Fall2011-HW10-soln

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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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).
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## 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.

### Page1 / 3

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online