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Predictorcorrector method

In mathematics, particularly numerical analysis, a predictorcorrector 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.

Predictorcorrector methods for solving ODEs

When considering the numerical solution of ordinary differential equations (ODEs), a predictorcorrector method

typically uses an explicit method for the predictor step and an implicit method for the corrector step.

Example: Euler method with the trapezoidal rule

A simple predictorcorrector method (known as Heun's method) can be constructed from the Euler method (an

explicit method) and the trapezoidal rule (an implicit method).

Consider the differential equation

and denote the step size by .

First, the predictor step: starting from the current value , calculate an initial guess value via the Euler

method,

Next, the corrector step: improve the initial guess using trapezoidal rule,

That value is used as the next step.

PEC mode and PECE mode

There are different variants of a predictorcorrector method, depending on how often the corrector method is

applied. The PredictEvaluateCorrectEvaluate (PECE) mode refers to the variant in the above example:

It is also possible to evaluate the function f only once per step by using the method in Predict EvaluateCorrect

(PEC) mode:

Additionally, the corrector step can be repeated in the hope that this achieves an even better approximation to the

true solution. If the corrector method is run twice, this yields the PECECE mode:

The PECEC mode has one fewer function evaluation. More generally, if the corrector is run k times, the method is in

P(EC)k

or P(EC)kE mode. If the corrector method is iterated until it converges, this could be called PE(CE)

.

http://en.wikipedia.org/w/index.php?title=Trapezoidal_rule_%28differential_equations%29http://en.wikipedia.org/w/index.php?title=Euler_methodhttp://en.wikipedia.org/w/index.php?title=Heun%27s_methodhttp://en.wikipedia.org/w/index.php?title=Explicit_and_implicit_methodshttp://en.wikipedia.org/w/index.php?title=Numerical_methods_for_ordinary_differential_equationshttp://en.wikipedia.org/w/index.php?title=Algorithmhttp://en.wikipedia.org/w/index.php?title=Numerical_analysishttp://en.wikipedia.org/w/index.php?title=Mathematics

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Predictorcorrector method 2

Notes

References

Butcher, John C. (2003),Numerical Methods for Ordinary Differential Equations, New York: John Wiley &

Sons, ISBN 978-0-471-96758-3.

Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 17.6. Multistep, Multivalue, and

Predictor-Corrector Methods" (http://apps.nrbook.com/empanel/index.html#pg=942).Numerical Recipes:

The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.

External links

Weisstein, Eric W., " Methods.html Predictor-Corrector Methods (http://mathworld.wolfram.com/

PredictorCorrector)",MathWorld.

Predictorcorrector methods (http://www.fisica.uniud. it/~ercolessi/md/md/node22.html) for differential

equations

http://www.fisica.uniud.it/~ercolessi/md/md/node22.htmlhttp://en.wikipedia.org/w/index.php?title=MathWorldhttp://mathworld.wolfram.com/Predictor%E5%83%AFrrectorhttp://mathworld.wolfram.com/Predictor%E5%83%AFrrectorhttp://en.wikipedia.org/w/index.php?title=Eric_W._Weissteinhttp://en.wikipedia.org/w/index.php?title=Special:BookSources/978-0-521-88068-8http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://apps.nrbook.com/empanel/index.html#pg=942http://en.wikipedia.org/w/index.php?title=Special:BookSources/978-0-471-96758-3http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=John_Wiley_%26_Sonshttp://en.wikipedia.org/w/index.php?title=John_Wiley_%26_Sonshttp://en.wikipedia.org/w/index.php?title=John_C._Butcher

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Article Sources and Contributors 3

Article Sources and ContributorsPredictorcorrector method Source: http://en.wikipedia.org/w/index.php?oldid=580394365 Contributors: Beetstra, Felix116, Jeffareid, Jitse Niesen, Katieh5584, Lunch, Michael Hardy,

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