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Homework7

# Homework7 - Department of Mathematics University of Houston...

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Department of Mathematics University of Houston Numerical Analysis I Dr. Ronald H.W. Hoppe Numerical Mathematics I (7. Homework Assignment) Exercise 25 ( 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 Δ 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 , 0 i n . (i) Show that for x [0 , h ] there holds | f ( x ) - s ( x ) | ≤ 3 x h ( h Z 0 | f 00 ( t ) | 2 dt ) 1 / 2 . [Hint: Use the integral representation of the remainder in the Taylor expansion, the Cauchy Schwarz inequality, and the fact that for a continuously differentiable function Φ there holds Φ( x ) - Φ(0) = x R 0 Φ 0 ( t ) dt .]

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Homework7 - Department of Mathematics University of Houston...

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