$S$ composed by the successive solutions that are computed during inner iterations.
At each $s$ iterations, the minimization step is applied in order to
-compute a new solution $x$. For that, the previous residuals are computed with
-$(b-AS)$. The minimization of the residuals is obtained by
+compute a new solution $x$. For that, the previous residuals of $Ax=b$ are computed by
+the inner iterations with $(b-AS)$. The minimization of the residuals is obtained by
\begin{equation}
\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2
\label{eq:01}
\State $S_{k \mod s}=x^k$ \label{algo:store}
\If {$k \mod s=0$ {\bf and} error$>\epsilon_{kryl}$}
\State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul}
- \State Solve least-square problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$ \label{algo:}
+ \State $\alpha=Solve\_Least\_Squares(R,b,max\_iter_{ls})$ \label{algo:}
\State $x^k=S\alpha$ \Comment{compute new solution}
\EndIf
\EndFor
