Algebraic Numerical Computing Seminar
Professor Pan
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 
programs).

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 
algebra.
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 
root-finding. 
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.