Towards non-linear quadrature formulae
This work provides a theoretical generalization of quadrature methods for numerical integration, offering a new formalism that may benefit applied mathematics and computational science, though the practical impact is incremental.
The paper introduces non-linear quadrature formulae that exactly integrate families of functions generated by scaling or affine transformations, achieving accuracy comparable to Newton-Cotes formulae with explicit error bounds.
Prompted by an observation about the integral of exponential functions of the form $f(x)=λe^{αx}$, 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.