On Some Pivotal Strategies in Gaussian Elimination by Sparse Technique | doi.page

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On Some Pivotal Strategies in Gaussian Elimination by Sparse Technique

SIAM Journal on Numerical Analysis1980Vol. 17(1), pp. 18–30

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Abstract

Pivotal interchanges are commonly used in the solution of large and sparse systems of linear algebraic equations by Gaussian elimination (in order to preserve the sparsity of the matrix and to prevent the appearance of large roundoff errors during the computations). The Markowitz strategy (see [H. M. Markowitz, The elimination form of inverse and its applications to linear programming, Management Sci., 3 (1957), pp. 255–269]) is often used to determine the pivotal sequence. An efficient implementation of this strategy is given by Curtis and Reid (see [A. R. Curtis and J. K. Reid, Fortran subroutines for the solution of sparse sets of linear equations, A.E.R.E., Report R.6844, HMSO, London, 1971]) and improved by Duff (see [I. S. Duff, MA28—a set of Fortran subroutines for sparse unsymmetric matrices, A.E.R.E., Report R.8730, HMSO, London, 1977]). In this paper it is shown how the classical Markowitz idea can be generalized. Consider the following parameters: u —the stability factor and $p(s)$—the number of rows that will be searched for a pivotal element at stage s of the elimination ($1 \leqq s \leqq n - 1,n$-the number of equations). Pivotal strategies depending on these two parameters are defined. The choice of the parameters is discussed. A comparison between the pivotal strategy used in [I. S. Duff, MA28—a set of Fortran subroutines for sparse unsymmetric matrices, A.E.R.E., Report R.8730, HMSO, London, 1977] (where $p(s) = n - s + 1$) and a pivotal strategy with $p(s) \leqq 3$ is carried out. The results indicate that pivotal strategies with small $p(s)$ may be more profitable than the original Markowitz strategy.

Citations Over TimeTop 10% of 1980 papers

Abstract

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