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# solhw8 - Solutions for Homework 8 Foundations of...

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Unformatted text preview: Solutions for Homework 8 Foundations of Computational Math 2 Spring 2011 Problem 8.2 Consider the two quadrature formulas I 2 ( f ) = 2 3 [2 f (- 1 / 2)- f (0) + 2 f (1 / 2)] I 4 ( f ) = 1 4 [ f (- 1) + 3 f (- 1 / 3) + 3 f (1 / 3) + f (1)] • What is the degree of exactness when I 2 ( f ) is used to approximate I ( f ) on [- 1 , 1]? • What is the degree of exactness when I 2 ( f ) is used to approximate I ( f ) on [- 1 / 2 , 1 / 2]? • What is the degree of exactness when I 4 ( f ) is used to approximate I ( f ) on [- 1 , 1]? Solution: We have I = integraldisplay 1 − 1 x k dx = braceleftBigg 0 if k is odd 2 k +1 if k is even I 2 ( f ) = braceleftBigg 4 3 bracketleftbig (- 1 / 2) k- 1 + (1 / 2) k bracketrightbig = 2 if k = 0 4 3 bracketleftbig (- 1 / 2) k + (1 / 2) k bracketrightbig if k ≥ 1 So clearly I = I 2 ( f ) = 0 if k ≥ 1 is odd. For k ≥ 2 and even we must have 3 × 2 k = 4( k + 1) which is equal for k = 2 and not equal for k = 4. Therefore the degree of exactness for= 4....
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solhw8 - Solutions for Homework 8 Foundations of...

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