## Introduction

Adams' methods are a subset of the family of multi-step methods used for the numerical integration of initial value problems in Ordinary Differential Equations (ODE's). Multi-step 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.

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