Unformatted text preview: g theorem. Theorem 3.3.3. Methods that lead to a diagonal or one of the rst two subdiagonals
of the Pade table for ez are Astable. 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 lefthalf of the h plane hence, they are
Astable.
2. The coe cients of the highestorder 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 Astable (Table 3.3.1).
3. For the subdiagonal (1,0) and (2,1) Pade approximations, jR(z)j ! 0 as jzj ! 1
and these methods will also be Astable.
It is quite di cult to nd highorder Astable methods. Implicit RungeKutta 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 Lstable if it is Astable and if jR(z)j ! 0 as
jzj ! 1.
The backward Euler method and,...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, yn, Tn, Numerical ordinary differential equations

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