To contact the instructor: j.rhodes@alaska.edu

Course syllabus: M614syl.pdf

Required Textbook: Numerical Linear Algebra, by Lloyd N. Trefethen and David Bau, III, 1997 / xii + 361 pages / Softcover / ISBN: 978-0-898713-61-9 / List Price $67.00 / SIAM Member Price $46.90

Note: Students may join SIAM (the Society for Industrial and Applied Mathematics) for $25, or for free if they are among the first two to ask the instructor for a nomination. Even at $25, the cost is almost offset by the discount price for the text, and comes with lots of other benefits as well. See SIAM Student Membership.

The dates below are when problems were assigned. Unless otherwise noted, problems are always due the following Monday.

Homework should be typed, in LaTeX,following the template HWtemplate.tex.

Problems are from T&B, unless otherwise stated.

- 8/28 1.1, 1.2, #1 Let A=[2 1 1; -2 0 1; 4 4 6], b=[1; -4; -4]. By hand solve Ax=b. Compute rank, determinant, inverse(if possible), and eigenvalues of A. Verify your results with MATLAB
- 8/30 none
- 9/1 1.4
- 9/4 LABOR DAY
- 9/6 2.1, 2.2, 2.3, 2.4, #1 Show the product of unitary matrices is unitary.
- 9/8 2.6, 2.7, 3.1, 3.2
- 9/11 3.3, 3.4, 3.5, #1 Show any matrix norm induced from vector norms is actually a norm (i.e., satisfies the defiinition given in equations 3.1 (or 3.15) of the text.
- 9/13 4.1, #2 Use appropriate MATLAB command to find the singular value and eigen decompositions of A=[3 1 0; 0 2 1; 0 0 1] and B=[1 1 0; 0 1 1; 0 0 1]. One of these 4 results should be wrong or produce an error message. Which one, and why?
- 9/15 4.2, 4.4
- 9/18 5.1, 5.3, 5.4
- 9/20 6.1, 6.2, 6.3
- 9/22 6.4, 6.5
- 9/25 7.1, 7.2, 7.5, #1 Continue orthonormalizing Legendre polynomials as in class, finding Q_2, P_3, and Q_3.
- 9/27 9.1
- 9/29 8.3, 10.1, 10.2 (turn in MATLAB code and example of run on the matrix in 10.3)
- 10/2 none
- 10/4 11.3, #1(a) If Sigma is diagonal and mxn of full rank n, what is the pseudoinverse Sigma^+? Justify your answer. (b) If A is mxn of full rank n and has svd U\Sigma V*, what is the svd of A^+? Justify your answer.
- 10/6 12.1, 12.2
- 10/9 none
- 10/11 none
- 10/13 none
- 10/16 MIDTERM EXAM
- 10/18 13.2ab, 13.3, 14.1a-c,e-g
- 10/20 15.1a-d,15.2
- 10/23 #1 Show that taking the inner product with a fixed vector v in C^n is component-wise backward stable, and hence backward stable. In particular, show that \tilde f(w)=f(\tilde w) where |w_i-\tilde w_i|/|w_i| =O(n\epsilon_m). #2 Show that taking the product of a matrix A with a fixed vector x is backward stable.
- 10/25 16.2
- 10/27 17.2, #3 What is the operation count for solving an nxn upper triangular system by back substitution? Explain your answer.
- 10/30 none
- 11/1 #1 Solve Ax=b by finding A=LU and solving 2 triangular systems, for A=[-2 1 -1; -4 1 0;2 -4 10], b=[3 9 12].', 20.1 20.3a, 20.4, 21.1
- 11/3 21.4, 22.2
- 11/6 23.1, 23.2
- 11/8 24.1, 24.4a
- 11/10 #1 For 2x2 and 10x10 matrices, experimentally investigate power iteration to determine eignenvectors/values. Specifically, for 5 random matrices and a random vector x_0 of each size, determine how many iterations are needed to obtain the dominant eigenvector and eigenvalue to at least 12 digits of accuracy. (You may assume MATLAB's eig command gives the "true" values.) For the same random matrices, with the dominant eigenvector already in hand, determine how many iterations are needed to obtain another eigenvector and eigenvalue to 12 digits of accuracy. #2 By hand, determine exact eigenvectors and eigenvalues of A=[5 1; -9 -1], and explain why investigating power iteration on A and A+10^{-10}*randn(2) should be interesting. Then do so, as in the previous problem.
- 11/13 25.1, 25.2
- 11/15 #1Write MATLAB code to obtain an upper Hessenberg form of a square matrix; i.e., given A, find Q and H with A=QHQ^*. (You may be able to modify your Householder QR code from earlier in the course.) Demonstrate it works correctly on two 5x5 random matrices, one Hermitian and one not. #2 Implement a "simple" qr scheme to find eigenvalues (i.e., iterate [Q R]=qr(A); A=R*Q), and experiment to see whether it converges faster when applied to a matrix A or an upper Hessenberg form of A.
- 11/17 27.3
- 11/20 27.5, 27.6 (due 12/4)
- 11/22 28.1, 28.2 a,b,c(Householder only)
- 11/24 THANKSGIVING BREAK
- 11/27 29.1 b,c,d,e (use MATLAB's hess, qr)
- 11/29 30.6, 31.4
- 12/1 none
- 12/4 none
- 12/6 none
- 12/8 none
- 12/13 Final Exam 10:15-12:15