An Introduction to Scientific Computing with MATLAB® and Python Tutorials

398 Pages·2022·4.67 MB·English

by  Sheng Xu

An Introduction to
Scientific Computing
®
with MATLAB and
Python Tutorials

An Introduction to
Scientific Computing
®
with MATLAB and
Python Tutorials
Sheng Xu
MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks
does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion
of MATLAB® software or related products does not constitute endorsement or sponsorship by The
MathWorks of a particular pedagogical approach or particular use of the MATLAB® software.
First edition published 2022
by CRC Press
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and by CRC Press
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© 2022 Sheng Xu
CRC Press is an imprint of Taylor & Francis Group, LLC
Reasonable efforts have been made to publish reliable data and information, but the author and pub-
lisher cannot assume responsibility for the validity of all materials or the consequences of their use.
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ISBN: 978-1-032-06315-7 (hbk)
ISBN: 978-1-032-06318-8 (pbk)
ISBN: 978-1-003-20169-4 (ebk)
DOI: 10.1201/9781003201694
Typeset in Nimbus Roman
by KnowledgeWorks Global Ltd.
Publisher’s note: This book has been prepared from camera-ready copy provided by the authors
To my family and friends.

Contents
Preface xiii
Author xv
1 AnOverviewofScientificComputing 1
1.1 WhatIsScientificComputing? . . . . . . . . . . . . . . . . . . . . 1
1.2 ErrorsinScientificComputing . . . . . . . . . . . . . . . . . . . . 3
1.2.1 AbsoluteandRelativeErrors . . . . . . . . . . . . . . . . . 3
1.2.2 UpperBounds . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 SourcesofErrors . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 AlgorithmProperties . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Taylor’sTheorem 9
2.1 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 PolynomialEvaluation . . . . . . . . . . . . . . . . . . . . 10
2.2 Taylor’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 TaylorPolynomials . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 TaylorSeries . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.3 Taylor’sTheorem . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 AlternatingSeriesTheorem . . . . . . . . . . . . . . . . . . . . . 19
2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 ProgrammingProblems . . . . . . . . . . . . . . . . . . . . . . . 22
3 RoundoffErrorsandErrorPropagation 25
3.1 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.1 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Floating-PointNumbers . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.1 ScientificNotationandRounding . . . . . . . . . . . . . . 27
3.2.2 DPFloating-PointRepresentation . . . . . . . . . . . . . . 29
3.3 ErrorPropagation . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3.1 CatastrophicCancellation . . . . . . . . . . . . . . . . . . 32
3.3.2 AlgorithmStability . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5 ProgrammingProblems . . . . . . . . . . . . . . . . . . . . . . . 37
vii
viii Contents
4 DirectMethodsforLinearSystems 43
4.1 MatricesandVectors . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 TriangularSystems . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3 GEandA=LU . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3.1 ElementaryMatrices . . . . . . . . . . . . . . . . . . . . . 51
4.3.2 A=LU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3.3 SolvingAx=bbyA=LU . . . . . . . . . . . . . . . . . . 55
4.4 GEPPandPA=LU . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4.1 GEPP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4.2 PA=LU . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4.3 SolvingAx=bbyPA=LU . . . . . . . . . . . . . . . . . . 65
4.5 TridiagonalSystems . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.6 ConditioningofLinearSystems . . . . . . . . . . . . . . . . . . . 69
4.6.1 VectorandMatrixNorms. . . . . . . . . . . . . . . . . . . 72
4.6.2 ConditionNumbers . . . . . . . . . . . . . . . . . . . . . . 76
4.6.3 ErrorandResidualVectors . . . . . . . . . . . . . . . . . . 77
4.7 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.9 ProgrammingProblems . . . . . . . . . . . . . . . . . . . . . . . 82
5 RootFindingforNonlinearEquations 85
5.1 RootsandFixedPoints . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2 TheBisectionMethod . . . . . . . . . . . . . . . . . . . . . . . . 90
5.3 Newton’sMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3.1 ConvergenceAnalysisofNewton’sMethod . . . . . . . . . 96
5.3.2 PracticalIssuesofNewton’sMethod . . . . . . . . . . . . . 99
5.4 SecantMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.5 Fixed-PointIteration . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.6 Newton’sMethodforSystemsofNonlinearEquations . . . . . . . 109
5.6.1 Taylor’sTheoremforMultivariateFunctions . . . . . . . . 110
5.6.2 Newton’sMethodforNonlinearSystems . . . . . . . . . . 114
5.7 UnconstrainedOptimization . . . . . . . . . . . . . . . . . . . . . 117
5.8 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.10 ProgrammingProblems . . . . . . . . . . . . . . . . . . . . . . . 127
6 Interpolation 131
6.1 TerminologyofInterpolation . . . . . . . . . . . . . . . . . . . . . 132
6.2 PolynomialSpace . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.2.1 ChebyshevBasis . . . . . . . . . . . . . . . . . . . . . . . 135
6.2.2 LegendreBasis . . . . . . . . . . . . . . . . . . . . . . . . 138
6.3 MonomialInterpolation . . . . . . . . . . . . . . . . . . . . . . . 141
6.4 LagrangeInterpolation . . . . . . . . . . . . . . . . . . . . . . . . 143
6.5 Newton’sInterpolation . . . . . . . . . . . . . . . . . . . . . . . . 145
6.6 InterpolationError . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Contents ix
6.6.1 ErrorinPolynomialInterpolation . . . . . . . . . . . . . . 150
6.6.2 BehaviorofInterpolationError . . . . . . . . . . . . . . . 153
6.6.2.1 Equally-SpacedNodes. . . . . . . . . . . . . . . 153
6.6.2.2 ChebyshevNodes . . . . . . . . . . . . . . . . . 156
6.7 SplineInterpolation . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.7.1 PiecewiseLinearInterpolation . . . . . . . . . . . . . . . . 159
6.7.2 CubicSpline . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.7.3 CubicSplineInterpolation . . . . . . . . . . . . . . . . . . 163
6.8 DiscreteFourierTransform(DFT) . . . . . . . . . . . . . . . . . . 167
6.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.10 ProgrammingProblems . . . . . . . . . . . . . . . . . . . . . . . 177
7 NumericalIntegration 183
7.1 DefiniteIntegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.2 NumericalIntegration . . . . . . . . . . . . . . . . . . . . . . . . 187
7.2.1 ChangeofIntervals . . . . . . . . . . . . . . . . . . . . . . 189
7.3 TheMidpointRule . . . . . . . . . . . . . . . . . . . . . . . . . . 191
7.3.1 DegreeofPrecision(DOP) . . . . . . . . . . . . . . . . . . 192
7.3.2 ErroroftheMidpointRule . . . . . . . . . . . . . . . . . . 195
7.4 TheTrapezoidalRule . . . . . . . . . . . . . . . . . . . . . . . . . 200
7.5 Simpson’sRule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
7.6 Newton-CotesRules . . . . . . . . . . . . . . . . . . . . . . . . . 209
7.7 GaussianQuadratureRules . . . . . . . . . . . . . . . . . . . . . . 209
7.8 OtherNumericalIntegrationTechniques . . . . . . . . . . . . . . . 216
7.8.1 IntegrationwithSingularities. . . . . . . . . . . . . . . . . 216
7.8.2 AdaptiveIntegration . . . . . . . . . . . . . . . . . . . . . 217
7.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
7.10 ProgrammingProblems . . . . . . . . . . . . . . . . . . . . . . . 221
8 NumericalDifferentiation 225
8.1 DifferentiationUsingTaylor’sTheorem . . . . . . . . . . . . . . . 225
8.1.1 TheMethodofUndeterminedCoefficients . . . . . . . . . 227
8.2 DifferentiationUsingInterpolation . . . . . . . . . . . . . . . . . 229
8.2.1 DifferentiationUsingDFT . . . . . . . . . . . . . . . . . . 231
8.3 RichardsonExtrapolation . . . . . . . . . . . . . . . . . . . . . . 232
8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
8.5 ProgrammingProblems . . . . . . . . . . . . . . . . . . . . . . . 235
9 InitialValueProblemsandBoundaryValueProblems 237
9.1 InitialValueProblems(IVPs) . . . . . . . . . . . . . . . . . . . . 238
9.1.1 Euler’sMethod . . . . . . . . . . . . . . . . . . . . . . . . 239
9.1.1.1 LocalTruncationErrorandGlobalError . . . . . 242
9.1.1.2 Consistency,ConvergenceandStability . . . . . . 244
9.1.1.3 ExplicitandImplicitMethods . . . . . . . . . . . 247
9.1.2 TaylorSeriesMethods . . . . . . . . . . . . . . . . . . . . 248