A. Bjorck. Stability Analysis of the Method of Seminormal Equations for Linear Least Squares Problems. Linear Alg. and Its Applic., 88/89:31--48, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....vector d. If iterative refinement is applied, errors in the computed corrections will again be governed by Econd(K) and the accuracy of the right hand side) Refinement is not e#ective with the associated normal equations unless the Cholesky factors are obtained from a QR factorization of A [2], which would usually be less e#cient. 4 Regularized Linear Programs Barrier methods for linear programming (e.g. 14] give rise to sequences of increasingly ill conditioned least squares or sqd systems. Here we focus on the regularized LP problem discussed in [10, 11] minimize x, s c x ....

A. Bjorck, Stability analysis of the method of seminormal equations for linear least squares problems, Linear Alg. and its Appl., 88/89 (1987), pp. 31--48.

....between these two systems are the seminormal equations (SNE) R b; since they do not involve Q, they are attractive for multiple right hand side problems where Q is too large to store. The SNE method has been in use since the early 1970s, and its stability is explained by analysis of Bjorck [18], who makes the assumption that R is obtained from a stable QR factorization method. Bjorck obtains a forward error bound proportional to 2 (A) just like for the normal equations method that Cholesky factorizes A A; hence the SNE method is not backward stable. To improve the stability a ....

Ake Bjorck. Stability analysis of the method of seminormal equations for linear least squares problems. Linear Algebra and Appl., 88/89:31--48, 1987.

....is that two solves with R are needed to solve Ax = b. Thus the forward error in x behaves like (A) 2 rather than (A) for LIU factors, so the method is more seriously affected by ill conditioning. For this reason this method is little used, although there is some evidence (Bjorck [5]) that iterative refinement can improve matters. Dense Factors of Sparse Matrices 19 7 Numerical results and Conclusions To quantify the relative costs and numerical stability of the different implicit factorizations, a range of calculations on eleven LP problems from the SOL test set with n up ....

Bjorck A. (1987), Stability analysis of the method of semi-normal equations for least squares problems, Linear Algebra Applns., 88/89, 31-48. Dense Factors of Sparse Matrices 21

....The solution obtained when the iterative method converges sufficiently quickly (it will normally satisfy some accuracy requirement) will be called the iterative solution. The preconditioned conjugate gradients (PCG) method is applied to the system A T Ax = A T b of semi normal equations [3], where the approximate factor R is used as a preconditioner, in a similar way as in [30] This means that the following preconditioned system of linear algebraic equations is treated: Cz = d ; 6) where C = R Gamma1 ) T P T A T A P R Gamma1 z = R P T ....

A. Bjorck, Stability analysis of the method of semi-normal equations for least squares problems, Lin. Alg. Appl., Vol. 88/89 1987, pp. 31--48.

....all matrix elements and fill ins that are relatively small (in absolute value) according to a certain criterion. As a result we obtain an incomplete QR factorization and an inaccurate initial solution. The incomplete factor R can be used as a preconditioner in the system of seminormal equations [2], as it is shown in [19] for the Givens Gentleman method. For the classical Givens method we have: C = R T ) Gamma1 A T AR Gamma1 z = Rx d = R T ) Gamma1 A T b (4) From (2) and (4) we obtain the linear system Cz = d (5) with symmetric and positive definite matrix C. It ....

A. Bjorck, Stability analysis of the method of semi-normal equations for least squares problems, Linear Algebra Appl. 1988/89, pp. 31--48.

....problem. Using (3.19) kr n k = min y kv 1 Gamma ( V n 1 H n 1;n F n )yk; which can be considered as a perturbation of the following least squares problem k r n k = min y kv 1 Gamma V n 1 H n 1;n yk: 5. 10) Applying the perturbation theorem proved by Wedin [53] see also [6], to the problem (5.10) we obtain k r n Gamma r n k kF n k k V n 1 H n 1;n k fk V n 1 H n 1;n kk y n k ( V n 1 H n 1;n )k r n k= g; 37 which, considering k y n k 1=oe n ( V n 1 H n 1;n ) and k r n k gives k r n Gamma r n k i 17 kF n k oe n ( V n 1 H ....

A. Bjorck, Stability analysis of the method of seminormal equations for linear least squares problems, Linear Algebra Appl. 88/89 (1987), pp. 31-48.

....hand side vector b. To treat least squares problems with right hand sides which are not available at factorization time, the corrected seminormal equations (CSNE) can be used. This approach is usually numerically stable, but is not suitable for problems with widely different row norms, see Bjorck [6]. Problems where there is a need to operate with Q and Q T arise also from mixed finite element discretizations. These can often be formulated as an equality constrained quadratic optimization problems min y 1 2 y T Hy Gamma b T y; A T y = c; 3) Computing sparse orthogonal factors ....

A. Bjorck. Stability analysis of the method of semi-normal equations for least squares problems. Linear Algebra Appl., 88/89:31--48, 1987.

....of a certain (possibly rank deficient) matrix after row or column additions or deletions. Our implementation (in sixteen digit precision) seems to be robust and (more than) accurate enough for NDO applications. Hence we have had no reason for employing more stable orthogonal decompositions (cf. Bjo87, Bjo88, Bjo90, BjP92, Fle91, GMSW91, GMW91, ScS79] that would require much more storage and work. The paper is organized as follows. Our dual method is described in Sect. 2 and its matrix factorizations in Sect. 3. Some modifications and extensions are Numerische Mathematik Electronic Edition ....

....and requires less storage than that of [Kiw89] unless mL p 2n) In fact no algorithm seems to be competitive with our method in terms of workspace and work per iteration, especially in large scale applications. Alternative implementations of our algorithm might use the techniques of [ADdR89, Bjo87, Bjo88, Bjo90, BjP92, GGM 84, Hig91, Pan90] These papers also discuss stability questions relevant to our approach. In general, the accuracy of matrix decompositions may be increased by using (partial) orthogonal factors and or iterative refinement (related to reorthogonalization) Yet, ....

Bjorck, A (1987): Stability analysis of the method of seminormal equations for linear least squares problems. Linear Algebra Appl. 88/89, 31--48

....than the normal equations method because it use an orthogonal factorization to compute R. However, a stability analysis of seminormal equations method by Bjorck shows that the error of the solution of the seminormal equations method is of the same order as the solution of the normal equations [10]. Under mild conditions, a correction step can be added to yield a solution as accurate as the QR method. As with the normal equations, if there are a few dense rows in A, severe fill in occurs in R. Usually, such rows are treated separately by using technique of updating a QR factorization ....

A. Bjorck. Stability analysis of the method of seminormal equations for linear least squares problems. Linear Algebra and its Applications, 88/89:31--48, 1987.

....for the least squares problem. Using (1.5) kr nk = min y kv 1 Gamma (Vn 1 Hn 1;n Fn )yk; which can be considered as a perturbation of the following least squares problem k r nk = min y kv 1 Gamma Vn 1Hn 1;n yk: 3. 1) Applying the perturbation theorem proved by Wedin [10] see also [3], to the problem (3.1) we obtain k r n Gamma r nk kFn k kVn 1Hn 1;nk fkVn 1Hn 1;nkk yn k (Vn 1Hn 1;n )k r n k= g ; which, considering k y n k 1=oe n (Vn 1 Hn 1;n ) and k r nk= 1 gives k r n Gamma r nk i 3 kFnk oe n (Vn 1 Hn 1;n ) 1 i 3 2; or k r n Gamma r nk i 3 i 1 ....

A. Bjorck, Stability Analysis of the Method of Seminormal Equations for Linear Least Squares Problems, Linear Algebra Appl. 88/89, pp. 31 - 48, 1987

....not seem very attractive because of the fill in to the factor Q. Additionally, dense rows in A will cause the factor R to be full. The storage of the denser Q can be avoided, at the cost of possible instability, by using the semi normal equations (SNE) R T R = A T b: Moreover, analysis by Bjorck (1987) has shown that, in most cases, numerically satisfactory results can be obtained by using the corrected semi normal equations (CSNE) where one step of iterative refinement is used. Sparse QR factorization uses the observation exploited by George and Heath (1980) that the factor R is the same as ....

Bjorck, A. (1987), `Stability analysis of the method of semi-normal equations for least squares problems', Linear Algebra and its Applications 88/89, 31--48.

....be undesirable from the standpoint of numerical stability. We stress that equations (1. 5) are different from the equations R T Rx = A T b for an overdetermined least squares problem, where A = Q [ R T 0 ] T 2 IR m Thetan with m n, yet these are also referred to as semi normal equations [4]. In this paper we are solely concerned with underdetermined systems so no confusion should arise. Other methods for obtaining minimal 2 norm solutions of underdetermined systems are surveyed in [5] Existing perturbation theory for the minimum norm solution problem, and error analysis for the ....

....not in general the minimum norm solution. For the SNE method it is not even possible to derive a residual bound of the form (4.1) The method of solution guarantees only that the seminormal equations themselves have a small residual. Thus, as in the context of overdetermined least squares problems [4] the SNE method is not backward stable. A possible way to improve the backward stability of the SNE method is to use iterative refinement in fixed precision, as advocated in the overdetermined case in [4] Some justification for this approach can be given using the analysis for an arbitrary linear ....

[Article contains additional citation context not shown here]

A. Bjorck, Stability analysis of the method of seminormal equations for linear least squares problems, Linear Algebra and Appl., 88/89 (1987), pp. 31--48.

No context found.

A. Bjorck. Stability Analysis of the Method of Seminormal Equations for Linear Least Squares Problems. Linear Alg. and Its Applic., 88/89:31--48, 1987.

No context found.

A. Bjorck. Stability Analysis of the Method of Seminormal Equations for Linear Least Squares Problems. Linear Algebra and Its Applications, 88/89:31--48, 1987.

No context found.

A. Bjorck, Stability analysis of the method of seminormal equations for linear least squares problems, Linear Algebra Appl. 88/89 (1987), pp. 31-48.

No context found.

A. Bjorck, "Stability analysis of the method of semi-normal equations for linear least squares problems", Linear Alg. Appl. 88/89 (1987), 31--48.

No context found.

A. Bjorck, "Stability analysis of the method of semi-normal equations for linear least squares problems", Linear Alg. Appl.

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