A Gaussian method for the operator square root

TL;DR

This paper introduces a rational approximation method using Gauss-Legendre quadrature for computing the inverse square root of accretive operators in Hilbert spaces, with proven error bounds and exponential convergence in finite dimensions.

Contribution

It develops a new rational approximation technique based on integral formulations and Gauss-Legendre rule, with sharp error estimates and exponential convergence analysis.

Findings

Derived sharp error estimates using the numerical range.

Demonstrated exponential convergence in finite-dimensional cases.

Provided numerical experiments validating the theoretical results.

Abstract

We consider the approximation of the inverse square root of regularly accretive operators in Hilbert spaces. The approximation is of rational type and comes from the use of the Gauss-Legendre rule applied to a special integral formulation of the problem. We derive sharp error estimates, based on the use of the numerical range, and provide some numerical experiments. For practical purposes, the finite dimensional case is also considered. In this setting, the convergence is shown to be of exponential type.