Talk:Predictor–corrector method

Proposed example

Example of a trapezoidal rule predictor-corrector method

$y'=f(t,y),\quad y(t_{0})=y_{0}.$

first calculate an initial guess value via Euler: ${\tilde {y}}_{g}$

${\tilde {y}}_{g}=y_{i}+hf(t_{i},y_{i})$

next improve the initial guess through iteration of the trapezoid rule:

${\tilde {y}}_{g+1}=y_{i}+{\frac {h}{2}}(f(t_{i},y_{i})+f(t_{i+1},{\tilde {y}}_{g})).$

${\tilde {y}}_{g+2}=y_{i}+{\frac {h}{2}}(f(t_{i},y_{i})+f(t_{i+1},{\tilde {y}}_{g+1})).$

...

${\tilde {y}}_{g+n}=y_{i}+{\frac {h}{2}}(f(t_{i},y_{i})+f(t_{i+1},{\tilde {y}}_{g+n-1})).$

until the guesses converge to within some error tolerance e:

$|{\tilde {y}}_{g+n}-{\tilde {y}}_{g+n-1}|<=e$

Once convergence is reached, then use the final guess as the next step:

$y_{i+1}={\tilde {y}}_{g+n}.$

If the guesses don't converge within some number of steps, such as $n=16$ reduce h and repeat the step. To optimize this, if the steps converge too soon, such as 4 steps, then increase h. If I remember correctly, the iterative process converges quadratically.

Jeffareid (talk) 04:14, 21 July 2009 (UTC)