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

Machine Precision Evaluation of Singular and Nearly Singular Potential Integrals by Use of Gauss Quadrature Formulas for Rational Functions

A new technique for machine precision evaluation of singular and nearly singular potential integrals with 1/R singularities is presented. The numerical quadrature scheme is based on a new rational expression for the integrands, obtained by a cancellation procedure. In particular, by using library routines for Gauss quadrature of rational functions readily available in the literature, this new expression permits the exact numerical integration of singular static potentials associated with polynomial source distributions. The rules to achieve the desired numerical accuracy for singular and nearly singular static and dynamic potential integrals are presented and discussed, and several numerical examples are provide

Design of quadrature rules for Müntz and Müntz-logarithmic polynomials using monomial transformation

A method for constructing the exact quadratures for Müntz and Müntz-logarithmic polynomials is presented. The algorithm does permit to anticipate the precision (machine precision) of the numerical integration of Müntz-logarithmic polynomials in terms of the number of Gauss-Legendre (GL) quadrature samples and monomial transformation order. To investigate in depth the properties of classical GL quadrature, we present new optimal asymptotic estimates for the remainder. In boundary element integrals this quadrature rule can be applied to evaluate singular functions with end-point singularity, singular kernel as well as smooth functions. The method is numerically stable, efficient, easy to be implemented. The rule has been fully tested and several numerical examples are included. The proposed quadrature method is more efficient in run-time evaluation than the existing methods for Müntz polynomial

The use of rational functions in numerical quadrature

Abstract Quadrature problems involving functions that have poles outside the interval of integration can proÿtably be solved by methods that are exact not only for polynomials of appropriate degree, but also for rational functions having the same (or the most important) poles as the function to be integrated. Constructive and computational tools for accomplishing this are described and illustrated in a number of quadrature contexts. The superiority of such rational=polynomial methods is shown by an analysis of the remainder term and documented by numerical examples

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