MODES 621 Numerical Linear Algebra
Course Syllabus, Fall 2009-2010
MODES 621 Numerical Linear Algebra (3-0)3
Catalogue Data
Floating Point Computations. Vector and Matrix Norms. Direct Methods for The
Solution of Linear Systems. Least Squares Problems. Eigenvalue Problems.
Singular Value Decomposition. Iterative Methods for Linear Systems.
Textbook(s)
1. L.N. Trefethen and D. Bau, III, Numerical Linear Algebra, SIAM, 1997.
2. J.W.Demmel, Applied Numerical Linear Algebra, SIAM, 1997.
Reference(s)
1. G.H. Golub and C.F. van Loan. Matrix Computations, John Hopkin’s University
Press, 3rd edition, 1996.
2. A. Greenbaum, Iterative Methods for Solving Linear Systems, SIAM, 1997.
3. C.D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, 2000.
4. O. Axelsson, Iterative Solution Methods, Cambridge University Press, 1996.
5. D.S. Watkins, Fundamentals on Matrix Computations, John Wiley and Sons, 1991.
6. K.E.Atkinson, An Introduction to Numericall Analysis, John Wiley and Sons, 1999.
Prerequisite(s) by Topic(s)
Introductory level knowledge in linear algebra (Math275 Linear Algebra or equivalent)
and computing environments such as Matlab or Maple.
Goals
This course is designed to give engineering, mathematics and science students the
expertise necessary to understand
and use computational methods for the
approximate solution of linear algebra problems that arise in many different fields of
science like electrical networks, solid mechanics, signal analysis and optimisation.
The emphasis is on methods for linear algebra problems such as solutions of linear
systems, least squares problems and eigenvalue-eigenvector problems, the effect of
roundoff on algorithms and the citeria for choosing the best algorithm for the
mathematical structure of the problem under consideration.
Objectives
The successful student will be able to
1. choose an efficient method to solve (large) linear systems, eigenvalue problems
and least squares problems coming from a certain application field,
2. implement the methods and/or algorithms as computer code and use them to
solve applied problems,
3. discuss the numerical methods and/or algorithms with respect to stability,
applicability, reliability, conditioning, accuracy, computational complexity and
efficiency.
Topics
1. Introduction to Numerical Computations. Vector and matrix norms, Condition
numbers and conditioning, Stability, Propogation of roundoff errors. (3/2 weeks)
2. Direct methods for linear systems, Gaussian elimination, Pivoting, Stability, LU and
Cholesky decompositions, Operation counts, Error analysis, Perturbation theory,
Special linear systems. (5/2 weeks)
3. Least Squares. Orthogonal matrices, Normal equations, QR factorization, GramSchmidt orthogonalization, Householder triangularization, Least squares problems.
(3/2 weeks)
4. Eigenproblem. Eigenvalues and eigenvectors, Gersgorin’s circle theorem, Iterative
methods for eigenvalue problems, Power, inverse power and shifted power
methods, Rayleigh quotients, Similarity transformations, Reduction to Hessenberg
and tridiagonal forms, QR algorithm for eigenvalues and eigenvectors, Other
eigenvalue algorithms. (3 weeks)
5. Singular Value Decomposition. Connection with least squares problem, Computing
the SVD using the QR algorithm. (3/2 weeks)
6. Iterative Methods for Linear Systems. Basic iterative methods, Richardson, Jacobi,
Gauss-Seidel
and SOR methods, Convergence analysis, Krylov subspace
Methods, Preconditioning and preconditioners. (4 weeks)
Computer Usage: Homework assignments including computer implementation of the
methods will be prepared using computing environments such as Matlab or Maple.
Homeworks: There will be 6 homework assignments. No late homework will be
accepted but, the lowest grade on 2 homeworks will be dropped. You may work
together on homeworks but, you must write up your homework assignments by
yourself independent of others.
Exams: There will be two Midterm Exams and one Final Exam during the semester.
No collaboration is allowed on these exams.
Grading: Homework (and program) : 24%, Midterm Exams: 20%+20%, Final: 36%.
Contribution of Professional Component
1. Math or Basic Science
1 credit
2. Engineering Topics and Design
1 credit
3. General Education
1 credit
Instructor Assoc. Prof. Dr. Ahmet Yaşar Özban