Fall2011-HW10-soln - 2 3. (Piecewise Linear Interpolation)...

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Homework 10 Solutions Math 371, Fall 2011 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. 1
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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 Chebyshev interpolation, n=32 Function Interpolant The interpolant of the sign function is converging pointwise (but not uniformly). The interpolant of the absolute value function is converging uniformly (hence pointwise as well).
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Unformatted text preview: 2 3. (Piecewise Linear Interpolation) Use the error theorem on p. 390 to compute an upper bound on the pointwise error from interpolating f ( x ) = e-x 2 on the interval [-4 , 4] with piecewise linear functions at the points-4,-2, 0, 2, and 4. max x ∈ [-4 , 4] | e-x 2-s ( x ) | ≤ 1 8 h 2 max x ∈ [-4 , 4] | f 00 ( x ) | = 1 . 4. (Cubic Spline Interpolation) P. 403 #10, P. 404 #15 • P. 403 #10a j a j b j c j d j 0.500 1.49 0.286-0.153 1 1.43 2.18 0.579-0.153 2 2.64 2.64 0.350-0.330 3 4.01 2.75-0.144-0.330 • P. 403 #10b j a j b j c j d j 0.500 1.50 0.756-0.106 1 1.43 2.18 0.600-0.175 2 2.64 2.64 0.334-0.287 3 4.01 2.76-0.0961-0.478 • P. 404 #15 j a j b j c j d j 0.500 1.72 0.523 1 1.43 2.11 0.784-0.297 2 2.64 2.67 0.338-0.426 3 4.01 2.69-0.301 0.200 3...
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This note was uploaded on 12/16/2011 for the course MATH 371 taught by Professor Krasny during the Fall '08 term at University of Michigan.

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Fall2011-HW10-soln - 2 3. (Piecewise Linear Interpolation)...

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