hw9_solutions - Math 471 Fall 2006 Homework 9 Solution...

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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 θ to obtain - 2 Z π/ 2 - π/ 2 cos( ) · cos( ) d θ. Apply the trig identity 2 · cos( ) · cos( ) = cos(( m + n ) θ ) - cos(( m - n ) θ ) Complete the integration with elementary techniques.
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2. (Convergence of functions) Recall that the n th order set of Chebyshev nodes consists of the points x j = cos((2 n - j ) π/ 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 ). -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x y Chebyshev interpolation, n=32 Function Interpolant -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 3.5 4 x y Chebyshev interpolation, n=32 Function Interpolant The interpolant of the sign function is converging pointwise (but not uniformly). The
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