Lecture 30: Absolute Stability and Relative Stability
Choosing , we obtain
(8.35)
or
on dividing by , we obtain
Since the roots of (8.34) are , we may write
and set to obtain
(8.36)
As .
Thus the first factor on the left hand side of (8.36) tends to zero as no other factor may do so, since, by zero stability. is a simple zero of . Hence we may conclude that exp establishing the result.
Remark: It now follows that for sufficiently small whenever . Thus every consistent zero stable method is absolutely unstable for small positive .
, we obtain 
, we obtain 
, we may write 
to obtain 
. 
no other factor may do so, since, by zero stability. 
is a simple zero of 
. Hence we may conclude that exp 
establishing the result. 
whenever 
. Thus every consistent zero stable method is absolutely unstable for small positive 
. 
