Pole Allocation for Rational Gauss Quadrature Rules for Matrix Functionals Defined by a Stieltjes Function

Abstract

This paper considers the computation of approximations of matrix functionals of form F(A):=vTf(A)v, where A is a large symmetric positive definite matrix, v is a vector, and f is a Stieltjes function. The functional F(A) is approximated by a rational Gauss quadrature rule with poles on the negative real axis (or part thereof) in the complex plane, and we focus on the allocation of the poles. Specifically, we propose that the poles, when considered positive point charges, be allocated to make the negative real axis (or part thereof) approximate an equipotential curve. This is easily achieved with the aid of conformal mapping.