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Introduction to Computational Mathematics Textbooks in Mathematics Series editors: Al Boggess, Kenneth H. Rosen ModelingChangeandUncertainty MachineLearningandOtherTechniques WilliamP.FoxandRobertE.Burks AbstractAlgebra AFirstCourse,SecondEdition StephenLovett MultiplicativeDifferentialCalculus SvetlinGeorgiev,KhaledZennir AppliedDifferentialEquations ThePrimaryCourse VladimirA.Dobrushkin IntroductiontoComputationalMathematics: AnOutline WilliamC.Bauldry https://www.routledge.com/Textbooks-in-Mathematics/book-series/CANDHTEXBOOMTH Introduction to Computational Mathematics: An Outline William C. Bauldry Firsteditionpublished2022 byCRCPress 6000BrokenSoundParkwayNW,Suite300,BocaRaton,FL33487-2742 andbyCRCPress 4ParkSquare,MiltonPark,Abingdon,Oxon,OX144RN ©2023WilliamC.Bauldry CRCPressisanimprintofTaylor&FrancisGroup,LLC Reasonableeffortshavebeenmadetopublishreliabledataandinformation,buttheauthorandpublisher cannotassumeresponsibilityforthevalidityofallmaterialsortheconsequencesoftheiruse. Theauthors andpublishershaveattemptedtotracethecopyrightholdersofallmaterialreproducedinthispublication andapologizetocopyrightholdersifpermissiontopublishinthisformhasnotbeenobtained. Ifany copyright material has not been acknowledged please write and let us know so we may rectify in any futurereprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted,orutilizedinanyformbyanyelectronic,mechanical,orothermeans,nowknownorhereafter invented,includingphotocopying,microfilming,andrecording,orinanyinformationstorageorretrieval system,withoutwrittenpermissionfromthepublishers. Forpermissiontophotocopyorusematerialelectronicallyfromthiswork,accesswww.copyright.comor contacttheCopyrightClearanceCenter, Inc. (CCC),222RosewoodDrive, Danvers, MA01923, 978- 750-8400. ForworksthatarenotavailableonCCCpleasecontactmpkbookspermissions@tandf.co.uk Trademarknotice: Productorcorporatenamesmaybetrademarksorregisteredtrademarksandareused onlyforidentificationandexplanationwithoutintenttoinfringe. ISBN:9781003299257(ebk) DOI:10.1201/9781003299257 Publisher’snote: Thisbookhasbeenpreparedfromcamera-readycopyprovidedbytheauthors. Coverphotos: MountFieldNationalPark,TallTreesWalk,Tasmania. ©WmCB,2018 Introduction to Computational Mathematics: An Outline “1+1 = 3 for large enough values of 1.” Photo©WmCB,2015 Introduction to Computational Mathematics: An Outline Contents Preface ...................................................viii I. Computer Arithmetic .......................................1 II. Control Structures .........................................29 S I. Special Topics: Computation Cost and Horner’s Form .......64 III. Numerical Differentiation ..................................70 IV. Root-Finding Algorithms ...................................84 S II. Special Topics: Modified Newton’s Method ................113 V. Numerical Integration ....................................117 VI. Polynomial Interpolation ..................................140 S III. Case Study: TI Calculator Numerics .......................162 VII. Projects .................................................175 VIII. References ...............................................200 Note: Light blue text links inside this Outline; grey text links to a web page. ICMvi Introduction to Computational Mathematics: An Outline Photo©WmCB,2015 On two occasions I have been asked, “Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?” ... I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question. — Charles Babbage Passages from the Life of a Philosopher, p. 67. ICMvii Preface: Is Computation Important? A simple computational error can cause very serious problems. The Mars Climate Orbiter Crash Two teams used different units of measure when writing the Orbiter’s software: NASA used the metric system, and Lockheed Martin used the Imperial system. Read NASA’s Mars Climate Orbiter page. The Ariane 5 Explosion The European Space Agency’s Ariane 5 rocket exploded 37 seconds after takeoff; the explosion was caused by an integer overflow in the software used for launching the rocket. Watch the launch video. The Gulf War Dhahran “Scud” Missile Attack The Patriot missile software measured time in binary and decimal based on the system clock that used 10th’s of a second, a non-terminating fraction in binary. After running for over 100 hours, the system was inaccurate. Twenty-eight U.S. soldiers died when the system failed to track a Scud missile fired from Iraq. Read Michael Barr’s “Lethal Software Defects: Patriot Missile Failure.” ICMviii Preface: Three Factors of Computation There are three aspects to keep in mind while studying computation. The first two are always in tension with each other: Accuracy versus Efficiency Accuracy concerns how much error occurs Accuracy Efficiency and how to control the error. Efficiency concerns how much computation Error is needed to produce a result. Balance Stability is the third aspect to always keep in mind. In a stable computation, a small change in inputs produces only a small change in outputs. Asa counterpoint, investigateinstability ina “Lorenz attractor.” Startingat[1,1,1]. Startingat[ 1, 2, 3]. − − − ICMix