Editing Rate of convergence (section)

== Non-asymptotic rates of convergence ==
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Non-asymptotic rates of convergence do not have the common, standard definitions that asymptotic rates of convergence have. Among formal techniques, [[Lyapunov theory]] is one of the most powerful and widely applied frameworks for characterizing and analyzing non-asymptotic convergence behavior.

For [[Iterative method|iterative methods]], one common practical approach is to discuss these rates in terms of the number of iterates or the [[CPU time|computer time]] required to reach close [[Neighbourhood (mathematics)|neighborhoods]] of a limit from starting points far from the limit. The non-asymptotic rate is then an inverse of that number of iterates or computer time. In practical applications, an iterative method that required fewer steps or less computer time than another to reach target accuracy will be said to have converged faster than the other, even if its asymptotic convergence is slower. These rates will generally be different for different starting points and different error thresholds for defining the neighborhoods. It is most common to discuss summaries of [[statistical distributions]] of these single point rates corresponding to distributions of possible starting points, such as the "average non-asymptotic rate," the "median non-asymptotic rate," or the "worst-case non-asymptotic rate" for some method applied to some problem with some fixed error threshold. These ensembles of starting points can be chosen according to parameters like initial distance from the eventual limit in order to define quantities like "average non-asymptotic rate of convergence from a given distance."

For [[Discretization|discretized approximation]] methods, similar approaches can be used with a discretization scale parameter such as an inverse of a number of [[Regular grid|grid]] or [[Polygon mesh|mesh]] points or a [[Fourier series]] [[cutoff frequency]] playing the role of inverse iterate number, though it is not especially common. For any problem, there is a greatest discretization scale parameter compatible with a desired accuracy of approximation, and it may not be as small as required for the asymptotic rate and order of convergence to provide accurate estimates of the error. In practical applications, when one discretization method gives a desired accuracy with a larger discretization scale parameter than another it will often be said to converge faster than the other, even if its eventual asymptotic convergence is slower.