This equation may be written in terms of first and second characteristic polynomial as

(8.34)

where

The polynomial is frequently referred to as the characteristic polynomial of the method. However, we shall call the stability polynomial of the method defined by and .

Definition: The linear multistep method (8.26) is said to be absolutely stable for a given if, for that, all the roots of satisfy , and to be absolutely unstable for thatotherwise. An interval of the real line is said to be an interval of absolute stability if the method is absolutely stable for all . If the method is absolutely unstable for all it is said to have no interval of absolute stability.

Definition: The linear multistep method (8.26) is said to be relatively stable for a given if for that, the roots of satisfy and to be relatively unstable otherwise. An interval of the real line is said to be an interval of relative stability if the method is relatively stable for all .