Using R for Numerical Analysis in Science and Engineering

356 Pages·2014·4.945 MB·English

by  Victor A. Bloomfield

The R Series
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• Explains how to statistically analyze and fit data to linear and i
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K13976 l
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K13976_Cover.indd 1 3/18/14 12:29 PM
Using R for Numerical
Analysis in Science
and Engineering
Victor A. Bloomfield
University of Minnesota
Minneapolis, USA
K13976_FM.indd 1 3/24/14 11:19 AM
Chapman & Hall/CRC
The R Series
Series Editors
John M. Chambers Torsten Hothorn
Department of Statistics Division of Biostatistics
Stanford University University of Zurich
Stanford, California, USA Switzerland
Duncan Temple Lang Hadley Wickham
Department of Statistics RStudio
University of California, Davis Boston, Massachusetts, USA
Davis, California, USA
Aims and Scope
This book series reflects the recent rapid growth in the development and application
of R, the programming language and software environment for statistical computing
and graphics. R is now widely used in academic research, education, and industry.
It is constantly growing, with new versions of the core software released regularly
and more than 5,000 packages available. It is difficult for the documentation to
keep pace with the expansion of the software, and this vital book series provides a
forum for the publication of books covering many aspects of the development and
application of R.
The scope of the series is wide, covering three main threads:
• Applications of R to specific disciplines such as biology, epidemiology,
genetics, engineering, finance, and the social sciences.
• Using R for the study of topics of statistical methodology, such as linear and
mixed modeling, time series, Bayesian methods, and missing data.
• The development of R, including programming, building packages, and
graphics.
The books will appeal to programmers and developers of R software, as well as
applied statisticians and data analysts in many fields. The books will feature
detailed worked examples and R code fully integrated into the text, ensuring their
usefulness to researchers, practitioners and students.
K13976_FM.indd 2 3/24/14 11:19 AM
Published Titles
Using R for Numerical Analysis in Science and Engineering , Victor A. Bloomfield
Event History Analysis with R, Göran Broström
Computational Actuarial Science with R, Arthur Charpentier
Statistical Computing in C++ and R, Randall L. Eubank and Ana Kupresanin
Reproducible Research with R and RStudio, Christopher Gandrud
Displaying Time Series, Spatial, and Space-Time Data with R,
Oscar Perpiñán Lamigueiro
Programming Graphical User Interfaces with R, Michael F. Lawrence
and John Verzani
Analyzing Baseball Data with R, Max Marchi and Jim Albert
Growth Curve Analysis and Visualization Using R, Daniel Mirman
R Graphics, Second Edition, Paul Murrell
Customer and Business Analytics: Applied Data Mining for Business Decision
Making Using R, Daniel S. Putler and Robert E. Krider
Implementing Reproducible Research, Victoria Stodden, Friedrich Leisch,
and Roger D. Peng
Dynamic Documents with R and knitr, Yihui Xie
K13976_FM.indd 3 3/24/14 11:19 AM
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.
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Contents
ListofFigures xiii
Preface xix
1 Introduction 1
1.1 ObtainingandinstallingR 1
1.2 LearningR 1
1.3 Learningnumericalmethods 1
1.4 Findinghelp 2
1.5 AugmentingRwithpackages 3
1.6 LearningmoreaboutR 5
1.6.1 Books 5
1.6.2 Onlineresources 5
2 Calculating 7
2.1 Basicoperatorsandfunctions 7
2.2 Complexnumbers 8
2.3 Numericaldisplay,round-offerror,androunding 9
2.4 Assigningvariables 11
2.4.1 Listingandremovingvariables 12
2.5 Relationaloperators 12
2.6 Vectors 13
2.6.1 Vectorelementsandindexes 13
2.6.2 Operationswithvectors 14
2.6.3 Generatingsequences 15
2.6.3.1 Regularsequences 15
2.6.3.2 Repeatingvalues 16
2.6.3.3 Sequencesofrandomnumbers 16
2.6.4 Logicalvectors 17
2.6.5 Speedofforminglargevectors 18
2.6.6 Vectordotproductandcrossproduct 19
2.7 Matrices 21
2.7.1 Formingmatrices 21
2.7.2 Operationsonmatrices 24
2.7.2.1 Arithmeticoperationsonmatrices 24
2.7.2.2 Matrixmultiplication 25
v
vi CONTENTS
2.7.2.3 Transposeanddeterminant 26
2.7.2.4 Matrixcrossproduct 26
2.7.2.5 Matrixexponential 27
2.7.2.6 Matrixinverseandsolve 27
2.7.2.7 Eigenvaluesandeigenvectors 29
2.7.2.8 Singularvaluedecomposition 31
2.7.3 TheMatrixpackage 33
2.7.4 Additionalmatrixfunctionsandpackages 34
2.8 Timeanddatecalculations 34
3 Graphing 37
3.1 Scatterplots 37
3.2 Functionplots 39
3.3 Othercommonplots 40
3.3.1 Barcharts 40
3.3.2 Histograms 42
3.3.3 Box-and-whiskerplots 43
3.4 Customizingplots 44
3.4.1 Pointsandlines 44
3.4.2 Axes,ticks,andpar() 44
3.4.3 Overlayingplotswithgraphicelements 46
3.5 Errorbars 48
3.6 Superimposingvectorsinaplot 49
3.7 Modifyingaxes 50
3.7.1 Logarithmicaxes 51
3.7.2 Supplementaryaxes 51
3.7.3 Incompleteaxisboxes 52
3.7.4 Brokenaxes 52
3.8 Addingtextandmathexpressions 54
3.8.1 Makingmathannotationswithexpression() 55
3.9 Placingseveralplotsinafigure 56
3.10 Two-andthree-dimensionalplots 58
3.11 Theplotrixpackage 60
3.11.1 radial.plot andpolar.plot 60
3.11.2 Triangleplot 61
3.11.3 Errorbarsinplotrix 62
3.12 Animation 63
3.13 Additionalplottingpackages 64
4 Programmingandfunctions 65
4.1 Conditionalexecution:ifandifelse 65
4.2 Loops 66
4.2.1 forloop 66
4.2.2 Loopingwithwhileandrepeat 68
4.3 User-definedfunctions 69
CONTENTS vii
4.4 Debugging 72
4.5 Built-inmathematicalfunctions 73
4.5.1 Besselfunctions 73
4.5.2 Betaandgammafunctions 74
4.5.3 Binomialcoefficients 75
4.6 Specialfunctionsofmathematicalphysics 75
4.6.1 Thegslpackage 75
4.6.2 Specialfunctionsinotherpackages 75
4.7 Polynomialfunctionsinpackages 78
4.7.1 PolynomF package 79
4.7.2 orthopolynompackage 83
4.8 Casestudies 86
4.8.1 Two-dimensionalrandomwalk 86
4.8.2 Eigenvaluesofapolymerchain 87
5 Solvingsystemsofalgebraicequations 91
5.1 Findingthezerosofapolynomial 91
5.2 Findingthezerosofafunction 92
5.2.1 Bisectionmethod 92
5.2.2 Newton’smethod 93
5.2.3 unirootanduniroot.all 94
5.3 Systemsoflinearequations:matrixsolve 96
5.4 Matrixinverse 97
5.5 Singularmatrix 97
5.6 Overdeterminedsystemsandgeneralizedinverse 98
5.7 Sparsematrices 99
5.7.1 Tridiagonalmatrix 99
5.7.2 Bandedmatrix 101
5.7.3 Blockmatrix 102
5.8 Matrixdecomposition 104
5.8.1 QRdecomposition 105
5.8.2 Singularvaluedecomposition 106
5.8.3 Eigendecomposition 107
5.8.4 LUdecomposition 107
5.8.5 Choleskydecomposition 108
5.8.6 Schurdecomposition 109
5.8.7 backsolveandforwardsolve 109
5.9 Systemsofnonlinearequations 109
5.9.1 multirootintherootSolvepackage 109
5.9.2 nleqslv 111
5.9.3 BBsolve() in the BB package 112
5.10 Casestudies 117
5.10.1 Spectroscopicanalysisofamixture 117
5.10.2 vanderWaalsequation 120
5.10.3 Chemicalequilibrium 122
viii CONTENTS
6 Numericaldifferentiationandintegration 125
6.1 Numericaldifferentiation 125
6.1.1 NumericaldifferentiationusingbaseR 125
6.1.1.1 Usingthefundamentaldefinition 125
6.1.1.2 diff() 126
6.1.2 NumericaldifferentiationusingthenumDerivpackage 127
6.1.2.1 grad() 128
6.1.2.2 jacobian() 128
6.1.2.3 hessian 129
6.1.3 Numericaldifferentiationusingthepracmapackage 129
6.1.3.1 fderiv() 129
6.1.3.2 numderiv()andnumdiff() 130
6.1.3.3 grad()andgradient() 131
6.1.3.4 jacobian() 131
6.1.3.5 hessian 132
6.1.3.6 laplacian() 133
6.2 Numericalintegration 133
6.2.1 integrate:BasicintegrationinR 134
6.2.2 Integratingdiscretizedfunctions 136
6.2.3 Gaussianquadrature 137
6.2.4 Moreintegrationroutinesinpracma 140
6.2.5 Functionswithsingularities 142
6.2.6 Infiniteintegrationdomains 144
6.2.7 Integralsinhigherdimensions 146
6.2.8 MonteCarloandsparsegridintegration 148
6.2.9 Complexlineintegrals 150
6.3 SymbolicmanipulationsinR 152
6.3.1 D() 152
6.3.2 deriv() 152
6.3.3 Polynomialfunctions 154
6.3.4 Interfacestosymbolicpackages 155
6.4 Casestudies 155
6.4.1 Circumferenceofanellipse 155
6.4.2 IntegrationofaLorentzianderivativespectrum 156
6.4.3 Volumeofanellipsoid 157
7 Optimization 159
7.1 One-dimensionaloptimization 159
7.2 Multi-dimensionaloptimizationwithoptim() 162
7.2.1 optim()with“Nelder–Mead”default 163
7.2.2 optim()with“BFGS”method 165
7.2.3 optim()with“CG”method 167
7.2.4 optim()with“L-BFGS-B”methodtofindalocalminimum 167
7.3 Otheroptimizationpackages 169
7.3.1 nlm() 169
CONTENTS ix
7.3.2 ucminfpackage 171
7.3.3 BBpackage 171
7.3.4 optimx()wrapper 172
7.3.5 Derivative-freeoptimizationalgorithms 172
7.4 Optimizationwithconstraints 173
7.4.1 constrOptimtooptimizefunctionswithlinearconstraints 173
7.4.2 ExternalpackagesalabamaandRsolnp 175
7.5 Globaloptimizationwithmanylocalminima 177
7.5.1 Simulatedannealing 178
7.5.2 Geneticalgorithms 181
7.5.2.1 DEoptim 181
7.5.2.2 rgenoud 183
7.5.2.3 GA 183
7.6 Linearandquadraticprogramming 183
7.6.1 Linearprogramming 183
7.6.2 Quadraticprogramming 186
7.7 Mixed-integerlinearprogramming 189
7.7.1 Mixed-integerproblems 189
7.7.2 Integerprogrammingproblems 190
7.7.2.1 Knapsackproblems 191
7.7.2.2 Transportationproblems 191
7.7.2.3 Assignmentproblems 192
7.7.2.4 Subsetsumproblems 193
7.8 Casestudy 194
7.8.1 MonteCarlosimulationofthe2DIsingmodel 194
8 Ordinarydifferentialequations 199
8.1 Eulermethod 200
8.1.1 Projectilemotion 201
8.1.2 Orbitalmotion 203
8.2 ImprovedEulermethod 205
8.3 deSolvepackage 208
8.3.1 lsoda()andlsode() 210
8.3.2 “adams”andrelatedmethods 211
8.3.3 Stiffsystems 213
8.4 MatrixexponentialsolutionforsetsoflinearODEs 214
8.5 Eventsandroots 215
8.6 Differenceequations 220
8.7 Delaydifferentialequations 221
8.8 Differentialalgebraicequations 224
8.9 rootSolveforsteadystatesolutionsofsystemsofODEs 227
8.10 bvpSolvepackageforboundaryvalueODEproblems 230
8.10.1 bvpshoot() 230
8.10.2 bvptwp() 231
8.10.3 bvpcol() 232