-\end{equation}
-where $\rho$ is the spectral radius of the square matrix. The different linear splittings~(\ref{eq:03}) arising from the multisplitting of matrix $A$can be solved exactly with a direct method or approximated with an iterative method. When the inner method used to solve the linear sub-systems is iterative, the multisplitting method is called {\it inner-outer iterative method} or {\it two-stage multisplitting method}.
-
-In this paper we are focused on two-stage multisplitting methods where the well-known iterative method GMRES is used as an inner iteration.
+\end{equation}
+At each $s$ outer iterations, the algorithm computes a new approximation $\tilde{x}=S\alpha$ which minimizes the residual
+\begin{equation}
+\min_{\alpha\in\mathbb{R}^s}{\|b-AS\alpha\|_2}.
+\label{eq:06}
+\end{equation}
+The algorithm in Figure~\ref{alg:02} includes the procedure of the residual minimization and the outer iteration is restarted with a new approximation $\tilde{x}$ at every $s$ iterations. The least-squares problem~(\ref{eq:06}) is solved in parallel by all clusters using CGLS method~\cite{Hestenes52} such that $\MIC$ is the maximum number of iterations and $\TOLC$ is the tolerance threshold for this method (line~\ref{cgls} in Figure~\ref{alg:02}).
+
+\begin{figure}[t]
+%\begin{algorithm}[t]
+%\caption{Krylov two-stage method using block Jacobi multisplitting}
+\begin{algorithmic}[1]
+ \Input $A_\ell$ (sparse matrix), $b_\ell$ (right-hand side)
+ \Output $x_\ell$ (solution vector)\vspace{0.2cm}
+ \State Set the initial guess $x^0$
+ \For {$k=1,2,3,\ldots$ until convergence}
+ \State $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m^{k-1}$
+ \State $x^k_\ell=Solve_{gmres}(A_{\ell\ell},c_\ell,x^{k-1}_\ell,\MIG,\TOLG)$
+ \State $S_{\ell,k\mod s}=x_\ell^k$
+ \If{$k\mod s = 0$}
+ \State $\alpha = Solve_{cgls}(AS,b,\MIC,\TOLC)$\label{cgls}
+ \State $\tilde{x_\ell}=S_\ell\alpha$
+ \State Send $\tilde{x_\ell}$ to neighboring clusters
+ \Else
+ \State Send $x_\ell^k$ to neighboring clusters
+ \EndIf
+ \State Receive $\{x_m^k\}_{m\neq\ell}$ from neighboring clusters
+ \EndFor
+\end{algorithmic}
+\caption{Krylov two-stage method using block Jacobi multisplitting}
+\label{alg:02}
+%\end{algorithm}
+\end{figure}
