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1、AnIntroductiontotheQualityofComputedSolutionsSvenHammarlingJuly2005MIMSEPrint:2005.29ManchesterInstituteforMathematicalSciencesSchoolofMathematicsTheUniversityofManchesterReportsavailablefrom:http://www.manchester.ac.uk/mims/eprintsAndbycontacting:TheMIMSSecretarySchoolofMathemati
2、csTheUniversityofManchesterManchester,M139PL,UKISSN1749-9097AnIntroductiontotheQualityofComputedSolutions1SvenHammarlingTheNumericalAlgorithmsGroupLtdWilkinsonHouseJordanHillRoadOxford,OX28DR,UKsven@nag.co.ukDecember5,20051Basedon[Hammarling,2005],Chapter4of[Einarsson,2005],whichi
3、savailableforpur-chasefromSIAMathttp://ec-securehost.com/SIAM/SE18.html.TheroyaltiesgototheSIAMStudentTravelFund.CONTENTS1Contents1Introduction22FloatingPointNumbersandIEEEArithmetic23WhyWorryaboutComputedSolutions?54Condition,StabilityandErrorAnalysis94.1Condition................
4、.......................94.2Stability........................................144.3ErrorAnalysis.....................................195FloatingPointErrorAnalysis236PosingtheMathematicalProblem297ErrorBoundsandSoftware308OtherApproaches349Summary34Bibliography351INTRODUCTION21Introd
5、uctionThisreportisconcernedwiththequalityofthecomputednumericalsolutionsofmathematicalproblems.Forexample,supposewewishtosolvethesystemoflinearequationsAx=businganumericalsoftwarepackage.Thepackagewillreturnacomputedsolution,sayx˜,andwewishtojudgewhetherornotx˜isareasonablesolutio
6、ntotheequations.Sadly,alltoooftensoftwarepackagesreturnpoor,orevenincorrect,numericalresultsandgivetheusernomeansbywhichtojudgethequalityofthenumericalresults.In1971,LeslieFoxcommented[Fox,1971,p.296]“Ihavelittledoubtthatabout80percent.ofalltheresultsprintedfromthecomputerareinerr
7、ortoamuchgreaterextentthantheuserwouldbelieve,...”Morethanthirtyyearsonthatpaperisstillveryrelevantandworthreading.AnotherveryreadablearticleisForsythe[1970].Thequalityofcomputedsolutionsisconcernedwithassessinghowgoodacomputedsolutionisinsomeappropriatemeasure.Qualitysoftwareshou
8、ldimplementreliablealgorithmsands
