# On the Robustness of Gaussian Elimination with Partial Pivoting

- 90 Downloads
- 2 Citations

## 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.

## Preview

Unable to display preview. Download preview PDF.

### 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.Google Scholar - 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).Google Scholar - 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.Google Scholar - 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.Google Scholar - 13.R. Motwani and P. Raghavan
*Randomized Algorithms*, Cambridge University Press, 1995.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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