# Proof cf butcher 7 well be able to construct several

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Unformatted text preview: g theorem. Theorem 3.3.3. Methods that lead to a diagonal or one of the rst two sub-diagonals of the Pade table for ez are A-stable. Proof. The proof appears in Ehle 13]. Without introducing additional properties of Pade approximations, we'll make some observations using the results of Table 3.3.1. 1. We have shown that the regions of absolute stability of the backward Euler method and the midpoint rule include the entire left-half of the h plane hence, they are A-stable. 2. The coe cients of the highest-order terms of Ps(z) and Qs(z) are the same for diagonal Pade approximations Rss(z) hence, jRss(z)j ! 1 as jzj ! 1 and these methods are A-stable (Table 3.3.1). 3. For the sub-diagonal (1,0) and (2,1) Pade approximations, jR(z)j ! 0 as jzj ! 1 and these methods will also be A-stable. It is quite di cult to nd high-order A-stable methods. Implicit Runge-Kutta methods provide the most viable approach. Examining Table 3.3.1, we see that we can introduce another stability notion. De nition 3.3.3. A numerical method is L-stable if it is A-stable and if jR(z)j ! 0 as jzj ! 1. The backward Euler method and,...
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