\Input $A$ (sparse matrix), $b$ (right-hand side)
\Output $x$ (solution vector)\vspace{0.2cm}
\State Set the initial guess $x^0$
- \For {$k=1,2,3,\ldots$ until convergence} \label{algo:conv}
+ \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon$)} \label{algo:conv}
\State $x^k=Solve(A,b,x^{k-1},m)$ \label{algo:solve}
+ \State retrieve error
\State $S_{k~mod~s}=x^k$ \label{algo:store}
- \If {$k$ mod $s=0$ {\bf and} not convergence}
+ \If {$k$ mod $s=0$ {\bf and} error$>\epsilon$}
\State $R=AS$ \Comment{compute dense matrix}
- \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$
+ \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$ \label{algo:}
\State $x^k=S\alpha$ \Comment{compute new solution}
\EndIf
\EndFor
Algorithm~\ref{algo:01} summarizes the principle of our method. The outer
iteration is inside the for loop. Line~\ref{algo:solve}, the Krylov method is
called for a maximum of $m$ iterations. In practice, we suggest to choose $m$
-equals to the restart number of the GMRES like method. Line~\ref{algo:store},
-$S_{k~ mod~ s}=x^k$ consists in copying the 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.
+equals to the restart number of the GMRES-like method. Moreover, a tolerance
+threshold must be specified for the solver. In practise, this threshold must be
+much smaller than the convergence threshold of the TSARM algorithm
+(i.e. $\epsilon$). Line~\ref{algo:store}, $S_{k~ mod~ s}=x^k$ consists in
+copying the 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. % à continuer Line
+
+To summarize, the important parameters of are:
+\begin{itemize}
+\item $\epsilon$ the threshold to stop the method
+\item $m$ the number of iterations for the krylov method
+\item $s$ the number of outer iterations before applying the minimization step
+\end{itemize}
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