A Gauss Laguerre approach for the resolvent of fractional powers

TL;DR

This paper presents a rapid numerical method using Gauss-Laguerre quadrature to compute the resolvent of fractional powers of self-adjoint positive operators, with error estimates for precision control.

Contribution

It introduces a novel Gauss-Laguerre based approach for efficiently computing resolvents of fractional operator powers, including error analysis for parameter selection.

Findings

Method achieves high accuracy with fewer nodes.

Error estimates enable precise control of computational tolerance.

Applicable to unbounded self-adjoint positive operators in Hilbert spaces.

Abstract

This paper introduces a very fast method for the computation of the resolvent of fractional powers of operators. The analysis is kept in the continuous setting of (potentially unbounded) self adjoint positive operators in Hilbert spaces. The method is based on the Gauss-Laguerre rule, exploiting a particular integral representation of the resolvent. We provide sharp error estimates that can be used to a priori select the number of nodes to achieve a prescribed tolerance.