Definition:

The linear multistep method defined by (8.1) is called Convergent , if the following statement is true for all functions satisfying the existence and uniqueness conditions and all values of If denotes the solution of the initial value problem

then

(8.9)

holds for all and all solutions of the difference equation (8.1) having starting values satisfying

(8.10)

It should be noted that this definition requires that condition (8.9) be satisfied not only for the sequence defined with the exact starting values- for these (8.10) is certainly satisfied- but also for all sequences whose starting values tend to the right value as . This more stringent condition is imposed because in practice it is almost never possible to start a computation with mathematically exact values.