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Unformatted text preview: Math 471, Fall 2006 Homework 9 Solution Assigned: Friday, November 9, 2007. Due: Friday, November 19, 2007 . 1. (Chebyshev polynomials) Do problem #3 on page 385 of Bradie. Let T n ( x ) = cos( n cos- 1 x ). We must compute the integral Z 1- 1 T n ( x ) T m ( x ) 1- x 2 dx . First, substitute x = cos to obtain- 2 Z / 2- / 2 cos( n ) cos( m )d . Apply the trig identity 2 cos( n ) cos( m ) = cos(( m + n ) )- cos(( m- n ) ) Complete the integration with elementary techniques. 2. (Convergence of functions) Recall that the n th order set of Chebyshev nodes consists of the points x j = cos((2 n- j ) / 2 n ) for j = 1 ,...,n . (a) Write a Matlab program that computes the n th degree polynomial interpolant p n of an input function f at the Chebyshev nodes and plots p n on the interval [- 2 , 2]. (b) Apply your program to the functions f ( x ) = abs( x ) and g ( x ) = sign( x )....
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