Ake Bjorck and Chris C. Paige. Loss and recapture of orthogonality in the modified Gram--Schmidt algorithm. SIAM J. Matrix Anal. Appl., 13(1):176--190, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the better stability properties. The MGS method is widely used, for example within iterative methods such as the Arnoldi method and GMRES. A remarkable connection between the MGS method and Householder QR factorization has been known since the 1960s but has only recently been fully exploited [20]: the MGS method applied to A 2 IR is equivalent, both mathematically and numerically, to Householder QR factorization of the padded matrix Theta 0n (m n) Thetan . This connection brings two benefits. First, it leads to shorter and more insightful error analysis for the MGS method. ....

....mathematically and numerically, to Householder QR factorization of the padded matrix Theta 0n (m n) Thetan . This connection brings two benefits. First, it leads to shorter and more insightful error analysis for the MGS method. Second, it leads to new stable algorithms. Bjorck and Paige [20], 21] derive a new backward stable MGS based algorithm for solving the augmented system I A y x b c This system characterizes the solutions to the following two problems: min x kb Gamma Axk 2 2c x; min y ky Gamma bk 2 subject to A y = c: Note that these problems ....

Ake Bjorck and C. C. Paige. Loss and recapture of orthogonality in the modified Gram--Schmidt algorithm. SIAM J. Matrix Anal. Appl., 13(1): 176--190, 1992.

....precision. 1. Previous Results We consider the Modified Gram Schmidt (MGS) algorithm applied to a matrix A # R mn with full rank n # m and singular values : s 1 # . # s n 0, we define the condition number of A as k = s 1 s n . Using results from Bjorck and Paige in [1] and [2], we know that MGS computes Q 1 # R mn and R # R nn so that there exists E # R mn , E # R mn and Q 1 # R mn , where A E = Q 1 R and # E# 2 # c 1 u#A# 2 , 1.1) #I Q T 1 Q 1 # 2 # c 2 ku, 1.2) if cuk 1 then A E = Q 1 R , Q 1 T Q 1 = I ....

....on k, if A is well conditioned then Q 1 is orthogonal to machine precision. Equations (1.3) say that R solves the QR factorization problem in a backward stable sense ; that is, there exists an exact orthonormal matrix Q 1 so that Q 1 R is a QR factorization of a slightly perturbed A. In [2], it is shown that the result (1.3) holds under the assumption that cuk 1. 1.4) In fact, 1.4) enables R to be singular. Under this assumption and defining h = 1 1 cuk , 1.5) Bjorck and Paige obtain an upper bound for # R 1 # 2 as #A# 2 # R 1 # 2 # hk. 1.6) L. Giraud: ....

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A. Bjorck and C. C. Paige. Loss and recapture of orthogonality in the modified Gram-Schmidt Algorithm. SIAM J. Matrix Analysis and Applications, 13(1):176--190, January 1992.

....T X) 2I, tril(Y T X) T = triu(Y T X) For algorithms based on projection operations, it is convenient in algorithm analysis to use nonsingular transformations that embed the projections. Such a technique is used, for example, in the analysis of the Modified Gram Schmidt(MGS) method [6]. An aggregated transformation G = I X [I tril(Y T X) 1 Y T can be augmented with an approach illustrated in the following: G def = U X [tril(Y T X) I] 1 (V T , Y T ) where U and V are chosen, for example, so that U T V = I. If diag(Y T X) is nonsingular, so is ....

....that the computational procedure for the WY form in Section 2.1 amounts to the merge of the explicit inverse in factor form and the multiplication of X and L. The factor form of aggregation kernels for embedding Householder transformations was first seen in the analysis work by Bjorck and Page [6] on the loss and recapture of orthogonality in the MGS method, although it is not 9 explicitly related to the YTY compact form. The factor form presented in Theorem 3 is for a much larger class of triangular matrices, especially, for aggregation kernels of general elementary matrices. It can be ....

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Ake Bjorck and C. C. Page. Loss and recapture of orthogonality in the modified Gram-Schmidt algorithm. SIAM J. Matrix Anal. Appl., 13(1):176--190, 1992.

....grows quadratically with m. Modified Gram Schmidt (MGS) improves the numerical stability of the GS by orthogonalizing individual pairs of vectors rather than a vector against a block. For MGS, the error in the orthogonality of Q can be bounded by ffl(W ) where (W ) is the condition number of W [1, 3]. The operation count for MGS is the same with GS, but working on individual vectors allows only for level 1 BLAS kernels, which impairs significantly the actual performance on cache based computers. Finally, the number of synchronization points grows quadratically with m. In the context of the ....

A. Bjorck and C. Paige. Loss and recapture of orthogonality in the modified gram-schmidt algorithm. SIAM J. Matrix Anal. Appl., 13(1):176--190, 1992.

....of rounding errors, the quantities computed either by the Gram Schmidt orthogonalization or the Householder orthogonalization do not satisfy the exact recurrence (2. 11) In the following two sections, based on the analysis of Wilkinson, Bjorck and Paige for the QR decomposition [57] 5] and [8], we find the analogue of (2.11) for the finite precision results. It is an extension of Paige s formula for the 3 term Lanczos recurrence in the symmetric case [38] 3.1 Implementation based on the Householder transformations In the section 2.2, the computation of the Arnoldi basis was ....

....by the extremal singular values of the original matrix A. 3.2 Modified Gram Schmidt implementation Up to now, we considered the Householder Arnoldi recurrence. The rounding error analysis of the modified Gram Schmidt QR factorization has been presented by Bjorck in [5] and Bjorck and Paige in [8] (see also [7] Using [5] p.10 or [8] p. 180, we have the recurrence formula for the computed quantities [v 1 ; f l(Av 1 ) f l(Av n ) V n 1 R n 1 F 0;n ; 3.17) where kF 0;n k O(nN) k[v 1 ; f l(Av 1 ) f l(Av n ) k F : 3.18) In a finite precision computation, the ....

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A. Bjorck, C. C. Paige, Loss and recapture of orthogonality in the modified Gram-Schmidt algorithm, SIAM J. Matrix Anal. Appl. 13, 1 (1992), pp. 176-190.

....to the relative advantages of factorization using Householder reflections, Givens rotations, or modified Gram Schmidt [8, Sec. 2. 4] The first two alternatives were known to have similar desirable error properties, and modified Gram Schmidt was finally shown stable in a paper of Bjorck and Paige [10] by exploiting the fact, known to many early practitioners such as Sheffield, that modified GramSchmidt is numerically equivalent to Householder QR applied to the matrix 0 M : If the problem is difficult in the sense that M is ill conditioned, then more refined tools are needed. The QR ....

Ake Bjorck and C. C. Paige. Loss and recapture of orthogonality in the modified Gram--Schmidt algorithm. SIAM J. Matrix Anal. Appl., 13:176-- 190, 1992.

....in terms of the condition number (v 1 ; AV n ) Assuming that nN 2 (v 1 ; AV n ) 1, we can write (2:5) MGSA) kI Gamma V T n V n k O(n 2 N) v 1 ; AV n ) i 3 n 2 N (v 1 ; AV n ) 8 J. DRKO SOV A, A. GREENBAUM, M. ROZLO ZN IK AND Z. STRAKO S Moreover, it follows from [30] and [4], see also [3] that for both the HHA and MGSA implementations there exists an exactly orthonormal matrix V n 1 such that (2:6) AV n = V n 1 H n 1;n F n ; 2:7) k F n k i 4 j(n; l; N) kAk i 0 4 nN 3=2 kAk: For the HHA implementation, the matrix V n 1 is close to V n 1 , 2:8) ....

....that nN 3=2 (v 1 ; AV n ) 1, we have (2:9) MGSA) kV n 1 Gamma V n 1 k i 6 j(n; l; N) v 1 ; AV n ) We defer the discussion of the IMGSA, ICGSA analogues of (2.5) 2.9) to Section 3. The positive constants i 1 ; i 2 ; i 6 depend on the details of the arithmetic ( 30] 1] [4]) We assume, for simplicity, i j 1; j = 1; 2; 6: Formulas (2.2) and (2.3) suggest an explanation for the deterioration effects of rounding errors to the Krylov subspace methods which were observed for very ill conditioned matrices in [10] 7] for example. Suppose that for some actual ....

A. Bjorck, C. C. Paige, Loss and recapture of orthogonality in the modified Gram-Schmidt algorithm, SIAM J. Matrix Anal. Appl. 13, 1 (1992), pp. 176-190.

.... Several techniques are currently used to produce a QR decomposition of a matrix [1, 2, 3, 4] They are mathematically equivalent, but, in finite precision arithmetic, it can be shown that some of these algorithms are unstable [5] Bjorck has studied the behavior of the Gram Schmidt algorithms [6, 7, 8]. For the modified version, it was able to prove that the loss of orthogonality of the matrix Q is Scientific Computing and Computational Mathematics, Gates 2B, Stanford University CA 94306. Email: dv sccm.Stanford.edu This work is partially supported by the Fonds National de la Recherche ....

A. Bjorck and C. C. Paige. Loss and recapture of orthogonality in the modified Gram-Schmidt algorithm. SIAM J. Mat. Anal. Appl., No. 1, pp. 176--190, 1992.

....T X) 2I, tril(Y T X) T = triu(Y T X) For algorithms based on projection operations, it is convenient in algorithm analysis to use nonsingular transformations that embed the projections. Such a technique is used, for example, in the analysis of the Modified Gram Schmidt(MGS) method [6]. An aggregated transformation G = I Gamma X [I tril(Y T X) Gamma1 Y T can be augmented with an approach illustrated in the following: G def = U X [tril(Y T X) I] Gamma1 (V T ; Y T ) where U and V are chosen, for example, so that U T V = I. If diag(Y T X) is ....

....that the computational procedure for the WY form in Section 2.1 amounts to the merge of the explicit inverse in factor form and the multiplication of X and L. The factor form of aggregation kernels for embedding Householder transformations was first seen in the analysis work by Bjorck and Page [6] on the loss and recapture of orthogonality in the MGS method, although it is not explicitly related to the YTY compact form. The factor form presented in Theorem 3 is for a much larger class of triangular matrices, especially, for aggregation kernels of general elementary matrices. It can be ....

[Article contains additional citation context not shown here]

Ake Bjorck and C. C. Page. Loss and recapture of orthogonality in the modified Gram-Schmidt algorithm. SIAM J. Matrix Anal. Appl., 13(1):176--190, 1992.

....formulation corresponding to computing r n via the modified Gram Schmidt, namely, r n = I Gamma q n q T n ) I Gamma q n Gamma1 q T n Gamma1 ) I Gamma q 1 q T 1 )r 0 ; 22) which would guarantee that the optimality condition (4) is satisfied to much higher accuracy. Following [2], 1] we denote q j = I Gamma q 1 q T 1 ) I Gamma q j Gamma1 q T j Gamma1 )q j ; j = 2; n; q 1 = q 1 (23) and Qn = q 1 ; q n ] Then (I Gamma q n q T n ) I Gamma q n Gamma1 q T n Gamma1 ) I Gamma q 1 q T 1 ) I Gamma Qn Q T n : 24) ....

Bjorck, A, Paige, C.C.: Loss and recapture of orthogonality in the modified GramSchmidt algorithm SIAM J. Matrix Anal. Appl. 13, 1 (1992) 176-190

....corresponding to computing r n via the modified Gram Schmidt process, namely, r n = I Gamma q n q T n ) I Gamma q n Gamma1 q T n Gamma1 ) I Gamma q 1 q T 1 )r 0 ; 3. 3) which would guarantee that the optimality condition (1. 6) is satisfied to much higher accuracy. Following [6], 5] we denote q j = I Gamma q 1 q T 1 ) I Gamma q j Gamma1 q T j Gamma1 )q j ; j = 2; n; q 1 = q 1 (3. 4) and Q n = q 1 ; q n ] Then (I Gamma q n q T n ) I Gamma q n Gamma1 q T n Gamma1 ) I Gamma q 1 q T 1 ) I Gamma Q n Q T n : 3. ....

A. Bjorck, C.C. Paige, Loss and Recapture of Orthogonality in the Modified GramSchmidt Algorithm, SIAM J. Matrix Anal. Appl., Vol. 13, No. 1, pp. 176-190, 1992

....= 1; 2; i h k;i = v T k w w = w Gamma h k;i v k end NUMERICAL BEHAVIOUR OF THE MGS GMRES IMPLEMENTATION 3 h i 1;i = kwk v i 1 = w=h i 1;i end . Here, the upper Hessenberg matrix Hn 1;n is still computed in a backward stable way, i.e. in finite precision arithmetic we have (see [2] [4], 1] 6] AVn = Vn 1Hn 1;n Fn ; AVn = Vn 1Hn 1;n Fn ; 1.5) where kFnk i 1 j(n; l; N ) kAk; k Fnk i 1 j(n; l; N ) kAk; V T n 1 Vn 1 = I n 1 ; j(n; l; N ) n 3=2 N n 1=2 lN 1=2 ; Vn 1 is the closest orthonormal matrix to Vn 1 in any unitarily invariant norm, is ....

.... and l is the maximal number of nonzero entries per row of the matrix A (as in [6] by r 0 , Vn , Hn 1;n , yn and xn we denote from now on the computed quantities) The orthogonality of Vn 1 may gradually deteriorate, but assuming nN 3=2 ( v 1 ; AVn ] 1, it can be shown, see [2] [4], 6] that kI Gamma V T n 1 Vn 1 k i 2 j(n; l; N ) v 1 ; AVn ] 1.6) and also k Vn 1 Gamma Vn 1k i 2 j(n; l; N ) v 1 ; AVn ] 1.7) for a moderate size constant i 2 . Constant factors i 1 , i 2 (as the other constants i j ; j = 3; introduced in a number of places) are ....

A. Bjorck, C.C. Paige, Loss and Recapture of Orthogonality in the Modified GramSchmidt Algorithm, SIAM J. Matrix Anal. Appl. 13, pp. 176-190, 1992

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Ake Bjorck and Chris C. Paige. Loss and recapture of orthogonality in the modified Gram--Schmidt algorithm. SIAM J. Matrix Anal. Appl., 13(1):176--190, 1992.

No context found.

A. Bjorck and C. Paige. Loss and Recapture of Orthogonality in the Modified Gram-Schmidt Algorithm. SIAM J. Matrix Anal. Appl. 13(1) (1992), 176-190.

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A. Bj orck and C. C. Paige, Loss and recapture of orthogonality in the modified GramSchmidt algorithm, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 176--190.

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