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2007.13: Stability of Block Algorithms with Fast Level-3 BLAS

2007.13: James W. Demmel and Nicholas J. Higham (1992) Stability of Block Algorithms with Fast Level-3 BLAS. ACM Transactions on Mathematical Software, 18 (3). pp. 274-291. ISSN 0098-3500

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DOI: 10.1145/131766.131769


Block algorithms are becoming increasingly popular in matrix computations. Since their basic unit of data is a submatrix rather than a scalar they have a higher level of granularity than point algorithms, and this makes them well-suited to high-performance computers. The numerical stability of the block algorithms in the new linear algebra program library LAPACK is investigated here. It is shown that these algorithms have backward error analyses in which the backward error bounds are commensurate with the error bounds for the underlying level 3 BLAS (BLAS3). One implication is that the block algorithms are as stable as the corresponding point algorithms when conventional BLAS3 are used. A second implication is that the use of BLAS3 based on fast matrix multiplication techniques affects the stability only insofar as it increases the constant terms in the normwise backward error bounds. For linear equation solvers employing $LU$ factorization it is shown that fixed precision iterative refinement helps to mitigate the effect of the larger error constants. Despite the positive results presented here, not all plausible block \alg s are stable; we illustrate this with the example of \LUf\ with block triangular factors, and we describe how to check a block \alg\ for stability without doing a full error analysis.

Item Type:Article
Uncontrolled Keywords:block algorithm, LAPACK, level 3 BLAS, iterative refinement, LU factorization, QR factorization,, backward error analysis.
Subjects:MSC 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number:2007.13
Deposited By:Nick Higham
Deposited On:05 February 2007

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