ORTHOGONALIZATION WITH A NON-STANDARD INNER PRODUCT WITH THE APPLICATION TO PRECONDITIONING
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1 ORTHOGONALIZATION WITH A NON-STANDARD INNER PRODUCT WITH THE APPLICATION TO PRECONDITIONING Mroslav Rozložník jont work wth Jří Kopal, Alcja Smoktunowcz and Mroslav Tůma Insttute of Computer Scence, Czech Academy of Scences, Prague, Czech Republc 10th IMACS Internatonal Symposum on Iteratve Methods n Scentfc Computng, Marrakech, Morocco, May 18-21, 2011 Mroslav Rozložník (ICS CAS) Orthogonalzaton wth a non-standard nner product 10th IMACS 1 / 20
2 Orthogonalzaton wth a non-standard nner product A symmetrc postve defnte m m matrx Z (0) = [z (0) 1,..., z (0) n ] full column rank m n matrx A-orthogonal bass of span(z (0) ): Z = [z 1,..., z n] - m n matrx havng orthogonal columns wth respect to the nner product, A U upper trangular n n matrx Z (0) = ZU, Z T AZ = I (Z (0) ) T AZ (0) = U T U Mroslav Rozložník (ICS CAS) Orthogonalzaton wth a non-standard nner product 10th IMACS 2 / 20
3 Condton number of orthogonal and trangular factor Z T AZ = (A 1/2 Z) T (A 1/2 Z) = I = A 1/2 Z s orthogonal wth respect to the standard nner product A 1/2 Z (0) = (A 1/2 Z)U s a standard QR factorzaton κ(z) κ 1/2 (A) κ(u) = κ(a 1/2 Z (0) ) κ 1/2 (A)κ(Z (0) ) partcular case Z (0) = I: Z = U 1 upper trangular m n matrx κ(u) = κ(z) Mroslav Rozložník (ICS CAS) Orthogonalzaton wth a non-standard nner product 10th IMACS 3 / 20
4 Approxmaton propertes of orthogonal factor Z T AZ = I ZZ T = Z (0) U 1 U T (Z (0) ) T = Z (0) [(Z (0) ) T AZ (0) ] 1 (Z (0) ) T AZZ T : orthogonal projector onto R(AZ (0) ) and orthogonal to R(Z (0) ) ZZ T A : orthogonal projector onto R(Z (0) ) and orthogonal to R(AZ (0) ) mportant case Z (0) square and nonsngular: nverse factorzaton ZZ T = A 1 Mroslav Rozložník (ICS CAS) Orthogonalzaton wth a non-standard nner product 10th IMACS 4 / 20
5 Important applcaton: approxmate nverse precondtonng Z gves the approxmate nverse Z Z T A 1 Ax = b Z T A Zy = Z T b, where x = Zy Ū approxmates the Cholesky factor of (Z (0) ) T AZ (0) the loss of orthogonalty Z T A Z I the factorzaton error Z (0) ZŪ Cholesky factorzaton error (Z (0) ) T AZ (0) ŪT Ū Mroslav Rozložník (ICS CAS) Orthogonalzaton wth a non-standard nner product 10th IMACS 5 / 20
6 Problem 1a (nverse Hlbert matrx) 10 5 A A+1e 14 A *rand(n,n) A+1e 12 A *rand(n,n) A+1e 10 A *rand(n,n) A+1e 8 A *rand(n,n) Loss of orthogonalty I Z T AZ = 0 = 1e 14 = 1e 12 = 1e 10 = 1e 8 u * A Z 2 X T X resdual norm r e 14 * A Z 2 X T X e 12 * A Z 2 X T X 1e 10 * A Z 2 X T X 1e 8 * A Z 2 X T X condton number (A) teraton n Mroslav Rozložník (ICS CAS) Orthogonalzaton wth a non-standard nner product 10th IMACS 6 / 20
7 Egenvalue based mplementaton - EIG 1. spectral decomposton A = VΛV T 2. QR factorzaton Λ 1/2 V T Z (0) = QU 3. orthogonal-dagonal-orthogonal matrx multplcaton Z = VΛ 1/2 Q backward stable egendecomposton + backward stable QR: Z T A Z I O(u) A Z 2 Mroslav Rozložník (ICS CAS) Orthogonalzaton wth a non-standard nner product 10th IMACS 7 / 20
8 Gram-Schmdt orthogonalzaton z (j) = z (j 1) α j z j z = z ( 1) /α, α = z 1 modfed Gram-Schmdt (MGS) algorthm SAINV algorthm: α j = z (j 1), z j A classcal Gram-Schmdt (CGS) algorthm: α j = z (0), z j A AINV algorthm: oblque projectons α j = z (j 1), z (0) j A /α jj A Mroslav Rozložník (ICS CAS) Orthogonalzaton wth a non-standard nner product 10th IMACS 8 / 20
9 Local errors n the (modfed) Gram-Schmdt process z (j) z (j) = z (j 1) α j z j α j = z (j 1), z j A, z j A = (1 z j 2 A ) z(j 1), z j A ( z (j), z j A = z (j) = z (j 1) ᾱ j z j ᾱ j = fl[ z (j 1), z j A ] fl[ z (j 1) ), z j A ] z (j 1), z j A z j 2 A Mroslav Rozložník (ICS CAS) Orthogonalzaton wth a non-standard nner product 10th IMACS 9 / 20
10 Loss of orthogonalty n the MGS algorthm: O(u)κ(A)κ(A 1/2 Z (0) ) < 1 I Z T A Z O(u) A Z 2 κ(a 1/2 Z (0) ) 1 O(u) A Z 2 κ(a 1/2 Z (0) ) Mroslav Rozložník (ICS CAS) Orthogonalzaton wth a non-standard nner product 10th IMACS 10 / 20
11 Loss of orthogonalty n the CGS and AINV algorthms O(u)κ(A)κ(A 1/2 Z (0) )κ(z (0) ) < 1 I Z T A Z O(u) A 1/2 Z κ(a 1/2 Z (0) )κ 1/2 (A)κ(Z (0) ) 1 O(u) A 1/2 Z κ(a 1/2 Z (0) )κ 1/2 (A)κ(Z (0) ) Mroslav Rozložník (ICS CAS) Orthogonalzaton wth a non-standard nner product 10th IMACS 11 / 20
12 Classcal Gram-Schmdt (CGS2) wth reorthogonalzaton z (1) = z (0) z (2) = z (1) P α (1) j z j, α (1) j = z (0), z j A 1 j=1 P α (2) j z j, α (2) j = z (1), z j A 1 j=1 z = z (2) /α, α = z (2) A O(u)κ 1/2 (A)κ(A 1/2 Z (0) ) < 1 I Z T A Z O(u) A Z 2 Mroslav Rozložník (ICS CAS) Orthogonalzaton wth a non-standard nner product 10th IMACS 12 / 20
13 Local errors n the nner product and normalzaton general postve defnte A : fl[ z (j 1), z j A ] z (j 1), z j A O(u) A z (j 1) z j 1 z j 2 A O(u) A z j 2 dagonal (weght matrx) A : fl[ z (j 1), z j A ] z (j 1), z j A O(u) z (j 1) A z j A 1 z j 2 A O(u) Mroslav Rozložník (ICS CAS) Orthogonalzaton wth a non-standard nner product 10th IMACS 13 / 20
14 A dagonal smlar to orthogonalzaton wth the standard nner product MGS algorthm: CGS and AINV algorthms: CGS wth reorthogonalzaton: O(u)κ(A 1/2 Z (0) ) < 1 I Z T A Z O(u)κ(A1/2 Z (0) ) 1 O(u)κ(A 1/2 Z (0) ) O(u)κ 2 (A 1/2 Z (0) ) < 1 I Z T A Z O(u)κ2 (A 1/2 Z (0) ) 1 O(u)κ 2 (A 1/2 Z (0) ) O(u)κ(A 1/2 Z (0) ) < 1 I Z T A Z O(u) [standard nner product A = I; MGS: Björck 1967, Björck and Page 1992, CGS: Graud, Langou, R, van den Eshof 2005, Barlow, Langou, Smoktunowcz 2006, CGS2: Graud, Langou, R, van den Eshof 2005] [weghted least squares problem; MGS: Gullksson, Wedn 1992, Gullksson 1995] Mroslav Rozložník (ICS CAS) Orthogonalzaton wth a non-standard nner product 10th IMACS 14 / 20
15 Numercal experments - four extremal cases 1. κ 1/2 (A) κ(a 1/2 Z (0) ) 2. κ(a 1/2 Z (0) ) κ 1/2 (A) 3. A postve dagonal 4. Z (0) = I Mroslav Rozložník (ICS CAS) Orthogonalzaton wth a non-standard nner product 10th IMACS 15 / 20
16 10 0 Problem 9 (Hlbert matrx), κ(a) = 1.2e Loss of orthogonalty I Z T AZ MGS CGS CGS2 AINV EIG u κ(a) u κ(a) κ(a 1/2 Z (0) ) u κ(a) κ(a 1/2 Z (0) ) κ(z (0) ) condton number (A 1/2 Z (0) ) Mroslav Rozložník (ICS CAS) Orthogonalzaton wth a non-standard nner product 10th IMACS 16 / 20
17 10 0 Problem 11 (Hlbert matrx) Loss of orthogonalty I Z T AZ MGS CGS CGS2 AINV EIG u κ(a) u κ(a) κ(a 1/2 Z (0) ) u κ(a) κ(a 1/2 Z (0) ) κ(z 0 ) condton number (A) Mroslav Rozložník (ICS CAS) Orthogonalzaton wth a non-standard nner product 10th IMACS 17 / 20
18 10 0 Problem 3 (dagonal matrx) Loss of orthogonalty I Z T AZ MGS CGS CGS2 AINV EIG u κ(a 1/2 Z (0) ) u κ 2 (A 1/2 Z (0) ) condton number (A) Mroslav Rozložník (ICS CAS) Orthogonalzaton wth a non-standard nner product 10th IMACS 18 / 20
19 10 0 Problem 6 (Hlbert matrx) Loss of orthogonalty I Z T AZ MGS CGS CGS2 AINV EIG u κ(a) u κ(a) κ(a 1/2 Z (0) ) u κ(a) κ(a 1/2 Z (0) ) κ(z 0 ) condton number (A) Mroslav Rozložník (ICS CAS) Orthogonalzaton wth a non-standard nner product 10th IMACS 19 / 20
20 Thank you for your attenton!!! Reference: J. Kopal, R, A. Smoktunowcz, and M. Tůma: Roundng error analyss of orthogonalzaton wth a non-standard nner product, submtted Mroslav Rozložník (ICS CAS) Orthogonalzaton wth a non-standard nner product 10th IMACS 20 / 20
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