Fast orthogonal decomposition of rank deficient Toeplitzmatrices

Hansen, Christian Per; Gesmar, Henrik

doi:10.1007/bf02142744

Cited by 6 publications

(3 citation statements)

References 12 publications

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“…We do not know any other methods which have been proved to be stable or weakly stable and have worst-case time bound O(n 2 ). Algorithms which involve pivoting and/or look-ahead [18,27,41,43,76] may work well in practice, but seem to require worst-case overhead O(n 3 ) to ensure stability.…”

Section: Resultsmentioning

confidence: 99%

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Stability analysis of a general toeplitz systems solver

1995

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We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that R T R is close to A T A. Thus, when the algorithm is used to solve the semi-normal equations R T Rx = A T b, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem min Ax − b 2 .

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Section: Resultsmentioning

confidence: 99%

“…The result also applies to the solution of full-rank Toeplitz or Hankel least squares problems. For a discussion of the rank-deficient case and a "look-ahead" modification of Cybenko's algorithm, see [43].…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of a general toeplitz systems solver

1995

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“…Algorithms for Vandermonde and many other classes of structured matrices are not considered here -we refer to the extensive survey by Olshevsky [60]. Also, we have omitted any discussion of fast "lookahead" algorithms [18,19,30,31,42,43,44,45,72] because, although such algorithms often succeed in practice, in the worst case they require of order n 3 operations.…”

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confidence: 99%

4. Stability of Fast Algorithms for Structured Linear Systems

Brent¹

1999

Fast Reliable Algorithms for Matrices With Structure

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We survey the numerical stability of some fast algorithms for solving systems of linear equations and linear least squares problems with a low displacement-rank structure. For example, the matrices involved may be Toeplitz or Hankel. We consider algorithms which incorporate pivoting without destroying the structure, and describe some recent results on the stability of these algorithms. We also compare these results with the corresponding stability results for the well known algorithms of Schur/Bareiss and Levinson, and for algorithms based on the semi-normal equations.

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