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1.3.Conditionnumber
21
whichcanbederivedsubstitutingtheequalities(1.11)and(1.12)totheVi´
ete’s
formulafortheproductoftheroots(x1·x2=c/a).Lookingfortherootwith
asmallerabsolutevalue,weusethenonlythisfromtheequalities(1.13)and(1.14)
whichhasthedenominatorwithalargerabsolutevalue(thefunctiondefiningthis
roothassmallerconditionnumber).
I
Ifthedataerrorsresultonlyfromthemachinenumberrepresentation,i.e.,
rd(dź)=dź(1+5ź),∆dź=dź5ź,|5ź|≤eps,ź=1,...,n,thenforeachof
themostusefulvectornorms,i.e.,"·"
1,"·"
2and"·"
∞(seeChapter2),wehave
∥∆d∥
∥d∥≤eps.Forinstance,fortheEuclideannorm
"∆d"
"d"
=
√(d151)2+(d252)2+···+(dn5n)2
"d"
≤
"d"eps
"d"
=eps.
Thefollowing(approximate)inequalityfollowsdirectlyfrom(1.7):
"∆w"
"w"
ścondϕ(d)·
"∆d"
"d"
,
(1.15)
(1.16)
andthesmallerthedataerrorsthebettertheapproximation.Iftheproblemcannot
bedefinedinaknownfunctionform,thenitisusuallydifficulttoevaluatethecon-
ditionnumberdirectlyfromthedefinition(1.7).Then,wecantakeasthecondition
numberthesmallestnumberforwhichtherelation
"∆w"
"w"
≤condϕ(d)·
"∆d"
"d"
(1.17)
istrue,forallpossibleandsmalldataperturbations∆d(comp.(1.16)).Ifthedata
errorsareonlythemachinenumberrepresentationerrors,thentakingintoaccount
(1.15),therelation(1.17)canbewrittenintheform
"∆w"
"w"
≤condϕ(d)·eps.
(1.18)
Thepresentedwayofreasoningwillbeillustratedbyaproblemofthescalarproduct
calculation,forwhichtheconditionnumber(1.8)wasevaluatedinExample1.3.
n
Example1060Lettheproblembe0(a,b)=
∑
aźbź.Thedataareperturbed,i.e.,
ź11
rd(aź)=aź(1+oź),rd(bź)=bź(1+Bź),whereoźandBźarerelativeerrors
ofthedatarepresentation,|oź|≤epsand|Bź|≤eps.Weshallevaluatean
upperboundoftheconditionnumber,doingsubsequentestimations(andomitting
