Numerical Methods in Engineering With Python

433 Pages·2005·1.781 MB·English

by  Jaan Kiusalaas

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Numerical Methods in Engineering with Python
NumericalMethodsinEngineeringwithPythonisatextforengineer-
ingstudentsandareferenceforpracticingengineers,especiallythose
whowishtoexplorethepowerandefficiencyofPython.Thechoiceof
numericalmethodswasbasedontheirrelevancetoengineeringprob-
lems.Everymethodisdiscussedthoroughlyandillustratedwithprob-
lems involving both hand computation and programming. Computer
codeaccompanieseachmethodandisavailableonthebookwebsite.
Thiscodeismadesimpleandeasytounderstandbyavoidingcomplex
book-keepingschemes,whilemaintainingtheessentialfeaturesofthe
methodPythonwaschosenastheexamplelanguagebecauseitisele-
gant,easytolearnanddebug,anditsfacilitiesforhandlingarraysare
unsurpassed.Moreover,itisanopen-sourcesoftwarepackagethatcan
bedownloadedfreelyontheweb.Pythonisagreatlanguageforteaching
scientificcomputation.
JaanKiusalaasisaProfessorEmeritusintheDepartmentofEngineer-
ingScienceandMechanicsatthePennsylvaniaStateUniversity.Hehas
taughtcomputermethods,includingfiniteelementandboundaryele-
mentmethods,forover30years.Heisalsotheco-authoroffourother
books—EngineeringMechanics:Statics,EngineeringMechanics:Dynam-
ics,MechanicsofMaterials,andanalternateversionofthisworkwith
®
MATLAB code.
i
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NUMERICAL METHODS IN
ENGINEERING WITH
Python
Jaan Kiusalaas
ThePennsylvaniaStateUniversity
iii
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge  , UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridg e.org /9780521852876
©JaanKiusalaas2005
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2005
- ---- eBook (NetLibrary)
- --- eBook (NetLibrary)
- ---- hardback
- --- hardback
Cambridge University Press has no responsibility for the persistence or accuracy of s
for external or third-party internet websites referred to in this publication, and does not
guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate.
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Contents
Preface.........vii
1. IntroductiontoPython.............................. 1
2. SystemsofLinearAlgebraicEquations............. 27
3. InterpolationandCurveFitting.....................103
4. RootsofEquations...................................142
5. NumericalDifferentiation...........................181
6. NumericalIntegration...............................198
7. InitialValueProblems...............................248
8. Two-PointBoundaryValueProblems...............295
9. SymmetricMatrixEigenvalueProblems............324
10. IntroductiontoOptimization.......................381
Appendices....409
Index...........419
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Preface
This book is targeted primarily toward engineers and engineering students of ad-
vanced standing (sophomores, seniors and graduate students). Familiarity with a
computerlanguageisrequired;knowledgeofbasicengineeringmechanicsisuseful,
butnotessential.
Thetextattemptstoplaceemphasisonnumericalmethods,notprogramming.
Mostengineersarenotprogrammers,butproblemsolvers.Theywanttoknowwhat
methodscanbeappliedtoagivenproblem,whataretheirstrengthsandpitfallsand
howtoimplementthem.Engineersarenotexpectedtowritecomputercodeforbasic
tasksfromscratch;theyaremorelikelytoutilizefunctionsandsubroutinesthathave
beenalreadywrittenandtested.Thusprogrammingbyengineersislargelyconfined
toassemblingexistingpiecesofcodeintoacoherentpackagethatsolvestheproblem
athand.
The“piece”ofcodeisusuallyafunctionthatimplementsaspecifictask.Forthe
userthedetailsofthecodeareunimportant.Whatmattersistheinterface(whatgoes
inandwhatcomesout)andanunderstandingofthemethodonwhichthealgorithm
isbased.Sincenonumericalalgorithmisinfallible,theimportanceofunderstanding
theunderlyingmethodcannotbeoveremphasized;itis,infact,therationalebehind
learningnumericalmethods.
This book attempts to conform to the views outlined above. Each numerical
methodisexplainedindetailanditsshortcomingsarepointedout.Theexamplesthat
followindividualtopicsfallintotwocategories:handcomputationsthatillustratethe
innerworkingsofthemethodandsmallprogramsthatshowhowthecomputercodeis
utilizedinsolvingaproblem.Problemsthatrequireprogrammingaremarkedwith(cid:1).
The material consists of the usual topics covered in an engineering course on
numericalmethods:solutionofequations,interpolationanddatafitting,numerical
differentiationandintegration,solutionofordinarydifferentialequationsandeigen-
valueproblems.Thechoiceofmethodswithineachtopicistiltedtowardrelevance
toengineeringproblems.Forexample,thereisanextensivediscussionofsymmetric,
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viii Preface
sparsely populated coefficient matrices in the solution of simultaneous equations.
Inthesamevein,thesolutionofeigenvalueproblemsconcentratesonmethodsthat
efficientlyextractspecificeigenvaluesfrombandedmatrices.
Animportantcriterionusedintheselectionofmethodswasclarity.Algorithms
requiringoverlycomplexbookkeepingwererejectedregardlessoftheirefficiencyand
robustness.Thisdecision,whichwastakenwithgreatreluctance,isinkeepingwith
theintenttoavoidemphasisonprogramming.
Theselectionofalgorithmswasalsoinfluencedbycurrentpractice.Thisdisqual-
ifiedseveralwell-knownhistoricalmethodsthathavebeenovertakenbymorerecent
developments. For example, the secant method for finding roots of equations was
omittedashavingnoadvantagesoverBrent’smethod.Forthesamereason,themul-
tistepmethodsusedtosolvedifferentialequations(e.g.,MilneandAdamsmethods)
wereleftoutinfavoroftheadaptiveRunge–KuttaandBulirsch–Stoermethods.
Notablyabsentisachapteronpartialdifferentialequations.Itwasfeltthatthis
topicisbesttreatedbyfiniteelementorboundaryelementmethods,whichareoutside
thescopeofthisbook.Thefinitedifferencemodel,whichiscommonlyintroduced
in numerical methods texts, is just too impractical in handling multidimensional
boundaryvalueproblems.
Asusual,thebookcontainsmorematerialthancanbecoveredinathree-credit
course.Thetopicsthatcanbeskippedwithoutlossofcontinuityaretaggedwithan
asterisk(*).
TheprogramslistedinthisbookweretestedwithPython2.2.2and2.3.4under
WindowsXPandRedHatLinux.Thesourcecodecanbedownloadedfromthebook’s
websiteat
www.cambridge.org/0521852870
The author wishes to express his gratitude to the anonymous reviewers and
Professor Andrew Pytel for their suggestions for improving the manuscript. Credit
isalsoduetotheauthorsofNumericalRecipes(CambridgeUniversityPress)whose
presentationofnumericalmethodswasinspirationalinwritingthisbook.
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1 Introduction to Python
1.1 General Information
Quick Overview
This chapter is not a comprehensive manual of Python. Its sole aim is to provide
sufficientinformationtogiveyouagoodstartifyouareunfamiliarwithPython.Ifyou
knowanothercomputerlanguage,andpresumablyyoudo,itisnotdifficulttopick
uptherestasyougo.
Python is an object-oriented language that was developed in late 1980s as a
scripting language (the name is derived from the British television show Monty
Python’s Flying Circus). Although Python is not as well known in engineering cir-
cles as some other languages, it has a considerable following in the programming
community—infact,PythonisconsiderablymorewidespreadthanFortran.Python
may be viewed as an emerging language, since it is still being developed and re-
fined. In the current state, it is an excellent language for developing engineering
applications—itpossessesasimpleelegancethatotherprogramminglanguagescan-
notmatch.
Pythonprogramsarenotcompiledintomachinecode,butarerunbyaninter-
preter1.Thegreatadvantageofaninterpretedlanguageisthatprogramscanbetested
anddebuggedquickly,allowingtheusertoconcentratemoreontheprinciplesbe-
hindtheprogramandlessonprogrammingitself.Sincethereisnoneedtocompile,
linkandexecuteaftereachcorrection,Pythonprogramscanbedevelopedinamuch
shortertimethanequivalentFortranorCprograms.Onthenegativeside,interpreted
programsdonotproducestand-aloneapplications.ThusaPythonprogramcanbe
runonlyoncomputersthathavethePythoninterpreterinstalled.
1 ThePythoninterpreteralsocompilesbytecode,whichhelpstospeedupexecutionsomewhat.
1