Numerical Linear Algebra
Math-6316 and CSE-7366
Spring 2009

• Course description:
  This course will cover the development and analysis of numerical linear algorithms used in the solution of linear equations, eigenvalue problems, and linear least-square problems. It will also cover the Conditioning (related to the problems) and Stability (related to the algorithms) issues. Both dense methods and sparse methods will be discussed. The emphasis will be dense methods.
  [ Prerequisites:   Basic knowledge of matrix theory and linear algebra; basic knowledge of computer programming. ]

• Educational outcomes:
  Students will be able to master theoretical properties related to important matrix decompositions including QR, LU, Eigen-decomposition, and SVD. They will also be able to understand why these decompositions are important, what are their applications, and what are the main ideas behind the standard numerical algorithms used to compute these decompositions. Students will be able to correctly code certain standard numerical algorithms for solving linear equations, least square problems, and eigenvalue problems.

• Textbooks:
  Lloyd N. Trefethen and David Bau , Numerical Linear Algebra,   SIAM, 1997. ISBN: 0-89871-361-7;

• Supplementary textbooks:    
  • Gene Golub and Charles Van Loan , Matrix computations, 3rd Ed., Johns Hopkins University Press, 1996. ISBN: 0-8018-5414-8
  • G. W. Stewart , Matrix algorithms I: Basic decompositions, SIAM, 1998. ISBN: 0-89871-414-1
    G. W. Stewart, Matrix algorithms II: Eigensystems, SIAM, 2001. ISBN: 0-89871-503-2
  • Yousef Saad , Iterative methods for sparse linear systems , SIAM, 2003. ISBN-13: 978-0-898715-34-7.   [1st edition]
  • J. Demmel , Applied numerical linear algebra, SIAM, 1997. ISBN: 0-89871-389-7
  • Henk A. Van der Vorst ,   Iterative Krylov methods for large linear systems, Cambridge University Press, 2003, ISBN: 0521818281.
    Lecture notes on large linear systems and eigenvalue problems
  • D. S. Watkins , Fundamentals of matrix computations, 2nd Ed., Wiley-Interscience, 2002. ISBN: 0-471-21394-2
  • Nicholas J. Higham , Accuracy and Stability of Numerical Algorithms, SIAM, 2002. SBN: 0-89871-521-0.
  • C. Moler, Numerical Computing with MATLAB, SIAM, 2004. ISBN-13: 978-0-898715-60-6 [electronic version]
  • C. D. Meyer , Matrix Analysis and Applied Linear Algebra, SIAM, 2000. [online version]

• Software:    the commercial   Matlab   and/or the open sourceware   Scilab , Octave .  
• Matlab primer (a bit outdated, but still a good starting point. Do a google search for other online tutorials)
• Scilab tutorial
• Further Readings of interest:
  • NLA at Rice
  • NLA at Berkeley
  • NLA for systems and control
  • Top ten algorithms .   [ the QR algorithm ,   matrix decompositions ,   Krylov methods ]
  • Templates for the Solution of Algebraic Eigenvalue Problems
  • Templates for the Solution of Linear Systems
  • Numerical Computing with IEEE Floating Point Arithmetic
  • Floating Point Arithmetic
    
  • Understanding Search Engines: Mathematical Modeling and Text Retrieval
  • Google's Pagerank and Beyond: The Science of Search Engine Rankings
  • The $25 billion eigenvector: The Linear Algebra Behind Google
    

• Course Policy:
  • Class attendance is required (unless with permission from the instructor)
  • Homework due dates are specified on the webpage
  • Homework counts 65% in total
  • Midterm and final count 35%