Math 6643 - Numerical Linear Algebra. Fall 2004
Time and location
Weber, SST II, room 2, Tuesdays and Thrusdays from 12:05pm to 1:25am
Instructor and office hours
Guillermo Goldsztein
Skiles Building, Room 226
Phone: 894-2286
e-mail: ggold@math.gatech.edu
Office hours: Tuesdays and Thrusdays from 11:00am to 12:00pm.
Prerequisites
Math 2402, Math 4305 or any introductory course in
linear algebra.
Goal of this course
Introduce the student to numerical methods to solve linear algebra problems and important concepts in numerical analysis such as stability of algorithms and condition of problems.
Material to be covered
We will cover the following set of topics:
Gaussian elimination: Triangular systems. LU factorization. Complexity of the Gaussian elimination algorithm.
Finite precision calculations: Floating point arithmetic. Round-off errors. Error analysis.
Condition of problems and stability of algorithms.
More Gaussian elimination: Pivoting. Banded systems. Cholesky decomposition.
Condition numbers and sensitivity of linear systems: Vector and matrix norms. Geometric interpretation of the condition number. Estimating the condition number.
Singular value decomposition.
Least square problems: Method of normal equations for least square problems.
Reflections.
QR decomposition: QR decomposition with reflections. Solving least square problems with QR factorizations. Rank deficient least squares. QR with pivoting.
Orthonormal vectors and Gram-Schmidt: Modified Gram-Schmidt. Sensitivity of least square problems.
Eigenvalues and eigenvectors: The power method and the inverse power method. Shifts for the power method and the Rayleigh Quotient. The QR algorithm. Hessenberg matrices. Unitary reduction to Hessenberg matrices. Sensitivity analysis.
Text books
You are not required to buy any books. The following books contain most of the material to be covered in class:
Fundamentals of Matrix Computations Matrix by D. S. Watkins.
Matrix Computations by Golub and Van Loan.
Numerical Linear Algebra by L. N. Trefethen and D. Bau, III.
Grading policy
There will be two exams in class (open book). The date of the first exam will be announced during the term. You will have at least one week notice. Each of these exam will be worth 35% of the total grade.
There will also be homeworks which will include some computational projects. You can program in any language you want. The homework will be worth 30% of the grade. You are suppose to work on them alone.
A grade of 80% or more is an A. A grade of 60% to 79% is a B. A grade below 60% is a C.