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Unformatted text preview: 0 10 9 0:13 10 10 0:27 10 12 0:60 10 14 0:13 10 14 Table 9.2.2: Maximum pointwise errors in the solution of Example 9.2.1 using collocation at two Gauss-Legendre points, three Lobatto points, and three Gauss-Legendre points. ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; three-point Gauss-Legendre collocation at r 1 3 1 = 2 (1 ; 5 ) r 1 3 3 = 2 (1 + 5 ): 2 = 1=2 The Gauss-Legendre methods do not need function evaluations at the endpoints of subintervals and, thus, the singular coe cient in the di erential equation at x = 0 poses no problem. However, the Lobatto method must be modi ed to account for the singularity. Using L'Hopital's rule, lim y (x) = y (0): 0 0 x! 00 x Using this result in the di erential equation yields y (0) = 1=2 thus, 00 1 lim y (x) = 2 : 0x 0 x! The maximum pointwise errors in y for the three methods are shown in Table 9.2.2. The two-point Gauss-Legendre, three-point Lobatto, and three-point Gauss-Legendre are converging at their expected rates of O(N 4), O(N 4), and O(N 6), respecti...
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