Math 128B, Numerical Analysis, Spring 2009.

• Class Information
  • General information about the course can be found at /~mgu/MA128BSp2009/general.html.
  • The class syllabus will be discussed in class. My current plan is to cover most of the material from Chapter 7 to Chapter 12 in Burden and Faires, plus one or two topics of interest to the class. We will probably discuss Unconventional ODE schemes by W. Kahan and R.-C. Li before Chapter 11 (boundary value ODEs.)
• Quiz and Midterm Schedule
  • Feb. 1 (quiz)
  • Feb. 22 (quiz)
  • March 8 (Midterm)
  • Mar. 22 (quiz)
  • Apr. 12 (quiz)
  • Apr. 26 (quiz)
• Term Final Projects
  • Non-symmetric QR Algorithm : In this project, we extend the symmetric QR algorithm to solve the non-symmetric eigenvalue problem. This will involve Hessenberg reduction, and single and double shift QR algorithm.
  • Preconditioned Conjugate Gradient Method : In this project, we will develop the preconditioned version of the CG method, and test the effectiveness of various preconditioners. See Cojugate gradient method on Wikipedia.
  • Variations of the Fast Foruier Transform : The FFT comes in many different forms and variations. In this project, we study these variations and develop corresponding matlab code for doing these transforms. See Discrete cosine transform and Discrete sine transform on Wikipedia.
  • Quasi-Newton Methods: There is a broad class of Quasi-Newton methods for solving nonlinear optimization problems, with the Broyden method being one of them. In this project, we will study such methods and develop corresponding matlab code. See Quasi-Newton method on Wikipedia.
  • Levenberg-Marquardt Method : This is another class of methods for solving nonlinear least squares problems. In this project, we will study this method and develop matlab code. See Levenberg-Marquardt method on Wikipedia.
• Exams and Solutions
  • The mean on the midterm was 54/100.
• Homework and Programming Assignment
  • Homework #1, due Feb. 2 in discussion.
  • Homework #2, due Feb. 9 in discussion.
  • Homework #3, due Feb. 18 in class.
  • Homework #4, due Feb. 23 in discussion.
  • Homework #5, due March 2 in discussion.
  • Homework #6, due March 9 in discussion. Note that problems from Section 9.3 have been deleted (March 3, 2009.)
  • Homework #7, due March 16 in discussion. Note that you can use the planerot function in matlab to compute Givens rotations.
  • Homework #8, due March 30 in discussion.
  • Homework #9, due April 6 in discussion.
  • Homework #10, due April 13 in discussion.
  • Homework #11, due April 20 in discussion.
  • Homework #12, due April 27 in discussion.
  • Homework #13, due May 4 in discussion.
  • Programming Assignment #1, due Feb. 16.
  • Programming Assignment #2, due March 18.
  • Programming Assignment #3, due April 13.
  • Programming Assignment #4, due May 4. Note that the basic Anadromic scheme for this project should be the one in this letter from Prof. Li.
• Matlab Codes
  • SunEarthSat.m . This code was written by Eric Keldrauk in the Fall of 2008 as his final project.
  • Here is an example with Newton's Method to find roots of non-linear equations.
  • Here is the QR algorithm for finding eigenvalues of a symmetric tridiagonal matrix. Last modified April 9, 2009 to fix a bug in deflation.
  • Here is an unstable version of the Householder Transformation.
  • The Power Method and the Inverse Power Method codes.
  • Redix 2 and 3 FFT .
  • The FFT and the Inverse FFT codes.
  • This is a demo code for performing Chebyshev rational approximation on an example function. Here is the Chebyshev polynomial evaluation function.
  • This code converts a power series into a Chebyshev series. For the function f(x) = 1/(1-x/2) on [-1, 1], the Chebyshev series converges much faster than the power series, requiring almost 40% fewer terms in the expansion for double precision accuracy. This matlab plot illustrates this point.
  • Conjugate Gradient with long recursions. This matlab code performs an iteration that is mathematically equivalent to Conjugate Gradient, but does so by explicitly performing A-conjugation against all previous A-conjudate vectors at every step. It also plots the non-zero patterns of the conjudate coefficients.
• Web Links
  • Guide to Available Mathematical Software.
  • A brief history of numerical analysis.
  • Prof. Kahan's Math 128 webpage .
  • Prof. Holtz's Math 128B webpage .
  • Unconventional ODE schemes by W. Kahan and R.-C. Li. Here is an email from R.-C. on the same subject.
  • Prof. Parlett's Notes on the QR algorithm .
  • Lecture notes on Chebyshev Alternation Theoreym at Cambridge University Numerical Analysis Group .
  • Wikipedia Entry on the Conjugate Gradient Method.
  • Top 10 Numerical Algorithms invented in 20th Century.