By E. Gekeler

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N) is not n e c e s s a r i l y the c h a r a c t e r i s t i c polynomial of a consistent method. But f o r every a l g e b r a i c polynomial ~(~,n) the shape o f the s t a b i l i t y region S near n = 0 is determined by the behavior of those roots ~i(q) which become unimodular in n = 0. I f a 'method' with the general polynomial ~(~,n) is A0-stable near q = 0 with a possible exception of the p o i n t n = 0 i t s e l f then C o r o l l a r y ( A . I . 2 1 ) implies t h a t a l l roots ~i(n) with I ~ i ( 0 ) I = I have near q = 0 the form ( A .

3) be linear, convergent, strongly D-stable in n : O, and Ao-stable. Then it is stiffly stable iff it is asymptotically A(7/2)stable. 7. 4. The f o l l o w i n g n o t a t i o n has become customary in the meanwhile here. 1) Definition. A multistep multiderivative method is Ir-stable if {in, - r < n < r} C S, 0 < r ~ ~. Recently, Jeltsch and Nevanlinna [81 , 82a, 82b] have developed an a l g e b r a i c comparison theory f o r numerical methods with respect to t h e i r s t a b i l i t y regions which allows the treatment of I r - s t a b l e methods from a r a t h e r general p o i n t of view.

O,Ok(At A ) Zi=O T Lj=1oij ~aL " Ja~ "n-k J " The Uniform Boundedness Theorem (cf. Appendix) yields by assumption ( i i i ) sent case In-IF~ (At2A2)nl : max1_-

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