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{\large Jos\'e Mas \\
{\bf Inverse Sherman-Morrison factorization and the NBIF preconditioner}}

Cami de Vera 14 \\ Edifici 3I \\ Universitat Politecnica de Valencia \\ Valencia \\ 46022 SPAIN
\\
{\tt jmasm@mat.upv.es}\\
Rafael Bru\\
Jos\'e Marin\\
Miroslav Tuma\end{center}


The Inverse Sherman-Morrison factorization of an
invertible matrix $A$, for a positive parameter $s$, is given by
\begin{equation*}
s^{-1}I - {A}^{-1} = s^{-2} Z_s D_s^{-1} V_s^T
\end{equation*}
where
\begin{equation*}
z_k=e_k-\sum_{i=1}^{k-1}\frac{v_i^T e_k}{s r_i} z_i
\ v_k=y_k-\sum_{i=1}^{k-1}\frac{y^T_k z_i}{sr_i}v_i,
\ \text{and} \ r_k=1+y^T_kz_k/s=1+v^T_k e_k/s
\end{equation*}
for $k=1,2,\ldots,n$, where
$e_k$ and $y_k$ are
the columns of the identity matrix $I$ and $Y=A^T-sI$, respectively.

It is known that from this factorization some factors of the LDU factorization
can be recovered, at the same time the inverse factors of the LDU factorization
are already computed, as can be deduced from the following relations
\begin{equation*}
{D}=s^{-1}D_s, \quad {U}= Z^{-1},
\quad V_s = {U}^T {D} - s {L}^{-T},
\end{equation*}

Since the ISM factorization is
highly parametrizable we analyze also some relations of the factors for
different choices of the parameter $s$, and relations with scaling of $A$.


We also present an algorithm (NBIF) to compute a preconditioner using the
ISM factorization and
the dropping strategies of Bollh\"ofer, taking advantage of more deep
relations between both factorizations.
We also prove existence properties of these preconditioners and present some numerical
results.

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\begin{footnotesize}
This work was supported by Spanish grant MTM 2007-64477,
also by the project No. IAA100300802 of the Grant Agency of the Academy
of Sciences of the Czech Republic
and partially by the international collaboration support M100300902
of AS CR.
\end{footnotesize}


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