Towards non-linear quadrature formulae

TL;DR

This paper develops nonlinear quadrature formulae that exactly integrate certain function families, generalizing Newton-Cotes methods and providing explicit error bounds.

Contribution

It introduces a framework for nonlinear quadrature formulae, extending classical methods and deriving explicit error bounds for a broad class of functions.

Findings

Explicit nonlinear quadrature formulae can be constructed for many functions.

These formulae match Newton-Cotes accuracy when linear, with explicit error bounds.

Nonlinear generalizations include and extend classical quadrature methods.

Abstract

Prompted by an observation about the integral of exponential functions of the form $f(x)=\lambda e^{\alpha x}$, we investigate the possibility to exactly integrate families of functions generated from a given function by scaling or by affine transformations of the argument using nonlinear generalizations of quadrature formulae. The main result of this paper is that such formulae can be explicitly constructed for a wide class of functions, and have the same accuracy as Newton-Cotes formulae based on the same nodes, with the latter emerging as the linear case of our general formalism. We also derive explicit bounds on the error of the nonlinear quadrature formulae, which in the linear case devolve into the well-known bounds for Newton-Cotes formulae.