Table Of Content

Richard Khoury Douglas Wilhelm Harder Numerical Methods and Modelling for Engineering Numerical Methods and Modelling for Engineering Richard Khoury (cid:129) Douglas Wilhelm Harder Numerical Methods and Modelling for Engineering RichardKhoury DouglasWilhelmHarder LakeheadUniversity UniversityofWaterloo ThunderBay,ON,Canada Waterloo,ON,Canada ISBN978-3-319-21175-6 ISBN978-3-319-21176-3 (eBook) DOI10.1007/978-3-319-21176-3 LibraryofCongressControlNumber:2016933860 ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodologynowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland Conventions Throughout this textbook, the following conventions are used for functions and variables: Object Format Example Scalarnumber Lowercase x¼5 (realor italics complex) Vector Lowercase v¼½x0;x1;...;xi;...;xN(cid:2)1(cid:3) bold 2 3 2 3 Matrix bUoplpdercase 6 x0,0 x0,1 ... x0,j ... x0,N(cid:2)1 7 6 v0 7 6 x1,0 x1,j x1,N(cid:2)1 7 6 v1 7 6 ... ... ... 7 6 ... 7 M¼6664 x.i.,.0 xi,1 ... x.i.,.j ... xi,.N.(cid:2).1 7775¼6664 .v.i. 7775 xM(cid:2)1,0 xM(cid:2)1,1 ... xM(cid:2)1,j ... xM(cid:2)1,N(cid:2)1 vM(cid:2)1 Scalar-valued Lowercase fðxÞ¼5xþ2 functionof italicswith scalar lowercase italics Scalar-valued Lowercase fðvÞ¼fðx0;x1;...;xi;...;xN(cid:2)1Þ functionof italicswith ¼5x0þ2x1þ(cid:4)(cid:4)(cid:4)þ7xiþ(cid:4)(cid:4)(cid:4)þ4xN(cid:2)1þ3 vector lowercase bold Vector- Lowercase fðxÞ¼½f0ðxÞ,f1ðxÞ,...,fiðxÞ,...,fN(cid:2)1ðxÞ(cid:3) valuedfunction boldwith ofscalar lowercase italic Vector-valued Lowercase fðvÞ¼½f0ðvÞ,f1ðvÞ,...,fiðvÞ,...,fN(cid:2)1ðvÞ(cid:3) functionof boldwith vector lowercase bold (continued) v vi Conventions Object Format Example 2 3 Mfsucanaltcartirixo-nvaolfued Ubloopwlpdeerwrccaiatshsee 666 ff01,,.00.ðð.xxÞÞ f0,1ðxÞ ... ff01.,,jj.ðð.xxÞÞ ... ff01,,NN.(cid:2)(cid:2).11.ððxxÞÞ 777 italics MðxÞ¼6664 fi,.0.ð.xÞ fi,1ðxÞ ... fi,.j.ð.xÞ ... fi,N.(cid:2).1.ðxÞ 7775 fM(cid:2)1,0ðxÞ fM(cid:2)1,1ðxÞ ... fM(cid:2)1,jðxÞ ... fM(cid:2)1,N(cid:2)1ðxÞ 2 3 Mfvuenactctrotiirxo-nvaolfued Ubloopwlpdeerwrccaiatshsee 666 ff01,,.00.ðð.vvÞÞ f0,1ðvÞ ... ff01,,.jj.ðð.vvÞÞ ... ff01,,NN.(cid:2)(cid:2).11.ððvvÞÞ 777 bold MðvÞ¼6664 fi,.0.ð.vÞ fi,1ðvÞ ... fi,.j.ð.vÞ ... fi,N.(cid:2).1.ðvÞ 7775 fM(cid:2)1,0ðvÞ fM(cid:2)1,1ðvÞ ... fM(cid:2)1,jðvÞ ... fM(cid:2)1,N(cid:2)1ðvÞ Acknowledgements Thanks to the following for pointing out mistakes, providing suggestions, or helpingtoimprovethequalityofthistext: (cid:129) KhadijehBayat (cid:129) DanBusuioc (cid:129) TimKuo (cid:129) AbbasAttarwala (cid:129) PrashantKhanduri (cid:129) MatthewChan (cid:129) ChristopherOlekas (cid:129) JaroslawKuszczak (cid:129) ChenHe (cid:129) HansJohannesPetrusVanleeuwen (cid:129) DavidSmith (cid:129) JeffTeng (cid:129) RomanKogan (cid:129) MohamedOussamaDamen (cid:129) RudkoVolodymyr (cid:129) VladimirRutko (cid:129) GeorgeRizkalla (cid:129) AlexandreJames (cid:129) ScottKlassen (cid:129) BradMurray (cid:129) BrendanBoese (cid:129) AaronMacLennan vii Contents 1 ModellingandErrors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 SimulationandApproximation. . . . . . . . . . . . . . . . . . . . . . . 2 1.3 ErrorAnalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.1 PrecisionandAccuracy. . . . . . . . . . . . . . . . . . . . . . 5 1.3.2 AbsoluteandRelativeError. . . . . . . . . . . . . . . . . . . 6 1.3.3 SignificantDigits. . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.4 BigONotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 NumericalRepresentation. . .. . . . .. . . . .. . . .. . . . .. . . . .. . . .. 13 2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 DecimalandBinaryNumbers. . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.1 DecimalNumbers. . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.2 BinaryNumbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.3 BaseConversions. . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 NumberRepresentation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.1 Fixed-PointRepresentation. . . . . . . . . . . . . . . . . . . . 19 2.3.2 Floating-PointRepresentation. . . . . . . . . . . . . . . . . . 20 2.3.3 Double-PrecisionFloating-PointRepresentation. . . .. 22 2.4 LimitationsofModernComputers. . . . . . . . . . . . . . . . . . . . . 24 2.4.1 UnderflowandOverflow. . . . . . . . . . . . . . . . . . . . . 24 2.4.2 SubtractiveCancellation. . . . . . . . . . . . . . . . . . . . . . 25 2.4.3 Non-associativityofAddition. . . . . . . . . . . . . . . . . . 27 2.5 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ix