Homework10 - Department of Mathematics University of...

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Unformatted text preview: Department of Mathematics University of Houston Numerical Analysis I Dr. Ronald H.W. Hoppe Numerical Mathematics I (10. Homework Assignment) Exercise 34 ( L 2-estimate for piecewise linear interpolation ) The interpolation error estimates presented in class provide upper bounds for the pointwise error in polynomial interpolation. Often, it is of interest to have estimates of the error in specific norms as, for instance, the L 2-norm k f k L 2 ([ a,b ]) := ( b Z a | f ( x ) | 2 dx ) 1 / 2 . Let h := { x i := ih } n i =0 , h := b/n, n N be a uniform partition of the intervale [0 ,b ]( b > 0). For f C 2 ([0 ,b ]) we want to estimate the L 2-norm of the interpo- lation error f- s , where s represents the polygon which interpolates f at the nodes x i , i n . (i) Show that for x [0 ,h ] there holds | f ( x )- s ( x ) | 3 x h ( h Z | f 00 ( t ) | 2 dt ) 1 / 2 ....
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Homework10 - Department of Mathematics University of...

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