By E. Gekeler
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Extra info for Discretization Methods for Stable Initial Value Problems
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K-1. > O, Proof. I f n E S ~ C then a l l roots of ~(z,q) l i e in the l e f t (a) i = a(a + 1 ) ' " ( a + i - I), h a l f - p l a n e , Rez < O. Let i E IN, then by a repeated a p p l i c a t i o n of Theorem ( A . I . 4 6 ) we f i n d t h a t a l l roots z* of ~l~Iz ~Z 1 k ~ = (1)isi(q) ~q) + ( 2 ) i s i + 1 ( n ) z + . . + (k - i + 1)iSk(~)z k-i s a t i s f y Rez* < O, too. Therefore s i ( q ) m O, i = O , . . , k , and the sum of the reciprocals of a l l roots z* has negative real part.
18) Lemma. ) If a convergent linear method ( 1 . 1 . 17)(i) then it is Ao-stable hence stiffly stable. Proof. We have to show t h a t (-b, O) c ~ and reconsider the polynomial ~(z,n) = r ( z ) - ns(z) introduced above. As the method is convergent and s t i f f l y stable r ( z ) and s(z) have only roots with Rez ~ O, and ak_ I = 2b k. For an n* E (-b, 0), ~ ( z , n * ) i s a poly- nomial with p o s i t i v e c o e f f i c i e n t s and we have to show that i t s roots l i e in the l e f t h a l f - p l a n e , Rez < O.
K} is the region of relative stability. 2g Notice t h a t l~1(n) I is not n e c e s s a r i l y bounded by one i n ~ , h e n c e r e l a t i v e s t a b i l i t y deals also with unstable d i f f e r e n t i a l equations. Obviously, a necessary and s u f f i c i e n t c o n d i t i o n f o r a c o n s i s t e n t method ( 1 . 1 . 3 ) to be r e l a t i v e l y hood o f n = 0 is t h a t i t stable in a ( f u l l ) neighbor- is ' s t r o n g l y D-stable' in n = 0 which means t h a t a l l roots of ~ ( ~ , 0 ) / ( ~ - I) = po(~)/(~ - I) are less than one in absolute value.