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Equivalence Theorems in Numerical Analysis : Integration, Differentiation and Interpolation
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.Comment: 18 page
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