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## 21.7 GMRES

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 x0 is an initial guess for the solution, and r0 = bAx0 is the corresponding residual. The GMRES method looks for a solution of the form xm= x0 + Qmym, ymmwhere the columns of Qmare an n-dimensional orthogonal basis for the Krylov subspace Km(A, r0) = {r0, Ar0,…, Am−1r0}. The vector ymis chosen so the residual

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

has minimal norm over Km(A, r0). This is a least-squares problem. We must find ...

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