19.num/BNP.fernando3.e
.m From demmel@zil.CS.Berkeley.EDU Thu Sep 23 12:45:04 1993
.ls 2
.na
.LP
.EQ
delim $$
gfont Roman
.EN
Sharp Inclusion Regions Using tr($M$) and tr($M sup 2$)

Dr. V. Fernando*

(Professor B. Parlett) 

(NSF) ASC-90-05933

Suppose that all zeros of a polynomial \fIp\fR of degree \fIn\fR are real and
suppose that, for some other value \fIx\fR, the values p(x), p(x),
and p(x) are known. The problem of finding strict outer bounds
on the extreme zeros was solved by Laguerre, and the related problem of
finding strict inner bounds on the extreme zeros was solved by Fejer.
In applications concerning a symmetric matrix \fIM\fR more information
is often available; for example, one may know the number of eigenvalues
greater than \fIx\fR or one may know that all eigenvalues are simple.
We present best possible bounds in such cases and show how much the
classical bounds may be improved.

Our method produces formulae for the loci of all eigenvalues of any
given multiplicity \fIm\fR. Beyond that our approach yields representations
of various curves in a certain parameter on which eigenvalues must lie.

*Numerical Algorithms Group, Oxford, England