Adams-Bashforth and Adams-Moulton MethodsGiven an initial value problem: y ' = f(x,y), y(x0) = y0 together with additional starting values y1 = y(x0 + h), . . . , yk-1 = y(x0 + (k-1) h) the k-step Adams-Bashforth method is an explicit linear multistep method that approximates the solution, y(x) at x = x0+kh, of the initial value problem by
yk = yk - 1 + h * ( a0 f(xk - 1,yk - 1) + a1 f(xk - 2,yk - 2) + . . . + ak - 1 f(x0,y0) )
where a0, a1, . . . , ak - 1 are constants.
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
yk = yk - 1 + h * ( b0 f(xk,yk) + b1 f(xk - 1,yk - 1) + . . . + bk - 1 f(x1,y1) )
where b1, . . . , bk - 1 are constants.
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.
The Adams-Bashforth method forms the predictor and Adams-Moulton method forms the corrector for a predictor-corrector multistep procedure for approximating the solution of a differential equation given historical values.
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.