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Unformatted text preview: Homework #7 Due Thursday, 03/06 1. (a) What system of nonlinear equations would you solve in order to find w 1 ,w 2 ,w 3 ,x 1 ,x 2 and x 3 such that the integration formula I 3 ( f ) = w 1 f ( x 1 ) + w 2 f ( x 2 ) + w 3 f ( x 3 ) is exactly equal to R 1- 1 f ( x ) dx for every polynomial f ( x ) of degree ≤ 5. (b) Check that the nodes and weights from the table I gave you in class do satisfy this system. 2. Gaussian Quadrature VS Simpson’s rule Let f ( x ) = 1 x + 1 . 5 (a) Sketch or plot with matlab f ( x ) on [- 1 , 1]. (b) Compute R 1- 1 f ( x ) dx by hand. (c) Write a code to compute R 1- 1 f ( x ) dx using Gaussian Quadrature with 7 nodes points x 1 ,x 2 ,...,x 7 (hint: store the nodes and weights of the table in two vectors “node(i)” and “w(i)”). What is the error? (d) Write a code to compute R 1- 1 f ( x ) dx using Simpson rule with nodes points x ,x 1 ,...,x 6 (so that you have a total of 7 nodes points, as in question (b)). What is the error?...
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