Complexity of multiplication with vectors for structured matrices

Gohberg, Israel; Olshevsky, Vadim

doi:10.1016/0024-3795(94)90189-9

Cited by 97 publications

(87 citation statements)

References 8 publications

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“…Following [34], we turn a Vandermonde-like system Au = v into a Toeplitzlike one. Our reduction follows that of [19]. However, that reference requires the entries of x to be pairwise distinct, i.e., that V(x, n) be invertible; else, the preprocessing step in [19,Section 2] fails.…”

Section: The Vandermonde Casementioning

confidence: 99%

“…Our reduction follows that of [19]. However, that reference requires the entries of x to be pairwise distinct, i.e., that V(x, n) be invertible; else, the preprocessing step in [19,Section 2] fails. Similarly, the reduction in [36, Example 4.8.4] does not solve the problem when V(x, n) is singular.…”

Section: The Vandermonde Casementioning

confidence: 99%

“…We use a transformation from [19], generalizing it by taking into account the possibility of repetitions in the diagonal component of the operator. Again, the main technical tool is to multiply (submatrices of) a Vandermonde-like matrix, given by its generator, by several vectors.…”

Section: Introductionmentioning

confidence: 99%

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Solving toeplitz- and vandermonde-like linear systems with large displacement rank

Bostan

Jeannerod

Schost

2007

Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation

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Linear systems with structures such as Toeplitz-, Vandermonde-or Cauchy-likeness can be solved in O˜(α 2 n) operations, where n is the matrix size, α is its displacement rank, and O˜denotes the omission of logarithmic factors. We show that for Toeplitz-like and Vandermonde-like matrices, this cost can be reduced to O˜(α ω−1 n), where ω is a feasible exponent for matrix multiplication over the base field. The best known estimate for ω is ω < 2.38, resulting in costs of order O˜(α 1.38 n). We also present consequences for Hermite-Padé approximation and bivariate interpolation.

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Section: The Vandermonde Casementioning

confidence: 99%

Section: The Vandermonde Casementioning

confidence: 99%

See 1 more Smart Citation

Solving toeplitz- and vandermonde-like linear systems with large displacement rank

Bostan

Jeannerod

Schost

2007

Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation

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“…To obtain (22) given these results, it thus only remains to find a computationally efficient way of forming (24), which is directly associated with the structure of the matrix P N involved, formed by (16) as the product of Toeplitz like matrices, since [R further use, we briefly present the basics from the displacement representation theory of matrices (see also [32][33][34]). Consider a matrix Q N ∈ C N ×N , and define the lower shifting matrix…”

Section: Msc Estimation Using Trigonometric Polynomialsmentioning

confidence: 99%

“…Given the displacement representation (33), the coefficients of the trigonometric polynomial associated…”

Section: Msc Estimation Using Trigonometric Polynomialsmentioning

confidence: 99%

Computationally efficient sparsity-inducing coherence spectrum estimation of complete and non-complete data sets

2013

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The magnitude squared coherence (MSC) spectrum is an often used frequency-dependent measure for the linear dependency between two stationary processes, and the recent literature contain several contributions on how to form high-resolution data-dependent and adaptive MSC estimators, and on the efficient implementation of such estimators. In this work, we further this development with the presentation of computationally efficient implementations of the recent iterative adaptive approach (IAA) estimator, present a novel sparse learning via iterative minimization (SLIM) algorithm, discuss extensions to two-dimensional data sets, examining both the case of complete data sets and when some of the observations are missing. The algorithms further the recent development of exploiting the estimators' inherently low displacement rank of the necessary products of Toeplitz-like matrices, extending these formulations to the coherence estimation using IAA and SLIM formulations. The performance of the proposed algorithms and implementations are illustrated both with theoretical complexity measures and with numerical simulations.

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Newton's Iteration for Inversion of Cauchy-Like and Other Structured Matrices

Pan

Zheng

Huang

et al. 1997

Journal of Complexity

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We specify some initial assumptions that guarantee rapid refinement of a rough initial approximation to the inverse of a Cauchy-like matrix, by means of our new modification of Newton's iteration, where the input, output, and all the auxiliary matrices are represented with their short generators defined by the associated scaling operators. The computations are performed fast since they are confined to operations with short generators of the given and computed matrices. Because of the known correlations among various structured matrices, the algorithm is immediately extended to rapid refinement of rough initial approximations to the inverses of Vandermonde-like, ChebyshevVandermonde-like, and Toeplitz-like matrices, where again the computations are confined to operations with short generators of the involved matrices.

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Complexity of multiplication with vectors for structured matrices

Cited by 97 publications

References 8 publications

Solving toeplitz- and vandermonde-like linear systems with large displacement rank

Solving toeplitz- and vandermonde-like linear systems with large displacement rank

Computationally efficient sparsity-inducing coherence spectrum estimation of complete and non-complete data sets

Newton's Iteration for Inversion of Cauchy-Like and Other Structured Matrices

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