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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Fast Gaussian elimination with partial pivoting for matrices with displacement structure
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by I. Gohberg, T. Kailath and V. Olshevsky PDF
Math. Comp. 64 (1995), 1557-1576 Request permission

Abstract:

Fast $O({n^2})$ implementation of Gaussian elimination with partial pivoting is designed for matrices possessing Cauchy-like displacement structure. We show how Toeplitz-like, Toeplitz-plus-Hankel-like and Vandermonde-like matrices can be transformed into Cauchy-like matrices by using Discrete Fourier, Cosine or Sine Transform matrices. In particular this allows us to propose a new fast $O({n^2})$ Toeplitz solver GKO, which incorporates partial pivoting. A large set of numerical examples showed that GKO demonstrated stable numerical behavior and can be recommended for solving linear systems, especially with nonsymmetric, indefinite and ill-conditioned positive definite Toeplitz matrices. It is also useful for block Toeplitz and mosaic Toeplitz (Toeplitz-block) matrices. The algorithms proposed in this paper suggest an attractive alternative to look-ahead approaches, where one has to jump over ill-conditioned leading submatrices, which in the worst case requires $O({n^3})$ operations.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Math. Comp. 64 (1995), 1557-1576
  • MSC: Primary 15A06; Secondary 15A09, 15A23, 15A57, 65F05
  • DOI: https://doi.org/10.1090/S0025-5718-1995-1312096-X
  • MathSciNet review: 1312096