Fast Gaussian Elimination with Partial Pivoting for Matrices with Displacement Structure

Gohberg, Israel; Kailath, T.; Olshevsky, Vadim

doi:10.2307/2153371

Cited by 92 publications

(77 citation statements)

References 23 publications

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“…The former four papers approximate the solution of Toeplitz, Hankel, Toeplitz-like, and Hankel-like linear systems of equations in nearly linear arithmetic time, versus the cubic time of the classical numerical algorithms and the previous record quadratic time of [GKO95]. All five papers [GKO95], [MRT05], [CGS07], [XXG12], and [XXCB14] begin with the transformation of an input matrix into a Cauchy-like one, by specializing the cited technique of [P90]. Then [GKO95] continued by exploiting the invariance of the Cauchy structure in row interchange, while the other four papers apply the numerically stable FMM in order to operate efficiently with HSS approximations of the basic Cauchy matrix.…”

Section: Related Work and Our Techniquesmentioning

confidence: 99%

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Fast approximate computations with Cauchy matrices and polynomials

Pan

2017

Math. Comp.

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Multipoint polynomial evaluation and interpolation are fundamental for modern symbolic and numerical computing. The known algorithms solve both problems over any field of constants in nearly linear arithmetic time, but the cost grows to quadratic for numerical solution. We fix this discrepancy: our new numerical algorithms run in nearly linear arithmetic time. At first we restate our goals as the multiplication of an n × n Vandermonde matrix by a vector and the solution of a Vandermonde linear system of n equations. Then we transform the matrix into a Cauchy structured matrix with some special features. By exploiting them, we approximate the matrix by a generalized hierarchically semiseparable matrix, which is a structured matrix of a different class. Finally we accelerate our solution to the original problems by applying Fast Multipole Method to the latter matrix. Our resulting numerical algorithms run in nearly optimal arithmetic time when they perform the above fundamental computations with polynomials, Vandermonde matrices, transposed Vandermonde matrices, and a large class of Cauchy and Cauchy-like matrices. Some of our techniques may be of independent interest.

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Section: Related Work and Our Techniquesmentioning

confidence: 99%

“…We incorporate the powerful FMM/HSS techniques, but extend them nontrivially. The papers [GKO95], [MRT05], [CGS07], [XXG12], and [XXCB14] handle just the special Cauchy matrix C = ( 1 si−tj ) m−1,n−1 i,j=0 for which m = n, {s 0 , . .…”

Section: Related Work and Our Techniquesmentioning

confidence: 99%

Fast approximate computations with Cauchy matrices and polynomials

Pan

2017

Math. Comp.

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“…(The paper [P89/90] has revealed a more general link among the matrix structures of Vandermonde, Cauchy, Toeplitz and Hankel types, based on their displacement representation. The paper has also pointed out potential algorithmic applications, and this work has indeed become the springboard for devising the highly efficient algorithms of the subsequent papers [GKO95], [G98], [MRT05], [R06], [XXG12], [XXCB14], [P15], and [Pa]). ) Restatement of our original task in terms of Cauchy matrices has open new opportunities for our study of Vandermonde matrices, and we obtained the desired progress by extending the known formula for the Cauchy determinant to the following simple expression for all entries of the inverse of a Cauchy matrix C s,t ,…”

Section: Introductionmentioning

confidence: 96%

How Bad Are Vandermonde Matrices?

Pan¹

2016

SIAM J. Matrix Anal. & Appl.

111

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The work on the estimation of the condition numbers of Vandermonde matrices, motivated by applications to interpolation and quadrature, can be traced back at least to the 1970s. Empirical study has shown consistently that Vandermonde matrices tend to be badly ill-conditioned, with a narrow class of notable exceptions, such as the matrices of the discrete Fourier transform (hereafter referred to as DFT). So far formal support for this empirical observation, however, has been limited to the matrices defined by the real set of knots. We prove that, more generally, any Vandermonde matrix of a large size is badly illconditioned unless its knots are more or less equally spaced on or about the circle C(0, 1) = {x : |x| = 1}. The matrices of DFT are perfectly conditioned, being defined by a cyclic sequence of knots, equally spaced on that circle, but we prove that even a slight modification of the knots into the so called quasi-cyclic sequence on this circle defines badly ill-conditioned Vandermonde matrices. Likewise we prove that the half-size leading block of a large DFT matrix is badly ill-conditioned. (This result was motivated by an application to pre-conditioning of an input matrix for Gaussian elimination with no pivoting.) Our analysis involves the Ekkart-Young theorem, the Vandermonde-to-Cauchy transformation of matrix structure, our new inversion formula for a Cauchy matrix, and low-rank approximation of its large submatrices.

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“…Existing approaches for solving Cauchy or Cauchy‐like linear systems associated with points in

double-struckR

mainly rely on some variants of Gaussian elimination with pivoting techniques. For example, fast O ( n 2 ) algorithms for solving Cauchy linear systems can be found in the works of Boros et al, Gohbert et al, and Gu; a superfast

O false(n \log^{3} n false)

algorithm based on a sequential block Gaussian elimination process was proposed in the work of Pan et al The performance of most existing methods depends on the the distribution of point sets x and y . As pointed out in the work of Boros et al, if two sets of points x and y cannot be separated, for example, when they are interlaced, existing algorithms (e.g., BP‐type algorithm of Boros et al) suffer from backward stability issues.…”

Section: Numerical Examplesmentioning

confidence: 99%

SMASH: Structured matrix approximation by separation and hierarchy

Cai

Chow

Erlandson

et al. 2018

Numerical Linear Algebra App

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Summary This paper presents an efficient method to perform structured matrix approximation by separation and hierarchy (SMASH), when the original dense matrix is associated with a kernel function. Given the points in a domain, a tree structure is first constructed based on an adaptive partition of the computational domain to facilitate subsequent approximation procedures. In contrast to existing schemes based on either analytic or purely algebraic approximations, SMASH takes advantage of both approaches and greatly improves efficiency. The algorithm follows a bottom‐up traversal of the tree and is able to perform the operations associated with each node on the same level in parallel. A strong rank‐revealing factorization is applied to the initial analytic approximation in the separation regime so that a special structure is incorporated into the final nested bases. As a consequence, the storage is significantly reduced on one hand and a hierarchy of the original grid is constructed on the other hand. Due to this hierarchy, nested bases at upper levels can be computed in a similar way as the leaf level operations but on coarser grids. The main advantages of SMASH include its simplicity of implementation, its flexibility to construct various hierarchical rank structures, and a low storage cost. The efficiency and robustness of SMASH are demonstrated through various test problems arising from integral equations, structured matrices, etc.

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Fast Gaussian Elimination with Partial Pivoting for Matrices with Displacement Structure

Cited by 92 publications

References 23 publications

Fast approximate computations with Cauchy matrices and polynomials

Fast approximate computations with Cauchy matrices and polynomials

How Bad Are Vandermonde Matrices?

SMASH: Structured matrix approximation by separation and hierarchy

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