A matricial computation of rational quadrature formulas on the unit circle

Bultheel, Adhemar; Cantero, Maŕıa José

doi:10.1007/s11075-008-9257-9

Cited by 12 publications

(12 citation statements)

References 22 publications

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“…We show theoretically and by a numerical example that convergence holds for wider class of unbounded functions when the matrix elements are nonnegative. 4. We also show the convergence of improper integrals on a finite endpoint for functions bounded by an operator convex function by using Jensen's operator inequality.…”

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confidence: 98%

Numerical integration as a finite matrix approximation to multiplication operator

Sarmavuori

Särkkä

2019

Journal of Computational and Applied Mathematics

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We study the convergence of a family of numerical integration methods where the numerical integral is formulated as a finite matrix approximation to a multiplication operator. For bounded functions, the convergence has already been established using the theory of strong operator convergence. In this article, we consider unbounded functions and domains which pose several difficulties compared to the bounded case. A natural choice of method for this study is the theory of strong resolvent convergence which has previously been mostly applied to study the convergence of approximations of differential operators. The existing theory already includes convergence theorems that can be used as proofs as such for a limited class of functions and extended for wider class of functions in terms of function growth or discontinuity. The extended results apply to all self-adjoint operators, not just multiplication operators. We also show how Jensen's operator inequality can be used to analyze the convergence of an improper numerical integral of a * Corresponding author function bounded by an operator convex function.

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confidence: 98%

Numerical integration as a finite matrix approximation to multiplication operator

Sarmavuori

Särkkä

2019

Journal of Computational and Applied Mathematics

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“…Therefore it is essential to select poles at strategic places much closer to the circle to put more emphasis on regions where the eigenvalues are more important. The effect of the location of the poles can be observed for example in the numerical experiments reported in [6,7,10].…”

Section: Relation With Direct and Inverse Eigenvalue Problemsmentioning

confidence: 89%

Orthogonal Rational Functions on the Unit Circle with Prescribed Poles not on the Unit Circle

Bultheel¹,

Cruz-Barroso²,

Lasarow³

2017

SIGMA

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Abstract. Orthogonal rational functions (ORF) on the unit circle generalize orthogonal polynomials (poles at infinity) and Laurent polynomials (poles at zero and infinity). In this paper we investigate the properties of and the relation between these ORF when the poles are all outside or all inside the unit disk, or when they can be anywhere in the extended complex plane outside the unit circle. Some properties of matrices that are the product of elementary unitary transformations will be proved and some connections with related algorithms for direct and inverse eigenvalue problems will be explained.

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“…and the matrix representation of T µ becomes U =ζ A (C ) with C as in the polynomial case, but again replacing the Verblunski coefficients with the δ 's of (2.4). See [2,24]. Now let us truncate all matrices to n rows and n columns which we indicate with a subscript n. Let δ 1 , .…”

Section: Remarkmentioning

confidence: 99%

“…Because of δ n ∈ T, we call these unitary truncations. Then we have the following theorem that can be found in [2].…”

Section: Remarkmentioning

confidence: 99%

Computation of rational Szegő–Lobatto quadrature formulas

Bultheel

González-Vera

Hendriksen³

et al. 2010

Applied Numerical Mathematics

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Szegő quadrature formulas are analogs of Gauss quadrature rules when dealing with the approximate integration of periodic functions, since they exactly integrate trigonometric polynomials of as high degree as possible, or more generally Laurent polynomials which can be viewed as rational functions with poles at the origin and infinity. When more general rational functions with prescribed poles on the extended complex plane not on the unit circle are considered to be exactly integrated, the so called "Rational Szegő Quadrature Formulas" appear. In this paper, some computational aspects concerning these quadratures are analyzed when one or two nodes are previously fixed on the unit circle.

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A matricial computation of rational quadrature formulas on the unit circle

Cited by 12 publications

References 22 publications

Numerical integration as a finite matrix approximation to multiplication operator

Numerical integration as a finite matrix approximation to multiplication operator

Orthogonal Rational Functions on the Unit Circle with Prescribed Poles not on the Unit Circle

Computation of rational Szegő–Lobatto quadrature formulas

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