124 thus the global and pointwise errors are both oh2

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Unformatted text preview: x1 < < x = bg converges to a higher order than it does globally. N Now, let us resume the search for special collocation points. Consider the case when = x , j = 0 1 : : : N , so that g(x x) is smooth and expand it in a Taylor's series on the subinterval (x 1 x ) to obtain g(x x) = g(x x 1 2 ) + g (x x 1 2 )(x ; x 1 2 ) + 1 g (x )(x ; x 1 2 )2 2 2 (x 1 x ): j j i; j j i i; = x j i; = i; = xx i j i; i i; = i Substitute this expansion into (9.1.18a) to obtain e (x ) = i Z xi j ;1 xi (x ; 1)(x ; 2) g(x x 1 2 ) + g (x x 1 2 )(x ; x 1 2 )+ i i j i; = x j i; = 1 g (x )(x ; x )2]dx: 12 2 The choice of 1 and 2 is now clear. We should select them so that xx i i Z xi Z xi xi ;1 i i; = (9.1.25) (x ; 1)(x ; 2)dx = 0 (9.1.26a) (x ; 1)(x ; 2)(x ; x 1 2 )dx = 0: (9.1.26b) xi and j i; = ;1 i i i i 9 i; = Assuming that this were possible, for the moment, then (9.1.25) would become e (x ) = i Z 1 (x ; )(x ; )g (x 1 2 2 i;1 xi i j i xx j i x We bound this as 1Z 2 je (x )j i j xi )(x ; x 1 2 )2 ]dx: jx ; 1jjx ; 2jjg (x i ;1 xi i xx j i; = i )jjx ; x 1 2 j2dx i; = or je (x )j 1 h5kg (x )k 2 i j i xx j (9.1.27...
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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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