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We show that if a numerical method is posed as a sequence of operators actingon data and depending on a parameter, typically a measure of the size ofdiscretization, then consistency, convergence and stability can be related by aLax-Richtmyer type equivalence theorem -- a consistent method is convergent ifand only if it is stable. We define consistency as convergence on a densesubspace and stability as discrete well-posedness. In some applicationsconvergence is harder to prove than consistency or stability since convergencerequires knowledge of the solution. An equivalence theorem can be useful insuch settings. We give concrete instances of equivalence theorems forpolynomial interpolation, numerical differentiation, numerical integrationusing quadrature rules and Monte Carlo integration.

Equivalence Theorems in Numerical Analysis : Integration, Differentiation and Interpolation