The structure of matrices in rational Gauss quadrature

Jagels, Carl; Reichel, Lothar

doi:10.1090/s0025-5718-2013-02695-6

Cited by 10 publications

(8 citation statements)

References 25 publications

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“…When the measure has discrete support, a situation that arises when evaluating inner products involving matrix functions, we show the existence of a rational Gauss quadrature rule with linear algebra techniques by exploiting the structure of matrices determined by the recursion relations. This results extends the discussions by Golub and Meurant [11,12] and that in [15] to rational Gauss rules determined by several distinct finite poles.…”

Section: Discussionsupporting

confidence: 89%

“…Our analysis extends the discussion in [15] on Laurent polynomials to rational functions with several distinct finite poles.…”

Section: Miroslav S Pranić and Lothar Reichelsupporting

confidence: 64%

“…Rational Gauss rules based on orthogonal Laurent polynomials, i.e., on orthogonal rational functions with poles at zero and infinity only, have received considerable attention; see, e.g., [5,15,16]. This paper generalizes some of the results available for this kind of quadrature rules to rational Gauss rules with several distinct finite poles.…”

Section: Introductionmentioning

confidence: 88%

“…The fact that rational Gauss quadrature rules can give much more accurate approximations of an integral than standard Gauss rules with the same number of nodes has been well demonstrated in the literature; see, e.g., [7,9,15,18]. We therefore omit numerical illustrations of the benefits of rational Gauss quadrature over standard Gauss quadrature.…”

Section: ) and Introduce The Projection Of A Onto The Rational Krylovmentioning

confidence: 99%

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Rational Gauss Quadrature

Pranić¹,

Reichel²

2014

SIAM J. Numer. Anal.

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The existence of (standard) Gauss quadrature rules with respect to a nonnegative measure dµ with support on the real axis easily can be shown with the aid of orthogonal polynomials with respect to this measure. Efficient algorithms for computing the nodes and weights of an n-point Gauss rule use the n × n symmetric tridiagonal matrix determined by the recursion coefficients for the first n orthonormal polynomials. Many rational functions that are orthogonal with respect to the measure dµ and have real or complex conjugate poles also satisfy a short recursion relations. This paper describes how banded matrices determined by the recursion coefficients for these orthonormal rational functions can be used to efficiently compute the nodes and weights of rational Gauss quadrature rules.

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Section: Discussionsupporting

confidence: 89%

“…Our analysis extends the discussion in [15] on Laurent polynomials to rational functions with several distinct finite poles.…”

Section: Miroslav S Pranić and Lothar Reichelsupporting

confidence: 64%

Section: Introductionmentioning

confidence: 88%

Section: ) and Introduce The Projection Of A Onto The Rational Krylovmentioning

confidence: 99%

See 2 more Smart Citations

Rational Gauss Quadrature

Pranić¹,

Reichel²

2014

SIAM J. Numer. Anal.

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“…In the present work, we also consider Krylov techniques related to rational Gauss-Radau quadrature formulae. These quadrature formulae integrate rational function r = p/|q| 2 exactly, where p is a polynomial of degree ≤ 2m − 2, q is the given denominator, and one of the m quadrature nodes is preassigned, see also [LLRW08,JR13]. For rational Gauss-Radau quadrature formulae in a more general setting see also [Gau04,DBVD10,DB12].…”

Section: Introduction and Historical Contextmentioning

confidence: 99%

A review of the Separation Theorem of Chebyshev-Markov-Stieltjes for polynomial and some rational Krylov subspaces

Jawecki¹

2022

Preprint

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The accumulated quadrature weights of Gaussian quadrature formulae constitute bounds on the integral over the intervals between the quadrature nodes. Classical results in this concern date back to works of Chebyshev, Markov and Stieltjes and are referred to as Separation Theorem of Chebyshev-Markov-Stieltjes (CMS Theorem). Similar separation theorems hold true for some classes of rational Gaussian quadrature. The Krylov subspace for a given matrix and initial vector is closely related to orthogonal polynomials associated with the spectral distribution of the initial vector in the eigenbasis of the given matrix, and Gaussian quadrature for the Riemann-Stieltjes integral associated with this spectral distribution. Similar relations hold true for rational Krylov subspaces. In the present work, separation theorems are reviewed in the context of Krylov subspaces including rational Krylov subspaces with a single complex pole of higher multiplicity and some extended Krylov subspaces. For rational Gaussian quadrature related to some classes of rational Krylov subspaces with a single pole, the underlying separation theorems are newly introduced here.

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Functions of rational Krylov space matrices and their decay properties

Pozza

Simoncini

2021

Numer. Math.

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Rational Krylov subspaces have become a fundamental ingredient in numerical linear algebra methods associated with reduction strategies. Nonetheless, many structural properties of the reduced matrices in these subspaces are not fully understood. We advance in this analysis by deriving bounds on the entries of rational Krylov reduced matrices and of their functions, that ensure an a-priori decay of their entries as we move away from the main diagonal. As opposed to other decay pattern results in the literature, these properties hold in spite of the lack of any banded structure in the considered matrices. Numerical experiments illustrate the quality of our results.

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The structure of matrices in rational Gauss quadrature

Cited by 10 publications

References 25 publications

Rational Gauss Quadrature

Rational Gauss Quadrature

A review of the Separation Theorem of Chebyshev-Markov-Stieltjes for polynomial and some rational Krylov subspaces

Functions of rational Krylov space matrices and their decay properties

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