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§1.1ErrorsandSignificantDigits1.1.1Truncationerrorandroundofferror

Truncationerror:madebynumericalalgorithms,arisefromtakingfinitenumberofstepsincomputation

Chapter1

Errors2022/11/241§1.1ErrorsandSignificantDcomputation

sinx,where

Accordingtotheexpansionofsin

x(1.1)

Butwehavetouseitsfiniteitemstoapproximatesinx,forexample,computesin0.5,setn=3,Eg.1.1xisaradian,notdegree.2022/11/242computationsinx,whereAc

AccordingtotheTaylor’sremainder,(1.2)Thisresultisveryaccurate.2022/11/243AccordingtotheTaylor’sEg.1.2TakingonlyafewtermsofaMaclaurinseriestoapproximateIfonly3termsareused,2022/11/244Eg.1.2TakingonlyafewtermsEg.1.3(Secantline)UsingafinitetoapproximatePQsecantlinetangentlineFigure1.ApproximatederivativeusingfiniteΔx2022/11/245Eg.1.3(Secantline)UsingafEg.1.4(Differentiation)FindforusingandTheactualvalueisTruncationerroristhen,Canyoufindthetruncationerrorwith2022/11/246Eg.1.4(Differentiation)FindEg.1.4(Integration)Usetworectanglesofequalwidthtoapproximatetheareaunderthecurveforovertheinterval2022/11/247Eg.1.4(Integration)UsetwoIntegrationexample(cont.)Choosingawidthof3,wehaveActualvalueisgivenbyTruncationerroristhenCanyoufindthetruncationerrorwith4rectangles?2022/11/248Integrationexample(cont.)ChoRoundofferror:usingfiniteprecisionfloating-pointnumbersoncomputerstorepresentrealnumbers

2022/11/249Roundofferror:usingfinite1.1.2Absoluteerrorandrelativeerror

Definition1.1let

x*betheaccuratevalue(unknown),andxbeanapproximationtox*,then

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

Definition1.2Apositivenumber?iscalledtheabsoluteerrorboundofx*if｜x*-x｜≤ε.2022/11/24101.1.2Absoluteerrorandrelat

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

Definition1.3Ifxisanapproximationtox*,then

iscalledtherelativeerrorofx*.2022/11/2411Remark.Ingeneral,x*isEg.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=-.3SignificantDigitsDefintion1.4Supposeistheapproximationtox*.Ifthenthenumberxissaidtoapproximatex*tolsignificantdigits.Machinerepresentationofnumbers2022/11/2412Eg.1.5Supposex=9999,x*=10Eg.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.2022/11/2413Eg.1.6Supposex*=20.03173,aSometimes,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)2022/11/2414Sometimes,wecutalongnumTheorem1.1Suppose

withnsignificantdigits,thentherelativeerrorboundofx*Theorem1.2Iftherelativeerrorboundofisthenxhasatleastnsignificantdigits.2022/11/2415Theorem1.1Suppose§1.2PropagationofErrorsWhenweuseinaccuratenumberstocalculate,howdotheseinaccuraciespropagatethroughthecalculations?2022/11/2416§1.2PropagationofErrorsWheEg.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≤4.99.2022/11/2417Eg.1.8FindtheboundsforthePropagationofErrorsInFormulasIfisafunctionofseveralvariablesthenthemaximumpossiblevalueoftheerrorinis2022/11/2418PropagationofErrorsInFormuEg.1.9Thestraininanaxialmemberofasquarecross-sectionisgivenbyGivenFindthemaximumpossibleerrorinthemeasuredstrain.2022/11/2419Eg.1.9ThestraininanaxialSolution

2022/11/2420Solution 2022/11/2220ThusHence2022/11/2421ThusHence2022/11/2221Eg.10Subtractionofnumbersthatarenearlyequalcancreateunwantedinaccuracies.Usingtheformulaforerrorpropagation,showthatthisistrue.SolutionLetThenSotherelativechangeis2022/11/2422Eg.10SubtractionofnumbersthEg.11Forexampleif=0.6667=66.67%2022/11/2423Eg.11Forexampleif=0.6667201.3.1Avoidthesubtractionofnearlyequalnumbers

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§3HowtoAvoidtheLossofAccuracy2022/11/24241.3.1AvoidthesubtractionofAbsoluteerror

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

Relativeerror2022/11/2425②Theapproximationswith4-dFindabettersolution:①moresignificantdigits;②usingabettercomputationform,forexample1.3.2Avoidbignumbers“swallowing”importantsmallnumbersEg.1.1.13Tocalculate1-cos0.1usingtwoways.(4significantdigit)1-cos0.1=1-0.9950=0.00502sin2(0.05)=2*0.04498*0.04498=0.004496.Truevalue:0.004496.2022/11/2426Findabettersolution:①moresEputingrootsofaquadraticequationUsing4digitsroundingarithmetic,applytheformulaTotheequationSolution.Inthisequation,b2isamuchlargerthan4ac,whichinducesWehave2022/11/2427EputingrootsofaButtheaccuratesolutionisWecanfindaverygreaterrorforx1.Abetterwayis2022/11/2428ButtheaccuratesolutionisWIfsetε=10-5，wehavetocomputeuntiln>105Let

x=1/3，resultcansatisfytheprecisionbyonlycomputing5times.Eg.AvoidabigNumberDividingbyasmallnumber1.3.4Reducingcomputations2022/11/2429Ifsetε=10-5，wehavetocompu§1.1ErrorsandSignificantDigits1.1.1Truncationerrorandroundofferror

Truncationerror:madebynumericalalgorithms,arisefromtakingfinitenumberofstepsincomputation

Chapter1

Errors2022/11/2430§1.1ErrorsandSignificantDcomputation

sinx,where

Accordingtotheexpansionofsin

x(1.1)

Butwehavetouseitsfiniteitemstoapproximatesinx,forexample,computesin0.5,setn=3,Eg.1.1xisaradian,notdegree.2022/11/2431computationsinx,whereAc

AccordingtotheTaylor’sremainder,(1.2)Thisresultisveryaccurate.2022/11/2432AccordingtotheTaylor’sEg.1.2TakingonlyafewtermsofaMaclaurinseriestoapproximateIfonly3termsareused,2022/11/2433Eg.1.2TakingonlyafewtermsEg.1.3(Secantline)UsingafinitetoapproximatePQsecantlinetangentlineFigure1.ApproximatederivativeusingfiniteΔx2022/11/2434Eg.1.3(Secantline)UsingafEg.1.4(Differentiation)FindforusingandTheactualvalueisTruncationerroristhen,Canyoufindthetruncationerrorwith2022/11/2435Eg.1.4(Differentiation)FindEg.1.4(Integration)Usetworectanglesofequalwidthtoapproximatetheareaunderthecurveforovertheinterval2022/11/2436Eg.1.4(Integration)UsetwoIntegrationexample(cont.)Choosingawidthof3,wehaveActualvalueisgivenbyTruncationerroristhenCanyoufindthetruncationerrorwith4rectangles?2022/11/2437Integrationexample(cont.)ChoRoundofferror:usingfiniteprecisionfloating-pointnumbersoncomputerstorepresentrealnumbers

2022/11/2438Roundofferror:usingfinite1.1.2Absoluteerrorandrelativeerror

Definition1.1let

x*betheaccuratevalue(unknown),andxbeanapproximationtox*,then

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

Definition1.2Apositivenumber?iscalledtheabsoluteerrorboundofx*if｜x*-x｜≤ε.2022/11/24391.1.2Absoluteerrorandrelat

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

Definition1.3Ifxisanapproximationtox*,then

iscalledtherelativeerrorofx*.2022/11/2440Remark.Ingeneral,x*isEg.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=-.3SignificantDigitsDefintion1.4Supposeistheapproximationtox*.Ifthenthenumberxissaidtoapproximatex*tolsignificantdigits.Machinerepresentationofnumbers2022/11/2441Eg.1.5Supposex=9999,x*=10Eg.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.2022/11/2442Eg.1.6Supposex*=20.03173,aSometimes,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)2022/11/2443Sometimes,wecutalongnumTheorem1.1Suppose

withnsignificantdigits,thentherelativeerrorboundofx*Theorem1.2Iftherelativeerrorboundofisthenxhasatleastnsignificantdigits.2022/11/2444Theorem1.1Suppose§1.2PropagationofErrorsWhenweuseinaccuratenumberstocalculate,howdotheseinaccuraciespropagatethroughthecalculations?2022/11/2445§1.2PropagationofErrorsWheEg.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≤4.99.2022/11/2446Eg.1.8FindtheboundsforthePropagationofErrorsInFormulasIfisafunctionofseveralvariablesthenthemaximumpossiblevalueoftheerrorinis2022/11/2447PropagationofErrorsInFormuEg.1.9Thestraininanaxialmemberofasquarecross-sectionisgivenbyGivenFindthemaximumpossibleerrorinthemeasuredstrain.2022/11/2448Eg.1.9ThestraininanaxialSolution

2022/11/2449Solution 2022/11/2220ThusHence2022/11/2450ThusHence2022/11/2221Eg.10Subtractionofnumbersthatarenearlyequalcancreateunwantedinaccuracies.Usingtheformulaforerrorpropagation,showthatthisistrue.SolutionLetThenSotherelativechangeis2022/11/2451Eg.10SubtractionofnumbersthEg.11Forexampleif=0.6667=66.67%2022/11/2452Eg.11Forexampleif=0.6667201.3.1Avoidthesubtractionofnearlyequalnumbers

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

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