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The Gohberg Anniversary Collection pp 359–389Cite as

Fast and Efficient Parallel Inversion of Toeplitz and Block Toeplitz Matrices

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Part of the book series: Operator Theory: Advances and Applications ((OT,volume 40))

Abstract

We call an n×n matrix A well-conditioned if log(cond A) = O(log n). We compute the inverse of any n×n well-conditioned and diagonally dominant Hermitian Toeplitz matrix A (with errors 1/2N, N = nc for a constant c) by a numerically stable algorithm using O(log2 log log n) parallel arithmetic steps and n log2n/log log n processors. This dramatically improves the previous results. We also compute the inverse and all the coefficients of the characteristic polynomial of any n×n nonsingular Toeplitz matrix A filled with integers (and possibly ill-conditioned) by a distinct algorithm using O(log2n) parallel arithmetic steps, O(n2) processors, and the precision of O(n log(n∥A∥1) binary digits. The results have several modifications, extensions, and further applications.

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Authors and Affiliations

  1. Dept. of Math., Lehman College, CUNY, Bronx, NY, 10468, USA

    Victor Pan

  2. Computer Science Dept., SUNY Albany, Albany, New York, 12222, USA

    Victor Pan

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  1. Victor Pan
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Editors and Affiliations

  1. Department of Mathematics and Computer Science, Vrije Universiteit, Amsterdam, The Netherlands

    H. Dym , S. Goldberg , M. A. Kaashoek  & P. Lancaster , ,  & 

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Dedicated to Prof. I.C. Gohberg

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© 1989 Birkhäuser Verlag Basel

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Pan, V. (1989). Fast and Efficient Parallel Inversion of Toeplitz and Block Toeplitz Matrices. In: Dym, H., Goldberg, S., Kaashoek, M.A., Lancaster, P. (eds) The Gohberg Anniversary Collection. Operator Theory: Advances and Applications, vol 40. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9276-6_14

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  • DOI: https://doi.org/10.1007/978-3-0348-9276-6_14

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-0348-9974-1

  • Online ISBN: 978-3-0348-9276-6

  • eBook Packages: Springer Book Archive

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