A Björck–Pereyra-type algorithm for Szegö–Vandermonde matrices based on properties of unitary Hessenberg matrices

Bella, T.; Eidelman, Yuli; Gohberg, Israel; Koltracht, I.; Olshevsky, Vadim

doi:10.1016/j.laa.2006.08.032

Cited by 17 publications

(19 citation statements)

References 19 publications

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“…However, already at this point the results of all of our experiments are fully consistent with the conclusions made in [BP70], [CF88], [H90], [RO91], [BKO99], [O03] and [BEGKO07] for similar algorithms. Specifically, there are examples, even in the most generic H-q.s.…”

Section: Special Choices Of Generatorssupporting

confidence: 91%

“…In particular, if (4.7) is inserted into the factors (3.2) in (4.3), then the result is exactly that derived in [BEGKO07,(3.10) and (3.15)], where the nice properties of the matrix C Φ # (φ # n ) were used to provide a computational speedup. Specifically, the algorithm is made fast by the factorization…”

Section: 2mentioning

confidence: 56%

“…Szegö polynomials. The [BEGKO07] algorithm. If the system of quasiseparable polynomials are the Szegö polynomials Φ # = {φ # 0 (x), .…”

Section: 2mentioning

confidence: 99%

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A Fast Björck–Pereyra-Type Algorithm for Solving Hessenberg-Quasiseparable-Vandermonde Systems

Bella¹,

Eidelman²,

Gohberg³

et al. 2009

SIAM J. Matrix Anal. & Appl.

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In this paper we derive a fast O(n 2) algorithm for solving linear systems where the coefficient matrix is a polynomial-Vandermonde matrix V R (x) = [r j−1 (x i)] with polynomials {r k (x)} related to a Hessenberg quasiseparable matrix. The result generalizes the well-known Björck-Pereyra algorithm for classical Vandermonde systems involving monomials. It also generalizes the algorithms of [RO91] for V R (x) involving Chebyshev polynomials, of [H90] for V R (x) involving real orthogonal polynomials, and of [BEGKO07] for V R (x) involving Szegö polynomials. The new algorithm applies to a fairly general class of H-k-q.s.-polynomials (Hessenberg order k quasiseparable) that includes (along with the above mentioned classes of real orthogonal and Szegö polynomials) several other important classes of polynomials. Preliminary numerical experiments are presented comparing the algorithm to standard structure-ignoring methods.

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Section: Special Choices Of Generatorssupporting

confidence: 91%

Section: 2mentioning

confidence: 56%

See 1 more Smart Citation

A Fast Björck–Pereyra-Type Algorithm for Solving Hessenberg-Quasiseparable-Vandermonde Systems

Bella¹,

Eidelman²,

Gohberg³

et al. 2009

SIAM J. Matrix Anal. & Appl.

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show abstract

“…This justifies the nomenclature recurrence matrices used for T n and U n . 2 The structure of the Toeplitz matrix is deduced similarly from (1.2) and the fact that on the unit circle we have x j = e ijθ = e −ijθ = x −j , so that each moment m kj now depends only on the difference of indices, which yields the Toeplitz structure. Many nice results originally derived only for Hankel and Toeplitz moment matrices (e.g., fast Levinson and Schur algorithms, fast multiplication algorithms, explicit inversion formulas, etc.)…”

Section: Classical Polynomial Families and Their Recurrence Matricesmentioning

confidence: 97%

“…Hence the matrix M has a Hankel structure. 2 The shift-invariant structure of H and T implies that these two square arrays are structured, i.e., they are defined by only O(n) parameters each.…”

Section: Classical Polynomial Families and Their Moment Matricesmentioning

confidence: 99%

Computations with quasiseparable polynomials and matrices

Bella

Eidelman

Gohberg

et al. 2008

Theoretical Computer Science

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In this paper, we survey several recent results that highlight an interplay between a relatively new class of quasiseparable matrices and univariate polynomials. Quasiseparable matrices generalize two classical matrix classes, Jacobi (tridiagonal) matrices and unitary Hessenberg matrices that are known to correspond to real orthogonal polynomials and Szegö polynomials, respectively. The latter two polynomial families arise in a wide variety of applications, and their short recurrence relations are the basis for a number of efficient algorithms. For historical reasons, algorithm development is more advanced for real orthogonal polynomials. Recent variations of these algorithms tend to be valid only for the Szegö polynomials; they are analogues and not generalizations of the original algorithms. Herein, we survey several recent results for the ''superclass'' of quasiseparable matrices, which includes both Jacobi and unitary Hessenberg matrices as special cases. The interplay between quasiseparable matrices and their associated polynomial sequences (which contain both real orthogonal and Szegö polynomials) allows one to obtain true generalizations of several algorithms. Specifically, we discuss the Björck-Pereyra algorithm, the Traub algorithm, certain new digital filter structures, as well as QR and divide and conquer eigenvalue algorithms.

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Fast Inversion of Polynomial-Vandermonde Matrices for Polynomial Systems Related to Order One Quasiseparable Matrices

Bella

Eidelman

Olshevsky

et al. 2013

Advances in Structured Operator Theory and Related Areas

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A Björck–Pereyra-type algorithm for Szegö–Vandermonde matrices based on properties of unitary Hessenberg matrices

Cited by 17 publications

References 19 publications

A Fast Björck–Pereyra-Type Algorithm for Solving Hessenberg-Quasiseparable-Vandermonde Systems

A Fast Björck–Pereyra-Type Algorithm for Solving Hessenberg-Quasiseparable-Vandermonde Systems

Computations with quasiseparable polynomials and matrices

Fast Inversion of Polynomial-Vandermonde Matrices for Polynomial Systems Related to Order One Quasiseparable Matrices

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