How Bad Are Vandermonde Matrices?

Pan, Victor Y.

doi:10.1137/15m1030170

Cited by 111 publications

(76 citation statements)

References 30 publications

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“…The issue remains open for the elimination preprocessed with various other random structured multipliers. We refer the reader to [27,40], and the references therein for our further results.…”

Section: Overviewmentioning

confidence: 99%

Random multipliers numerically stabilize Gaussian and block Gaussian elimination: Proofs and an extension to low-rank approximation

Pan

Qian

Yan

2015

Linear Algebra and its Applications

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We study two applications of standard Gaussian random multipliers. At first we prove that with a probability close to 1 such a multiplier is expected to numerically stabilize Gaussian elimination with no pivoting as well as block Gaussian elimination. Then, by extending our analysis, we prove that such a multiplier is also expected to support low-rank approximation of a matrix without customary oversampling. Our test results are in good accordance with this formal study. The results remain similar when we replace Gaussian multipliers with random circulant or Toeplitz multipliers, which involve fewer random parameters and enable faster multiplication. We formally support the observed efficiency of random structured multipliers applied to approximation, but we still continue our research in the case of elimination. We ✩ Some results of this paper have been presented at 203 Gaussian elimination Pivoting Block Gaussian elimination Low-rank approximation SRFT matrices Random circulant matrices specify a narrow class of unitary inputs for which Gaussian elimination with no pivoting is numerically unstable and then prove that, with a probability close to 1, a Gaussian random circulant multiplier does not fix numerical stability problems for such inputs. We also prove that the power of the random circulant preprocessing increases if we also include random permutations.

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Section: Overviewmentioning

confidence: 99%

Random multipliers numerically stabilize Gaussian and block Gaussian elimination: Proofs and an extension to low-rank approximation

Pan

Qian

Yan

2015

Linear Algebra and its Applications

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“…In Section 4.4 we describe how to use the matrices P m , P n in specific situations. Since these matrices are obtained via V m , V n in (2.4)-(2.5) and real-valued Vandermonde matrices are usually highly ill-conditioned [4,5,48], care is needed when computing their null spaces, as extracting the orthogonal factors in QR (or SVD) is susceptible to numerical errors. Berrut and Mittelmann [9] suggest a careful elimination process to remedy this (for a slightly different problem).…”

Section: 1mentioning

confidence: 99%

Rational Minimax Approximation via Adaptive Barycentric Representations

Filip¹,

Nakatsukasa²,

Trefethen³

et al. 2018

SIAM J. Sci. Comput.

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Computing rational minimax approximations can be very challenging when there are singularities on or near the interval of approximation -precisely the case where rational functions outperform polynomials by a landslide. We show that far more robust algorithms than previously available can be developed by making use of rational barycentric representations whose support points are chosen in an adaptive fashion as the approximant is computed. Three variants of this barycentric strategy are all shown to be powerful: (1) a classical Remez algorithm, (2) a "AAA-Lawson" method of iteratively reweighted least-squares, and (3) a differential correction algorithm. Our preferred combination, implemented in the Chebfun MINIMAX code, is to use (2) in an initial phase and then switch to (1) for generically quadratic convergence. By such methods we can calculate approximations up to type (80, 80) of |x| on [−1, 1] in standard 16-digit floating point arithmetic, a problem for which Varga, Ruttan, and Carpenter required 200-digit extended precision.Key words. barycentric formula, rational minimax approximation, Remez algorithm, differential correction algorithm, AAA algorithm, Lawson algorithm AMS subject classifications. 41A20, 65D151. Introduction. The problem we are interested in is that of approximating functions f ∈ C([a, b]) using type (m, n) rational approximations with real coefficients, in the L ∞ setting. The set of feasible approximations is(1.1)Given f and prescribed nonnegative integers m, n, the goal is to computewhere · ∞ denotes the infinity norm over [a, b], i.e., f − r ∞ = max x∈ [a,b] |f (x) − r(x)|. The minimizer of (1.2) is known to exist and to be unique [58, Ch. 24]. Let the minimax (or best) approximation be written r * = p/q ∈ R m,n , where p and q have no common factors. The number d = min {m − deg p, m − deg q} is called the defect of r * . It is known that there exists a so-called alternant (or reference) set consisting of ordered nodes a x 0 < x 1 < · · · < x m+n+1−d b, where f − r * takes

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“…We consider in this section the numerical solution of Cauchy‐like matrices. It is known that Cauchy‐like matrices are related to other types of structured matrices including Toeplitz matrices, Vandermonde matrices, Hankel matrices, and their variants . Consider the kernel

κ false(x, y false) = 1 false/ false(x - y false), x \neq y \in double-struckC

.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…It is known that Cauchy-like matrices are related to other types of structured matrices including Toeplitz matrices, Vandermonde matrices, Hankel matrices, and their variants. [55][56][57][58][59][60] Consider the kernel (x, ) = 1∕(x − ), x ≠ ∈ C. Let x i , y j (i, j = 1 ∶ n) be 2n pair-wise distinct points in C. The Cauchy matrix is then given by  = [ (x i , )] i, =1∶n , which is known to be invertible. 61 Given two matrices w, v ∈ C n×p , the (i, j) entry of a Cauchy-like matrix A associated with generators w and v is defined by 62…”

Section: Cauchy-like Matricesmentioning

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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How Bad Are Vandermonde Matrices?

Cited by 111 publications

References 30 publications

Random multipliers numerically stabilize Gaussian and block Gaussian elimination: Proofs and an extension to low-rank approximation

Random multipliers numerically stabilize Gaussian and block Gaussian elimination: Proofs and an extension to low-rank approximation

Rational Minimax Approximation via Adaptive Barycentric Representations

SMASH: Structured matrix approximation by separation and hierarchy

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