Talk:Predictor–corrector method: Difference between revisions

Revision as of 22:27, 22 July 2009

Latest revision as of 03:43, 9 March 2024

(9 intermediate revisions by 5 users not shown)

Line 1:

{{WikiProject banner shell|class=Start|

{{maths rating}}

{{WikiProject Mathematics|priority=Mid}}

}}

__TOC__

== Proposed example ==

Example of a trapezoidal predictor-corrector method.

In this example ''h'' = <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>

first calculate an initial guess value <math>\tilde{y}_{g}</math> via Euler ( note in this example ''h'' = <math>\Delta{t} </math>, <math> t_{i+1} = t_{i} + \Delta{t} = t_{i} + h </math> ) :

: <math>\tilde{y}_{g} = y_i + h f(t_i,y_i)</math>

Line 20 ⟶ 24:

: <math>\tilde{y}_{g+n} = y_i + \frac{h}{2}(f(t_i, y_i) + f(t_{i+1},\tilde{y}_{g+n-1})).</math>

until some fixed value ''n'' or until the guesses converge to within some error tolerance ''e'' :

: <math> | \tilde{y}_{g+n} - \tilde{y}_{g+n-1} | <= e </math>

Line 30 ⟶ 34:

If I remember correctly, the iterative process converges quadratically. 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. Also see [[stiff equation]]

[[User:Jeffareid|Jeffareid]] ([[User talk:Jeffareid|talk]]) 2202:2742, 2225 July 2009 (UTC)

:The relation to the https://de.wikipedia.org/wiki/Picard-Iteration might be a worthwhile refernce. I am dubious on the quadratic convergence claim as it looks more like a type of gradient descent to me. [[Special:Contributions/2001:638:904:FFC8:3433:CB4E:3261:66DB|2001:638:904:FFC8:3433:CB4E:3261:66DB]] ([[User talk:2001:638:904:FFC8:3433:CB4E:3261:66DB|talk]]) 23:32, 11 March 2023 (UTC)