On a novel numerical quadrature based on cycle index of symmetric group for the Hadamard finite-part integrals

TL;DR

This paper introduces a new numerical quadrature method for Hadamard finite-part integrals using cycle index of symmetric groups, improving accuracy and efficiency with proven convergence and error estimates.

Contribution

It presents a novel interpolatory quadrature based on symmetric group cycle index, incorporating divided differences for high-order derivatives, with comprehensive convergence analysis.

Findings

Numerical results confirm the method's high accuracy across various weight functions.

The proposed quadrature demonstrates superior performance compared to existing methods.

Convergence and error bounds are rigorously established.

Abstract

To evaluate the Hadamard finite-part integrals accurately, a novel interpolatory-type quadrature is proposed in this article. In our approach, numerical divided difference is utilized to represent the high order derivatives of the integrated function, which make it possible to reduced the numerical quadrature into a concise formula based on the cycle index for symmetric group. In addition, convergence analysis is presented and the error estimation is given. Numerical results are presented on cases with different weight functions, which substantiate the performance of the proposed method.