THE EQUIVALENCE OF GRAM-SCHMIDT AND QR FACTORIZATION (page 227) Gram-Schmidt provides another way to compute a QR decomposition: n
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1 HE EQUIVAENCE OF GRA-SCHID AND QR FACORIZAION (page 7 Ga-Schdt podes anothe way to copute a QR decoposton: n gen ectos,, K, R, Ga-Schdt detenes scalas j such that o [ ] [ ] hs s a QR factozaton of the atx [ ] O Snce, n the full an case, thee exsts a unue QR factozaton (f the > 0, the aboe factos Q and R ust be dentcal (n the absence of oundoff eo to those that would be detened by, say, coputng the factozaton usng Householde atces Conesely, any eans of coputng a QR factozaton A QR s an othonoalzaton algoth: the colun ectos of Q fo an othonoal set that spans the sae subspace as the colun ectos of A Note: unueness eues soe noalzaton such as all > 0 Wthout ths, thee s no unueness snce, fo exaple, any can be eplaced by NUERICA SABIIY (page 8 he (classcal Ga-Schdt pocedue s nuecally unstable -- oundoff eos ay cause the coputed ectos to be fa fo othogonal Howee, a slght odfcaton of the pocedue esults n the followng stable coputatonal schee 5
2 he odfed Ga-Schdt Pocedue In the followng, we use a pe to dstngush between entes coputed by the classcal and the odfed Ga-Schdt pocedues ( As befoe: ( Subtact ultples of all othogonal to : K ae ( fo,, so that the esultng ectos, K, ( ( ( (,,, ( ( ( all ae othogonal to hen ( ( ( Copute ( ( ( ( ( (,,, ( ( ( ( ( ( all ae othogonal to and hen ( ( and so on 5
3 Copason of the two pocedues: CASSICA G - S step ectos used ( ( ( ODIFIED G - S step ( ( ( ectos used Execse 0 on page 8: an outlne of the steps n a poof that the odfed Ga- Schdt and the Classcal Ga-Schdt pocedues ae the sae (n the absence of oundoff eos In fact, n the absence of oundoff, and Page 0: a dscusson of why the odfed G-S pocedue s bette nuecally In the Classcal pocedue, one coputes ˆ he uantty s called the coponent of n the decton of Subtactng ths fo aes ˆ othogonal to In the Classcal pocedue, all of the tes (fo,, K, ae subtacted fo at the sae te 55
4 In the odfed pocedue, these coponents ( n the absence of oundoff eo ae subtacted fo n dffeent steps of the algoth Fo exaple, at step (: at step (: at step (: (, ( ( ( ( (, ( ( ( (, wheeas n step ( of the Classcal pocedue, ae coputed ~ (, he coputed alue of ( ( (, (, ay be uch oe accuate than (, because the coponents n the decton of and ae aleady subtacted out In geneal, fl ( fl( and the "nceental" o "gadual" subtacton of the coponents n the odfed pocedue s nuecally bette than the sultaneous subtacton n the Classcal pocedue An EXAPE of ths -- see Execse 6, page A poof of the stablty of the odfed Ga-Schdt pocedue was fst gen by Bjoc n 967 COPARISON OF HE ODIFIED GRA-SCHID PROCEDURE AND QR FACORIZAION (usng eflectos -- see pages - 56
5 hese pocedues ae copaed wth espect to fo two dffeent pobles: -- effcency (flop count -- stablty -- solng l (dscete least-suaes pobles -- calculatng a set of othonoal ectos fo a gen set { } of lnealy ndependent ectos Note: both ethods sole both of these pobles, but they dffe n the aount of coputaton one has to do o sole l pobles usng QR factozaton, one doesn't hae to explctly asseble the poduct Q Q Q Q, wheeas fo the second poble that s what s eued SABIIY RESUS odfed G-S s less stable, especally f the gen ectos ae nealy lnealy dependent (that s, f they ae ll-condtoned If Q [ ] s exactly an soety, then Q I Q and I Q Q 0 hus, I Q Q can be used as a easue of the deaton of the coputed ectos fo othonoalty Usng eflectos, the coputed ectos (wth oundoff eos ae ey nealy othogonal, n that t can be shown that I Q Q cu, whee u s the unt oundoff and c s a constant In contast, the coputed ectos poduced by the odfed Ga-Schdt pocedue deate fo othonoalty n popoton to the condton nube assocated wth the gen lnealy ndependent ectos { } he condton nube of a nonsuae atx wth lnealy ndependent colun ectos can be defned by It can be shown that f ax Vx x axag( V κ ( V n Vx nag( V,, x Schdt pocedue and [ ] K ae the ectos coputed by the odfed Ga- Q, then 57
6 I Q Q uκ V ( Note that ( V ae nealy lnealy dependent, and n ths case, the odfed Ga-Schdt ectos ay deate sgnfcantly fo othonoalty See Exaple 7 on page κ wll be lage f the gen lnealy ndependent ectos { } Reothogonalzaton (page Note that thee s a faly sple way to poe the othonoalty of ectos ~ coputed usng odfed Ga-Schdt If Q [ ~ ~ ~ ] denotes the output fo odfed Ga-Schdt, use Q ~ as the nput fo a second un of the odfed Ga- sn t too ll-condtoned, t ust be ey well condtoned ( axag( Q ~ / nag( Q ~ so the ~ output of the second un wll satsfy I Q Q uκ ( Q u Usually one such un of the eothogonalzaton s suffcent, but t does double the flop count Schdt pocedue Snce Q ~ s nealy an soety (as long as V [ ] As dscussed on page, ths eothogonalzton can actually be done dung the fst un of odfed Ga-Schdt, by eothogonalzng the ectos as they ae coputed EFFICIENCY RESUS ( Fo l pobles, eflectos ae cheape, eung only n flops esus n flops fo odfed G-S (and the Classcal G-S See Execse (page and Execse (page 9 Note, howee, that these flop counts dffe by ey lttle f n >> ( Fo coputng an othonoal set of ectos, f eflectos ae used, then the cost of asseblng the atx Q Q Q Q s also n flops, fo a total cost of n flops See Execse on page 9 But the odfed G-S pocedue ges the othonoal set, so the cost s just n flops CONCUSION: the use of eflectos s about twce as expense as odfed G-S, but s oe stable; so t s stll ecoended f the poble s ll-condtoned (that s nealy lnealy dependent s, f the { } 58
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