J. W. Demmel, N. J. Higham, and R. S. Schreiber. Block LU factorization. Numerical Linear Algebra with Applications, 2(2), 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... to the cost of the cube dag method (see Section 2) Figure 1: Iterative block Gaussian elimination Figure 2: Recursive block Gaussian elimination A lower communication cost for LU decomposition can be achieved by applying the block algorithm recursively (see e.g. [8, 7]) This standard method was suggested as a means of reducing the communication cost in [1] for the transitive closure problem) The BSP cost of block Gauss Jordan elimination was analysed in [17] we summarise the results here for completeness. Given a nonsingular matrix A, the algorithm produces ....

J. W. Demmel, N. J. Higham, and R. S. Schreiber. Block LU factorization. Numerical Linear Algebra with Applications, 2(2), 1995.

....system for the pressure and Lagrangian vector components. This system is then frequently solved by the iterative conjugate gradient type method. Whereas this approach is well known considerable less attention has been paid to the numerical stability aspects of such transformation. It was shown in [5] that block LU factorization can be unstable even if the system matrix is symmetric positive de nite. In this paper we examine this type of conditional stability for a particular application in the underground water ow modelling. We show that the actual error of the computed approximate solution ....

....matrix and its Schur complement system from a simple problem obtained in the mixed hybrid nite element approximation of the potential uid problem stability analysis of such approach. Thorough rounding error analysis of the block LU factorization has been given by Demmel, Higham and Schreiber in [5]. They showed that such block method is stable for symmetric positive systems only if the system matrix is well conditioned. In this section we follow their approach for our particular inde nite system (1.1) We estimate the backward error associated with the approximate solution in terms of the ....

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J.W. Demmel, N.J. Higham, R.S. Schreiber. Block LU Factorization. Numerical Linear Algebra with Applications, 2(2): 173-190, 1995.

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J. W. Demmel, N. J. Higham, and R. S. Schreiber, Stability of block LU factorization, Numerical Linear Algebra with Applications, 2 (1995), 173-190.

....system solution, an exception being block LU factorization. How and when to pivot in block LU factorization are key questions. A positive result is that if the matrix is diagonally dominant by columns in either the point or the block sense, then appropriate implementations are perfectly stable [41]. On the other hand, for a symmetric positive definite matrix stability is assured only if the matrix is well conditioned (this being a rare example where symmetric positive definiteness is not the most desirable property a matrix can have) 3.2 Iterative Refinement If y is an approximate ....

James W. Demmel, Nicholas J. Higham, and Robert S. Schreiber. Stability of block LU factorization. Numerical Linear Algebra with Applications, 2 (2):173--190, 1995.

....Analysis That the growth factor is nicely bounded does not, by itself, imply that computation of the block LDL factorization is a numerically stable process; see [10] for a discussion in the case of block LDL factorization of general symmetric matrices. From results on block LU factorization [6], numerical stability could be deduced if we could show that kLk=kAk is suitably bounded. We therefore examine the size of the block CE of L in (2.2) For s = 1 we have k1 = ja 11 j and the bound is sharp. It follows that kLk=kAk can be arbitrarily large. A parametrized example is given ....

James W. Demmel, Nicholas J. Higham, and Robert S. Schreiber. Stability of block LU factorization. Numerical Linear Algebra with Applications, 2(2):173--190, 1995.

....the diagonal pivoting method. A sufficient condition for stability can be obtained by regarding the block LDL factorization computed by the diagonal pivoting method as a special case of a block LU factorization. Error analysis for block LU factorization is given by Demmel, Higham and Schreiber [8], and a suitable modification of this analysis gives the following result: if linear systems involving 2 Theta 2 pivots are solved in a normwise backward stable fashion then the condition kLk1kDk1kL k1 c n kAk1 ; 2.2) for a modest constant c n , is sufficient to ensure that the diagonal ....

James W. Demmel, Nicholas J. Higham, and Robert S. Schreiber, Stability of block LU factorization, Numerical Linear Algebra with Applications, 2 (1995), pp. 173--190.

....L N. J. HIGHAM AND A. POTHEN 7 is ill conditioned, it shows that if L is well conditioned then the partitioned inverse method is guaranteed to be normwise backward stable. It is interesting to note that dependence of the backward error on the condition number occurs also in block LU factorization [5]. Another example of this dependence is a parallel triangular system solver analysed by Sameh and Brent [21] for which a backward error result with k DeltaLk c n u 2 (L)kLk is obtained. It seems to be a rule of thumb that if we attempt to improve the parallelism of Gaussian elimination or ....

J. W. Demmel, N. J. Higham, and R. S. Schreiber, Stability of block LU factorization, Numerical Linear Algebra with Applications, 2 (1995), pp. 173--190.

....That the growth factor is nicely bounded does not, by itself, imply that computation of the block LDL T factorization is a numerically stable process; see [10] for a discussion in the case of block LDL T factorization of general symmetric matrices. From results on block LU factorization [6], numerical stability could be deduced if we could show that kLk=kAk is suitably bounded. We therefore examine the size of the block CE Gamma1 of L in (2.2) For s = 1 we have kCE Gamma1 k1 = ja 21 j ja 11 j oe ffja 21 j ; and the bound is sharp. It follows that kLk=kAk can be ....

James W. Demmel, Nicholas J. Higham, and Robert S. Schreiber. Stability of block LU factorization. Numerical Linear Algebra with Applications, 2(2):173--190, 1995.

....the diagonal pivoting method. A sufficient condition for stability can be obtained by regarding the block LDL T factorization computed by the diagonal pivoting method as a special case of a block LU factorization. Error analysis for block LU factorization is given by Demmel, Higham and Schreiber [8], and a suitable modification of this analysis gives the following result: if linear systems involving 2 Theta 2 pivots are solved in a normwise backward stable fashion then the condition kLk1kDk1kL T k1 c n kAk1 ; 2.2) for a modest constant c n , is sufficient to ensure that the diagonal ....

James W. Demmel, Nicholas J. Higham, and Robert S. Schreiber, Stability of block LU factorization, Numerical Linear Algebra with Applications, 2 (1995), pp. 173--190.

....the diagonal pivoting method. A sufficient condition for stability can be obtained by regarding the block LDL T factorization computed by the diagonal pivoting method as a special case of a block LU factorization. Error analysis for block LU factorization is given by Demmel, Higham and Schreiber [8], and a suitable modification of this analysis gives the following result: if linear systems involving 2 Theta 2 pivots are solved in a normwise backward stable fashion then the condition kLk1 kDk1 kL T k1 c n kAk1 ; 2.2) for a modest constant c n , is sufficient to ensure that the diagonal ....

James W. Demmel, Nicholas J. Higham, and Robert S. Schreiber. Stability of block LU factorization. Numerical Linear Algebra with Applications, 2(2):173-- 190, 1995.

No context found.

J. W. Demmel, N. J. Higham, and R. S. Schreiber. Block LU factorization. Numerical Linear Algebra with Applications, 2(2), 1995.

No context found.

J. W. Demmel, N. J. Higham, and R. S. Schreiber. Block LU factorization. Numerical Linear Algebra with Applications, 2(2), 1995.

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