The constants ai can be determined by assuming that the linear expression is exact for polynomials in x of degree k - 1 or less, in which case the order of the Adams-Bashforth method is k. The major advantage of the Adams-Bashforth method over the Runge-Kutta methods is that only one evaluation of the integrand f(x,y) is performed for each step.
The (k-1)-step Adams-Moulton method is an implicit linear multistep method that iteratively approximates the solution, y(x) at x = x0+kh, of the initial value problem by
The constants bi can be determined by assuming that the linear expression is exact for polynomials in x of degree k - 1 or less, in which case the order of the Adams-Moulton method is k.
In order to start the Adams-Moulton iterative method, the Adam-Bashforth method is used to generate an initial estimate for yk. Applications of the left-hand side Adams-Moulton formula is then used to generate successive estimates for yk. The process is converges providing that the step size h is chosen so that |h f,y(x,y) | < 1 over the region of interest, where f,y denotes the partial derivative of f with respect to y.
Usually a k-step Adams-Bashforth method is paired with a (k-1)-step Adams-Moulton method but this is not necessary it is possible to pair any k-step Adams-Bashforth method with any l-step Adams-Moulton method.