Convergence and computation of simultaneous rational quadrature formulas

Abstract

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.