# Is a c 1 nujxjt jyj ou2 nujxjt jyj1 ou c nujxjt

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Unformatted text preview: jyj { Golub and van Loan (1996): C = 1:01 if nu < 0:01 We have jfl(xT y) ; xT yj jfl(xT y) ; xT yj jxT yj jxjT jyj C nu jxT yj (5) { The relative error may not be small since jxT yj can be much less than jxjT jyj 6 Backward Error Analysis Prior analysis is called forward error analysis { Relate rounding errors to the solution Backward error analysis: { Relate rounding errors to the original data Consider the dot-product computation { Write (4a) as ^^ fl(xT y) = xT y (6a) ^^ x, y are perturbed values of x, y The rounded dot product fl(xT y) is the exact dot ^^ product xT y of perturbed vectors { Using (4a) 2p x1 6 x2p1 + 1+ ^6 x=6 4 p .. xn 1 + 3 1 7 27 7 5 2p y1 1 + 6 y2p1 + ^6 y=6 4 p .. yn 1 + n { Let x, y be perturbations, i.e., ^ x=x+ x 7 ^ y=y+ y 3 1 7 27 7 (6b) 5 n Backward Error Analysis Observe that Thus, p 1+ k 2 = 1 + k =2 + O( k ) 2 3 2 x ( =2 + O( )) 6 x1( 1=2 + O( 12)) 7 6 7 2 x=6 2 2 . 7 . 4 5 2 xn( n=2 + O( n)) But j kj nu + O(u2), so j xj nu jxj + O(u2) (7a) j yj nu jyj + O(u2) (7b) and 2 2 These bounds are very conservative 8...
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## This document was uploaded on 03/16/2014 for the course CSCI 6800 at Rensselaer Polytechnic Institute.

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