%% \end{algorithmic}
%% \end{algorithm}
+\begin{algorithm}
+\caption{Standard GMRES\index{GMRES} algorithm.} \label{alg:GMRES}
+ \KwIn{ $ A \in \mathbb{R}^{N \times N}$, $b \in \mathbb{R}^N$,
+ and initial guess $x_0 \in \mathbb{R}^N$}
+ \KwOut{ $x \in \mathbb{R}^N$: $\| b - A x\|_2 < tol$}
+
+ $r = b - A x_0$\;
+ $h_{1,0}=\left \| r \right \|_2$\;
+ $m=0$\;
+
+ \While{$m < max\_iter$} {
+ $m = m+1$;
+ $v_{m} = r / h_{m,m-1}$\;
+ \label{line:mvp} $r = A v_m$\;
+ \For{$i = 1\ldots m$} {
+ $h_{i,m} = \langle v_i, r \rangle$\;
+ $r = r - h_{i,m} v_i$\;
+ }
+ $h_{m+1,m} = \left\| r \right\|_2$\label{line:newnorm} \;
+ %\STATE Generate Givens rotations to triangularize $\tilde{H}_m$
+ %\STATE Apply Givens rotations on $h_{1,0}e_1$ to get residual $\epsilon$
+ Compute the residual $\epsilon$\;
+ \If{$\epsilon < tol$} {
+ Solve the problem: minimize $\|b-Ax_m\|_2$\;
+ Return $x_m = x_0 + V_m y_m$\;
+ }
+ }
+\end{algorithm}
+
+
At a first glance, the cost of using standard GMRES directly to
solve the Newton update in Eq.~\eqref{eq:Newton}
seems to come mainly from two parts: the
