Adams methods

Adams methods are a subset of the general family of multistep methods used for the numerical integration of initial value problems based on odes. Multistep methods benefit from the fact that the computation has been going on for a while 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,

:<math>
y^\prime = f(t,y)
</math>


:<math>
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
</math>

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