First Steps in Numerical Analysis

232 Pages·1998·0.679 MB·English

by  R. J. Hosking, S. Joe, D. C. Joyce, J. C. Turner

CONTENTS
Prefacetothesecondedition xi
Prefacetothefirstedition xiii
Prologue 1
1 Historical 1
2 NumericalAnalysistoday 2
3 Thisbook 3
ERRORS
Step1 Sourcesoferror 4
1 Example 4
Step2 Approximationtonumbers 7
1 Numberrepresentation 7
2 Round-offerror 8
3 Truncationerror 8
4 Mistakes 8
5 Examples 8
Step3 Errorpropagationandgeneration 10
1 Absoluteerror 10
2 Relativeerror 10
3 Errorpropagation 11
4 Errorgeneration 11
5 Example 12
Step4 Floatingpointarithmetic 14
1 Additionandsubtraction 14
2 Multiplication 14
3 Division 15
4 Expressions 15
5 Generatederror 15
6 Consequences 15
vi CONTENTS
Step5 Approximationtofunctions 18
1 TheTaylorseries 18
2 Polynomialapproximation 20
3 Otherseriesexpansions 20
4 Recursiveprocedures 20
NONLINEAREQUATIONS
Step6 Nonlinearalgebraicandtranscendentalequations 23
1 Atranscendentalequation 23
2 Locatingroots 24
Step7 Thebisectionmethod 27
1 Procedure 27
2 Effectiveness 28
3 Example 28
Step8 Methodoffalseposition 30
1 Procedure 30
2 Effectivenessandthesecantmethod 31
3 Example 32
Step9 Themethodofsimpleiteration 34
1 Procedure 34
2 Example 34
3 Convergence 35
Step10 TheNewton-Raphsoniterativemethod 37
1 Procedure 37
2 Example 38
3 Convergence 39
4 Speedofconvergence 40
5 Thesquareroot 41
SYSTEMSOFLINEAREQUATIONS
Step11 Solutionbyelimination 42
1 Notationanddefinitions 42
2 Theexistenceofsolutions 43
3 Gaussianeliminationmethod 44
4 Thetransformationoperations 45
5 Generaltreatmentoftheeliminationprocess 45
6 Numericalexample 48
CONTENTS vii
Step12 Errorsandill-conditioning 51
1 Errorsinthecoefficientsandconstants 51
2 Round-offerrorsandnumbersofoperations 52
3 Partialpivoting 52
4 Ill-conditioning 53
Step13 TheGauss-Seideliterativemethod 56
1 Iterativemethods 56
2 TheGauss-Seidelmethod 56
3 Convergence 57
Step14 Matrixinversion* 59
1 Theinversematrix 59
2 Methodforinvertingamatrix 59
3 Solutionoflinearsystemsusingtheinversematrix 61
Step15 UseofLUdecomposition* 64
1 Procedure 64
2 Example 65
3 EffectinganLUdecomposition 66
Step16 Testingforill-conditioning* 69
1 Norms 69
2 Testingforill-conditioning 70
THEEIGENVALUEPROBLEM
Step17 Thepowermethod 73
1 Powermethod 74
2 Example 74
3 Variants 75
4 Otheraspects 77
FINITEDIFFERENCES
Step18 Tables 79
1 Tablesofvalues 79
2 Finitedifferences 80
3 Influenceofround-offerrors 80
Step19 Forward,backward,andcentraldifferencenotations 83
1 Theshiftoperator E 83
2 Theforwarddifferenceoperator(cid:49) 83
3 Thebackwarddifferenceoperator∇ 84
viii CONTENTS
4 Thecentraldifferenceoperatorδ 84
5 Differencedisplay 85
Step20 Polynomials 88
1 Finitedifferencesofapolynomial 88
2 Example 89
3 Approximationofafunctionbyapolynomial 89
INTERPOLATION
Step21 Linearandquadraticinterpolation 92
1 Linearinterpolation 92
2 Quadraticinterpolation 94
Step22 Newtoninterpolationformulae 96
1 Newton’sforwarddifferenceformula 96
2 Newton’sbackwarddifferenceformula 96
3 UseofNewton’sinterpolationformulae 97
4 Uniquenessoftheinterpolatingpolynomial 98
5 AnalogywithTaylorseries 99
Step23 Lagrangeinterpolationformula 101
1 Procedure 101
2 Example 102
3 Notesofcaution 103
Step24 Divideddifferences* 104
1 Divideddifferences 104
2 Newton’sdivideddifferenceformula 105
3 Example 105
4 Errorininterpolatingpolynomial 106
5 Aitken’smethod 107
Step25 Inverseinterpolation* 110
1 Linearinverseinterpolation 110
2 Iterativeinverseinterpolation 110
3 Divideddifferences 111
CURVEFITTING
Step26 Leastsquares 114
1 Theproblemillustrated 114
2 Generalapproachtotheproblem 115
3 Errors‘assmallaspossible’ 116
CONTENTS ix
4 Theleastsquaresmethodandnormalequations 116
5 Example 117
Step27 Leastsquaresandlinearequations* 122
1 Pseudo-inverse 122
2 Normalequations 123
3 QRfactorization 124
4 TheQRfactorizationprocess 126
Step28 Splines* 129
1 Constructionofcubicsplines 129
2 Examples 132
NUMERICALDIFFERENTIATION
Step29 Finitedifferences 135
1 Procedure 135
2 Errorinnumericaldifferentiation 136
3 Example 137
NUMERICALINTEGRATION
Step30 Thetrapezoidalrule 139
1 Thetrapezoidalrule 139
2 Accuracy 140
3 Example 141
Step31 Simpson’srule 143
1 Simpson’srule 143
2 Accuracy 144
3 Example 145
Step32 Gaussianintegrationformulae 146
1 Gausstwo-pointintegrationformula 146
2 OtherGaussformulae 147
3 ApplicationofGaussianquadrature 148
ORDINARYDIFFERENTIALEQUATIONS
Step33 Single-stepmethods 149
1 Taylorseries 149
2 Runge-Kuttamethods 150
3 Example 151
x CONTENTS
Step34 Multistepmethods 153
1 Introduction 153
2 Stability 154
Step35 Higherorderdifferentialequations* 156
1 Systemsoffirst-orderinitialvalueproblems 156
2 Numericalmethodsforfirst-ordersystems 157
3 Numericalexample 158
AppliedExercises 160
Appendix: Pseudo-code 163
AnswerstotheExercises 173
Bibliography 216
Index 217
PREFACE TO THE SECOND EDITION
FirstStepsinNumericalAnalysis,originallypublishedin1978,isnowinitstwelfth
impression. It has been widely used in schools, polytechnics, and universities
throughouttheworld. However,wedecidedthatafteralifeofseventeenyearsin
the classroom and lecture theatre, the contents of the book should be reviewed.
Feedbackfrommanyusers,bothteachersandstudents,couldbeincorporated;and
thedevelopmentofthesubjectsuggestedthatsomenewtopicsmightbeincluded.
This Second Edition of the book is the outcome of our consideration of these
matters.
Thechangeswehavemadearenotveryextensive,whichreflectsourviewthat
the syllabus for a first course in Numerical Analysis must continue to include
mostofthebasictopicsintheFirstEdition. However,theresultofrapidchanges
in computer technology is that some aspects are obviously less important than
they were, and other topics have become more important. We decided that less
shouldbesaidaboutfinitedifferences,forexample,butmoreshouldbesaidabout
systemsoflinearequationsandmatrices. Newmaterialhasbeenaddedoncurve
fitting(forexample, useofsplines), andmorehasbeengivenonthesolutionof
differentialequations. ThetotalnumberofStepshasincreasedfrom31to35.
Forthebenefitofbothteachersandstudents,additionalexerciseshavebeenset
at the end of many of the Steps, and brief answers again supplied. Also, a set
ofAppliedExerciseshasbeenincluded,tochallengestudentstoapplynumerical
methods in the context of ‘real world’ applications. To make it easier for users
to implement the given algorithms in a computer program, the flowcharts in the
AppendixoftheFirstEditionhavebeenreplacedbypseudo-code. Themethodof
organizingthematerialintoSTEPS(ofalengthsuitableforpresentationinoneor
twohours)hasbeenretained,forthishasbeenapopularfeatureofthebook.
We hope that these changes and additions, together with the new typesetting
used, will be found acceptable, enhancing and attractive; and that the book will
continuetobewidelyused. Manyoftheideaspresentedshouldbeaccessibleto
studentsinmathematicsatthelevelofSeventhForminNewZealand,Year12in
Australia,orGCEAlevelintheUnitedKingdom. Theadditionofmore(optional)
starredStepsinthisEditionmakesthisbookalsosuitableforfirstandsecondyear
introductoryNumericalAnalysiscoursesinpolytechnicsanduniversities.
R.J.Hosking
S.Joe
D.C.Joyce
J.C.Turner
1995
xii
PREFACE TO THE FIRST EDITION
Asitstitlesuggests,thisbookisintendedtoprovideanintroductiontoelementary
conceptsandmethodsofNumericalAnalysisforstudentsmeetingthesubjectfor
thefirsttime. Inparticular,theideaspresentedshouldbeaccessibleatthelevelof
SeventhFormAppliedMathematicsinNewZealandoratAdvancedLevelG.C.E.
intheUnitedKingdom. Weexpectthisbookwillalsobefoundusefulformany
coursesinpolytechnicsanduniversities.
For ease of teaching and learning, the material is divided into short ‘Steps’,
mostofwhichwouldbeincludedinanyfirstcourse. Adiscussionofthecontent
andplanofthebookisgiveninsection3ofthePrologue.
R.J.Hosking
D.C.Joyce
J.C.Turner
1978
xiv