On GMRES-equivalent bounded operators (Q2706250)

The author studies the generalized minimal residual (GMRES) method applied to some operator equation \(Ax= r\) in a Hilbert space \(H\), where the operator \(A\in L(H)\) is supposed to be linear and bounded. At the \(k\)th step, the GMRES produces an approximate solution which minimizes the residual norm \(\|Ax-r\|\) over the Krylov subspace \(K^k(A,r):= \text{span}\{A^0 r,A^1r,\dots, A^{k-1}r\}\).NEWLINENEWLINENEWLINEThe author constructs a unitary operator \(U\), a nonnegative definite, bounded selfadjoint operator \(P\) and an indefinite, bounded selfadjoint operator \(T\) such that all corresponding Krylov subspace \(K^k(A, R)\), \(K^k(U,r)\), \(K^k(P, r)\) and \(K^k(T, r)\) of the same dimension are equal. It is clear that this result can be heavily used in the analysis of the convergence properties of the GMRES.NEWLINENEWLINENEWLINEThe author reformulates all these results for the matrix case that is of most practical interest from the point of view of solving systems of linear algebraic equations.