On the Robustness of Gaussian Elimination with Partial Pivoting Paola Favati Mauro Leoncini Angeles Martinez Article DOI :
10.1023/A:1022314201484

Cite this article as: Favati, P., Leoncini, M. & Martinez, A. BIT Numerical Mathematics (2000) 40: 62. doi:10.1023/A:1022314201484
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.

Gaussian elimination stability pivoting

REFERENCES 1.

E. Anderson et al.,

Lapack User's Guide , Society for Industrial and Applied Mathematics, Philadelphia, PA, 1992.

Google Scholar 2.

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.

3.

L. M. Delves and J. I. Mohamed,

Computational Methods for Integral Equations , Cambridge University Press, Cambridge, 1985.

Google Scholar 4.

J. W. Demmel, Trading off parallelism and numerical accuracy , Tech. Report CS–92–179, University of Tennessee, June 1992 (Lapack Working Note 52).

5.

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.

Google Scholar 6.

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.

Google Scholar 7.

L. V. Foster,

Gaussian elimination with partial pivoting can fail in practice , SIAM J. Matrix Anal. Appl., 15 (1994), pp. 1354–1362.

Google Scholar 8.

L. V. Foster,

The growth factor and efficiency of Gaussian elimination with rook pivoting , J. Comp. Appl. Math., 86 (1997), pp. 177–194.

Google Scholar 9.

N. J. Higham,

Algorithm 694: A collection of test matrices in MATLAB , ACM Trans. Math. Software, 17:3 (1991), pp. 289–305.

Google Scholar 10.

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.

Google Scholar 11.

Using Matlab 5.1 , The MATHWORKS Inc., 1997.

12.

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.

13.

R. Motwani and P. Raghavan Randomized Algorithms , Cambridge University Press, 1995.

14.

L. Neal and G. Poole,

A geometric analysis of Gaussian Elimination II , Linear Algebra Appl., 173 (1992), pp. 239–264.

Google Scholar 15.

R. D. Skeel, Scaling for numerical stability in Gaussian Elimination , J. ACM, 26 (1979), pp. 494–526.

16.

L. N. Trefethen,

Three mysteries of Gaussian Elimination , ACM SIGNUM Newsletter, 20 (1985), pp. 2–5.

Google Scholar 17.

L. N. Trefethen and D. Bau, Numerical Linear Algebra , SIAM, Philadelphia, PA, 1997.

18.

L. N. Trefethen and R. S. Schreiber,

Average-case stability of Gaussian Elimination , SIAM J. Matrix Anal. Appl., 11 (1990), pp. 335–360.

Google Scholar 19.

J. H. Wilkinson,

Error analysis of direct methods of matrix inversion , J. ACM, 8 (1961), pp. 281–330.

Google Scholar 20.

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.

Google Scholar © Swets & Zeitlinger 2000

Authors and Affiliations Paola Favati Mauro Leoncini Angeles Martinez 1. Istituto di Matematica Computazionale del CNR Pisa Italy 2. Facoltà di Economia di Foggia Università di Bari Foggia Italy 3. IMC-CNR Pisa Italy. 4. Departamento de Matemática Aplicada Universidad Politécnica de Valencia Valencia Spain.