COS 323, Fall 2010, Assignment 4
README.txt

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General Questions
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1. Using Matlab, compute the value of each of the following functions,
sampled at locations x = 0:15
   y3 = cos(3/8*pi*x);
   y5 = cos(5/8*pi*x);
   y7 = cos(7/8*pi*x);
   y9 = cos(9/8*pi*x);
Compute the DFT of each, and plot the complex magnitude of the components.
(No need to hand in the plots.  Use the Matlab fft, abs, and plot functions.)

Explain the qualitative behavior of the plots.  Comment on the reason
behind the relationship between the FFTs of y7 and y9.


[your answer here]


2. [Heath, exercise 8.2]
(a) Using the midpoint method, compute (by hand) the approximate value for
the integral from 0 to 1 of x^3, using h = 1 and also h = 0.5.
(b) Based on the two approximate values computed in (a), use Richardson
extrapolation to compute a more accurate approximation to the integral.
(c) Would you expect the extrapolated result computed in (b) to be exact?
Why or why not?


[your answer here]


3. Implement the midpoint method, and use it, together with a suitable
change of variables and subdivision into intervals, to compute the
integral of the Gaussian function exp(-x*x/2) from 0 to positive infinity.

Compare to the analytic value of this integral, sqrt(0.5*pi), as a function
of the number of function evaluations.  How does accuracy depend on the
location you chose to subdivide the domain into a "standard" integral and
one with the change of variables?  (Use the same number of samples in both
sub-domains.)


[your answer here]



Programming assignment: Chaos
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1) How do your results in part (a) compare to a graph of y=exp(t)?
How does the size of the time steps and number of iterations affect
the accuracy of your results?  Similarly, for part (b), how do your
results compare to y=sin(t), and what is the effect of varying the
time step size and number of iterations?


[your answer here]


4) Compare the results you found with Runge-Kutta to those you found
with Euler's method in parts 1a, 1b, 3a, and 3b.


[your answer here]


Briefly describe the structure of your code, your method for finding
unique peaks (parts 2a and 2b), and any other implementation notes.


[your answer here]