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Prof. Anne Greenbaum
Department of Mathematics University of Washington Monday , March 19th, 103 Schaeffer Hall, 10:30 am - 11:20 am |
ABSTRACT:
Let A be a given n by n matrix. In order to estimate ||f(A)|| for
various functions f, it is helpful is one can associate A with some
set in the complex plane and relate ||f(A)|| to the size of f on this
set. For normal matrices, an appropriate set is the spectrum of A,
but for nonnormal matrices it is less clear what (if any) set(s) in
the complex plane should be associated with A. For a given positive
integer k, we identify the largest set Omega in the complex plane
satisfying ||p(A)|| >= max_{z in Omega} |p(z)| for all polynomials
p of degree k or less. We refer to this as the effective spectrum
of A for degree k, since, under the action of kth degree polynomials,
A behaves like a normal matrix with eigenvalues throughout this set
(or, at least, ||p(A)|| is greater than or equal to ||p(B)|| for any
such normal matrix B). For k=1, this set is just the field of values
of A, and for k >= m, where m is the degree of the minimal polynomial
of A, it is the spectrum of A. For 1 < k < m, these sets are intermediate
between the field of values and the spectrum and sometimes closely
resemble pseudospectra.
Prof. Anne Greenbaum is
currently a Professor in the Math Dept. at the University of
Washington. She received her PhD from UC Berkeley in 1981, worked
at Lawrence Livermore National Lab from 1974 to 1986,
and was at the Courant Institute from 1986-1997. She is ther author of
a book entitled "Iterative Methods for Solving Linear Systems,"
published by SIAM.