Numerical Integration as a Finite Matrix Approximation to Multiplication Operator

TL;DR

This paper presents a novel perspective on numerical integration by framing it as a finite matrix approximation of the multiplication operator, unifying Gaussian quadrature within this framework and analyzing convergence properties.

Contribution

It introduces a new matrix-based formulation of numerical integration, generalizes Gaussian quadrature, and studies node placement and convergence of the method.

Findings

Gaussian quadrature is a special case of the proposed matrix method

The method provides a discrete spectral approximation of the multiplication operator

Convergence and node placement are systematically analyzed

Abstract

In this article, numerical integration is formulated as evaluation of a matrix function of a matrix that is obtained as a projection of the multiplication operator on a finite-dimensional basis. The idea is to approximate the continuous spectral representation of a multiplication operator on a Hilbert space with a discrete spectral representation of a Hermitian matrix. The Gaussian quadrature is shown to be a special case of the new method. The placement of the nodes of numerical integration and convergence of the new method are studied.