Remark: Convergence assures that the exact solution can be approximated arbitrarily closely by making h smaller and smaller using greater precision. Stability is concerned with the effect of perturbation on the numerical solution.

Definition: A single-step method (5.1) is stable if for each differential equation satisfying a Lipschitz condition, there exist positive constants and such that the difference between two different numerical solutions and each satisfying (5.1) is such that

for all

Remark: Stability is nearly automatic for single–step methods as the following theorem shows:

Theorem: If the increment function satisfies a Lipschitz Condition

in , then the method given by (5.1) is stable.