On Convergence and Implementation of Minimal Residual KrylovSubspace Methods for Unsymmetric Linear Systems

Abstract

Consider linear algebraic systems $Ax = b$ with a general unsymmetric nonsingular matrix A. We study Krylov subspace methods for solving such systems that minimize the norm of the residual at each step. Such methods are often formulated in terms of a sequence of least squares problems of increasing dimension. Therefore we begin with an overdetermined least squares problem $Bu \approx c$ and present several basic identities and bounds for the least squares residual $r = c- By$.Then we apply these results to minimal residual Krylov subspace methods, and formulate several theoretical consequences about their convergence. We consider possible implementations, in particular various forms of the GMRES method [26], and discuss their numerical properties. Finally, we illustrate our findings by numerical examples and draw conclusions.