= Generalized minimal residual method
{title2=$x_k=\operatorname*{argmin}_{x\in\mathcal K_k(A,b)}\|Ax-b\|_2$}
= GMRES
{c}
{synonym}
With zero initial guess, the generalized minimal residual method minimizes the <Euclidean norm> of the <linear-system residual> over each <Krylov subspace>. The <Arnoldi decomposition> reduces this to $\min_y\|\widetilde H_ky-\|b\|_2e_1\|_2$, followed by $x_k=Q_ky$. <Givens rotations> solve the small problem in $O(k^2)$ operations once the decomposition is available. For an <invertible matrix> $A$, the same iterate minimizes the error in the <weighted norm> induced by $A^TA$, because $\|Ax-b\|_2=\|x-x_*\|_{A^TA}$.
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