MTH 451- 551 : NUMERICAL LINEAR ALGEBRA - Fall 2017
Links
General information
Assignments
Assignments and schedule
  1. 9/20/17: Introduction, objectives, notation. [Read Lecture 1; solve Pbm 1.1].
  2. 9/22/17: Norms of vectors and matrices. [Read Lectures 2-3; solve Pbms 2.1-4]
    Quiz 1.
  3. 9/25/17: Orthogonality, inner (dot) product, and characterization of 2-norm. [Solve Pbms 3.1-3.4]
    Office hours in MLC Kidd 108C computer lab. Come if you need a gentle introduction to MATLAB.
  4. 9/27/17: SVD: what, and why. [Read lectures 4-5; solve 4.1-5; 5.1, 5.3].
  5. 9/29/17: activity on SVD. Two worksheets with simple steps on calculating SVD.
    Assignment 1 due in class.
    Quiz 2: unitary and spd matrices.
  6. 10/2/17: Further activities on SVD: examples; tips and tricks.

    For students who need more practice with Review sheet for linear algebra, I will hold LA Clinics for MTH 451-551 students in week 2; if there is interest, we will continue in weeks 3, 4. Here is a link to the online Linear Algebra text used in MTH 341 (Chapters 1-8) that you can use to review. MTH 342 material is there as well.

    LA Clinic I, 5:00-, Kidd 292A.
  7. 10/4/17: Further activities on SVD: low rank approximations and applications. Introduction to computational complexity of numerical linear algebra (flops!).
  8. 10/6/17: Projections and orthogonalization [Read Lectures 6-8; solve 6.1, 6.3-5; 7.1, 7.5; 8.1]
    Assignment 2 due in class. You will need to download .

    NEW: some of you reported difficulties with displaying the approximation of your image. If you get a blank image, try replacing imshow(matrix) with imshow(uint8(matrix)), where "matrix" is the svd-derived approximation to your image. (This tip is from one of your classmates).


    Quiz 3: SVD and matrix norms.
  9. 10/9/17: Classical and modified Gram-Schmidt. QR decomposition. Worksheet with examples. LA Clinic II, 5:00-, Kidd 292A.
  10. 10/11/17: Householder reflections and Givens rotations: stable alternatives to GS. [Read lecture 10; solve 10.1 and 10.4].
  11. 10/13/17: Compare Householder and GS on simple examples. Least squares: "how-to". [Read Lecture 11; for practice, solve problems assigned in class.]
    Assignment 3 due in class
  12. 10/16/17: Continue least squares. LA Clinic III, 5:00-, Kidd 292A.
  13. 10/18/17: Review and problem solving.
  14. 10/20/17: MIDTERM. One page formula sheet will be allowed (write only on one side please). I will ask that you turn it in with your exam. No calculators!
  15. 10/23/17: Conditioning and stability in numerical calculations. [Read Chapters 12-17 lightly. Solve 14.1-2, 15.1].
  16. 10/25/17: Continue conditioning. Pseudoinverse and how to use in conditioning.
  17. 10/27/17: Quiz 4=worksheet. Stability and backward stabilityof algorithms: definition and arithmetic examples.
  18. 10/30/17: Error bounds on algorithms = conditioning of the problem + error bound from (backward) stability
    Assignment 4 due in class.
    In 1, you are looking for a lower bound. In 3, focus on the difference between the two methods of evaluation of the polynomial. You can also experiment further with the computational assignments, if you wish. (Instead of degree k=7, use a polynomial of degree 11, or 13. Narrow the range of plotting).
  19. 11/1/17: Solving linear systems: big picture. [Read Lectures 20-23; follow example with matrix (20.9), solve problems 20.1-3; 21.1, 21.4]
  20. 11/3/17: Quiz 5 (stability, conditioning, LSQ).
    Solving general (dense) linear systems.
    Example of (in)stability when A=LU with and without pivoting.
    Assignment 5 due in class. (For polynomial evaluation, you can use polyval).
  21. 11/6/17: Solving special linear systems. Cholesky for spd matrices. Examples of spd matrices: covariance matrix for Karhunen-Loeve representations. Discrete Laplacian.
  22. 11/8/17: Stationary iterative methods handout. Solving linear systems with sparse matrices by iteration. Estimating eigenvalues with Gerschgorin theorem [Solve 24.2, 24.4a].
  23. 11/13/17: Convergence of Jacobi method for strictly diagonally dominant matrices as an example of application of Gerschgorin theorem. Relationship between the spectral radius, matrix norms, and the convergence of simple iteration.
  24. 11/15/17: Examples of Jacobi, Gauss-Seidel iterations that converge or not. Further theory. [Same handout]. [Solve pbm 35.3]
  25. 11/17/17: Assignment 6 due in class. Extra credit can be turned in by Monday 11/20/17. Quiz 6 (positive definite matrices).
  26. 11/20/17: Conjugate gradient for spd systems and preconditioning. [Read lectures 38, 39, 40; solve 38.2-5]
  27. 11/22/17: Holiday special.
  28. 11/27/17: No class today (office hours cancelled as well). Please read basic algorithms for numerically finding eigenvalues. [Lecture 27.]
  29. 11/29/17: A word on preconditioning. Review iterative methods.
  30. 12/1/17: Review (puzzle worksheet) and problem solving. (Bring questions).
    Assignment 7 due in class. Note: this assignment will be worth a double credit. Since I plan to drop the two lowest HW scores, the assignment is essentially optional for those too busy to do it.
    Extra: In Pbm 1, draw the iterates on an ellipse corresponding to the contours of phi(x)=1/2 x^TAx-b^tx. Do this for both CG and steepest descent.
Final Exam is on Tuesday, Dec 5. Notes and books (hard copy) are allowed. No electronics or calculators Help session for Final Exam: Monday, Dec. 4, 2:30-4:30 in Kidd 280.