Abstract
It has been recently shown that large growth factors might occur in Gaussian Elimination with Partial Pivoting (GEPP) also when solving some plausibly natural systems. In this note we argue that this potential problem could be easily solved, with much smaller risk of failure, by very small (and low cost) modifications of the basic algorithm, thus confirming its inherent robustness. To this end, we first propose an informal model with the goal of providing further support to the comprehension of the stability properties of GEPP. We then report the results of numerical experiments that confirm the viewpoint embedded in the model. Basing on the previous observations, we finally propose a simple scheme that could be turned into (even more) accurate software for the solution of linear systems.
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REFERENCES
E. Anderson et al., Lapack User's Guide, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1992.
M. Blum, M. Luby, and R. Rubinfeld, Self-testing/correcting with applications to numerical problems, in Proc. 22nd ACM Symposium on Theory of Computing, ACM Press, 1990, pp. 73–83.
L. M. Delves and J. I. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge, 1985.
J. W. Demmel, Trading off parallelism and numerical accuracy, Tech. Report CS–92–179, University of Tennessee, June 1992 (Lapack Working Note 52).
A. Edelman. and W. Mascarenhas, On the complete pivoting conjecture for a Hadamard matrix of order 12, Linear and Multilinear Algebra, 38 (1995), pp. 181–188.
A. M. Erisman and J. K. Reid, Monitoring the stability of the triangular factorization of a sparse matrix, Numer. Math., 22 (1974), pp. 183–186.
L. V. Foster, Gaussian elimination with partial pivoting can fail in practice, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 1354–1362.
L. V. Foster, The growth factor and efficiency of Gaussian elimination with rook pivoting, J. Comp. Appl. Math., 86 (1997), pp. 177–194.
N. J. Higham, Algorithm 694: A collection of test matrices in MATLAB, ACM Trans. Math. Software, 17:3 (1991), pp. 289–305.
N. J. Higham and D. J. Higham, Large growth factors in Gaussian elimination with pivoting, SIAM J. Matrix Anal. Appl., 10 (1989), pp. 155–164.
Using Matlab 5.1, The MATHWORKS Inc., 1997.
J. M. D. Hill, W. F. McColl, D. C. Stefanescu, M. W. Goudreau, K. Lang, S. B. Rao, T. Suel, T. Tsantilas, and R. Bisseling, BSPlib: The BSP Programming Libarary, Tech. Report PRG-TR–29–9, Oxford University Computing Laboratory, May 1997.
R. Motwani and P. Raghavan Randomized Algorithms, Cambridge University Press, 1995.
L. Neal and G. Poole, A geometric analysis of Gaussian Elimination II, Linear Algebra Appl., 173 (1992), pp. 239–264.
R. D. Skeel, Scaling for numerical stability in Gaussian Elimination, J. ACM, 26 (1979), pp. 494–526.
L. N. Trefethen, Three mysteries of Gaussian Elimination, ACM SIGNUM Newsletter, 20 (1985), pp. 2–5.
L. N. Trefethen and D. Bau, Numerical Linear Algebra, SIAM, Philadelphia, PA, 1997.
L. N. Trefethen and R. S. Schreiber, Average-case stability of Gaussian Elimination, SIAM J. Matrix Anal. Appl., 11 (1990), pp. 335–360.
J. H. Wilkinson, Error analysis of direct methods of matrix inversion, J. ACM, 8 (1961), pp. 281–330.
S. J. Wright, A collection of problems for which Gaussian elimination with partial pivoting is unstable, SIAM J. Sci. Statist. Comput., 14 (1993), pp. 231–238.
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Favati, P., Leoncini, M. & Martinez, A. On the Robustness of Gaussian Elimination with Partial Pivoting. BIT Numerical Mathematics 40, 62–73 (2000). https://doi.org/10.1023/A:1022314201484
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DOI: https://doi.org/10.1023/A:1022314201484