Gram quadrature: numerical integration with Gram polynomials

TL;DR

This paper introduces a new numerical integration method using Gram polynomials, providing a stable, high-order quadrature formula with equidistant points and an efficient iterative algorithm.

Contribution

It develops a novel high-order quadrature formula based on Gram polynomials and presents an iterative algorithm that reduces space complexity.

Findings

Stable high-order quadrature formulas achieved with Gram polynomials

Iterative algorithm reduces space complexity significantly

Improved numerical integration accuracy with equidistant points

Abstract

The numerical integration of an analytical function $f(x)$ using a finite set of equidistant points can be performed by quadrature formulas like the Newton-Cotes. Unlike Gaussian quadrature formulas however, higher-order Newton-Cotes formulas are not stable, limiting the usable order of such formulas. Existing work showed that by the use of orthogonal polynomials, stable high-order quadrature formulas with equidistant points can be developed. We improve upon such work by making use of (orthogonal) Gram polynomials and deriving an iterative algorithm, together allowing us to reduce the space-complexity of the original algorithm significantly.