Convergence and computation of simultaneous rational quadrature formulas

22 pages, no figures.-- MSC2000 codes: Primary 41A55. Secondary 41A28, 65D32.MR#: MR2286008 (2008a:65049)Zbl#: Zbl 1168.65326We discuss the convergence and numerical evaluation of simultaneous quadrature formulas which are exact for rational functions. The problem consists in integrating a single function with respect to different measures using a common set of quadrature nodes. Given a multi-index n, the nodes of the integration rule are the zeros of the multi-orthogonal Hermite–Padé polynomial with respect to (S, α, n), where S is a collection of measures, and α is a polynomial which modifies the measures in S. The theory is based on the connection between Gauss-type simultaneous quadrature formulas of rational type and multipoint Hermite–Padé approximation. The numerical treatment relies on the technique of modifying the integrand by means of a change of variable when it has real poles close to the integration interval. The output of some tests show the power of this approach in comparison with other ones in use.The work of U.F.P. and G.L.L. was partially supported by Dirección General de Enseñanza Superior under grant BFM2003-06335-C03-02 and of G.L.L. by INTAS under Grant INTAS 03-51-6637. The work of J.R.I. was supported by a research grant from the Ministerio de Educación y Ciencia, project code MTM 2005-01320.Publicad

Gaussian rational quadrature formulas for ill-scaled integrands

AbstractA flexible treatment of Gaussian quadrature formulas based on rational functions is given to evaluate the integral ∫If(x)W(x)dx, when f is meromorphic in a neighborhood V of the interval I and W(x) is an ill-scaled weight function. Some numerical tests illustrate the power of this approach in comparison with Gautschi’s method

Convergence and computation of simultaneous rational quadrature formulas

We discuss the convergence and numerical evaluation of simultaneous quadrature formulas which are exact for rational functions. The problem consists in integrating a single function with respect to different measures using a common set of quadrature nodes. Given a multi-index n, the nodes of the integration rule are the zeros of the multi-orthogonal HermitePadé polynomial with respect to (S, , n), where S is a collection of measures, and is a polynomial which modifies the measures in S. The theory is based on the connection between Gauss-type simultaneous quadrature formulas of rational type and multipoint HermitePadé approximation. The numerical treatment relies on the technique of modifying the integrand by means of a change of variable when it has real poles close to the integration interval. The output of some tests show the power of this approach in comparison with other ones in use