Numerical Methods, Algorithms and Tools in C'

583 Pages·2009·2.55 MB·English

by  Waldemar Dos Passos

Numerical
Methods,
Algorithms
Tools
and in
C#
© 2010 by Taylor and Francis Group, LLC
Numerical
Methods,
Algorithms
Tools
and in
C#
Waldemar Dos Passos
Boca Raton London New York
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Library of Congress Cataloging‑in‑Publication Data
Dos Passos, Waldemar.
Numerical methods, algorithms, and tools in C# / Waldemar Dos Passos.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-8493-7479-1 (hardcover : alk. paper)
1. Numerical analysis--Data processing. 2. Algorithms. 3. C# (Computer program
language) I. Title.
QA297.D684 2010
518.0285’5133--dc22 2009031461
Visit the Taylor & Francis Web site at
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© 2010 by Taylor and Francis Group, LLC
Preface
Today,morethanatanyothertimeinthehistoryofmankind,computersareincreas-
inglyandsuccessfullybeingexploitedtogainabetterunderstandingofourphysical
worldandasaresult,alsodeepenourappreciationandreverenceforGod’sCreation.
Consequently,ascomputersevolve,somustthemeanstocontrolthemthroughad-
vancementsnotjustinhardwarebutalsoinsoftware.
Inordertosatisfythisdemandforbettersoftware,Microsoftreleasedanentirely
new programminglanguagecalled C# that incorporatesthe best features of all the
otherexistingpopularprogramminglanguagessuchasJava,C/C++,andVisualBa-
sic. Inspiteofconsiderableresistancebysomepeoplewhopersistonclingingonto
thepastandcontinuetoprogramcomputersthehardway,C#hasnowfirmlyestab-
lished itself worldwideas arguablythe preferredlanguageforsoftware application
development. Althoughmanyexcellentbooksonthetopicofgeneralprogramming
in C# have been written, there is still a considerablelack of publishedmaterial on
thetopicofnumericalmethodsinC#.
Accordingly,NumericalMethods,AlgorithmsandToolsinC# isabookcontain-
ingalargecollectionofveryusefulready-to-usemathematicalroutines,algorithms
andothercomputationaltoolsaimedatprogrammers,mathematicians,statisticians,
scientists, engineers and anyone else interested in developing mathematically ori-
entedcomputerapplicationsin the relativelynew andeasy-to-learnobject-oriented
C# programminglanguage from Microsoft. With a heavy emphasis on using well
established numerical methods, object-oriented techniques and the latest state-of-
the-artMicrosoft.NETprogrammingenvironment,thisbookprovidesreaderswith
workingC#codeincludingpracticalexamplesthatcanbeeasilycustomizedandim-
plementedto solvecomplexengineeringandscientific problemstypicallyfoundin
real-worldapplications.
ForthebenefitofthosereaderswhoarenotyetfamiliarwithC#,Chapter1pro-
videsabriefoutlineofthe.NETFramework,theC#programminglanguageandthe
basicconceptsofObjectOrientedProgramming(OOP).Specialattentionisgivento
topicsthatillustratehowtobestutilizetheseandothertoolstodevelopaccurateand
robustnumericalmethodsinC#.
Chapter2isentirelyfocusedonthe.NETFrameworkMathClassLibrarywhich
already comes built into Microsoft’s Visual Studio software development system.
Additional material is introduced where appropriate in order to supplement, com-
pleteorotherwiseenhancethefeaturesalreadyavailablewiththislibrary.
Chapter3introducesdatastructuresalongwiththeirassociatedfunctionsthatare
particularlyusefulforprogrammingandworkingwithvectorsandmatrices. These
iii
© 2010 by Taylor and Francis Group, LLC
iv NumericalMethods,AlgorithmsandToolsinC#
routinesareoftenusedinmoreadvancedapplicationsinlaterchapters.
Chapter 4 is entirely dedicated to the topic of complex numbers. Since timing
issuescansometimesposeasubstantialproblemwhendoingnumericalcalculations,
complexnumberfunctionsare presentedusingboth elegantstate-of-the-artobject-
oriented methodswhich, althoughslick, can at times carry some overheadand the
old fashioned but proven methods which at times have been found to actually run
fasteronsomecomputers. Inaddition,importantoverflowandunderflowissuesare
alsodiscussedandalternativesolutionstoavoidthoseproblemsareproposed.
Chapter 5 is devoted solely to sorting and searching algorithms. Computersare
often required to perform various types of data sorting for which many different
algorithms exist. Consequently, choosing the most efficient sorting algorithm is a
very important decision that developers frequently have to make. In this chapter,
readersareprovidedwithbothawideselectionofsortingandsearchingalgorithms
fromwhichtochoosealongwithabriefexplanationofhoweachalgorithmworks.
Chapter6 iscenteredonthe topicof bitmanipulationwhichis typicallyusedin
avarietyofprogrammingapplicationsranginganywherefromcomputerinterfacing
toimageprocessing.
Chapter 7 is focusedon interpolationmethods. Equationsthat cannotbe solved
analytically often need to be solved using some kind of interpolation scheme, and
thischapterhasplentyofpracticalexamplestoillustratehowonemighthandlethis
kindofproblem.
Chapter 8 centers on the numerical manipulation of linear algebraic equations.
Thisisactuallyahugetopicbyitself andquiteworthyofitsownbook. Neverthe-
less, asubstantialamountofusefulinformationcanbereadilyobtainedfromjusta
handfulofthesepowerfultools.
Chapter9isfocusedonnumericalmethodsforcalculatingapproximatesolutions
to nonlinearequationswhich often appearnaturally in variousbranchesof science
andengineering.
Chapter10isdevotedexclusivelytothetopicofrandomnumbers. AlthoughC#
comeswithitsowninternalrandomnumbergeneratorfunction,itisnotregardedto
be sufficientlyrobustforuse inadvancedsecuredapplicationsorin computersim-
ulationsthatrequirethousandsandsometimesevenmillionsofrandomnumbersin
order to producereliable and accurate results. Alternate ways to obtainboth com-
puter generated pseudo-randomnumbers and real random numbers obtained from
naturallyoccurringphysicalphenomenaarealsodiscussed. Inaddition,routinesare
also provided for generating random numbers that follow a particular probability
distributionfunction.
Chapter11describesvariousmethodsforapproximatingnumericaldifferentiation
offunctions.Thisisaverytrickyandcontroversialtopicwhoseapproximationscan
givefairlygoodtoatrociouslybadresults. Nevertheless,numericalmethodsdoexist
forcalculatingthesetypesoffunctions.Thetrickisreallyinlearningtorecognizethe
differencebetweengoodandbadresultsandinchoosingthebestavailablemethod
foruseinaparticularsituation.
Chapter12centersondevelopingnumericalmethodsforapproximatingintegrals
ofspecificfunctionsaswellasfromcollectionsofrawdatapoints.Othermoreexotic
© 2010 by Taylor and Francis Group, LLC
Preface v
waysofcalculatingintegrals,suchasbyusingMonteCarlomethods,arealsobriefly
discussed.
Chapter13containsaconsiderablenumberofroutinesforuseinperformingsta-
tisticalanalysisofdata.
Chapter14isdevotedtodevelopingnumericalmethodsforapproximatingspecial
functionswhicharetypicallyfoundinvariousbranchesofmathematics,physicsand
engineering.
Chapter 15 is focused on least squares and numericalcurve fitting methodsthat
are frequently used in analyzing experimental data. A brief discussion of the χ2
goodness-of-fittestisalsoincluded.
Chapter16centersondevelopingroutinestofindnumericalsolutionstoordinary
differential equations. Although this is really a huge topic, there are some basic
numericalmethodswhich can be used successfully to solve a lot of these types of
equationsinmanyreal-worldapplications.
Chapter 17 introduces some numerical methods for solving partial differential
equations. Although this is also a huge topic by itself and quite deserving of its
own book, there are some standardtypesof partialdifferentialequationsthat arise
naturallyinmanyareasofscienceandengineering,andwhosesolutionscanbeap-
proximatedbywellestablishednumericalmethods.
Chapter 18 focuses on optimization methods which are primarily aimed at the
minimizationor maximizationof functionsand thus have manypracticalscientific
andengineeringapplications.Sincethisisstillaveryactiveareaofongoingresearch,
the examples presented here are more narrowly focused on just a few established
topicswiththeexplicitpurposeofillustratinghowsuchmethodsmaybeindividually
customizedandthenappliedtowardssolvingmoreadvancedproblems.
Lastly,Iwouldliketopointoutthatmostofthenumericalmethodsdescribedin
thisbookhaveactuallybeenaroundinoneformoranotherforyears,andsometimes
even for centuries, and it is only their computer implementation in C# that makes
this bookuniquelydifferentfromsome other bookon the topic of numericalanal-
ysis. Accordingly,Ihavemadeeveryefforttotrackdownandgivepropercreditto
originalsources wheneverpossible as the size of this book’sreference section can
easily attest. Inaddition,I havealso madeeveryeffortto providemyreaderswith
accurate,reliableinformationtohelpthemintheireffortstosuccessfullycomplete
theirprogrammingprojects.Unfortunately,unwantedmistakesincludingtypograph-
icalerrorsmayinadvertentlycreepupsomewhereinthisbook. Asaresult,Iwould
greatlyappreciateifmyreaderswouldbesokindastobringtomyattentionifsuch
errorsareeverfoundsothatImaypromptlyhavetheproblemcorrectedforanyfu-
tureeditionsofthisbook. Also,aswithjustabouteverythingwedoinlife,thereis
alwaysroomforimprovement. Accordingly,Iwouldalsoverymuchwelcomeany
constructivecriticismthatmyreadersmayhaveregardingthisbooksothatIcanper-
hapsmakeappropriatechanges.Finally,thereisanoldsayingthatstates,“anauthor
neverfinishesabook,butmerelyabandonsit.” Ihavecertainlycometoappreciate
thatobservationafterworkingonthisprojectforsolongandmakingcountlessrevi-
sions. Nevertheless,thishascertainlybeenaveryenjoyableprojectwherejustabout
everyword was carefullychosen and everytopic was meticulouslyresearchedand
© 2010 by Taylor and Francis Group, LLC
vi NumericalMethods,AlgorithmsandToolsinC#
documented. Therefore,ifitisindeedtruethatIhavewillinglychosentoabandon
writingthisbook,itisonlywiththemodesthopethatitmaybeusefultomyreaders
inspiteofanypossibleshortcomings.
WaldemarDosPassos,Ph.D.
Concord,California
e-mail:[email protected]
website: www.waldemardospassos.com
Acknowledgements
It givesme greatpleasure to thank the manypeople who made this bookpossible.
First,Iwouldverymuchliketothankmypublisher,NoraKonopka,fornotonlyac-
ceptingthisbookforpublicationbutalsoforherexceptionalpatienceasIunderwent
aseriesofunforeseentumultuouseventsinmylifeduringthecourseofwritingthis
book which unfortunatelyled to some regrettabledelaysin its originalpublication
targetdate. Iwouldalsoliketoparticularlythankbothmyprojectdirector,Theresa
Delforn,andmyeditor,AmyRodriguez,fortheirexcellentexpertguidanceinvari-
ousaspectsofthisproject. IwouldalsoliketothankDawnSniderforherexcellent
artistic skills in designingthe coverfor this book. Many thanksto Ashley Gasque
forguidingmethroughthenecessarybureaucraticpaperworkandtoShashiKumar
for some expert LATEX tips he gave me. I would also like to thank all those other
wonderfulpeopleatTaylor&Franciswhohaveworkedtirelesslybehindthescenes
to make this project a success but whose exact names I may likely never come to
know.
IamalsoverygratefulforthesupportIreceivedfromtheH.E.MartinFoundation
undergrant13011938.Withouttheirmostkindandextraordinarygenerousfinancial
assistance,thewritingofthisbookwouldnothavebeenpossible.
I am especially grateful to my third grade teacher, Miss Daly, for all her help,
patience,kindness,andenthusiasmwhichultimatelysparkedmyinterestinmathe-
maticsandeventually,physics. Lookingbackoveralltheseyearsthathaveelapsed
since Iwasa studentinherclass, I cannowsay unequivocallythatMissDaly was
byfartheverybestandmostcaringteacher,professor,orinstructorIeverhad.
Lastly, I would also like to express my deepest and most heartfeltthanks to my
parents,HeleniceandWaldemarDosPassos(Sr.)
© 2010 by Taylor and Francis Group, LLC
Thisbookisdedicatedwithallmyloveandcaretomyparents,
HeleniceandWaldemarDosPassos(Sr.)
foralltheirhardwork,genuinelove,andselflesssacrifices
madeonmybehalfthroughoutmyentirelife.
“Inthislifewecannotdogreatthings;onlysmallthingswithgreatlove.”
MotherTeresa
AdMajorem Dei Gloriam
© 2010 by Taylor and Francis Group, LLC
Contents
1 Introduction 1
1.1 C#andthe.NETFramework . . . . . . . . . . . . . . . . . . . . 1
1.2 InstallingC#andthe.NETFramework . . . . . . . . . . . . . . . 3
1.3 OverviewofObject-OrientedProgramming(OOP) . . . . . . . . 3
1.4 YourFirstC#Program . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 OverviewoftheIDEDebugger . . . . . . . . . . . . . . . . . . . 9
1.6 OverviewoftheC#Language . . . . . . . . . . . . . . . . . . . 11
1.6.1 DataTypes . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6.2 ValueTypes . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6.3 ReferenceTypes . . . . . . . . . . . . . . . . . . . . . . 14
1.6.4 Type-ParameterTypes . . . . . . . . . . . . . . . . . . . 16
1.6.5 PointerTypes . . . . . . . . . . . . . . . . . . . . . . . . 17
1.6.6 VariableDeclaration . . . . . . . . . . . . . . . . . . . . 17
1.6.7 ConstantDeclaration . . . . . . . . . . . . . . . . . . . . 18
1.6.8 NullableTypes . . . . . . . . . . . . . . . . . . . . . . . 18
1.6.9 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6.10 Characters. . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6.11 Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.6.12 FormattingofOutputData . . . . . . . . . . . . . . . . . 19
1.6.13 TypeConversion . . . . . . . . . . . . . . . . . . . . . . 20
1.6.14 ReadingKeyboardInputData . . . . . . . . . . . . . . . 23
1.6.15 BasicExpressionsandOperators . . . . . . . . . . . . . . 24
1.6.16 ProgramFlowMechanisms. . . . . . . . . . . . . . . . . 27
1.6.17 JumpStatements . . . . . . . . . . . . . . . . . . . . . . 29
1.6.18 Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.6.19 Enumerations . . . . . . . . . . . . . . . . . . . . . . . . 32
1.6.20 Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.6.21 Exceptions . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.6.22 Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
ConstructorsandDestructors . . . . . . . . . . . . . . . . . 37
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.6.23 Indexers . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.6.24 OverloadingMethods,ConstructorsandOperators . . . . 42
1.6.25 Delegates . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.6.26 Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
ix
© 2010 by Taylor and Francis Group, LLC