Computer Science 106 (Fall 2010) Numerical Linear Algebra (site under construction)

Computer Science Dartmouth College

This course is about how to solve, computationally, linear algebra problems. It is not a course to introduce linear algebra (you are expected to already have some background in this — see prerequisites). However, we will review this material as needed and you might be asked to demonstrate understanding of this. It is also not a course on the applications of linear algebra: for that, see CS 36/136. Rather, it is on how to solve those linear algebra problems that you get in your applications: we will study the algorithms of linear algebra. Or, to quote one of the authors of our textbook (see the Appendix)

"Numerical analysis is the study of algorithms for the problems of continuous mathematics."

Please read the Appendix of the text book before the second class for a much more insightful and thorough discussion about what this course is about.

• Main text:  L. N. Trefethen and D. Bau. Numerical Linear Algebra, SIAM, 1997.
• We might also use supplementary material.

• orthogonal vectors, norms, singular value decomposition
• QR factorization, Gram-Schmidt orthogonalization, and least squares problems
• conditioning and stability
• Gaussian elimination and Cholesky factorization
• eigenvalue problems, Raleigh quotient, QR algorithm
• iterative methods: Arnoldi, Lanczos, conjugate gradient
• preconditioning

• Familiarity with linear algebra and with computers
• Mathematical maturity

• 60% Weekly homework assigments: a mix of problem solving, writing mathematical proofs, and programming in Matlab.
• 15% Midterm exam: in class, starting at the X-hour on Thu Oct 28, 2010. Details and rules here.
• 25% Final exam: take home, self-timed exam (due by 3:00pm on Dec 8, 2010)
  You may read the instructions on page 1, but do not look at any other pages until you are ready to start the exam!

In general, homework will be due each Wednesday before class. You may submit by either bringing it to me (Amit) in class, or putting it in my mailbox.

• HW 1, due Wed Sep 29, before class
• HW 2, due Wed Oct 6, before class
  (Matlab is available on department Unix servers and 0^th Floor machines, plus you can download it)
• HW 3, due Wed Oct 13, before class
  
• HW 4 due Wed Oct 20, before class
  
• HW 5 due Wed Oct 27, before class
  
• HW 6 due Fri Nov 5, before class
  
• HW 7 due Fri Nov 12, before class
  
• HW 8 due Fri Nov 19, before class
  
• HW 9 (details added, submission deadline extended), due Wed Dec 1, before class
  

• The now-classic textbook Matrix Computations, by Golub and van Loan has lots of technical detail that our textbook doesn't.
• For plenty more detail on backward errors and condition numbers for least squares problems, refer to the very detailed report "Optimal Sensitivity Analysis of Linear Least Squares", by Joseph F. Grcar.
• For more on smoothed analysis of Gaussian Elimination, with and without pivoting, refer to "Smoothed Analysis of Gaussian Elimination", the Ph.D. Thesis (MIT, 2004) of Arvind Sankar.
• Jonathan Shewchuk's 1994 paper An Introduction to the Conjugate Gradient Method Without the Agonizing Pain is a good read, despite the exaggerated negativity of the title.