Lecture 30: Absolute Stability and Relative Stability
O() approximation to roots of Stability Polynomial:
From
We see that when the roots coincide with the zeros of the first characteristic polynomial , which, by zero-stability, all lie in or the unit circle. Consistency and zero stability imply that has a simple zero at +1; we have labeled this zero . Let be the root of (8.34) which tends to as . We shall now show that
as
where is the order of the linear multistep method. By definition of order, for any sufficiently differentiable function .
) approximation to roots of Stability Polynomial: 
the roots 
coincide with the zeros 
of the first characteristic polynomial 
, which, by zero-stability, all lie in or the unit circle. Consistency and zero stability imply that 
. Let 
be the root of (8.34) which tends to 
as 
. We shall now show that 
as 
is the order of the linear multistep method. By definition of order, 
for any sufficiently differentiable function 
. 
