2 pq2 vj j 1 pqj 1 vj q1 q1 vj q2

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Unformatted text preview: IRm Pq = qq qq The complements are rank m − 1 orthogonal projections P⊥ q = I − qq qq By deﬁnition of Pj Pj = P⊥qj −1 · · · P⊥q2 P⊥q1 where P1 = I , and thus vj = P⊥qj −1 · · · P⊥q2 P⊥q1 xj 22 / 27 Modiﬁed Gram-Schmidt process (cont’d) For numerical stability, evaluating the following formulas in order (for now consider an algorithm is considered as stable if it is not too sensitive to the eﬀects of rounding errors) (1) vj vj = xj , (2) vj (3) vj = P⊥q1 vj (j ) = vj (1) = vj (1) (2) = vj . . . (2) = P⊥q2 vj . . . (j −1) = P⊥qj −1 vj − q1 q1 vj , − q2 q2 vj , (i ) (1) (1) (3) (2) (1) (2) (j −1) = vj Projection matrix P⊥qi can be applied to vj immediately after qi is known (1) (1) v1 = v1 = x1 , q1 = r1 v1 11 (2) (1) (j −1) − qj −1 qj −1 vj for each j &gt; i (1) (2) v2 = v2 = v2 − q1 q1 v2 = v2 − ( r1 v1 )r12 11 v3 = v3 = v3 − q2 q2 v3 , (2) (1) (1) (1) (1) v3 = v3 − q1 q1 v3 = v3 − ( r1 v1 )r13 11 23 / 27 Modiﬁed Gram-Schmidt algorithm Steps (1) vj vj (j ) = vj = xj , . . . (j −1) = P⊥qj −1 vj Algorithm: for i = 1 to n do vi = xi end for for i = 1 to n do rii = vi v qi = riii for j = i + 1 to n do rij = qi vj vj = vj − rij qi end for end for . . . (j −1) (j −1) = vj − qj −1 qj −1 vj 24 / 27 25 / 27 d I S S iY I Y Y I bI `d 1Q xR    q xpR aVVq wrS ¤V V' w q `YW e  I i f a i q © t  % %   £ S q ipxR Q U a I  Qb w  pdsi q  I d  w a p  4d Y a Q x rxR I Y g w q TP VU a I w S wS SS  h £ Q d  Y !  aUw d d w YW Sw w t¨I Y a d g b  w a &quot; U g S a d VI Y  a d S  Y w q `)Y cVI Y  Q &amp;wxY g  Y Yi S Rb VU a WpY V S &amp;Q  Yi Q af II S Qa Y  a RU RI xyY e d f S a Y I Q XS Vq d g S e 1V YeTw q `W b)  d I Y a xY  &amp;w  WpY I d a VI Y g VU  S a SR dQ I Q ' Y Q a i  w Y a Q S R d S P S W iYR S i  w Y Q YY  t SpyY e )w q T!VU a Y S q px!VU a Y a pd I Y a xY  w  ¥I Y  a d  ipd I p5VU a wxY a fb VU g d  h  wY S S  de w § pd w 3 x`W Vq Y \$I Y a d wrS  Y4VU a I I RU PI aI g a £ Q hwrS f pY  IbVf bSi Y  d S SR Si Yi w a t ¨I Y Y ¥VI d 1&amp;Q d FVI Y g S Q VU a Y a a xyY e d bp I d yg &amp;w      I £ SVU a I  Y bS Rq Y Q f IQ  f aS SVU a x q VU a &amp;Q d ¤&amp;Q eP¤VU a Y a VRI Y HQ &amp;wxY g xY &amp;a Y e RU 6a d x I d rg &amp;w  I £ VU  U Yw Q  a w S a Sa Si S Qf  S w aS     As m, n → ∞, the ﬁgure converges to a right triangular pri...
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This document was uploaded on 02/10/2014.

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