**Fields Institute Monographs**

Volume: 18;
2003;
136 pp;
Hardcover

MSC: Primary 30; 47; 65;

Print ISBN: 978-0-8218-3247-9

Product Code: FIM/18

List Price: $57.00

AMS Member Price: $45.60

MAA Member Price: $51.30

**Electronic ISBN: 978-1-4704-3145-7
Product Code: FIM/18.E**

List Price: $57.00

AMS Member Price: $45.60

MAA Member Price: $51.30

# Meromorphic Functions and Linear Algebra

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*Olavi Nevanlinna*

A co-publication of the AMS and Fields Institute

This volume describes for the first time in monograph form important
applications in numerical methods of linear algebra. The author presents new
material and extended results from recent papers in a very readable style.

The main goal of the book is to study the behavior of the resolvent
of a matrix under the perturbation by low rank matrices. Whereas the
eigenvalues (the poles of the resolvent) and the pseudospectra (the
sets where the resolvent takes large values) can move dramatically
under such perturbations, the growth of the resolvent as a
matrix-valued meromorphic function remains essentially unchanged. This
has practical implications to the analysis of iterative solvers for
large systems of linear algebraic equations.

First, the book introduces the basics of value distribution theory of
meromorphic scalar functions. It then introduces a new nonlinear tool for
linear algebra, the total logarithmic size of a matrix, which allows for a
nontrivial generalization of Rolf Nevanlinna's characteristic function from the
scalar theory to matrix- and operator-valued functions. In particular, the
theory of perturbations by low rank matrices becomes possible. As an example,
if the spectrum of a normal matrix collapses under a low rank perturbation,
there is always a compensation in terms of the loss of orthogonality of the
eigenvectors. This qualitative phenomenon is made quantitative by using the new
tool. Applications are given to rational approximation, to the Kreiss matrix
theorem, and to convergence of Krylov solvers.

The book is intended for researchers in mathematics in general and
especially for those working in numerical linear algebra. Much of the
book is understandable if the reader has a good background in linear
algebra and a first course in complex analysis.

Titles in this series are co-published with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

#### Readership

Graduate students and research mathematicians interested in numerical methods.