§1.1ErrorsandSignificantDigits1.1.1Truncationerrorandroundofferror

Truncationerror:madebynumericalalgorithms,arisefromtakingfinitenumberofstepsincomputation

Chapter1

Errors2023/11/101computation

sinx,where

Accordingtotheexpansionofsin

x(1.1)

Butwehavetouseitsfiniteitemstoapproximatesinx,forexample,

computesin0.5,setn=3,Eg.1.12023/11/102xisaradian,notdegree.

AccordingtotheTaylor’sremainder,(1.2)Thisresultisveryaccurate.2023/11/103Eg.1.2TakingonlyafewtermsofaMaclaurinseriestoapproximateIfonly3termsareused,2023/11/104Eg.1.3(Secantline)UsingafinitetoapproximatePQsecantlinetangentlineFigure1.ApproximatederivativeusingfiniteΔx2023/11/1056Eg.1.4(Differentiation)FindforusingandTheactualvalueisTruncationerroristhen,Canyoufindthetruncationerrorwith2023/11/107Eg.1.4(Integration)Usetworectanglesofequalwidthtoapproximatetheareaunderthecurveforovertheinterval2023/11/10Integrationexample(cont.)Choosingawidthof3,wehaveActualvalueisgivenbyTruncationerroristhenCanyoufindthetruncationerrorwith4rectangles?2023/11/108Roundofferror:usingfiniteprecisionfloating-pointnumbersoncomputerstorepresentrealnumbers

2023/11/1091.1.2Absoluteerrorandrelativeerror

Definition1.1let

x*betheaccuratevalue(unknown),andxbeanapproximationtox*,then

E=x-x*iscalledtheabsoluteerrorofx*.Ingeneral,wecan’tgettheabsoluteerrorbecausewedonotknowthetruevalueofx,butwecanestimatetheerrorwithabsoluteerrorbounddefinedasfollows:

Definition1.2Apositivenumber?iscalledtheabsoluteerrorboundofx*if｜x*-x｜≤ε.2023/11/1010

Remark.Ingeneral,x*isunknown,sowereplacex*byx,

Canyougivethereason?

Definition1.3Ifxisanapproximationtox*,then

iscalledtherelativeerrorofx*.2023/11/1011Eg.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=-0.1.1.1.3SignificantDigitsDefintion1.4Supposeistheapproximationtox*.Ifthenthenumberxissaidtoapproximatex*tolsignificantdigits.2023/11/1012MachinerepresentationofnumbersEg.1.6Supposex*=20.03173,andx1=20.03,x2=20.031,x3=20.032areitsapproximationsrespectively.Determinethenumbersofsignificantdigitsofthem.Solution.Rewritex1,x2andx3asx1=0.2003×102,x2=0.20031×102,x3=0.20032×102.Since

AccordingtoDefinition1.4,x1,x2andx3have4,4and5significantdigitsrespectively.2023/11/1013Sometimes,wecutalongnumberintoashortnumberthroughrounding.Eg.1.7Accordingtotheroundingrule,writethefollowingnumberswith5significantdigits.

①287.9325②0.03785551③8.000033④2.718281828459045⑤2.765450Solution.

①287.9325≈287.93(roundingdown)②0.03785551≈0.037856(roundingup)③8.000033≈8.0000(roundingdown)④2.718281828459045≈2.7183(roundingup)⑤2.765450≈2.7655(roundingup)2023/11/1014Theorem1.1Suppose

withnsignificantdigits,thentherelativeerrorboundofx*Theorem1.2Iftherelativeerrorboundofisthenxhasatleastnsignificantdigits.2023/11/1015§1.2PropagationofErrorsWhenweuseinaccuratenumberstocalculate,howdotheseinaccuraciespropagatethroughthecalculations?2023/11/1016Eg.1.8Findtheboundsforthepropagationinaddingtwonumbers.ForexampleifoneiscalculatingX+Ywhere

X=1.5±0.05

Y=3.4±0.04SolutionMaximumpossiblevalueofX=1.55andY=3.44MaximumpossiblevalueofX+Y=1.55+3.44=4.99MinimumpossiblevalueofX=1.45andY=3.36.MinimumpossiblevalueofX+Y=1.45+3.36=4.81Hence 4.81≤X+Y≤4.99.2023/11/1017PropagationofErrorsInFormulasIfisafunctionofseveralvariablesthenthemaximumpossiblevalueoftheerrorinis2023/11/1018Eg.1.9Thestraininanaxialmemberofasquarecross-sectionisgivenbyGivenFindthemaximumpossibleerrorinthemeasuredstrain.2023/11/1019Solution

2023/11/1020ThusHence2023/11/1021Eg.10Subtractionofnumbersthatarenearlyequalcancreateunwantedinaccuracies.Usingtheformulaforerrorpropagation,showthatthisistrue.SolutionLetThenSotherelativechangeis2023/11/1022Eg.11Forexampleif=0.6667=66.67%2023/11/10231.3.1Avoidthesubtractionofnearlyequalnumbers

Thesubtractionofnearlyequalnumberswilllosssignificantdigitslargely,sothisoperationleadstoalargerrelativeerror.Eg.1.12Computetheapproximationtox*－y*using4-digits.

①x=18.496;y=17.208②x=18.496;y=18.493Solution.①Theapproximationswith4-digitsto

x*andy*usingtheRoundingRuleareasfollows：

x=18.50;y=17.21

x－y=

18.50－17.21=1.29Infact

x*－y*＝18.496－17.208＝1.288§3HowtoAvoidtheLossofAccuracy2023/11/1024Absoluteerror

Relativeerror

②Theapproximationswith4-digitsto

x*andy*usingtheRoundingRuleareasfollows：

x=18.50;y=18.49

x－y=

18.50－18.49=0.01infact

x*－y*＝18.496－18.493＝0.003

Absoluteerror

Relativeerror2023/11/1025Findabettersolution:①moresignificantdigits;②usingabettercomputationform,forexample1.3.2Avoidbignumbers“swallowing”importantsmallnumbers2023/11/1026Eg.1.1.13Tocalculate1-cos0.1