Exponentially convergent trapezoidal rules to approximate fractional powers of operators

TL;DR

This paper develops and analyzes exponentially convergent trapezoidal rules for efficiently approximating fractional powers of self-adjoint positive operators, improving error estimates and demonstrating numerical reliability.

Contribution

It introduces refined parameter choices for faster convergence and extends error analysis of double exponential transforms from scalar functions to operators.

Findings

Improved error estimates for scalar case with double exponential transform

Extension of error analysis to operator case

Numerical experiments confirm theoretical predictions

Abstract

In this paper we are interested in the approximation of fractional powers of self-adjoint positive operators. Starting from the integral representation of the operators, we apply the trapezoidal rule combined with a single-exponential and a double-exponential transform of the integrand function. For the first approach our aim is only to review some theoretical aspects in order to refine the choice of the parameters that allow a faster convergence. As for the double exponential transform, in this work we show how to improve the existing error estimates for the scalar case and also extend the analysis to operators. We report some numerical experiments to show the reliability of the estimates obtained.