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2006.165: Stability of the diagonal pivoting method with partial pivoting

2006.165: Nicholas J. Higham (1997) Stability of the diagonal pivoting method with partial pivoting. SIAM Journal On Matrix Analysis And Applications, 18 (1). pp. 52-65. ISSN 1095-7162

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DOI: 10.1137/S0895479895290371

Abstract

LAPACK and LINPACK both solve symmetric indefinite linear systems using the diagonal pivoting method with the partial pivoting strategy of Bunch and Kaufman [Math. Comp., 31 (1977), pp. 163--179]. No proof of the stability of this method has appeared in the literature. It is tempting to argue that the diagonal pivoting method is stable for a given pivoting strategy if the growth factor is small. We show that this argument is false in general and give a sufficient condition for stability. This condition is not satisfied by the partial pivoting strategy because the multipliers are unbounded. Nevertheless, using a more specific approach we are able to prove the stability of partial pivoting, thereby filling a gap in the body of theory supporting LAPACK and LINPACK.

Item Type:Article
Uncontrolled Keywords:symmetric indefinite matrix, diagonal pivoting method, $LDL^T$ factorization, partial pivoting, growth factor, numerical stability, rounding error analysis, LAPACK, LINPACK
Subjects:MSC 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number:2006.165
Deposited By:Miss Louise Stait
Deposited On:03 July 2006

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