A. Bjorck and G. H. Golub. Iterative Refinement of Linear Least Squares Solutions by Householder Transformation. BIT, 7:322--337, 1967.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(higham ma.man.ac.uk, http: www.ma.man.ac.uk higham ) This work was supported by Engineering and Physical Sciences Research Council grant GR L76532. Solving this unconstrained problem by QR factorization completes the elimination method as originally presented by Bjorck and Golub [5] (see also [13, Chapter 21] It is instructive to think of the method in terms of transformations on the matrix B over A : 0 e A 2 Gamma e A 1 R 5 ; is upper triangular. Note that the penultimate transformation is simply the annihilation of A 1 by Gaussian elimination. We will ....

....The EG method produces exactly the same factorization; the difference in the methods lies in the intermediate quantities computed. Note that another way to derive both the EG method and Algorithm EH is to substitute the factorization into the associated augmented system; this is done in [4] and [5] for the EG method. 4. Numerical Stability. Now we consider the stability of Algorithm EH. First, we recall what is known about the stability of Householder QR factorization for solving the LS problem min x kb Gamma Axk 2 . The method is normwise backward stable. Moreover, when column pivoting ....

[Article contains additional citation context not shown here]

Ake Bjorck and Gene H. Golub. Iterative refinement of linear least squares solutions by Householder transformation. BIT, 7:322--337, 1967.

....: 3.1) The EG method produces exactly the same factorization; the difference in the methods lies in the intermediate quantities computed. Note that another way to derive both the EG method and Algorithm EH is to substitute the factorization into the associated augmented system; this is done in [4] and [5] for the EG method. 4. Numerical Stability. Now we consider the stability of Algorithm EH. First, we recall what is known about the stability of Householder QR factorization for solving the LS problem min x kb Gamma Axk 2 . The method is normwise backward stable. Moreover, when column ....

....as a single loop with the help of a parameter that specifies the extent of the Householder vector, yielding the succinct pseudocode shown in Figure 4.1. The EG method and a variant of it based on modified Gram Schmidt orthogonalization can be expressed in similarly concise fashions, as shown in [4], 5] How does the stability of Algorithm EH compare with that of the EG method The only published result for the EG method is one of Barlow and Handy [2] which bounds kC Gamma Rk 2 for the factorization (3.1) in terms of kCk 2 ; this result does not reveal the stability of the overall ....

Ake Bjorck. Iterative refinement of linear least squares solutions II. BIT, 8:8--30, 1968.

....all multiplied by 10 ; this choice of B is denoted small oe min (B 11 ) The vectors b and d have random elements from the normal(0,1) distribution. We solved the LSE problems using three different methods. Computed solutions are denoted by b x. 1. The elimination method of Bjorck and Golub [4], 3, Sec. 5.1.2] which uses a QR factorization of B to eliminate p of the unknowns, thus reducing the problem to an unconstrained LS problem; backward error analysis for the method is given in [7] It is well known that column pivoting needs to be used in the QR factorization in order to obtain ....

A. Bjorck and G. H. Golub. Iterative refinement of linear least squares solutions by Householder transformation. BIT, 7:322--337, 1967.

.... 2 ] e A 1 2 R m Thetap we reduce the LSE problem to the unconstrained problem e x2 k( e A 2 Gamma e A 1 R 1 R 2 )ex 2 Gamma (b Gamma e A 1 R d)k 2 : Solving this unconstrained problem by QR factorization completes the elimination method as originally presented by Bjorck and Golub [9] (see also [38, Chapter 21] It is instructive to think of the method in terms of transformations on the matrix B over A : p B 1 B 2 m A 1 A 2 e A 1 e A 2 0 e A 2 Gamma e A 1 R 0 0 5 ; where R 3 2 R (n Gammap) Theta(n Gammap) is upper triangular. Note that the penultimate ....

Ake Bjorck and Gene H. Golub. Iterative refinement of linear least squares solutions by Householder transformation. BIT, 7:322--337, 1967.

....above. Some column interchanges might be necessary. Note: Part (a) is a statement of Gram Schmidt orthogonalization (see any introductory linear algebra text, for example, Anton and Rorres (1987) but the Gram Schmidt algorithm is numerically unstable, while the Householder algorithm is stable. Bjorck (1968) has an alternative stable algorithm. 7.3 The L U factorization. Theorem 7.3.1 Under certain conditions (stated in proof) an n Theta n matrix A of rank n can be factored to the form A = LU ; where L is lower triangular with diagonal elements unity and U is upper triangular of rank n. Note ....

Bjorck A., Iterative refinement of linear least squares solutions II. BIT, 8, pp8-30, 1968.

....= B 2 Gamma B 1 C Gamma1 1 C 2 . Consequently, the computation of the solution of (3) can be reduced to solving the unconstrained linear least squares problem k B 2 w 2 Gamma TZ (f)k min w22R N (5) A practical way for computing the solution of (5) and thus of (3) is dating back to [1] (see also [6] Chapter 21) The starting point in [1] is a decomposition of C = Q T 1 [ C 1 ; C 2 ] where Q 1 2 R Q ThetaQ is orthogonal and C 1 2 R Q ThetaQ is upper triangular. Using the identity B 2 = B 2 Gamma (B 1 C Gamma1 1 ) Q 1 C 2 ) B 2 Gamma B 1 ....

.... computation of the solution of (3) can be reduced to solving the unconstrained linear least squares problem k B 2 w 2 Gamma TZ (f)k min w22R N (5) A practical way for computing the solution of (5) and thus of (3) is dating back to [1] see also [6] Chapter 21) The starting point in [1] is a decomposition of C = Q T 1 [ C 1 ; C 2 ] where Q 1 2 R Q ThetaQ is orthogonal and C 1 2 R Q ThetaQ is upper triangular. Using the identity B 2 = B 2 Gamma (B 1 C Gamma1 1 ) Q 1 C 2 ) B 2 Gamma B 1 C 2 the computation of B 2 requires to solve the ....

Ake Bjrk and G. H. Golub, Iterative Refinement of Linear Least Squares Solutions by Householder Transformation, BIT 7 (1967), 322--337.

....so that the weighted residual (d Gamma Cb) remains of a size with the residual y Gamma Xb. If is large enough, the residual d Gamma Cb will be so small that the constraint is effectively satisfied. The origin of the weighting method is unknown, but an early reference is a paper by Bjorck [3], who observed that as increases the solution of (1.2) approaches the constrained least squares solution. The Weighting Method The method has the appeal of simplicity weight the constraints and invoke a least squares solver. In principal, the weighted problem (1.2) can be solved by any ....

A. Bjorck. Iterative refinement of linear least squares solutions II. BIT, 8:8--30, 1968.

....performing m steps of Gaussian elimination on W. At that point we can shift to orthogonal triangularization followed by back substitution to solve the entire problem. This technique has been used by Shepherd and McWhirter [12] in recursive least squares applications. In a different spirit, Bjorck [2] uses an orthogonal decomposition of C to compute the Schur complement. The perturbation theory for the constrained problem [6] says that, among other things, the condition of the problem depends on the condition number of the constraint matrix C. Now it is possible for C to be well conditioned ....

A. Bjorck. Contribution no. 22. Iterative refinement of linear least squares solutions by Householder transformations. BIT, 7:322--337, 1967.

....by 10 Gamma8 ; this choice of B is denoted small oe min (B 11 ) The vectors b and d have random elements from the normal(0,1) distribution. We solved the LSE problems using three different methods. Computed solutions are denoted by b x. 1. The elimination method of Bjorck and Golub [4], 3, Sec. 5.1.2] which uses a QR factorization of B to eliminate p of the unknowns, thus reducing the problem to an unconstrained LS problem; backward error analysis for the method is given in [7] It is well known that column pivoting needs to be used in the QR factorization in order to obtain ....

A. Bjorck and G. H. Golub. Iterative refinement of linear least squares solutions by Householder transformation. BIT, 7:322--337, 1967.

....A = 0 b 0 1 A or 0 ffI A A T 0 1 A 0 ff Gamma1 r x 1 A = 0 b 0 1 A (2:3) which is the scaled form. This method was proposed by Bartels et al. and later considered for the sparse case by Hachtel. Bjorck used it in a study of iterative refinement for least square solutions [7]. Numerical experiments on this method and the comparison with other methods have been done [1] 20] If the pivots are chosen from the diagonal when Cholesky factorization is applied to the augmented system, then after m steps, the reduced system is exactly the normal equations. Expressing the ....

A. Bjorck. Iterative refinement of linear least squares solutions. BIT, 7:257--278, 1967.

....http: www.ma.man.ac.uk higham ) This work was supported by Engineering and Physical Sciences Research Council grant GR L76532. 2 A. J. COX AND N. J. HIGHAM Solving this unconstrained problem by QR factorization completes the elimination method as originally presented by Bjorck and Golub [5] (see also [13, Chapter 21] It is instructive to think of the method in terms of transformations on the matrix B over A : B A = p n Gammap p B 1 B 2 m A 1 A 2 R 1 R 2 e A 1 e A 2 R 1 R 2 0 e A 2 Gamma e A 1 R Gamma1 1 R 2 2 4 R 1 R 2 0 R 3 0 0 3 5 ; ....

....The EG method produces exactly the same factorization; the difference in the methods lies in the intermediate quantities computed. Note that another way to derive both the EG method and Algorithm EH is to substitute the factorization into the associated augmented system; this is done in [4] and [5] for the EG method. 4. Numerical Stability. Now we consider the stability of Algorithm EH. First, we recall what is known about the stability of Householder QR factorization for solving the LS problem min x kb Gamma Axk 2 . The method is normwise backward stable. Moreover, when column pivoting ....

[Article contains additional citation context not shown here]

Ake Bjorck and Gene H. Golub. Iterative refinement of linear least squares solutions by Householder transformation. BIT, 7:322--337, 1967.

....: 3.1) The EG method produces exactly the same factorization; the difference in the methods lies in the intermediate quantities computed. Note that another way to derive both the EG method and Algorithm EH is to substitute the factorization into the associated augmented system; this is done in [4] and [5] for the EG method. 4. Numerical Stability. Now we consider the stability of Algorithm EH. First, we recall what is known about the stability of Householder QR factorization for solving the LS problem min x kb Gamma Axk 2 . The method is normwise backward stable. Moreover, when column ....

....as a single loop with the help of a parameter that specifies the extent of the Householder vector, yielding the succinct pseudocode shown in Figure 4.1. The EG method and a variant of it based on modified Gram Schmidt orthogonalization can be expressed in similarly concise fashions, as shown in [4], 5] How does the stability of Algorithm EH compare with that of the EG method The only published result for the EG method is one of Barlow and Handy [2] which bounds kC Gamma b Q b Rk 2 for the factorization (3.1) in terms of kCk 2 ; this result does not reveal the stability of the ....

Ake Bjorck. Iterative refinement of linear least squares solutions II. BIT, 8:8--30, 1968.

No context found.

A. Bjorck and G. H. Golub. Iterative Refinement of Linear Least Squares Solutions by Householder Transformation. BIT, 7:322--337, 1967.

No context found.

A. Bjorck. Iterative Refinement of Linear Least Squares Solutions II. BIT, 8:8--30, 1968.

No context found.

A. Bjorck. Iterative Refinement of Linear Least Squares Solutions I. BIT, 7:257--278, 1967.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC