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The maximumorder coe cients identi ed above are the roots of the s thdegree
Legendre polynomial scaled to the interval (0 1). The rst six Legendre polynomials are
listed in Table 3.3.2. Additional polynomials and their roots appear in Abromowitz and
Stegun 1], Chapter 22.
p
Example 3.3.6. According to Table 3.3.2, the roots of P2 (x) are x1 2 = 1= 3 on
;1 1]. Mapping these to 0 1] by the linear transformation = (1 + x)=2, we obtain
the collocation points for the maximalorder twostage method as
1
1
1
c2 = 2 (1 + p ):
c1 = 1 (1 ; p )
2
3
3
34 Since this is our rst experience with these techniques, let us verify our results by a direct
evaluation of (3.3.22) using (3.3.20b) thus,
Z1
Z1
( ; c1 )( ; c2)d = 0
( ; c1 )( ; c2) d = 0:
Integrating 0 0 1 ; c 1 + c2 + c c = 0
1 ; c1 + c2 + c1 c2 = 0:
12
3
2
4
3
2
These may easily be solved to con rm the collocation points obtained by using the roots
of P2 (x). In this case, we recognize c1 and c2 as the evaluation points of the HammerHollingsworth formula of Example 3.3.2.
With th...
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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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