&lt;title&gt;Error analysis of a fast partial pivoting method for structured matrices&lt;/title&gt;

Sweet, D. R.; Brent, Richard P.

doi:10.1117/12.211404

Cited by 21 publications

(21 citation statements)

References 9 publications

(34 reference statements)

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“…Sweet and Brent [22] have done an error analysis of the GKO algorithm applied to a Cauchy-like matrix C. They point out that the error propagation depends not only on the magnitude of the triangular factors in the LU factorization of C (as is expected for ordinary Gaussian elimination), but also on the magnitude of the generators. In some cases, the generators can suffer large internal growth, even if the triangular factors do not grow too large, and therefore cause a corresponding growth in the backward and forward error.…”

Section: Modified Gko Algorithmmentioning

confidence: 99%

A Fast Algorithm for Approximate Polynomial GCD Based on Structured Matrix Computations

Бини¹,

Boito²

2010

Numerical Methods for Structured Matrices and Applications

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Abstract. An O(n 2 ) complexity algorithm for computing an -greatest common divisor (gcd) of two polynomials of degree at most n is presented. The algorithm is based on the formulation of polynomial gcd given in terms of resultant (Bézout, Sylvester) matrices, on their displacement structure and on the reduction of displacement structured matrices to Cauchy-like form originally pointed out by Georg Heinig. A Matlab implementation is provided. Numerical experiments performed with a wide variety of test problems, show the effectiveness of this algorithm in terms of speed, stability and robustness, together with its better reliability with respect to the available software. Mathematics Subject Classification (2000). Da mettere.

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Section: Modified Gko Algorithmmentioning

confidence: 99%

A Fast Algorithm for Approximate Polynomial GCD Based on Structured Matrix Computations

Бини¹,

Boito²

2010

Numerical Methods for Structured Matrices and Applications

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“…We adapted these strategies to our augmented matrix approach for solving a Cauchy-like linear system, keeping the storage linear. We implemented three variations of Algorithm 2, in the case there are no repetitions in s, to include: a) partial pivoting (Algorithm 4); b) Sweet & Brent's pivoting [29] (Algorithm 5); c) Gu's pivoting and generator scaling technique [11] (Algorithm 6).…”

Section: Pivoting Strategiesmentioning

confidence: 99%

A fast solver for linear systems with displacement structure

Aricò¹,

Rodriguez²

2010

Numer Algor

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We describe a fast solver for linear systems with reconstructible Cauchy-like structure, which requires O(rn2) floating point operations and O(rn) memory locations, where n is the size of the matrix and r its displacement rank. The solver is based on the application of the generalized Schur algorithm to a suitable augmented matrix, under some assumptions on the knots of the Cauchy-like matrix. It includes various pivoting strategies, already discussed in the literature, and a new algorithm, which only requires reconstructibility. We have developed a software package, written inMatlab and C-MEX, which provides a robust implementation of the above method. Our package also includes solvers for Toeplitz(+Hankel)-like and Vandermonde-like linear systems, as these structures can be reduced to Cauchy-like by fast and stable transforms. Numerical experiments demonstrate the effectiveness of the software

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“…At the same time our current numerical experience [and the private communications with R.Brent, D.Sweet, M.Gu and M.Stewart] suggests that although there constructed few examples of Toeplitz matrices with larger generator-growth-factor [SB95], [S96], nevertheless the size of this factor is typically almost invariable comparable with the size of the element-growth factor, and the GKO algorithm [i.e., with partial pivoting] seems to be quite reliable in practice. To sum up, we think that the comments made and cited in Sec.…”

Section: Hankel Matrix Into a Loewner Matrix Of The Form Ai−bj Xi−yjmentioning

confidence: 99%

“…An a-priori rounding error analysis for the GKO algorithm of [GKO95] has been performed in [SB95], yielding weak stability. Along with the usual elementgrowth-factor [appearing also in the error bounds of the usual structure-ignoring GE], the corresponding error bound for the GKO algorithm also involves one more term, called a generator-growth-factor.…”

Section: Hankel Matrix Into a Loewner Matrix Of The Form Ai−bj Xi−yjmentioning

confidence: 99%

Pivoting for structured matrices and rational tangential interpolation

Olshevsky

2003

Fast Algorithms for Structured Matrices: Theory and Applications

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Gaussian elimination is a standard tool for computing triangular factorizations for general matrices, and thereby solving associated linear systems of equations. As is well-known, when this classical method is implemented in finite-precision-arithmetic, it often fails to compute the solution accurately because of the accumulation of small roundoffs accompanying each elementary floating point operation. This problem motivated a number of interesting and important studies in modern numerical linear algebra; for our purposes in this paper we only mention that starting with the breakthrough work of Wilkinson, several pivoting techniques have been proposed to stabilize the numerical behavior of Gaussian elimination. Interestingly, matrix interpretations of many known and new algorithms for various applied problems can be seen as a way of computing triangular fac-torizations for the associated structured matrices, where different patterns of structure arise in the context of different physical problems. The special structure of such matrices [e.g., Toeplitz, Hankel, Cauchy, Vandermonde, etc.] often allows one to speed-up the computation of its triangular factorization, i.e., to efficiently obtain fast implementations of the Gaussian elimination procedure. There is a vast literature about such methods which are known under different names, e.g., fast Cholesky, fast Gaussian elimination, generalized Schur, or Schur-type algorithms. However, without further improvements they are efficient fast implementations of a numerically inaccurate [for indefinite matrices] method. In this paper we survey recent results on the fast implementation of various pivoting techniques which allowed us to improve numerical accuracy for a variety of fast algorithms. This approach led us to formulate new more accurate numerical methods for factorization of general and of J-unitary rational matrix functions, for solving various tangential interpolation problems, new Toeplitz-like and Toeplitz-plus-Hankel-like solvers, and new divided differences schemes. We beleive that similar methods can be used to design accurate fast algorithm for the other applied problems and a recent work of our colleagues supports this anticipation.

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<title>Error analysis of a fast partial pivoting method for structured matrices</title>

Cited by 21 publications

References 9 publications

A Fast Algorithm for Approximate Polynomial GCD Based on Structured Matrix Computations

A Fast Algorithm for Approximate Polynomial GCD Based on Structured Matrix Computations

A fast solver for linear systems with displacement structure

Pivoting for structured matrices and rational tangential interpolation

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