ИСТИНА

We consider prerequisites of the GMRES as a method of minimization of the residual on the Krylov subspaces generated by a given nonsingular matrix and by initial residual, the optimality results and convergence estimates starting from the pioneering paper [1], and some recent developments including the convergence theory for special classes of matrices [2,3]. [1] G. I. Marchuk, Yu. A. Kuznetsov, Iterative methods and quadratic functionals (in Russian). Translated as Methodes iteratives et fonctionnelles quadratiques, in Sur les methodes numeriques en sciences physique et economique, J. Kraus and U. Langer, eds., Nauka, Novosibirsk, Dunod, Paris, 1972, 1974, pp. 3-132. [2] B. Beckermann, S.A. Goreinov, E.E.Tyrtyshnikov, Some remarks on the Elman estimate for GMRES, SIMAX, Vol. 27, Issue 3 (2006), 772-778. [3] E. E. Tyrtyshnikov, N. L. Zamarashnkin, A. Yu. Yeremin, Clusters, preconditioners, convergence, Linear Algebra Appl., 263 (1997), pp. 25-48. The work is supported by the RFBR grant 14-01-00804.