Table Of ContentJoe Pitt-Francis (cid:2) Jonathan Whiteley
Guide to Scientific
Computing in C++
Dr.JoePitt-Francis Dr.JonathanWhiteley
DepartmentofComputerScience DepartmentofComputerScience
UniversityofOxford UniversityofOxford
Oxford,UK Oxford,UK
ISSN1863-7310 UndergraduateTopicsinComputerScience
ISBN978-1-4471-2735-2 e-ISBN978-1-4471-2736-9
DOI10.1007/978-1-4471-2736-9
SpringerLondonDordrechtHeidelbergNewYork
BritishLibraryCataloguinginPublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
LibraryofCongressControlNumber:2012931858
©Springer-VerlagLondonLimited2012
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Preface
ManybookshavebeenwrittenontheC++programminglanguage,varyingacross
aspectrumfromtheverypracticaltotheverytheoretical.This bookcertainlylies
at the practical end of this spectrum, and has a particular focus for the practical
treatmentofthislanguage:scientificcomputing.
Traditionally, Fortran and MATLAB®1 have been the languages of choice for
scientificcomputingapplications.Therecentdevelopmentofcomplexmathemati-
calmodels—infieldsasdiverseasbiology,finance,andmaterialsscience,toname
butafew—hasdrivenaneedforsoftwarepackagesthatallowcomputationalsim-
ulationsbasedonthesemodels.Thecomplexityoftheunderlyingmodels,together
withtheneedtoexchangecodebetweencoworkers,hasmotivatedprogrammersto
developobject-orientedcode(oftenwritteninC++)forthesesimulationpackages.
Thecomputationaldemandsofthesesimulationsmayrequiresoftwaretobewrit-
tenforparallelcomputingfacilities,typicallyusingtheMessagePassingInterface
(MPI).Theneedtotrainprogrammersintheskillstoprogramapplicationssuchas
theseledtothedevelopmentofagraduatelevelcourseC++forScientificComput-
ing,taughtbytheauthorsofthisbook,attheUniversityofOxford.
This book provides a guide to C++ programming in scientific computing. In
contrast to many other books on C++, features of the language are demonstrated
mainlyusingexamplesdrawnfromscientificcomputing.Object-orientationisfirst
mentioned in Chap. 1 where we briefly describe what this phrase—and other re-
lated terms such as inheritance—mean, before postponing any further discussion
ofobject-orientationorrelatedtopicsuntilChap.6.Intheinterveningchaptersun-
til object-orientation reappears, we present what is best described as “procedural
programminginC++”,coveringvariables,flowofcontrol,inputandoutput,point-
ers (including dynamic allocation of memory), functions and reference variables.
ArmedwiththisgroundinginC++wethenintroduceclassesinChaps.6and7.In
thesetwochapters,wherethemainfeaturesofobject-orientationareshowcased,we
initially,forthesakeofclarity,abandonourprincipleofusingexamplesdrawnfrom
scientificcomputing.Oncethetopicshavebeenpresentedhowever,weresumeour
strategy of demonstrating concepts through scientific computing examples. More
advancedC++featuressuchastemplatesandexceptionsareintroducedinChaps.8
and9.HavingintroducedthefeaturesofC++requiredforscientificcomputing,the
1MATLABisaregisteredtrademarkofTheMathWorks,Inc.
remainderofthebookfocusesontheapplicationofthesefeatures.InChap.10,we
begintodevelopacollectionofclassesforlinearalgebracalculations:theseclasses
are then developed further in the exercises at the end of this chapter. Chapter 11
presentsanintroductiontoparallelcomputingusingMPI.Finally,inChap.12,we
discuss how an object-oriented library for solving second order differential equa-
tionsmaybeconstructed.Theimportanceofaclearprogrammingstyletominimise
theintroductionoferrorsintocodeisstressedthroughoutthebook.
Thisbookisaimedatprogrammersofalllevelsofexpertisewhowishtowrite
scientific computing programs in C++. Experience with a computer to the level
wherefilescanbestoredandeditedisexpected.Abasicknowledgeofmathematics,
suchasoperationsbetweenvectorsandmatrices,andtheNewton–Raphsonmethod
forfindingtherootsofnonlinearequationswouldbeanadvantage.
Thematerialpresentedherehasbeenenhancedsignificantlybydiscussionsabout
C++withcolleagues,toonumeroustolisthere,intheDepartmentofComputerSci-
enceattheUniversityofOxford.Aspecialmentionmust,however,bemadeofthe
Chaste2 programming team: particular gratitude should be expressed to Jonathan
Cooper for readily sharing with us his impressively wide and deep knowledge of
theC++language.Othermembersoftheteamwhohavesignificantlyhelpedclarify
ourthoughtsontheC++languageareMiguelBernabeu,JamesOsborne,PrasPath-
manathanandJamesSouthern.WeshouldalsothankstudentsfromboththeM.Sc.
in Mathematical Modelling and Scientific Computing and the Doctoral Training
CentresattheUniversityofOxfordforunwittinglyaidingourunderstandingofthe
languagethroughaskingpertinentquestions.
Finally,itisalwaysimportanttoremember—especiallywhendebuggingapar-
ticularly tiresome code—that there is far more to life than C++ programming for
scientific computing. We would both like to thank our families for their love and
support,especiallyduringthewritingofthisbook.
Oxford,UK JoePitt-Francis
JonathanWhiteley
2TheCancer,HeartAndSoftTissueEnvironment(Chaste)isanobject-orientedpackage,written
in C++, for simulations in the field of biology. More details on this package may be found at
http://www.cs.ox.ac.uk/chaste/.
Contents
1 GettingStarted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 ABriefIntroductiontoC++ . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 C++is“Object-Oriented” . . . . . . . . . . . . . . . . . . 1
1.1.2 WhyYouShouldWriteScientificProgramsinC++ . . . . 2
1.1.3 WhyYouShouldNotWriteScientificProgramsinC++ . . 4
1.1.4 ScopeofThisBook . . . . . . . . . . . . . . . . . . . . . 4
1.2 AFirstC++Program . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 CompilingaC++Program. . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 IntegratedDevelopmentEnvironments . . . . . . . . . . . 6
1.3.2 CompilingattheCommandLine . . . . . . . . . . . . . . 7
1.3.3 CompilerFlags. . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.1 BasicNumericalVariables. . . . . . . . . . . . . . . . . . 10
1.4.2 OtherNumericalVariables . . . . . . . . . . . . . . . . . 12
1.4.3 MathematicalOperationsonNumericalVariables . . . . . 13
1.4.4 DivisionofIntegers . . . . . . . . . . . . . . . . . . . . . 15
1.4.5 Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.6 ASCIICharacters . . . . . . . . . . . . . . . . . . . . . . 17
1.4.7 BooleanVariables . . . . . . . . . . . . . . . . . . . . . . 17
1.4.8 Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 SimpleInputandOutput . . . . . . . . . . . . . . . . . . . . . . . 19
1.5.1 BasicConsoleOutput . . . . . . . . . . . . . . . . . . . . 19
1.5.2 KeyboardInput . . . . . . . . . . . . . . . . . . . . . . . 20
1.6 TheassertStatement . . . . . . . . . . . . . . . . . . . . . . . 21
1.7 Tips:DebuggingCode . . . . . . . . . . . . . . . . . . . . . . . . 22
1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 FlowofControl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1 TheifStatement . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.1 ASingleifStatement . . . . . . . . . . . . . . . . . . . 26
2.1.2 Example:CodeforaSingleifStatement . . . . . . . . . 27
2.1.3 if–elseStatements . . . . . . . . . . . . . . . . . . . . 27
2.1.4 MultipleifStatements . . . . . . . . . . . . . . . . . . . 27
2.1.5 NestedifStatements . . . . . . . . . . . . . . . . . . . . 28
2.1.6 BooleanVariables . . . . . . . . . . . . . . . . . . . . . . 28
2.2 LogicalandRelationalOperators . . . . . . . . . . . . . . . . . . 29
2.3 ThewhileStatement . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 LoopsUsingtheforStatement . . . . . . . . . . . . . . . . . . . 32
2.4.1 Example:CalculatingtheScalarProductofTwoVectors . . 34
2.5 TheswitchStatement . . . . . . . . . . . . . . . . . . . . . . . 34
2.6 Tips:LoopsandBranches . . . . . . . . . . . . . . . . . . . . . . 35
2.6.1 Tip1:ACommonNoviceCodingError . . . . . . . . . . 35
2.6.2 Tip2:CountingfromZero. . . . . . . . . . . . . . . . . . 36
2.6.3 Tip3:EqualityVersusAssignment . . . . . . . . . . . . . 37
2.6.4 Tip4:NeverEndingwhileLoops . . . . . . . . . . . . . 38
2.6.5 Tip5:ComparingTwoFloatingPointNumbers . . . . . . 39
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3 FileInputandOutput . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1 RedirectingConsoleOutputtoFile . . . . . . . . . . . . . . . . . 43
3.2 WritingtoFile . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.1 SettingthePrecisionoftheOutput . . . . . . . . . . . . . 46
3.3 ReadingfromFile . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4 ReadingfromtheCommandLine . . . . . . . . . . . . . . . . . . 49
3.5 Tips:ControllingOutputFormat . . . . . . . . . . . . . . . . . . 50
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 Pointers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1 PointersandtheComputer’sMemory . . . . . . . . . . . . . . . . 55
4.1.1 Addresses . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1.2 PointerVariables . . . . . . . . . . . . . . . . . . . . . . . 56
4.1.3 ExampleUseofPointers . . . . . . . . . . . . . . . . . . 56
4.1.4 WarningsontheUseofPointers. . . . . . . . . . . . . . . 57
4.2 DynamicAllocationofMemoryforArrays . . . . . . . . . . . . . 58
4.2.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.3 IrregularlySizedMatrices . . . . . . . . . . . . . . . . . . 61
4.3 Tips:Pointers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.1 Tip1:PointerAliasing. . . . . . . . . . . . . . . . . . . . 62
4.3.2 Tip2:SafeDynamicAllocation . . . . . . . . . . . . . . . 63
4.3.3 Tip3:EverynewHasadelete . . . . . . . . . . . . . . 63
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5 Blocks,FunctionsandReferenceVariables. . . . . . . . . . . . . . . 65
5.1 Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2.1 SimpleFunctions . . . . . . . . . . . . . . . . . . . . . . 66
5.2.2 ReturningPointerVariablesfromaFunction . . . . . . . . 69
5.2.3 UseofPointersasFunctionArguments . . . . . . . . . . . 70
5.2.4 SendingArraystoFunctions . . . . . . . . . . . . . . . . 71
5.2.5 Example:AFunctiontoCalculatetheScalarProductof
TwoVectors . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3 ReferenceVariables . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.4 DefaultValuesforFunctionArguments . . . . . . . . . . . . . . . 75
5.5 FunctionOverloading . . . . . . . . . . . . . . . . . . . . . . . . 76
5.6 DeclaringFunctionsWithoutPrototypes . . . . . . . . . . . . . . 78
5.7 FunctionPointers . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.8 RecursiveFunctions . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.9 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.10 Tips:CodeDocumentation . . . . . . . . . . . . . . . . . . . . . 83
5.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6 AnIntroductiontoClasses . . . . . . . . . . . . . . . . . . . . . . . . 87
6.1 TheRaisond’ÊtreforClasses . . . . . . . . . . . . . . . . . . . . 87
6.1.1 ProblemsThatMayAriseWhenUsingModules . . . . . . 88
6.1.2 Abstraction,EncapsulationandModularityPropertiesof
Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2 AFirstExampleSimpleClass:AClassofBooks . . . . . . . . . . 89
6.2.1 BasicFeaturesofClasses . . . . . . . . . . . . . . . . . . 89
6.2.2 HeaderFiles . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.2.3 SettingandAccessingVariables . . . . . . . . . . . . . . . 92
6.2.4 CompilingMultipleFiles . . . . . . . . . . . . . . . . . . 94
6.2.5 AccessPrivileges . . . . . . . . . . . . . . . . . . . . . . 96
6.2.6 IncludingFunctionImplementationsinHeaderFiles . . . . 97
6.2.7 ConstructorsandDestructors . . . . . . . . . . . . . . . . 98
6.2.8 PointerstoClasses . . . . . . . . . . . . . . . . . . . . . . 103
6.3 ThefriendKeyword . . . . . . . . . . . . . . . . . . . . . . . 103
6.4 ASecondExampleClass:AClassofComplexNumbers. . . . . . 105
6.4.1 OperatorOverloading . . . . . . . . . . . . . . . . . . . . 105
6.4.2 TheClassofComplexNumbers . . . . . . . . . . . . . . . 106
6.5 SomeAdditionalRemarksonOperatorOverloading . . . . . . . . 112
6.6 Tips:CodingtoaStandard. . . . . . . . . . . . . . . . . . . . . . 112
6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7 InheritanceandDerivedClasses . . . . . . . . . . . . . . . . . . . . 117
7.1 Inheritance,ExtensibilityandPolymorphism . . . . . . . . . . . . 117
7.2 Example:AClassofE-booksDerivedfromaClassofBooks . . . 118
7.3 AccessPrivilegesforDerivedClasses . . . . . . . . . . . . . . . . 120
7.4 ClassesDerivedfromDerivedClasses . . . . . . . . . . . . . . . 121
7.5 Run-TimePolymorphism . . . . . . . . . . . . . . . . . . . . . . 122
7.6 TheAbstractClassPattern . . . . . . . . . . . . . . . . . . . . . . 124
7.7 Tips:UsingaDebugger . . . . . . . . . . . . . . . . . . . . . . . 126
7.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8 Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.1 TemplatestoControlDimensionsandVerifySizes . . . . . . . . . 131
8.2 TemplatesforPolymorphism . . . . . . . . . . . . . . . . . . . . 133
8.3 ABriefSurveyoftheStandardTemplateLibrary . . . . . . . . . . 134
8.3.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.3.2 Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.4 Tips:TemplateCompilation . . . . . . . . . . . . . . . . . . . . . 139
8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
9 ErrorsandExceptions . . . . . . . . . . . . . . . . . . . . . . . . . . 141
9.1 Preconditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
9.1.1 Example:TwoImplementationsofaGraphicsFunction . . 142
9.2 ThreeLevelsofErrors . . . . . . . . . . . . . . . . . . . . . . . . 143
9.3 IntroducingtheException . . . . . . . . . . . . . . . . . . . . . . 144
9.4 UsingExceptions . . . . . . . . . . . . . . . . . . . . . . . . . . 145
9.5 Tips:Test-DrivenDevelopment . . . . . . . . . . . . . . . . . . . 146
9.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
10 DevelopingClassesforLinearAlgebraCalculations . . . . . . . . . 151
10.1 RequirementsoftheLinearAlgebraClasses . . . . . . . . . . . . 151
10.2 ConstructorsandDestructors . . . . . . . . . . . . . . . . . . . . 156
10.2.1 TheDefaultConstructor . . . . . . . . . . . . . . . . . . . 156
10.2.2 TheCopyConstructor . . . . . . . . . . . . . . . . . . . . 156
10.2.3 ASpecialisedConstructor . . . . . . . . . . . . . . . . . . 157
10.2.4 Destructor . . . . . . . . . . . . . . . . . . . . . . . . . . 157
10.3 AccessingPrivateClassMembers . . . . . . . . . . . . . . . . . . 157
10.3.1 AccessingtheSizeofaVector. . . . . . . . . . . . . . . . 158
10.3.2 OverloadingtheSquareBracketOperator . . . . . . . . . . 158
10.3.3 Read-OnlyAccesstoVectorEntries. . . . . . . . . . . . . 158
10.3.4 OverloadingtheRoundBracketOperator . . . . . . . . . . 158
10.4 OperatorOverloadingforVectorOperations . . . . . . . . . . . . 158
10.4.1 TheAssignmentOperator . . . . . . . . . . . . . . . . . . 159
10.4.2 UnaryOperators . . . . . . . . . . . . . . . . . . . . . . . 159
10.4.3 BinaryOperators. . . . . . . . . . . . . . . . . . . . . . . 159
10.5 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
10.5.1 MembersVersusFriends. . . . . . . . . . . . . . . . . . . 159
10.6 Tips:MemoryDebuggingTools . . . . . . . . . . . . . . . . . . . 160
10.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
11 AnIntroductiontoParallelProgrammingUsingMPI. . . . . . . . . 165
11.1 DistributedMemoryArchitectures . . . . . . . . . . . . . . . . . 165
11.2 InstallingMPI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
11.3 AFirstProgramUsingMPI . . . . . . . . . . . . . . . . . . . . . 167
11.3.1 EssentialMPIFunctions . . . . . . . . . . . . . . . . . . . 168
11.3.2 CompilingandRunningMPICode . . . . . . . . . . . . . 169
11.4 BasicMPICommunication . . . . . . . . . . . . . . . . . . . . . 171
11.4.1 Point-to-PointCommunication . . . . . . . . . . . . . . . 171
11.4.2 CollectiveCommunication . . . . . . . . . . . . . . . . . 174
11.5 ExampleMPIApplications . . . . . . . . . . . . . . . . . . . . . 180
11.5.1 SummationofSeries . . . . . . . . . . . . . . . . . . . . . 180
11.5.2 ParallelLinearAlgebra . . . . . . . . . . . . . . . . . . . 182
11.6 Tips:DebuggingaParallelProgram . . . . . . . . . . . . . . . . . 186
11.6.1 Tip1:MakeanAbstractProgram . . . . . . . . . . . . . . 186
11.6.2 Tip2:DatatypeMismatch . . . . . . . . . . . . . . . . . . 186
11.6.3 Tip3:IntermittentDeadlock. . . . . . . . . . . . . . . . . 187
11.6.4 Tip4:AlmostCollectiveCommunication . . . . . . . . . . 187
11.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
12 DesigningObject-OrientedNumericalLibraries . . . . . . . . . . . 193
12.1 DevelopingtheLibraryforOrdinaryDifferentialEquations . . . . 194
12.1.1 ModelProblems . . . . . . . . . . . . . . . . . . . . . . . 194
12.1.2 FiniteDifferenceApproximationtoDerivatives . . . . . . 195
12.1.3 ApplicationofFiniteDifferenceMethodstoBoundary
ValueProblems . . . . . . . . . . . . . . . . . . . . . . . 197
12.1.4 ConcludingRemarksonBoundaryValueProblemsinOne
Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 199
12.2 DesigningaLibraryforSolvingBoundaryValueProblems . . . . 200
12.2.1 TheClassSecondOrderOde . . . . . . . . . . . . . . . 200
12.2.2 TheClassBoundaryConditions . . . . . . . . . . . . 201
12.2.3 TheClassFiniteDifferenceGrid . . . . . . . . . . 202
12.2.4 TheClassBvpOde . . . . . . . . . . . . . . . . . . . . . 203
12.2.5 UsingtheClassBvpOde . . . . . . . . . . . . . . . . . . 205
12.3 ExtendingtheLibrarytoTwoDimensions . . . . . . . . . . . . . 205
12.3.1 ModelProblemforTwoDimensions . . . . . . . . . . . . 207
12.3.2 FiniteDifferenceMethodsforBoundaryValueProblems
inTwoDimensions . . . . . . . . . . . . . . . . . . . . . 207
12.3.3 SettingUptheLinearSystemfortheModelProblem . . . 209
12.3.4 DevelopingtheClassesRequired . . . . . . . . . . . . . . 210
12.4 Tips:UsingWell-WrittenLibraries . . . . . . . . . . . . . . . . . 210
12.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
AppendixA LinearAlgebra. . . . . . . . . . . . . . . . . . . . . . . . . 213
A.1 VectorsandMatrices . . . . . . . . . . . . . . . . . . . . . . . . . 213
A.1.1 OperationsBetweenVectorsandMatrices . . . . . . . . . 214
A.1.2 TheScalarProductofTwoVectors . . . . . . . . . . . . . 215
A.1.3 TheDeterminantandtheInverseofaMatrix . . . . . . . . 215
A.1.4 EigenvaluesandEigenvectorsofaMatrix. . . . . . . . . . 216
A.1.5 VectorandMatrixNorms . . . . . . . . . . . . . . . . . . 216
A.2 SystemsofLinearEquations . . . . . . . . . . . . . . . . . . . . 217
A.2.1 GaussianElimination . . . . . . . . . . . . . . . . . . . . 217
A.2.2 TheThomasAlgorithm . . . . . . . . . . . . . . . . . . . 222
A.2.3 TheConjugateGradientMethod . . . . . . . . . . . . . . 222
AppendixB OtherProgrammingConstructsYouMightMeet. . . . . . 225
B.1 CStyleOutput . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
B.2 CStyleDynamicMemoryAllocation . . . . . . . . . . . . . . . . 226
B.3 Ternary?:Operator . . . . . . . . . . . . . . . . . . . . . . . . . 226
B.4 UsingNamespace . . . . . . . . . . . . . . . . . . . . . . . . . . 227
B.5 Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
B.6 MultipleInheritance . . . . . . . . . . . . . . . . . . . . . . . . . 228
B.7 ClassInitialisers . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
AppendixC SolutionstoExercises . . . . . . . . . . . . . . . . . . . . . 231
C.1 MatrixandLinearSystemClasses. . . . . . . . . . . . . . . . . . 231
C.2 ODESolverLibrary . . . . . . . . . . . . . . . . . . . . . . . . . 240
FurtherReading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
MathematicalMethodsandLinearAlgebra . . . . . . . . . . . . . 245
C++Programming . . . . . . . . . . . . . . . . . . . . . . . . . . 245
TheMessage-PassingInterface(MPI) . . . . . . . . . . . . . . . . 245
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
