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1、§1.1ErrorsandSignificantDigits1.1.1TruncationerrorandroundofferrorTruncationerror:madebynumericalalgorithms,arisefromtakingfinitenumberofstepsincomputationChapter1Errors2021/7/301computationsinx,whereAccordingtotheexpansionofsinx(1.1)Butwehavetouseitsfiniteitem
2、stoapproximatesinx,forexample,computesin0.5,setn=3,Eg.1.1xisaradian,notdegree.2021/7/302AccordingtotheTaylor’sremainder,(1.2)Thisresultisveryaccurate.2021/7/303Eg.1.2TakingonlyafewtermsofaMaclaurinseriestoapproximateIfonly3termsareused,2021/7/304Eg.1.3(Secantl
3、ine)UsingafinitetoapproximatePQsecantlinetangentlineFigure1.ApproximatederivativeusingfiniteΔx2021/7/305Eg.1.4(Differentiation)FindforusingandTheactualvalueisTruncationerroristhen,Canyoufindthetruncationerrorwith2021/7/306Eg.1.4(Integration)Usetworectanglesofeq
4、ualwidthtoapproximatetheareaunderthecurveforovertheinterval2021/7/307Integrationexample(cont.)Choosingawidthof3,wehaveActualvalueisgivenbyTruncationerroristhenCanyoufindthetruncationerrorwith4rectangles?2021/7/308Roundofferror:usingfiniteprecisionfloating-point
5、numbersoncomputerstorepresentrealnumbers2021/7/3091.1.2AbsoluteerrorandrelativeerrorDefinition1.1letx*betheaccuratevalue(unknown),andxbeanapproximationtox*,thenE=x-x*iscalledtheabsoluteerrorofx*.Ingeneral,wecan’tgettheabsoluteerrorbecausewedonotknowthetruevalue
6、ofx,butwecanestimatetheerrorwithabsoluteerrorbounddefinedasfollows:Definition1.2Apositivenumberɛiscalledtheabsoluteerrorboundofx*if｜x*-x｜≤ε.2021/7/3010Remark.Ingeneral,x*isunknown,sowereplacex*byx,Canyougivethereason?Definition1.3Ifxisanapproximationtox*,theni
7、scalledtherelativeerrorofx*.2021/7/3011Eg.1.5Supposex=9999,x*=10000,y=9,y*=10.Pleaseshowtheabsoluteerrorandrelativeerrorofthem.Solution.Ex=9999-10000=-1,Ey=9-10=-1.er(x)=(9999-10000)/10000=-0.0001.er(y)=(9-10)/10=-0.1.1.1.3SignificantDigitsDefintion1.4Supposeis
8、theapproximationtox*.Ifthenthenumberxissaidtoapproximatex*tolsignificantdigits.Machinerepresentationofnumbers2021/7/3012Eg.1.6Supposex*=20.03173,andx1=20.03,x2=20.031,x3=20.
