Predictor–corrector method: Difference between revisions

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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.

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== Example ==

In approximating the solution to a first-order [[ordinary differential equation]], suppose one knows the solution points <math>y_0</math> and <math>y_1</math> at times <math>t_0</math> and <math>t_1</math>. By fitting a cubic polynomial to the points and their derivatives (gotten through the differential equation), one can predict a point <math>\tilde{y}_2</math> by [[Extrapolation|extrapolating]] to a future time <math>t_2</math>. Using the new value <math>\tilde{y}_2</math> and its derivative there <math>\tilde{y}^'_2</math> along with the previous points and their derivatives, one can then better [[Interpolation|interpolate]] the derivative between <math>t_1</math> and <math>t_2</math> to get a better approximation <math>y_2</math>. The interpolation and subsequent integration of the differential equation constitute the corrector step.

== Euler trapezoidal example ==

Example of an Euler - trapezoidal predictor-corrector method.

In this example <math>''h</math>'' = <math>\Delta{t} </math>, <math> t_{i+1} = t_i + \Delta{t} = t_i + h </math>

: <math> y' = f(t,y), \quad y(t_0) = y_0. </math>

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Note that the overall error is unrelated to convergence in the algorithm but instead to the step size and the core method, which in this example is a trapezoidal, (linear) approximation of the actual function. The step size ''h'' ( <math>\Delta{t} </math> ) needs to be relatively small in order to get a good approximation. See also [[stiff equation]].

== See also ==

* [[Backward differentiation formula]]

* [[Beeman's algorithm]]

* [[Heun's method]]

* [[Mehrotra predictor-corrector method]]

* [[Numerical continuation]]

== External links ==

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

* [http://www.fisica.uniud.it/~ercolessi/md/md/node22.html Predictor-corrector methods] for differential equations

[[Category:Algorithms]]

[[Category:Numerical analysis]]