An Algorithm Based on Orthogonal Polynominal Vectors for Toeplitz Least Squares Problems
NAA '00 Revised Papers from the Second International Conference on Numerical Analysis and Its Applications
Pivoting for structured matrices and rational tangential interpolation
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A lanczos-type algorithm for the QR factorization of cauchy-like matrices
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Structured matrix-based methods for polynomial ∈-gcd: analysis and comparisons
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A fast solver for linear systems with displacement structure
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Extended companion matrix for approximate GCD
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Recent research shows that structured matrices such as Toeplitz and Hankel matrices can be transformed into a different class of structured matrices called Cauchy-like matrices using the FFT or other trigonometric transforms. Gohberg, Kailath, and Olshevsky [Math. Comp., 64 (1995), pp. 1557--1576] demonstrate numerically that their fast variation of the straightforward Gaussian elimination with partial pivoting (GEPP) procedure on Cauchy-like matrices is numerically stable. Sweet and Brent [Adv. Signal Proc. Algorithms, 2363 (1995), pp. 266--280] show that the error growth in this variation could be much larger than would be encountered with straightforward GEPP in certain cases. In this paper, we present a modified algorithm that avoids such extra error growth and can perform a fast variation of Gaussian elimination with complete pivoting (GECP). Our analysis shows that it is both efficient and numerically stable, provided that the element growth in the computed factorization is not large. We also present a more efficient variation of this algorithm and discuss implementation techniques that further reduce execution time. Our numerical experiments show that this variation is highly efficient and numerically stable.