Method not absolutely stable but relatively stable :

Example: Consider, for example, Simpson's rule which, being an optimal method, turns out to have no interval of absolute stability. A method is said to be an optimal method, if its order is , where K is the step number of the method. The Simpson's rule is

.

For this method, we have

And the stability polynomial is

It is easily established that the roots and of this equation are real for all values of . Using approximation of the roots, we can take

. The spurious root of is and so we write,

. Substituting this value in the stability polynomial gives . For sufficiently small , we can ignore and have and .

This gives that the method is not absolutely stable but is relatively stable for .