Fall2011-HW11-soln - Homework 11 Solutions Math 371, Fall...

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Homework 11 Solutions Math 371, Fall 2011 (1) (Richardson Extrapolation) P. 454 #11 (a) Adding the Taylor series for f ( x 0 + h ) and f ( x 0 - h ) gives f ( x 0 + h ) + f ( x 0 - h ) = 2 f ( x 0 ) + h 2 f 00 ( x 0 ) + h 4 12 f (4) ( x 0 ) + h 8 2880 f (8) ( x 0 ) + h 8 2880 [ f (8) ( ξ ) - f (8) ( x 0 )] . Thus f 00 ( x 0 ) = f ( x 0 + h ) - 2 f ( x 0 ) + f ( x 0 - h ) h 2 - h 2 12 f (4) ( x 0 ) - h 4 360 f (6) ( x 0 ) - h 6 2880 f (8) ( x 0 ) + o ( h 6 ) . (b) h O ( h 2 ) O ( h 4 ) O ( h 6 ) 0.5 2.255252 0.25 2.062826 2.049998 0.125 2.015645 2.012500 2.010000 (2) (Newton–Cotes). Suppose that f is a function with four continuous derivatives on the interval [ a,b ]. Recall that the error bound for the composite trapezoidal rule T ( h ) with panel width h is T ( h ) - Z b a f ( x ) d x = ( b - a ) h 2 12 f 00 ( ξ ) for some ξ [ a,b ]. The error in the composite Simpson’s rule S ( h ) with panel width h is S ( h ) - Z b a f ( x ) d x = ( b - a ) h 4 180 f (4) ( ξ ) for some (different) ξ [ a,b ]. (a) Let f ( x ) = e - x sin x . For the composite trapezoidal rule and the composite Simpson’s rule, find the number of panels n required to integrate f on the interval [0 , 2 π ] with error at most 10 - 4 . Recall that h = 2 π/n . How many function evaluations are required in each case? The first five derivatives of f are 1
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2 f 0 ( x ) = e - x (cos x - sin x ) f 00 ( x ) = - 2e - x cos x f 000 ( x ) = 2e - x (sin x + cos x ) f (4) ( x ) = - 4e - x sin x f (5) ( x ) = 4e - x (sin x - cos x ) To develop the bound for the composite trapezoidal rule, we need to calculate the maximum value of | f 00 ( x ) | on the interval [0 , 2 π ]. The extremum must occur at x = 0, x = 2 π , or at a point where f 000 ( x ) = 0. A short calculation establishes that the maximum occurs when x = 0, so max x [0 , 2 π ] | f 00 ( x ) | = 2 . To ensure that the error is no more than 10 - 4 , the error bound requires 2 π h 2 12 max x [0 , 2 π ] | f 00 ( x ) | ≤ 10 - 4 . Solving this inequality for the panel width
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Fall2011-HW11-soln - Homework 11 Solutions Math 371, Fall...

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