Math. 556-01, Spring, 1998 Assignments

• 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

• 6.20, 6.21, 6.23

• 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.

1. 7.15, (a) and (b) only
2. Use Householder transformations to transform the matrix

[ 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

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.

• (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.

• 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.

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.)