Some Necessary Conditions

Theorem 1: A necessary condition for convergence of the linear multistep method (8.1) is that the modulus of no root of the associated polynomial exceeds one, and that the roots of modulus one be simple.

The condition thus imposed on is called the condition of zero-stability.

Proof: If the method is convergent, it is convergent for the initial value problem , whose exact solution is . For this problem (8.1) reduces to the difference equation with constant coefficients

(8.11)

If the method is convergent, then by (8.10), for any

(8.12)

for all solutions of (8.11) satisfying

(8.13)

where . Let be a root of . Then, the numbers

(8.14)