Algebraic Numerical Computing Seminar
Tuesdays, 6:30 pm - 8:30 pm, Rm. 3305
Algebraic and numerical computing is the cornerstone of modern
computations in sciences, engineering, and signal processing. It is a
huge cache of topics for study and research in both Computer Science and
Mathematics. Systematic comparison of algebraic and numerical techniques
for algorithm design and analysis simplifies the study in both areas. The
seminar is attended by the students in both Computer Science and
Mathematics. Besides just studying the subject, they have a chance
to participate in research projects leading to their dissertations.
(The seminar has lead to 17 PhD defenses in the last 8 years in both
The subjects in the seminar can be partly adjusted to the students'
interests and background. Recent topics considered include:
a) Matrices with displacement structure, such as Toeplitz, Hankel,
Cauchy, and Pick matrices. They are omnipresent in modern computations as
well as in many areas of math and are closely related to fundamental
algebraic computations with polynomials and rational functions. This
subject is treated both algebraically and numerically and has lead to
series of publications produced in this seminar. Important classes
of banded and semi-separable matrices were also studied.
b) Recent novel methods for some fundamental computations in linear
c) Solving a polynomial equation (tha is, polynomial root-finding). This
is the central and most influential problem in math for 4 millennia and
still highly important in algebraic and geometric computing and signal
processing. Some extensions to the fundamentals of the solution of systems
of multivariate polynomial equations have been studied.
d) Algebraic eigenproblem and application of eigen-solving to polynomial
e) Polynomial and rational interpolation.
f) Algebraic techniques for coding and cryptography.
The seminar resumes with new topics and new students every semester.
If needed, the students are divided into the entry level group and the
advanced group. The instructor meets separately for 2 hours per week with
each group. The students in the entry level group study the fundamentals
and eventually join the group of advanced students. Survey and research
papers are supplied as handouts by the instructor. The instructor
guides the Computer Science students who wish to implement new algorithms
devised in the seminar, and also guides the Mathematics students in
solving the relevant open problems in math.