4. Stability of Fast Algorithms for Structured Linear Systems

Brent, Richard P.

doi:10.1137/1.9781611971354.ch4

Cited by 11 publications

(9 citation statements)

References 59 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In table 2 we illustrate that the fast Gaussian elimination with partial pivoting performed on the factors G and B enjoys the same property, under the condition that (21) is not too small [1]. This is in fact an optimal result for a fast linear system solver.…”

Section: Stability and Reliabilitymentioning

confidence: 77%

“…For classical Gaussian elimination with partial pivoting performed on the full matrix T instead of on the factors G and B, the error inb, say the width diam(Ȇ) ofȆ, is typically of the order of the product of the condition number of T and the machine epsilon 1 2 β −t+1 where β and t, respectively, denote the radix and precision of the floating-point system in use. In table 2 we illustrate that the fast Gaussian elimination with partial pivoting performed on the factors G and B enjoys the same property, under the condition that (21) is not too small [1].…”

Section: Stability and Reliabilitymentioning

confidence: 99%

“…We use the definition as given in [14] where the {L, R}-displacement rank, or displacement rank for short, of an u × v matrix T is defined as the rank of the matrix ∇T = LT − T R with L and R being so-called left and right displacement operators. If T is a Toeplitz matrix as in (2) and L = Z (1) then the resulting matrix LT − T R is almost entirely filled with zeroes, except for its first row and last column. Hence T has displacement rank 2, irrespective of its size.…”

Section: Introductionmentioning

confidence: 99%

“…Here O(·) indicates that the floatingpoint operation count of the algorithm is bounded above by a constant multiple of (·). It has been indicated that such fast solvers are usually less stable [1,5]. Precisely for that reason we do not consider any so-called superfast algorithms.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the fast solution of Toeplitz-block linear systems arising in multivariate approximation theory

Becuwe

Cuyt

2006

Numer Algor

View full text Add to dashboard Cite

When constructing multivariate Padé approximants, highly structured linear systems arise in almost all existing definitions [10]. Until now little or no attention has been paid to fast algorithms for the computation of multivariate Padé approximants, with the exception of [17]. In this paper we show that a suitable arrangement of the unknowns and equations, for the multivariate definitions of Padé approximant under consideration, leads to a Toeplitz-block linear system with coefficient matrix of low displacement rank. Moreover, the matrix is very sparse, especially in higher dimensions. In Section 2 we discuss this for the so-called equation lattice definition and in Section 3 for the homogeneous definition of the multivariate Padé approximant. We do not discuss definitions based on multivariate generalizations of continued fractions [12,25], or approaches that require some symbolic computations [6,18]. In Section 4 we present an explicit formula for the factorization of the matrix that results from applying the displacement operator to the Toeplitz-block coefficient matrix. We then generalize the well-known fast Gaussian elimination procedure with partial pivoting developed in [14,19], to deal with a rectangular block structure where the number and size of the blocks vary. We do not aim for a superfast solver because of the higher risk for instability. Instead we show how the developed technique can be combined with an easy interval arithmetic verification step. Numerical results illustrate the technique in Section 5.

show abstract

Section: Stability and Reliabilitymentioning

confidence: 77%

Section: Stability and Reliabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the fast solution of Toeplitz-block linear systems arising in multivariate approximation theory

Becuwe

Cuyt

2006

Numer Algor

View full text Add to dashboard Cite

show abstract

“…Accordingly, there is growing interest in structured perturbation analysis; cf. [36,8,24,25,2,16,4,15,7,39,37,38,14]. Moreover, different kinds of structured perturbations are investigated in robust and optimal control, for example, the analysis of the µ-number or structured distances [11,13,34,41,35,29].…”

Section: But (A+∆a)mentioning

confidence: 99%

Structured Perturbations Part I: Normwise Distances

Rump¹

2003

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

Abstract. In this paper we study the condition number of linear systems, the condition number of matrix inversion, and the distance to the nearest singular matrix, all problems with respect to normwise structured perturbations. The structures under investigation are symmetric, persymmetric, skewsymmetric, symmetric Toeplitz, general Toeplitz, circulant, Hankel, and persymmetric Hankel matrices (some results on other structures such as tridiagonal and tridiagonal Toeplitz matrices, both symmetric and general, are presented as well). We show that for a given matrix the worst case structured condition number for all right-hand sides is equal to the unstructured condition number. For a specific right-hand side we give various explicit formulas and estimations for the condition numbers for linear systems, especially for the ratio of the condition numbers with respect to structured and unstructured perturbations. Moreover, the condition number of matrix inversion is shown to be the same for structured and unstructured perturbations, and the same is proved for the distance to the nearest singular matrix. It follows a generalization of the classical Eckart-Young theorem, namely, that the reciprocal of the condition number is equal to the distance to the nearest singular matrix for all structured perturbations mentioned above.

show abstract