I. S. Duff and J. K. Reid. MA47, a Fortran code for direct solution of indefinite sparse symmetric linear systems. to appear, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Markowitz, BLAS 3 Pub NETLIB MA38 (same as UMFPACK) Com HSL nonsym. MA48 [6] Anal: RL, Markowitz Com HSL Fact: LL, partial, BLAS 1, SD nonsym. SPARSE [14] RL, Markowitz, Scalar Pub NETLIB sympattern ) MUPS [1] MA42 [8] MF, threshold, BLAS 3 Frontal, BLAS 3 Com HSL Com HSL sym. MA27 [7] MA47 [5] MF, LDL T , BLAS 1 Com HSL s.p.d. Ng Peyton [16] LL, BLAS 3 Pub Author Shared Memory Algorithms nonsym. SuperLU LL, partial, BLAS 2.5 Pub UCB nonsym. PARASPAR [19, 20] RL, Markowitz, BLAS 1, SD Res Author sym MUPS [2] MF, threshold, BLAS 3 Res Author pattern nonsym. George Ng [9] RL, ....

I.S Duff and J. K. Reid. MA47, a Fortran code for direct solution of indefinite sparse symmetric linear systems. Technical Report RAL-95-001, Rutherford Appleton Laboratory, 1995.

....factorization to compute L; 4. Solution of Ly = b and L T x = y. In the first phase, although it is computationally expensive (NP hard) to find an optimal P in terms of minimizing fills, many heuristics have been used successfully in practice, such as variants of minimum degree orderings [3, 10, 38, 52] and various dissection orderings based on graph partitioning [15, 58, 92] or hybrid approaches [13, 18, 76] Two important data structures have been introduced in efficient implementations of the Cholesky factorization. One is the elimination tree and another is the supernode. The elimination ....

....Pub netlib MA38 (same as UMFPACK) Com HSL unsym. MA48 [39] Anal: RL, Markowitz Com HSL Fact: LL, partial BLAS 1, SD unsym. SPARSE [79] RL, Markowitz Scalar Pub netlib sympattern ) MA41 [4] MA42 [42] MF, threshold Frontal (eqn element) BLAS 3 BLAS 3 Com HSL Com HSL sym. MA27 [40] MA47 [38] MF, LDL T BLAS 1 BLAS 3 Com HSL Com HSL s.p.d. Ng Peyton [89] LL BLAS 3 Pub Author Shared Memory Algorithms unsym. SuperLU LL, partial BLAS 2.5 Pub UCB unsym. PARASPAR [112, 113] RL, Markowitz BLAS 1, SD Res Author sym MUPS [6] MF, threshold BLAS 3 Res Author pattern unsym. George Ng [57] ....

I.S Duff and J. K. Reid. MA47, a Fortran code for direct solution of indefinite sparse symmetric linear systems. Technical Report RAL-95-001, Rutherford Appleton Laboratory, 1995.

....into account and only the sparsity structure of the matrix is considered. We did start to design such an analysis phase but were not convinced of its utility because of the problems with perturbing the data structures that we mention in Section 6. However, we can apply recent work of Duff and Reid [17] based on algorithms developed by Duff, Gould, Reid, Scott, and Turner [11] to obtain a suitable analysis. We discuss the use of their algorithms in this section. Methods based on the elimination dag are presented in [18, 23] Duff et al. 11] design algorithms for factorizing symmetric ....

....will ensure that the off diagonal entries of the oxo pivot will be the same as the entry in B (and B T ) with the lowest Markowitz count. Thus we can use the symmetric minimum degree ordering of Duff et al. 11] to obtain a symbolic analysis of an unsymmetric matrix. The code of Duff and Reid [17] will also produce the equivalent of our directed acyclic graph (G) which could, after suitable modification, be used as input to the factor only algorithm of this paper. Because numerical values were not taken at all into account in the analysis phase, we do not, however, recommend this route ....

I. S. Duff and J. K. Reid. MA47, a Fortran code for direct solution of indefinite sparse symmetric linear systems. to appear, 1993.

....1028 with 504 ones and 524 zeros and A the 1028 x 524 matrix FFFFF800 from the collection in Gay (1985) an earlier multifrontal code which does not exploit the zero block structure (namely HSL code MA27) forecasts 1.5 million floating point operations but requires 16.5 million. The HSL code MA47 (Duff and Reid 1995, Duff and Reid 1996b) however, forecasts and requires only 7,954 flops. Unfortunately, the new code is much more complicated and involves more data movement since frontal matrices are not necessarily absorbed by the parent and can percolate up the tree. Additionally, the penalty for straying ....

Duff, I. S. and Reid, J. K. (1995), MA47, a Fortran code for direct solution of indefinite sparse symmetric linear systems, Technical Report RAL 95-001, Rutherford Appleton Laboratory.

....on developing parallel versions of multifrontal methods is considered in Section 7. In the first case, data movement can be reduced by continuing with one frontal matrix rather than stacking it and starting another. Taken to its extreme, this strategy would give a (uni )frontal scheme. Davis and Duff (1995) discuss the effect of this technique and have incorporated it in the HSL code MA38. Although one of the early multifrontal codes MA37 (Duff and Reid 1984) solved sets of unsymmetric equations, the analyse phase was performed on the pattern of the matrix with entries a ij if either a ij or a ji ....

....to the number of edges to nodes outside its clique. It is quite natural, in a finite element application, to perform the minimum degree ordering on a graph where nodes in the same clique are treated as a single node, and this was done by the minimum degree ordering algorithm in the HSL code MA47 (Duff and Reid 1995). Ashcraft (1995) has performed an exhaustive study of this and obtains speed ups of over 5 for some standard test matrices. The second improvement to pivot selection stems from the observation that, if two nodes of the same degree are not adjacent in the graph, they can be eliminated ....

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Duff, I. S. and Reid, J. K. (1995), MA47, a Fortran code for direct solution of indefinite sparse symmetric linear systems, Technical Report RAL 95-001, Rutherford Appleton Laboratory.

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