hw9 - x = x ln x Fix the two nodes x = 1 and x 1 = 3(So n =...

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Math 471, Fall 2007 Homework 9 Assigned: Friday, November 9, 2007. Due: Friday, November 16, 2007 . 1. (Chebyshev polynomials, Page 385, # 3) Show that Z 1 - 1 T n ( x ) T m ( x ) 1 - x 2 dx = 0 , m 6 = n c n π 2 , m = n , where c 0 = 2 and c n = 1 ( n 1). This implies that the Chebyshev polynomials form an orhogonal set on [ - 1 , 1] with respect to the weight function w ( x ) = (1 - x 2 ) - 1 / 2 . ( Hint: Make the substitution θ = cos - 1 x .) 2. (Convergence of functions) Recall that the n th order set of Chebyshev nodes consists of the points x j = cos( πj/n ) for j = 0 , 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]. You may use Matlab’s built-in interpolation routines. The first line of your code should read function interpolate( f, n ) (b) Apply your program to the functions f ( x ) = abs( x ) and g ( x ) = sign( x ) for each n = 4 , 8 , 16. For each function, report whether the interpolants appear to be converging on [ - 1 , 1]. If so, is the convergence pointwise or uniform? 3. (Error estimates) Consider the function
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Unformatted text preview: x ) = x ln x . Fix the two nodes x = 1 and x 1 = 3. (So n = 1.) (a) Use the linear interpolant and the Hermite interpolant to approximate the value of f (1 . 5). Which estimate is more accurate? (b) Use the error estimate theorems on pages 348 and 410 to obtain rigorous bounds on the error in each interpolant at x = 1 . 5. 4. (Other types of interpolation) Many types of interpolants can be computed by solving linear systems. This problem explores two examples. (a) Find a polynomial of least degree that satisfies the following five conditions: p (1) =-1 p (1) = 2 p 00 (1) = 0 p (2) = 1 p (2) =-2 . (b) A trigonometric polynomial has the form p ( x ) = a + N X j =1 [ a j cos( jx ) + b j sin( jx )] . The coefficients { a j } and { b j } are arbitrary real numbers. Interpolate the function f ( x ) = e-x sin x at the points x = 0 , 1 ,..., 10 using a trigonometric polynomial with N = 5. 2...
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