Numerical linear algebra
MAD6406-3217
Fall 2003
- Instructor
- Jay Gopalakrishnan
- Venue
- Little Hall 235
- Times
- MWF 3:00-3:50pm
- Office hours
-
Tuesdays 3:00-3:50pm or by appointment.
- Prerequisites
-
This is a graduate course on numerical linear algebra, and is one of
the courses leading to a qualifying examination in the mathematics
department. Accordingly, a solid undergraduate background in linear
algebra is expected. (Occasionally, we will borrow simple tools from
real and complex analysis.)
In the first two lectures, I will review some elementary concepts. If
you find the first lecture inaccessible, please don't take this
course. Students without prior knowledge of the MATLAB™
software are expected to learn it as the course proceeds (you may find
the tutorials at
ufl, The
Mathworks Inc., etc. helpful.)
- References
-
- Textbook: ``Numerical Linear Algebra'', by
Lloyd N. Trefethen and
David Bau III.
- Additional reference: ``Matrix Computations'', by Gene H. Golub
and Charles F. van Loan
- Additional reference: ``Applied Numerical Linear Algebra'', by
William W. Hager.
- Course objective
-
Every mathematical scientist needs to work with matrices and vectors
theoretically and numerically. The course aims to teach
often needed ideas on computations with linear operators.
- Course outline
-
The course will cover essential concepts of numerical linear algebra
including factorizations, conditioning, eigenvalue solvers, and
iterative techniques. We will cover most of the textbook following it
closely. The online
outline
of the textbook will get you an idea of how the course will
proceed.
- Core topics
- Singular value decomposition
- (Existence of SVD, conditions for
uniqueness, further properties.)
- Least squares
-
(Projectors, Gram-Schmidt orthogonalization,
Householder reflections, Givens rotations, QR factorization,
solving least squares problems using normal equations, QR, and
SVD, the Moore-Penrose pseudoinverse.)
- General linear systems
-
(LU factorization, backsubstitution, Gaussian
elimination, partial and complete pivoting, positive definite
systems, Cholesky factorization.)
- Conditioning and stability
-
(Condition number of a map, condition
numbers of matrices, floating point
number system and round off, generalities on accuracy,
stability, and backward stability, simple examples,
backward stability of backsubstitution,
basic perturbation theory for linear systems and
least squares.)
- Eigenvalue problems
-
(Schur factorization, power method,
Rayleigh quotient iteration, Jacobi methods,
basic perturbation theory for eigenvalues,
metric to measure convergence of spaces, simultaneous
iteration, QR iteration with and without shifts.)
- Krylov space methods
-
(Arnoldi iteration, Lanczos iteration,
conjugate gradients, GMRES, basic
preconditioning techniques.)
- Further topics (depending on time
available and class interest, only one or two of these
topics are covered)
- More on eigenproblems
(the divide & conquer
algorithm for eigenvalues, Courant-Fischer theorems,
Sturm sequences, pseudospectra, etc)
- Computing the SVD
(bidiagonalization and
allied techniques)
- Classical linear iterations revisited
- More perturbation theory
- More Krylov space techniques
- More on preconditioners
- Evaluation
-
Exercises will be given regularly. You are not required to turn in
solutions. Every couple of weeks, I'll devote one period to answering
questions on exercises. If no one asks any questions, this
session will revert to a lecture.
There will also be computational assignments. These will need to be
turned in, and will determine 20% of your total grades. The remaining
component of your grades are determined by examinations as follows:
- Two midterm examinations (20% each)
- Final examination (40%)
All examinations are take-home tests. All assignments are open to
discussion, but none of the examinations are.
- Mailing list
- f03-3217@clas.ufl.edu
This mailing list provides a forum for everyone to communicate to
entire class, raise questions on assignments, publicize typos,
advertise seminars or talks of interest to class etc.
Please sign up for this mailing list by sending email to
f03-3217-request@clas.ufl.edu (not f03-3217@clas.ufl.edu)
with the single word "subscribe"
(without quotes) as the message body (not subject). When you get a
reply, follow further instructions in it.
- Additional information
-
- Policy on class attendance: Class attendance is not mandatory.
- Policy related to make-up exams or other work:
There will be no opportunities to make up for work not
submitted. However, if a student provides a legitimate excuse well in
advance, scores will be prorated.
Work with due date should be turned in at the beginning of class
on the stated due date. Late work will not be graded and will
be deemed work not submitted.
- University's honesty policy: All students are expected to
know and abide by the University of Florida Honor Code. E.g., on all
work submitted for credit by University of Florida students, the
following pledge is either required or implied: ``On my honor, I have
neither given nor received unauthorized aid in doing this
assignment.'' Know your responsibilities (and rights) as detailed in
Student Guide.
- Copyrighted materials: All course materials handed out in
class or placed on the web are protected by Copyright laws, and are
for personal use only. Multiple copies or sale of these materials is
prohibited.
- Students with disabilities: Students requesting
classroom accommodation must first register with the Dean of Students
Office. The Dean of Students Office will provide documentation to the
student who must then provide this documentation to me
when requesting accommodation.
Jay Gopalakrishnan
November 20, 2003