From: "Neil Judell"
Newsgroups: sci.math.num-analysis
Subject: Re: Determinant of Toeplitz matrix
Date: Sat, 20 Dec 1997 22:13:55 -0500
[This was a multi-part message in MIME format; streamlined -- djr]
Uh, I guess I'm not clear on EXACTLY what you're asking. But, with
respect to determinants of Toeplitz matrices, the following two things
MIGHT help you in what you're looking for:
1) The determinant of a Toeplitz matrix is equal to the product of
(1-|k-sub-i|^2), where k-sub-i is the i'th reflection coefficient of the
Levinson recursion.
2) limit k->infinity |R_subk|^(1/k) is equal to exp((1/4*pi) loop
integral z^-1 ln(|H(z)|^2) dz), where H(z) is the generating polynomial
for the row of R.
An excellent reference book for all kinds of lovely properties of
Toeplitz Matrices is an old book "Toeplitz Matrices and their
Applications" by Grenander and Szego. I'm not sure, but it may be out
of print now.
Does this help any?
--neil
serena ng wrote in message <349C3316.D780D8DA@bc.edu>...
Hi
I need to evaluate the determinant of a k-th order Toeplitz matrix
to that of one of order k+1. This amounts to considering
A1 x1^k +A2 x2^k + ....
_____________
A1 x1^(k+1) + A2 x2 ^(k+1) + .....
I know as k goes to infinity this limit is 1. But does someone know
what this limit is as a function of k?
Thanks in advance.