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Unformatted text preview: e collocation points ci, i = 1 2 : : : s, determined, the coe cients aij and bj ,
i j = 1 2 : : : s, may be determined from (3.3.15a,b). These maximal order collocation
formulas are Astable since they correspond to diagonal Pade approximations (Theorem
3.3.3).
We may not want to impose the maximal order conditions to obtain, e.g., better
stability and computational properties. With Radau quadrature, we x one of the coefcients at an endpoint thus, we set either c1 = 0 or cs = 1. The choice c1 = 0 leads to
methods with bounded regions of absolute stability. Thus, the methods of choice have
cs = 1. They correspond to the subdiagonal Pade approximations and are, hence, Aand Lstable (Theorem 3.3.3). They have orders of p = 2s ; 1 17], Section IV.5. Such
excellent stability and accuracy properties makes these methods very popular for solving
sti systems.
The Radau polynomial of degree s on ;1 x 1 is
Rs(x) = Ps(x) ; 2s s 1 Ps 1(x):
;
The roots of Rs transformed to 0 1] (using = (1 + x)=2) are the ci, i + 1 2 : : :...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

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