Quadrature rules for rational functions

Gautschi, Walter; Gori, Laura; Cascio, Maria

doi:10.1007/pl00005412

Cited by 20 publications

(25 citation statements)

References 9 publications

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“…This has been largely studied when dealing with quadrature rules satisfying I σ (P) = I n (P), P being an arbitrary polynomial of degree as high as possible and σ (x) positive on [a, b] (for details, see the excellent survey [19]). On the other hand, in the last years, quadrature formulae exactly integrating rational functions with prescribed poles have been dealt with (see, for instance, [17,21]) and although numerical experiments have shown their effectiveness especially when the integrand f (x) possesses singularities close to [a, b] very few computer routines are still available (cf. [20]) and very few have been worked out in order to check error bounds for this type of quadrature rules (for a recent contribution on this topic, see [11]).…”

Section: Introductionmentioning

confidence: 99%

“…2) can be considered as rational functions with poles at the origin and at infinity. If we let those poles move and be placed so as to simulate the singularities of f great improvement in the approximation of the integral can be expected (see [21] for examples of this situation on the real line). This gave rise to the construction of quadrature formulae with nodes on the unit circle and exactly integrating rational functions with arbitrary poles not on T but fixed beforehand.…”

Section: Introductionmentioning

confidence: 99%

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Rational quadrature formulae on the unit circle with arbitrary poles

Ysern

González-Vera

2007

Numer. Math.

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Interpolatory quadrature rules exactly integrating rational functions on the unit circle are considered. The poles are prescribed under the only restriction of not lying on the unit circle. A computable upper bound of the error is obtained which is valid for any choice of poles, arbitrary weight functions and any degree of exactness provided that the integrand is analytic on a neighborhood of the unit circle. A number of numerical examples are given which show the advantages of using such rules as well as the sharpness of the error bound. Also, a comparison is made with other error bounds appearing in the literature. Mathematics Subject Classification (2000)65D32 · 41A20 · 41A80

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Rational quadrature formulae on the unit circle with arbitrary poles

Ysern

González-Vera

2007

Numer. Math.

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“…[9,10,11,14,15,22]). In what follows we will remark on some general features of the rational integration rules.…”

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confidence: 99%

“…The presence of difficult poles in the integrand causes slow convergence of the procedure, so we have to choose a suitable strategy as that of selecting some zeros of α N as difficult poles (cf. [9,10,11]). …”

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confidence: 99%

Convergence and computation of simultaneous rational quadrature formulas

Prieto¹,

González

Lagomasino

2007

Numer. Math.

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Abstract. We discuss the theoretical convergence and numerical evaluation of simultaneous interpolation quadrature formulas which are exact for rational functions. Basically, the problem consists in integrating a single function with respect to different measures by using a common set of quadrature nodes. Given a multi-index n, the nodes of the integration rule are the zeros of the multiorthogonal Hermite-Padé polynomial with respect to (S, α, n), where S is a collection of measures, and α is a polynomial which modifies S. The theory is based on the connection between Gausslike simultaneous quadrature formulas of rational type and multipoint Hermite-Padé approximation. As for the numerical treatment we present a procedure based on the technique of modifying the integrand by means of a change of variable when the integrand has real poles close to the integration interval. The output of some tests show the power of this approach when compared to other ones.Key words. simultaneous integration rule, Gauss-like rational quadrature formula, meromorphic integrand AMS subject classifications. Primary 41A55. Secondary 41A28, 65D321. Introduction. Simultaneous integration is a problem which arises in computer graphics to determinate the color of light which emanates from a given point on a surface toward the viewer (see Borges [4] and the bibliography therein). In an abstract setting, this problem was earlier studied by Nikishin [19] in connection with Hermite-Padé approximation. Simultaneous integration means that we are integrating a single function f : I → R, with respect to m distinct measures ds 1 , ..., ds m on I, respectively.A reference index Φ = (Ov + 1)/N u (performance ratio) is considered in [4] to state the efficiency of the procedure. Here Ov is the overall degree of exactness and N u is the number of integrand evaluations. Borges [4] remarked that when m ≥ 3 the use of the m corresponding Gaussian rules of polynomial type yields Φ < 1, which indicates a low performance. Instead he suggests quadrature rules whose nodes are the zeros of multi-orthogonal polynomials for which Φ > 1 holds.Geometric rate of convergence has been proved by the authors in [8] for polynomials methods when the integrand is analytic in a neighborhood of I. Nevertheless, instability shows up when the corresponding numerical method is applied to meromorphic integrands with poles close to I. Several methods to integrate functions with singularities on or near the integration interval have been developed in the past few years.An interesting approach, whose starting point seems to be [13], is that based on the use of rational Gaussian integration rules connected with multipoint Padé

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“…This is also true for the orthogonal rational functions, which have been under investigation with respect to rational Gauss-Chebyshev quadrature; see e.g. [15], [17], [18], [20], [29] and [30]. To get more general cases where explicit expressions are obtained, a technique of rational modifications of these Chebyshev weights are considered in [14].…”

Section: Introductionmentioning

confidence: 99%

Rational Szegő quadratures associated with Chebyshev weight functions

et al. 2008

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Abstract. In this paper we characterize rational Szegő quadrature formulas associated with Chebyshev weight functions, by giving explicit expressions for the corresponding para-orthogonal rational functions and weights in the quadratures. As an application, we give characterizations for Szegő quadrature formulas associated with rational modifications of Chebyshev weight functions. Some numerical experiments are finally presented.

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Quadrature rules for rational functions

Cited by 20 publications

References 9 publications

Rational quadrature formulae on the unit circle with arbitrary poles

Rational quadrature formulae on the unit circle with arbitrary poles

Convergence and computation of simultaneous rational quadrature formulas

Rational Szegő quadratures associated with Chebyshev weight functions

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