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32 legendre polynomials pdx of degree d 2 0 5 on 1 x

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Unformatted text preview: mations. The maximum-order coe cients identi ed above are the roots of the s th-degree 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 maximal-order two-stage 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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