===firstname:
Jos\'e
===firstname3:
Danny
===affil6:

===lastname3:
Guerrero
===email:
jmarinma@imm.upv.es
===keyword_other2:

===lastname6:

===affil5:
Institut de Matematica Multidisciplinar
Universitat Politecnica de Valencia
===lastname4:

===lastname7:

===affil7:

===postal:
Institut de Matematica Multidisciplinar
Universitat Politecnica de Valencia
Camí de Vera, 14
46022 Valencia
Spain
===ABSTRACT:
\newcommand{\reals}{\mathbb{R}}

In this work we study the iterative solution of nonsingular, nonsymmetric linear systems of $n$ equations
\begin{equation} Ax=b \end{equation}
where the skew-symmetric part of the coefficient matrix $A$ can be approximated by a low-rank matrix.  
Consider $A=H+K$ where  $H$ and $K$ are the symmetric and skew-symmetric parts of $A$, respectively.  It is assumed that the skew-symmetric matrix can be written as $K=PQ^T + E$ for some full rank $P,Q \in \reals^{n \times s}$ with $k \ll n$, and $\| E \| \ll 1$.  

Different strategies have been proposed when the skew-symmetric part $K$ has exactly rank $s \ll n$.  In \cite{Beckermann:2008} it is presented  a progressive GMRES method that allows for the short-term computation of an orthogonal Krylov subspace basis. As pointed out in \cite{ESSSX.2012}, although the method is mathematically equivalent to full GMRES \cite{sasc:86} in practice it may suffer from instabilities due to the loss of orthogonality between the vectors of the generated Krylov subspace basis. In the same paper, the authors propose a Schur complement method that also permits the application of short-term formulas. The method obtains an 
approximate solution by applying the MINRES method $s+1$ times. The authors also suggest that can be successfully applied as a preconditioner for GMRES for the problem considered in this paper.

We study a method based on the framework proposed in \cite{cmm:16}. Assuming that the matrix $H+PQ^T$ is nonsingular, our approach computes an approximate LU factorization of the matrix
\begin{equation}\label{eq:augmented}
\left[
\begin{array}{cc}
		H  &  P\\
	-Q^T & I	
\end{array}
\right]
\end{equation}
with the Balanced Incomplete Factorization (BIF) algorithm \cite{bmmt:08,bmmt:10}. Interestingly, the matrix in (\ref{eq:augmented}) is
similar to the one used in \cite{ESSSX.2012} to develop the Schur complement method, but in this work it is used to update a previously computed preconditioner for the symmetric part $H$. Then, the factorization is used as a preconditioner for the GMRES method. The results of the numerical experiments for different problems will be presented.

\begin{thebibliography}{1}

\bibitem{Beckermann:2008}
Bernhard Beckermann and Lothar Reichel.
\newblock The {A}rnoldi process and {GMRES} for nearly symmetric matrices.
\newblock {\em SIAM J. Matrix Anal. Appl.}, 30(1):102--120, February 2008.

\bibitem{bmmt:08}
R.~Bru, J.~Mar{\'{\i}}n, J.~Mas, and M.~T\accent23uma.
\newblock Balanced incomplete factorization.
\newblock {\em SIAM J. Sci. Comput.}, 30(5):2302--2318, 2008.

\bibitem{bmmt:10}
R.~Bru, J.~Mar{\'{\i}}n, J.~Mas, and M.~T\accent23uma.
\newblock Improved balanced incomplete factorization.
\newblock {\em SIAM J. Matrix Anal. Appl.}, 31(5):2431--2452, 2010.

\bibitem{cmm:16}
J.~Cerd\'an, J.~Mar\'{\i}n, and J.~Mas.
\newblock Low-rank updates of balanced incomplete factorization
  preconditioners.
\newblock {\em Submitted.}

\bibitem{ESSSX.2012}
Mark Embree, Josef~A. Sifuentes, Kirk~M. Soodhalter, Daniel~B. Szyld, and Fei
  Xue.
\newblock Short-term recurrence {K}rylov subspace methods for nearly hermitian
  matrices.
\newblock {\em SIAM. J. Matrix Anal. and Appl.}, 33-2:480--500, 2012.

\bibitem{sasc:86}
Y.~Saad and M.~H. Schulz.
\newblock {GMRES}: {A} generalized minimal residual algorithm for solving
  nonsymmetric linear systems.
\newblock {\em {SIAM} Journal on Scientific and Statistical Computing},
  7:856--869, 1986.

\end{thebibliography}
===affil3:
Institut de Matematica Multidisciplinar
Universitat Politecnica de Valencia
===title:
Preconditioners for nonsymmetric linear systems with low-rank skew-symmetric part
===affil2:
Institut de Matematica Multidisciplinar
Universitat Politecnica de Valencia
===lastname2:
Cerd\'an
===firstname4:

===keyword1:
Solvers for indefinite systems
===workshop:
no
===lastname:
Mar{\'{\i}}n
===firstname5:
Jos\'e
===keyword2:
NOT_SPECIFIED
===otherauths:

===affil4:

===competition:
no
===firstname7:

===firstname6:

===keyword_other1:

===lastname5:
Mas
===affilother:

===firstname2:
Juana