Predictor–corrector method: Difference between revisions

Revision as of 05:49, 16 November 2011

Revision as of 04:45, 17 November 2011

Line 19:

Next, improve the initial guess through iteration of the trapezoidal rule. This iteration process normally converges quickly.

: <math>

: <math>\tilde{y}_{[1]} = y_i + \frac{h}{2}(f(t_i, y_i) + f(t_{i+1},\tilde{y}_{[0]})).</math>

\begin{align}

: <math>\tilde{y}_{[21]} & = y_i + \frac{h}{2}(f(t_i, y_i) + f(t_{i+1},\tilde{y}_{[10]})).</math> \\

: <math>\tilde{y}_{[12]} & = y_i + \frac{h}{2}(f(t_i, y_i) + f(t_{i+1},\tilde{y}_{[01]})).</math> \\

...

& {}\ \vdots \\

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

\end{align}

</math>

This iteration process is repeated until some fixed value ''n'' or until the guesses converge to within some error tolerance ''e'' :