A constructive proof of Simpson's Rule

TL;DR

This paper provides a constructive proof of Simpson's Rule and related numerical analysis results, avoiding classical non-constructive theorems by using alternative ideas from Ampère and Genocchi.

Contribution

It introduces a constructive approach to prove the error term for Simpson's Rule, replacing classical theorems with constructive methods.

Findings

Constructive proof of Simpson's Rule error term

Avoids classical Rolle's theorem in proofs

Uses ideas from Ampère and Genocchi

Abstract

For most purposes, one can replace the use of Rolle's theorem and the mean value theorem, which are not constructively valid, by the law of bounded change. The proof of two basic results in numerical analysis, the error term for Lagrange interpolation and Simpson's rule, however seem to require the full strength of the classical Rolle's Theorem. The goal of this note is to justify these two results constructively, using ideas going back to Amp\`ere and Genocchi.