The Use Of Pivoting To Improve The Numerical Performance Of Toeplitz Solvers

Douglas R. Sweet

doi:10.1117/12.936869

4 April 1986 The Use Of Pivoting To Improve The Numerical Performance Of Toeplitz Solvers

Douglas R. Sweet

Proceedings Volume 0696, Advanced Algorithms and Architectures for Signal Processing I; (1986) https://doi.org/10.1117/12.936869
Event: 30th Annual Technical Symposium, 1986, San Diego, United States

Abstract

Levinson's Toeplitz solver breaks down if any leading principal minors are zero, and recently methods have been proposed to handle such cases by "jumping over" such singularities. It is shown that, as might be expected, the occurrence of small leading minors causes a serious loss of accuracy in solvers such as the Levinson algorithm. However, no Toeplitz algorithm has previously been proposed to handle such near-singularities. A pivoting scheme has been incorporated into the Toeplitz solver of Bareiss which allows near-singularities to be treated without significant loss of accuracy. As special cases, the pivoted Toeplitz solver also handles exact singularities and numerical singularities - singularities which appear as near-singularities when finite-precision floating point arithmetic is used. Bareiss's algorithm also yields the LU factors of a Toeplitz matrix, and it is shown that a restricted version of the pivoted Toeplitz solver produces an LU factorisation of a row and column permuted version of the original Toeplitz matrix. Finally, it is shown how the Toeplitz inverter of Trench and Zohar can be derived from the multipliers produced by the Bareiss algorithm, and this connection is used to derive a pivoted version of the Trench-Zohar algorithm from the pivoted Bareiss algorithm.

Citation Download Citation

Douglas R. Sweet "The Use Of Pivoting To Improve The Numerical Performance Of Toeplitz Solvers", Proc. SPIE 0696, Advanced Algorithms and Architectures for Signal Processing I, (4 April 1986); https://doi.org/10.1117/12.936869

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