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[GMRES2stage.git] / paper.tex

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 \section{Related works}
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 \section{Related works}
 \label{sec:02} 

-GMRES method is one of the most widely used iterative solvers chosen to deal with the sparsity and the large order of linear systems. It was initially developed by Saad \& al.~\cite{Saad86} to deal with non-symmetric and non-Hermitian problems, and indefinite symmetric problems too. The convergence of the restarted GMRES with preconditioning is faster and more stable than those of some other iterative solvers. 
+%GMRES method is one of the most widely used iterative solvers chosen to deal with the sparsity and the large order of linear systems. It was initially developed by Saad \& al.~\cite{Saad86} to deal with non-symmetric and non-Hermitian problems, and indefinite symmetric problems too. The convergence of the restarted GMRES with preconditioning is faster and more stable than those of some other iterative solvers. 

 
 

-The next two chapters explore a few methods which are considered currently to be among the
-most important iterative techniques available for solving large linear systems. These techniques
-are based on projection processes, both orthogonal and oblique, onto Krylov subspaces, which
-are subspaces spanned by vectors of the form p(A)v where p is a polynomial. In short, these
-techniques approximate A −1 b by p(A)b, where p is a “good” polynomial. This chapter covers
-methods derived from, or related to, the Arnoldi orthogonalization. The next chapter covers
-methods based on Lanczos biorthogonalization.
+%The next two chapters explore a few methods which are considered currently to be among the most important iterative techniques available for solving large linear systems. These techniques are based on projection processes, both orthogonal and oblique, onto Krylov subspaces, which are subspaces spanned by vectors of the form p(A)v where p is a polynomial. In short, these techniques approximate A −1 b by p(A)b, where p is a “good” polynomial. This chapter covers methods derived from, or related to, the Arnoldi orthogonalization. The next chapter covers methods based on Lanczos biorthogonalization.

 
 

-Krylov subspace techniques have inceasingly been viewed as general purpose iterative methods, especially since the popularization of the preconditioning techniqes.
+%Krylov subspace techniques have inceasingly been viewed as general purpose iterative methods, especially since the popularization of the preconditioning techniqes.

 
 

-Preconditioned Krylov-subspace iterations are a key ingredient in
-many modern linear solvers, including in solvers that employ support
-preconditioners. 
+%Preconditioned Krylov-subspace iterations are a key ingredient in many modern linear solvers, including in solvers that employ support preconditioners. 

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