Predictor–corrector method: Difference between revisions

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{{Short description|Algorithms in numerical analysis}}

In [[mathematics]], particularly [[numerical analysis]], a '''predictor–corrector method''' is an [[algorithm]] that proceeds in two steps. First, the prediction step calculates a rough approximation of the desired quantity. Second, the corrector step refines the initial approximation using another means.

In [[numerical analysis]], '''predictor–corrector methods''' belong to a class of [[algorithm]]s designed to integrate ordinary differential equations{{snd}}to find an unknown function that satisfies a given differential equation. All such algorithms proceed in two steps:

__TOC__

# The initial, "prediction" step, starts from a function fitted to the function-values and derivative-values at a preceding set of points to extrapolate ("anticipate") this function's value at a subsequent, new point.

# The next, "corrector" step refines the initial approximation by using the ''predicted'' value of the function and ''another method'' to interpolate that unknown function's value at the '''same''' subsequent point.

== Predictor–corrector methods for solving ODEs ==

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: <math> \begin{align}

\tilde{y}_{i+1} &= y_i + h f(t_i,y_i), \\

\hat{y}_{i+1} &= y_i + \tfrac12 h \bigl( f(t_i, y_i) + f(t_{i+1},\tilde{y}_{i+1}) \bigr)., \\

y_{i+1} &= y_i + \tfrac12 h \bigl( f(t_i, y_i) + f(t_{i+1},\hat{y}_{i+1}) \bigr).

\end{align} </math>

The PECEC mode has one fewer function evaluation. than PECECE mode.

More generally, if the corrector is run ''k'' times, the method is in P(EC)^''k''

or P(EC)^''k''E mode. If the corrector method is iterated until it converges, this could be called PE(CE)^∞.<ref>{{harvnb|Butcher|2003|p=104}}</ref>

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== External links ==

* {{MathWorld |title=Predictor-Corrector Methods |urlname=Predictor–Corrector MethodsPredictor-CorrectorMethods}}

* [https://web.archive.org/web/20080617035745/http://www.fisica.uniud.it/~ercolessi/md/md/node22.html Predictor–corrector methods] for differential equations

{{Numerical integrators}}

{{DEFAULTSORT:Predictor-corrector method}}