2 0 1 3320a where ms s y i1 ci 3320b di

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Unformatted text preview: ly, the quadrature rule can be designed to integrate polynomials exactly to as high a degree as possible. The actual ^ series expansion is complicated by the fact that F (s) is evaluated at ( ) in (3.3.21b). Isaacson and Keller 19] provide additional details on this matter however, we'll sidestep ^ the subtleties by assuming that all derivatives of ( ) are bounded so that F (s) has an expansion in powers of of the form ^ F (s)( ) = 0 + 1 +:::+ 33 r 1 ; r ; 1 + O( r ): d Pd(x) 01 1x 2 x2 ; 1 3 3 x3 ; 35x 3 4 x4 ; 6x2 + 35 7 5 x5 ; 109x3 + 5x 21 Table 3.3.2: Legendre polynomials Pd(x) of degree d 2 0 5] on ;1 x 1. The rst r terms of this series will be annihilated by (3.3.21b) if Ms ( ) is orthogonal to polynomials of degree r ; 1, i.e., if Z1 M ( ) q 1d = 0 q = 1 2 : : : r: (3.3.22) ; 0 Under these conditions, were we to transform the integrals in (3.3.21) and (3.3.22) back to t dependence using (3.3.13c), we would obtain the error of (3.3.18b) with p = s + r. With the s coe cients ci, i = 1 2 : : : s, we would expect the maximum value of r to be s. According to Theorem 3.3.6, this choice would lead to a collocation method of order 2s, i.e., a method having p = r + s = 2s and an O(h2s+1) local error. These are Butcher's maximal order formulas (Theorem 3.3.1) corresponding to the diagonal Pade approxi...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.

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