Assignment 3
Due Thursday March 28

No late penalty till April 3


1)  Read the paper "Handheld Calculator Evaluates Integrals"
    by W. Kahan, and write a digest of the paper (not more than
    two pages).  You may find a pdf copy of the paper at

    http://www.cs.berkeley.edu/~wkahan/Math128/INTGTkey.pdf

2)  The zeroth order Bessel Function has the following
    integral representation:

    J0(x) = int( cos(x*sin(t)), t=0..pi)/pi

    a) Write an m-file to evaluate this function given x.
       Use your function to plot J0(x) for 0 <= x <= 20.
       You may use quad or quadl to evaluate the integral.

    b) Use your m-file to find all the zeros of J0(x) on
       0 <= x <= 20.  You may use fzero.


3. Question 8, p. 670
    Older versions of the text may have the factor of 2 missing from the RHS
    of the equation.

4. Question 27, p. 672
   minor typo: "... for any input value \lambda T.", should read
               "... for any input value \lambda^* T".


5. The differential equation x y'' + y' + x y = 0 with the
   initial conditions y(0) = 1 and y'(0) = 0 has the solution J0(x).
   These initial conditions are a bit tricky to deal with numerically.

   a) Write the differential equation as a 1st order system.
   b) Use your program from question 2 to evaluate J0(x) at some
      other point, say x=1.
   c) Estimate the derivative at that point by numerical differentiation.
   d) Use ode45 to solve the initial value problem on 1 <= x <= 20
      using your initial points computed in (b) and (c).  Compare 
      your plot with that of question 2.

6. Question 22, p. 731

7. Question 26, p. 732--733.

8. One more question to be announced later.