Adams Open Formulas

The Adams Open Formulas are a class of muliistep formulas such that the first-order formula reproduces the Euler formula. The second-order Adams open formula is given by... 

This formula and the higher-order formulas are not self starting since they require f,, f etc. The common practice is to employ a Runge-Kutta formula of the same order to compute the first term(s) of y,. The general Adams open formula may be written as... 

A combination of open- and closed-type formulas is referred to as the predictor-corrector method. First the open equation (the predictor) is used to estimate a value for y,, this value is then inserted into the right side of the corrector equation (the closed formula) and iterated to improve the accuracy of y. The predictor-corrector sets may be the low-order modified (open) and improved (closed) Euler equations, the Adams open and closed formulas, or the Milne method, which gives the following system... 

---