MATH 497/597 Fall 2002
Final homework due December 19, 2002

Retrieve the matlab functions for this assignment from http://ucs.orst.edu/~weidemaj/differ.html . Hand in your Matlab code in addition to the requested graphs and tables, and please condense it onto as few pieces of paper as possible.

1.
For the function $u(x)=\cos (\pi x) $ calculate its derivative on $-1\leq x \leq 1$ at the Chebyshev points $x_j=\cos (\frac{\pi j}{N})$ $j=0,\ldots , N$ when N=8,16,32 using
(a)
polydif.m (differentiation matrices for arbitrary sets of points)
(b)
chebdif.m (differentiation matrices on at the Chebyshev points)
(c)
chebdifft.m (derivative approximation at Chebyshev points using Fast Fourier Transform)
Provide a table of values of $\displaystyle \vert\vert u'(x)-\hat{u}' \vert\vert _{\infty} = \max_{0\leq j \leq N} \vert u'(x_j)-\hat{u}_j'\vert$, where u'(x) is the exact derivative and $\hat{u}'$ is the computed derivative. Do this for each N and each method and present your results in a table of the form
  $\vert\vert u'(x)-\hat{u}' \vert\vert _{\infty}$
N polydif.m chebydif.m chebdifft.m
8      
16      
32      
Determine which method gives the most accurate representation of the derivative.

2.
For the function $u(x)=e^{\sin^3x} $ calculate u' and u'' on $0 \leq x \leq 2\pi $ at equally spaced points when N=32,64 using
(a)
fourdif.m (differentiation matrices)
(b)
fourdifft.m (derivative approximation using FFT)
Provide a table of $\vert\vert u'(x)-\hat{u}' \vert\vert _{\infty}$ and $\vert\vert u''(x)-\hat{u}'' \vert\vert _{\infty}$ for each method and each N, i.e.
  $\vert\vert u'(x)-\hat{u}' \vert\vert _{\infty}$ $\vert\vert u''(x)-\hat{u}'' \vert\vert _{\infty}$
N fourdif.m fourdifft.m fourdif.m fourdifft.m
32        
64