On computing rational Gauss-Chebyshev quadrature formulas

Deun, Joris Van; Bultheel, Adhemar; Vera, Pablo González

doi:10.1090/s0025-5718-05-01774-6

Cited by 28 publications

(39 citation statements)

References 20 publications

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“…For the second and last example, we consider the function f (x) = 1 π sin 1 (x 2 − J(β) 2 )(x 2 − J(iβ) 2 ) , β ∈ (0, 1), (5.2) which is analogous to the one in Example 5.4 from [19]. This function has essential singularities in x = J(i j β), j = 0, .…”

Section: [P ]mentioning

confidence: 99%

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The existence and construction of rational Gauss-type quadrature rules

Deckers

Bultheel

2012

Applied Mathematics and Computation

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Abstract. Consider a hermitian positive-definite linear functional F, and assume we have m distinct nodes fixed in advance anywhere on the real line. In this paper we then study the existence and construction of nth rational Gauss-Radau (m = 1) and Gauss-Lobatto (m = 2) quadrature formulas that approximate F{f }. These are quadrature formulas with n positive weights and with the n−m remaining nodes real and distinct, so that the quadrature is exact in a (2n−m)-dimensional space of rational functions.

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Section: [P ]mentioning

confidence: 99%

“…[1,Chapt. 11.6] and [2,7,8,10,11,12,14,15,17,18,19,20]. Here we can also consider more general rational Gauss-type quadrature rules obtained by fixing one or two nodes in the quadrature rule.…”

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confidence: 99%

The existence and construction of rational Gauss-type quadrature rules

Deckers

Bultheel

2012

Applied Mathematics and Computation

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“…Typically, the rule is required to be exact, that is, R m (f ) ≡ 0 for each element of a predefined linear function space L. Moreover, the rule is said to be Gaussian if m is the minimal number of nodes t i ∈ R n at which f has to be evaluated. There is an extensive number of various quadrature rules depending on n (f is univariate [15], bivariate [28,39], multivariate [16]), domain shape (disc, hypercube, simplex) [37], and the type of the linear space (polynomials [15], splines [4,6,29,33], rational functions [38], smooth non-polynomial [7,27]). For polynomial multivariate integration, the field is well studied and the reader is referred to [37].…”

Section: Introductionmentioning

confidence: 99%

On numerical quadrature for C1 quadratic Powell–Sabin 6-split macro-triangles

Bartoň

Kosinka

2019

Journal of Computational and Applied Mathematics

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The quadrature rule of Hammer and Stroud [16] for cubic polynomials has been shown to be exact for a larger space of functions, namely the C 1 cubic Clough-Tocher spline space over a macro-triangle if and only if the split-point is the barycentre of the macro-triangle [21]. We continue the study of quadrature rules for spline spaces over macro-triangles, now focusing on the case of C 1 quadratic Powell-Sabin 6-split macro-triangles. We show that the 3-node Gaussian quadrature(s) for quadratics can be generalised to the C 1 quadratic Powell-Sabin 6-split spline space over a macro-triangle for a two-parameter family of inner split-points, not just the barycentre as in [21]. The choice of the inner split-point uniquely determines the positions of the edge split-points such that the whole spline space is integrated exactly by a corresponding polynomial quadrature. Consequently, the number of quadrature points needed to exactly integrate this special spline space reduces from twelve to three.For the inner split-point at the barycentre, we prove that the two 3-node quadratic polynomial quadratures of Hammer and Stroud exactly integrate also the C 1 quadratic Powell-Sabin spline space if and only if the edge split-points are at their respective edge midpoints. For other positions of the inner and edge split-points we provide numerical examples showing that three nodes suffice to integrate the space exactly, but a full classification and a closed-form solution in the generic case remain elusive.

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“…In [VDBGV06] we constructed rational generalizations of the well-known classical Gauss-Chebyshev quadrature formula. These rational quadrature rules integrate functions with arbitrary real poles outside the interval [−1, 1], with respect to the different Chebyshev weight functions (1 − x) α (1 + x) β , with α and β belonging to {±1/2}.…”

Section: Introductionmentioning

confidence: 99%

“…Unlike the formulas from [VDBGV06], the quadrature rule constructed in the present paper is not a Gaussian rule, but instead a rational generalization of Fejér's rule. We use the nodes of the rational Gauss-Chebysh ev formula from [VDBGV06], but there is no weight function in the integral.…”

Section: Introductionmentioning

confidence: 99%

A quadrature formula based on Chebyshev rational functions

Deun

Bultheel

2006

IMA Journal of Numerical Analysis

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Several generalisations to the classical Gauss quadrature formulas have been made over the last few years. When the integrand has singularities near the interval of integration, formulas based on rational functions give more accurate results than the classical quadrature rules based on polynomials. In this paper we present one such generalisation which uses results from the theory of orthogonal rational functions. Compared to similar existing formulas, it has the advantage of improved stability and smaller quadrature weights.

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On computing rational Gauss-Chebyshev quadrature formulas

Cited by 28 publications

References 20 publications

The existence and construction of rational Gauss-type quadrature rules

The existence and construction of rational Gauss-type quadrature rules

On numerical quadrature for C1 quadratic Powell–Sabin 6-split macro-triangles

A quadrature formula based on Chebyshev rational functions

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