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%"Moro","Julio","jmoro@math.uc3m.es","
\section*{Structured H\""older condition numbers for eigenvalues under fully nongeneric perturbations} By {Mar\'{\i}a J.\ Pel\'aez and Julio Moro$^*$}.  

\medskip \noindent Let $\lambda$ be an eigenvalue of a matrix or operator $A$. The condition number $\kappa(A,\lambda)$  measures the sensitivity of $\lambda$ with respect  to arbitrary perturbations in $A$. If $A$ belongs to some relevant class, say ${\mathbb S}$, of structured operators, one can define the {\em structured} condition number $\kappa(A,\lambda;\mathbb{S})$, which measures the sensitivity of $\lambda$ to perturbations {\em within} the set ${\mathbb S}$. Whenever the structured condition number is much smaller than the unstructured one, thepossibility  opens for a structure-preserving spectral algorithm to be more accurate than a conventional one.  

\medskip  For multiple, possibly defective, eigenvalues the condition number is usually defined as a pair of nonnegative numbers, with the first component reflecting the worst-case asymptotic order which is to be expected from the perturbations in the eigenvalue. In this talk we adress the case when this asymptotic order differs for structured and for unstructured perturbations: if we denote $\kappa(A,\lambda)=(n,\alpha)$ and $\kappa(A,\lambda;\mathbb{S}) = (n_{{\mathbb S}},\alpha_{{\mathbb S}})$, we consider the case when $n\not= n_{{\mathbb S}}$, i.e., when structured perturbations induce a {\em qualitatively} different perturbation behavior than unstructured ones. If this happens, we say that the class ${\mathbb S}$ of perturbations is {\em fully nongeneric} for $\lambda$.  

\medskip  On one hand, full nongenericity is characterized in terms of the eigenvector matrices corresponding to $\lambda$, and it is shown that, for linear structures, this is related to the so-called skew-structure associated with ${\mathbb S}$.  On the other hand, we make use of Newton polygon techniques to obtain explicit formulas for structured condition numbers in the fully nongeneric case: both the asymptotic order and the largest possible leading coefficient are identified in the asymptotic expansion of perturbed eigenvalues for fully nongeneric perturbations.

%","eigenvalue problem, condition number, perturbation theory","65F15","15A18","this talk will be part of the minisymposium on ""Eigenproblems: theory and computation""","12:10:15","Tue Apr 01 2008","163.117.203.37"
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%"Sendov","Hristo","hssendov@stats.uwo.ca","
\section*{Spectral Manifolds} By {\sl A. Daniilidis, J. Malick, A. Lewis, H.S. Sendov$^*$}.  

\medskip \noindent 
It is well known that the set of all $n \times n$ symmetric matrices of rank $k$ is a smooth manifold. This set can be described as those symmetric matrices whose ordered vector of eigenvalues has exactly $n-k$ zeros. The set of all vectors in $\R^n$ with exactly $n-k$ zero entries is itself an analytic manifold.   In this work, we characterize the manifolds $M$ in $\R^n$ with the property that the set of all $n \times n$ symmetric matrices whose ordered vector of eigenvalues belongs to $M$ is a manifold. In particular, we show that if $M$ is a $C^2$, $C^{\infty}$, or $C^{\omega}$ manifold then so is the corresponding matrix set. We give a formula for the dimension of the matrix manifold in terms of the dimension of $M$.

%","eigenvalue, manifold, symmetric matrix","15A18","","","14:18:31","Tue Apr 01 2008","129.100.76.90"
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%"Vander Meulen","Kevin","kvanderm@cs.redeemer.ca","


\section*{Sparse Inertially Arbitrary Sign Patterns} By {\sl L. Vanderspek, M. Cavers, K.N. Vander Meulen$^*$}.  

\bigskip \noindent The inertia of a real matrix $A$ is an ordered triple $i(A)=(n_1,n_2,n_3)$ where $n_1$ is the number of eigenvalues of $A$ with positive real part, $n_2$ is the number of eigenvalues of $A$ with negative real part, and $n_3$ is the number of eigenvalues of $A$ with zero real part. A sign pattern is a matrix whose entries are in $\{ +,-,0\}$.  An order $n$ sign pattern $S$ is inertially arbitrary if for every ordered triple $(n_1,n_2,n_3)$ with $n_1+n_2+n_3=n$ there is a real matrix $A$ such that $A$ has sign pattern $S$ and $i(A)=(n_1,n_2,n_3)$. We describe some techniques in determining a pattern is inertially arbitrary. We present some irreducible inertially arbitrary patterns of order $n$ with less than $2n$ entries.
%","sign pattern, inertia, nilpotent","15A18","05C50","MS1 Combinatorial Matrix Theory","15:58:12","Tue Apr 01 2008","72.38.23.115"
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%"Frank","Martin","frank@mathematik.uni-kl.de","

\section*{An iterative method for transport equations in radiotherapy} By {\sl Bruno Dubroca \and Martin Frank$^*$}.  

\medskip \noindent Treatment with high energy ionizing radiation is one of the main methods in modern cancer therapy that is in clinical use. During the last decades two main approaches to dose calculation were used, Monte Carlo simulations and pencil-beam models. A third way to dose calculation has not attracted much attention in the medical physics community. This approach is based on deterministic transport equations of radiative transfer. In this work, we study a full discretization of the transport equation which yields a large linear system of equations. The computational challenge is that scattering is strongly forward-peaked, which means that traditional solution methods like source iteration fail in this case. Therefore we propose a new method, which combines an incomplete factorization of the scattering matrix and several iterative steps to obtain a fast and accurate solution. Numerical examples are given.
%","Iterative methods for linear systems; transport equations; radiotherapy","65F10","82C70","","06:59:33","Wed Apr 02 2008","131.246.168.20"

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