On Kosloff Tal-Ezer Least-Squares Quadrature Formulas

TL;DR

This paper introduces a stable and accurate quadrature method combining the Kosloff Tal-Ezer map with least-squares approximation, improving numerical integration of analytic functions on compact intervals, especially with non-optimal sampling nodes.

Contribution

It develops a novel KTL quadrature scheme that enhances stability and accuracy by using auxiliary mappings and least-squares, with practical implementation strategies.

Findings

Static map parameters improve quadrature over trapezoidal rule.

Dynamic parameter selection enhances stability and convergence.

Method is effective even with perturbed sampling nodes.

Abstract

In this work, we study a global quadrature scheme for analytic functions on compact intervals based on function values on quasi-uniform grids of quadrature nodes. In practice it is not always possible to sample functions at optimal nodes that give well-conditioned and quickly converging interpolatory quadrature rules at the same time. Therefore, we go beyond classical interpolatory quadrature by lowering the degree of the polynomial approximant and by applying auxiliary mapping functions that map the original quadrature nodes to more suitable fake nodes. More precisely, we investigate the combination of the Kosloff Tal-Ezer map and least-squares approximation (KTL) for numerical quadrature: a careful selection of the mapping parameter ensures stability of the scheme, a high accuracy of the approximation and, at the same time, an asymptotically optimal ratio between the degree of the polynomial and the spacing of the grid. We will investigate the properties of this KTL quadrature and focus on the symmetry of the quadrature weights, the limit relations for the mapping parameter, as well as the computation of the quadrature weights in the standard monomial and in the Chebyshev bases with help of a cosine transform. Numerical tests on equispaced nodes show that a static choice of the map's parameter improve the results of the composite trapezoidal rule, while a dynamic approach achieves larger stability and faster convergence, even when the sampling nodes are perturbed. From a computational point of view the proposed method is practical and can be implemented in a simple and efficient way.