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

- 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*) = 2*x*, 0 <=*x*<= 0.5,*f*(*x*) = 2- 2*x*, 0.5 <=*x*<= 1. - Compute the fast Fourier transform for this function, using 16 points in [0,1]. Graph the resulting Fourier approximation.
- Graph the resulting approximation.
- Graph the approximation you get by ignoring the highest-order terms.
- Compute the fast Haar transform for this function, using 16 points in [0,1].
- Graph the resulting approximation.
- Graph the approximation you get by ignoring the highest-order terms in the Haar expansion.

- 7.15, (a) and (b) only
- Use Householder transformations to transform the matrix
- 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*. - 7.6

[ 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.)*

- 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.)

- 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*}. - Solve the resulting system of equations to obtain a solution of the
form =
*a*1*v*1(*x*) +*a*2*v*2(*x*) +*a*3*v*3(*x*). - Graph the resulting approximate solution
*u*3(*x*). Also graph the exact solution on the same graph, so you can compare. - Also do the above, using the least squares technique. Also try both
the Galerkin technique and the least squares technique with basis functions
*v*1 =*x*(1-*x*),*v*2 =*x^*2(1-*x*),*v*3 =*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.)