Introduction:

Most of the methods discussed earlier (particularly for 1^st order equation) can be considered as a special case of the formula

(8.1)

where is a fixed integer,

and where and

denote real constants which do not depend on . We shall always assume that Equation (8.1) is said to define the General linear K-step method. The method is called linear because the values enter linearly in (8.1); it is not assumed that f is a linear function of y.

One way in which we were able to derive the coefficients of Adams method was by requiring that they are exact for polynomials of degree . There are unknowns in (8.1). There is an arbitrary normalizing factor so we set , leaving unknowns.

Consequently, we expect to be able to choose the and so that this method is exact for polynomials of degree upto . This is possible. However, it has been observed that such methods are never useful for , and only marginally useful when . If we were only concerned with local truncation error and the problem had well-behaved derivatives, we would be tempted to use K-step methods of maximal order . But it is known that for such methods cause the small truncation errors committed in one step to be unacceptably amplified in later steps due to instability. However, there are stable K-step methods of order (the Adams-Moulton method, for example) and order if K is even.