### Math. 556-01, Spring, 1998 Assignments

/ Assignments / Exams /
This list is updated as the assignments are made and exam dates are set.

1. For Wednesday, January 21:
• Hand in 5.10.
• Review Section 6.6 of the text..

2. For Monday, February 2:

• Hand in 6.2, 6.5, 6.9, 6.15, 6.17
3. For Friday, February 6:
• 6.20, 6.21, 6.23
4. For Friday, February 20:
This assignment deals with material from the article J, S. Walker, "Fourier Analysis and Wavelet Analysis, Notices of the AMS 44(6), July, 1997, pp. 658--670.
• Explain, in your own words and formulas, the role of the constant Cm in the formula for the discrete Haar series on page 662.
• Consider the function f(x) = 2x, 0 <= x <= 0.5, f(x) = 2- 2x, 0.5 <= x <= 1.
1. Compute the fast Fourier transform for this function, using 16 points in [0,1]. Graph the resulting Fourier approximation.
1. Graph the resulting approximation.
2. Graph the approximation you get by ignoring the highest-order terms.
2. Compute the fast Haar transform for this function, using 16 points in [0,1].
1. Graph the resulting approximation.
2. Graph the approximation you get by ignoring the highest-order terms in the Haar expansion.
5. For Monday, March 9:
1. 7.15, (a) and (b) only
2. Use Householder transformations to transform the matrix
1. `[ 2 -1  0  0`
` -1  2 -1  0`
`  0 -1  2 -1`
`  0  0 -1  2 ]`

to upper Hessenberg form. (Hint: refer to Proposition 6.12 on page 349.)

3. Simplify the statement of Theorem 7.49 when M is the identity matrix. In the restatement of the theorem, use A instead of K and lambda instead of mu. That is, state the Gershgorin theorem for the special case of the standard eigenvalue problem A v = lambda v.
4. 7.6
6. For Friday, March 13:
1. Use Theorem 9.4 and a degree 3 interpolating polynomial at x=-1, x=0, and x=1 to give a better bound on the error in Simpson's rule than that given in class.
7. For Friday, March 20, 1998:
Do the following to hand in: 9.2, 9.6 (use Fortran or Matlab).
8. For Monday, March 30, 1998:
• (A simple Galerkin approximation.)
1. Using the basis functions vi(x) = sin(i pi x), 1 <= i <= 3, use the Galerkin technique to discretize the boundary value problem u"(x) = -1, u(0) = u(1) = 0. As dot product, use <f,g> = {integral from 0 to 1 of f(x)g(x) dx}.
2. Solve the resulting system of equations to obtain a solution of the form = a1v1(x) + a2v2(x) + a3v3(x).
3. Graph the resulting approximate solution u3(x). Also graph the exact solution on the same graph, so you can compare.
4. Also do the above, using the least squares technique. Also try both the Galerkin technique and the least squares technique with basis functions v1 = x(1-x), v2 = x^2(1-x), v3 = x(1-x)^2.
9. For Monday, April 6:
Hand in 9.9.
10. For Wednesday, April 22:
• Hand in Computer Problem 9.1 from Kincaid / Cheney.
• Hand in Computer Problem 9.2 from Kincaid / Cheney, but use the initial condition froom Problem 9.1, rather than the one stated in the book.
• Do Problem 9.2, no. 1 from Kincaid / Cheney.
11. For Wednesday, April 22, 1998:
The following are from Section 9.2 of Kincaid / Cheney, first edition. (Descriptions will be given to help identify corresponding problems from the second edition. Ask if in doubt.)
• Hand in Problem 7 from section 9.6. (Show that the discriminant of the differential equation remains the same under the transformation to determine the characteristics, provided the Jacobian of the transformation is non-zero. Thus, such transformations do not change the type of the differential equation.)
12. For Wednesday, April 29, 1998:
This is a computer problem from the section on multigrid methods in Kincaid and Cheney: Program the V-cycle algorithm for the two-dimensional problem uxx + uyy = f(x,y), 0 < x < 1, 0 < y < 1, u(x,y) = 0 on the boundary. Test the program using f(x,y) = 2x(x-1) + 2y(y-1). (The true solution is u(x,y) = xy(1-x)(1-y).)

Final exam:
• The final exam is take-home, due on or before Monday, May 11, 1998 at 1:00 PM CDT. Click here for a Postscript copy.