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

Pole Allocation for Rational Gauss Quadrature Rules for Matrix Functionals Defined by a Stieltjes... 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Axioms Multidisciplinary Digital Publishing Institute

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

, Volume 12 (2) – Jan 19, 2023

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

, Volume 12 (2) – Jan 19, 2023

### 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.  Publisher
Multidisciplinary Digital Publishing Institute
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ISSN
2075-1680
DOI
10.3390/axioms12020105
Publisher site
See Article on Publisher Site

### 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.

### Journal

AxiomsMultidisciplinary Digital Publishing Institute

Published: Jan 19, 2023

Keywords: Stieltjes function; matrix function; rational Gauss quadrature