and detailed coding examples
• Suitable for undergraduates in mathematics, and other quantitative sciences
• Empowers readers to design and create their own Python tools.