Necessary and Sufficient condition for Convergence

We shall now study the necessary and sufficient condition for convergence of a general one-step method which is given in the following theorem:

Theorem: If is continuous in for and all , and if it satisfies a Lipschitz condition in in that region, a necessary and sufficient condition for convergence is that

(5.4)

Remark: The equation (5.4) is called the condition of consistency. Since, by suitable choice of initial conditions, can take on any value for a given , the equation (5.4) will hold for any in the form

Proof: Let

Since satisfies the conditions of existence and uniqueness of the solution, the IVP

(5.5)

has a unique differentiable solution. We shall show that the numerical solution given by (5.1) converges to , and hence is a necessary and sufficient condition.