Equivalence Theorems in Numerical Analysis : Integration, Differentiation and Interpolation

TL;DR

This paper establishes an equivalence theorem linking consistency, stability, and convergence for numerical methods in integration, differentiation, and interpolation, providing a unified framework to analyze and verify numerical algorithms.

Contribution

It introduces a Lax-Richtmyer type equivalence theorem for numerical analysis methods, connecting consistency, stability, and convergence in a general operator framework.

Findings

Equivalence theorem applies to polynomial interpolation, differentiation, and integration.

Consistency is defined as convergence on a dense subspace.

Stability is characterized as discrete well-posedness.

Abstract

We show that if a numerical method is posed as a sequence of operators acting on data and depending on a parameter, typically a measure of the size of discretization, then consistency, convergence and stability can be related by a Lax-Richtmyer type equivalence theorem -- a consistent method is convergent if and only if it is stable. We define consistency as convergence on a dense subspace and stability as discrete well-posedness. In some applications convergence is harder to prove than consistency or stability since convergence requires knowledge of the solution. An equivalence theorem can be useful in such settings. We give concrete instances of equivalence theorems for polynomial interpolation, numerical differentiation, numerical integration using quadrature rules and Monte Carlo integration.