Assume A is a real n × n matrix, b is an n × 1 vector, and we want to solve the system Ax = b. Assume that x_{0} is an initial guess for the solution, and r_{0} = b − Ax_{0} is the corresponding residual. The GMRES method looks for a solution of the form x_{m}= x_{0} + Q_{m}y_{m}, y_{m}∈ ^{m}where the columns of Q_{m}are an n-dimensional orthogonal basis for the Krylov subspace K_{m}(A, r_{0}) = {r_{0}, Ar_{0},…, A^{m−1}r_{0}}_{.} The vector y_{m}is chosen so the residual

${\Vert {r}_{m}\Vert}_{\text{2}}={\Vert b-A({x}_{0}+{Q}_{m}{y}_{m})\Vert}_{\text{2}}={\Vert {r}_{0}-A{Q}_{m}{y}_{m}\Vert}_{\text{2}}$

has minimal norm over K_{m}(A, r_{0}). This is a least-squares problem. We must find ...

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