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\title{Computation of the fundamental singular subspace of large matrices
for pseudospectrum determinations}
\author{\underline{B. Philippe} \thanks{INRIA-IRISA,
Campus de Beaulieu, 35042 RENNES Cedex, France ;
email : {\tt philippe@irisa.fr}} \and M. Sadkane
\thanks{Universit\'e de Bretagne Occidentale,
Facult\'e des Sciences et Techniques, 6, avenue V. Le Gorgeu, B.P. 809,
29285 Brest Cedex, France ;
email : {\tt sadkane@univ-brest.fr} }}
\date{\empty}


\maketitle
\begin{abstract}

The eigenanalysis of matrices or operators based only on the
knowledge of the spectrum may be misleading in the non-normal case. Instead
of the spectrum, one may fully characterize the spectral behavior of a
non-normal matrix by analyzing its pseudospectrum or
spectral portrait, i.e. the set of
its resolvent norm.
The construction of a spectral portrait relies on a heavy use of the
computation of the smallest singular value of a sequence of matrices. We
consider here the case of large sparse matrices.

In a first part, we present a theoretical comparison of
the block-Lanczos and the Davidson methods for computing
a basis of a singular subspace associated with the smallest singular values.
We introduce a simple modification on the preconditioning
step of Davidson's method which
appears to be efficient on a range of large sparse matrices \cite{svd}.

We then propose parallel versions of the
Davidson method for analyzing and plotting the spectral
portrait of large non-normal matrices. We report the performance results
obtained on the machine Paragon \cite{para}.
\end{abstract}

\begin{thebibliography}{10}

\bibitem{svd}
B. Philippe and M. Sadkane.
\newblock Computation of the fundamental singular subspace of a large matrix.
\newblock {\em  Lin. Alg. Applic.}, 257 (1997), 77-104.

\bibitem{para}{ V.~Heuveline, B.~Philippe, M.~Sadkane},
\newblock {Parallel Computation of Spectral Portrait of Large Matrices by
  Davidson type Methods},
\newblock  \emph{Numerical Algorithms}, To appear.

\end{thebibliography}

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