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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
2 x2 ; 1
3 x3 ; 35x
4 x4 ; 6x2 + 35
7 5 x5 ; 109x3 + 5x
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
M ( ) q 1d = 0
q = 1 2 : : : r:
; 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
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.
- Spring '14