% paper title
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\title{TSIRM: A Two-Stage Iteration with least-square Residual Minimization algorithm to solve large sparse linear systems}
-%où
-%\title{A two-stage algorithm with error minimization to solve large sparse linear systems}
-%où
-%\title{???}
+
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-\section{Two-stage algorithm with least-square residuals minimization}
+\section{Two-stage iteration with least-square residuals minimization algorithm}
\label{sec:03}
A two-stage algorithm is proposed to solve large sparse linear systems of the
form $Ax=b$, where $A\in\mathbb{R}^{n\times n}$ is a sparse and square
In practice, $R$ is a dense rectangular matrix belonging in $\mathbb{R}^{n\times s}$,
-with $s\ll n$. In order to minimize~(\eqref{eq:01}), a least-square method such as
+with $s\ll n$. In order to minimize~\eqref{eq:01}, a least-square method such as
CGLS ~\cite{Hestenes52} or LSQR~\cite{Paige82} is used. Remark that these methods are more
appropriate than a single direct method in a parallel context.
solution $x_k$ into the column $k~ mod~ s$ of the matrix $S$. After the
minimization, the matrix $S$ is reused with the new values of the residuals. To
solve the minimization problem, an iterative method is used. Two parameters are
-required for that: the maximum number of iteration and the threshold to stop the
+required for that: the maximum number of iterations and the threshold to stop the
method.
Let us summarize the most important parameters of TSIRM:
interesting to solve the least-square minimization, CGLS and LSQR.
In the following we remind the CGLS algorithm. The LSQR method follows more or
-less the same principle but it take more place, so we briefly explain the parallelization of CGLS which is similar to LSQR.
+less the same principle but it takes more place, so we briefly explain the parallelization of CGLS which is similar to LSQR.
\begin{algorithm}[t]
\caption{CGLS}
In each iteration of CGLS, there is two matrix-vector multiplications and some
-classical operations: dots, norm, multiplication and addition on vectors. All
+classical operations: dot product, norm, multiplication and addition on vectors. All
these operations are easy to implement in PETSc or similar environment.
