Math 221, Matrix Computations, Fall 2008.

• Class Information
  • General information. In particular, see this document for what is to be covered in the class this semester (updated Sept. 1, 2008.)
• Homework Sets
  • Homework Set #1, Due Sept. 3 Selected solutions . Here is a link to George Forsythe's paper on solving quadratic equations .
  • Homework Set #2, Due Sept. 10 Selected solutions .
  • Homework Set #3, Due Sept. 17 Selected solutions .
  • Homework Set #4, Due Sept. 24 Selected solutions .
  • Homework Set #5, Due Oct. 1
  • Homework Set #6, Due Oct. 13 Notice that it has been updated (Oct. 3, 2008) Selected solutions .
  • Homework Set #7, Due Oct. 20
  • Homework Set #8, Due Oct. 27
  • Homework Set #9, Due Nov. 3
  • Homework Set #10, Due Nov. 10
  • Homework Set #11, Due Nov. 17
  • Homework Set #12, Due Nov. 24
  • Homework Set #13, Due Dec. 1
  • Homework Set #14, Due Dec. 8
• Midterm Solutions and Sample Exams.
• Final Exam, Fall 2007.
• Midterm Exam, Fall 2007. Solutions.
• Class Projects
  • Here are the class projects . We will start to have weekly meetings on these projects next week (week of Oct. 27).
• Matlab codes
  • A stable way and and unstable way to generate the Householder vectors.
  • quadroot.m is a matlab program to find the two roots of a quadratic equation. However, it can easily overflow or underflow for very large or very tiny inputs. This problem can be much alleviated by
  • a simple modification involving a scaling trick.
  • bisect.m by Prof. J. Demmel.
  • pcg.m is a matlab program implementing the preconditioned conjugate gradient (pcg) method to solve linear system of equations.
  • housetest.m is a matlab program that tests the numerical stability of two slightly different ways of computing the Householder transformation.
  • Classicl Gram-Schmidt and modified Gram-Schmidt for computing QR factorization.
  • A simple matlab code for computing quadratic roots in two different ways, with its modified version to reduce overflow/underflow problems.
  • Hager's Condition Estimator hager.m . This version estimates the 1-norm of a given matrix B. However, Hager's condition estimator can sometimes fail badly. Run counterexample_hager.m to see the results.
  • Generalized Kahan Matrix as generated by counterexample_GEPP.m. This matrix causes very large element growth with GEPP.
  • Here is the Kahan Matrix. This one misses tiny singular values in QR with column pivoting.
  • Here is the QR method. It can be used to drive one eigenvalue to zero.
  • Here is SQR: the Symmetric QR method. It can be used to drive one eigenvalue to zero.
  • Here is the Rayleigh Quotient Iteration method. It is usually cubically convergent (to some unknown eigenvalue). But it does not always converge.
  • Here is BiSVD: the dqds method for the bidiagonal SVD problem, without shifts. It should always converge to the smallest singular value.
• Pictures
  • The modified Grcar matrix is a banded matrix with upper bandwidth 10 and lower bandwidth 3, where the non-zero entries on and above the main diagonal are all 1 and non-zero entries below the main diagonal are all -1. pseudospectra of a modified Grcar matrix with complex perturbations.
  • pseudospectra of a modified Grcar matrix with real perturbations.
  • Convergence Behavior of Power Method on random matrices.
  • Convergence Behavior of Inverse Power Method on random matrices. Here we used the shift -1 for an eigenvalue -0.994.
• External Link
  • Prof. Demmel has taught Math221 in the past, using his own text. See Demmel's Math 221 Website. Among other things, there is an errata of the text book, as well as a number of relevant matlab codes.