(Difference between revisions)
 Revision as of 03:23, 10 December 2005 (view source)Ted (Talk | contribs)m (John Couch Adams)← Older edit Revision as of 09:11, 10 December 2005 (view source)Praveen (Talk | contribs) (corrected equation)Newer edit → Line 9: Line 9: :$:[itex] - y(t_{N+1}) = y(t_{n}) + \int_{t_n}^{t_{n+1}} y^\prime (t) dt = \int_{t_n}^{t_{n+1}} f(t,y(t)) dt + y(t_{N+1}) = y(t_{n}) + \int_{t_n}^{t_{n+1}} y^\prime (t) dt = y(t_{n}) + \int_{t_n}^{t_{n+1}} f(t,y(t)) dt$ [/itex] Since $f$ is unknown in the interval $t_n$ to $t_{n+1}$ it is approximated by an interpolating [[polynomial]] $p(t)$ using the previously computed steps $t_{n},t_{n-1},t_{n-2} ...$ and the current step at $t_{n+1}$ if an implicit method is desired. Since $f$ is unknown in the interval $t_n$ to $t_{n+1}$ it is approximated by an interpolating [[polynomial]] $p(t)$ using the previously computed steps $t_{n},t_{n-1},t_{n-2} ...$ and the current step at $t_{n+1}$ if an implicit method is desired.

## Revision as of 09:11, 10 December 2005

Adams' methods are a subset of the family of multistep methods used for the numerical integration of initial value problems in ODEs. Multistep methods benefit from the fact that the computations have been going on for some time, and use previously computed values of the solution (BDF methods), or the right hand side (Adams' methods), to approximate the solution at the next step.

Adams' methods begin by the integral approach,

$y^\prime = f(t,y)$

$y(t_{N+1}) = y(t_{n}) + \int_{t_n}^{t_{n+1}} y^\prime (t) dt = y(t_{n}) + \int_{t_n}^{t_{n+1}} f(t,y(t)) dt$

Since $f$ is unknown in the interval $t_n$ to $t_{n+1}$ it is approximated by an interpolating polynomial $p(t)$ using the previously computed steps $t_{n},t_{n-1},t_{n-2} ...$ and the current step at $t_{n+1}$ if an implicit method is desired.