## Adams-Bashforth and Adams-Moulton Methods

Given an initial value problem:*y ' = f(x,y), y(x*together with additional starting values

_{0}) = y_{0}*y*the

_{1}= y(x_{0}+ h), . . . , y_{k-1}= y(x_{0}+ (k-1) h)*k*-step Adams-Bashforth method is an explicit linear multistep method that approximates the solution,

*y(x)*at

*x = x*, of the initial value problem by

_{0}+kh*y*

_{k}= y_{k - 1}+ h * ( a_{0}f(x_{k - 1},y_{k - 1}) + a_{1}f(x_{k - 2},y_{k - 2}) + . . . + a_{k - 1}f(x_{0},y_{0}) )where

*a*are constants.

_{0}, a_{1}, . . . , a_{k - 1}The constants

*a*can be determined by assuming that the linear expression is exact for polynomials in

_{i}*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 = x*, of the initial value problem by

_{0}+kh*y*

_{k}= y_{k - 1}+ h * ( b_{0}f(x_{k},y_{k}) + b_{1}f(x_{k - 1},y_{k - 1}) + . . . + b_{k - 1}f(x_{1},y_{1}) )where

*b*are constants.

_{1}, . . . , b_{k - 1}The constants

*b*can be determined by assuming that the linear expression is exact for polynomials in

_{i}*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

*y*. Applications of the left-hand side Adams-Moulton formula is then used to generate successive estimates for

_{k}*y*. The process is converges providing that the step size

_{k}*h*is chosen so that |

*h f,*| < 1 over the region of interest, where

_{y}(x,y)*f,*denotes the partial derivative of

_{y}*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.