Krylov subspace methods for solving large unsymmetric linear systems

Abstract

Some algorithms based upon a projection process onto the Krylov subspace K m = Span ⁡ ( r 0 , A r 0 , … , A m − 1 r 0 ) {K_m} = \operatorname {Span}({r_0},A{r_0}, \ldots ,{A^{m - 1}}{r_0}) are developed, generalizing the method of Conjugate gradients to unsymmetric systems. These methods are extensions of Arnoldi’s algorithm for solving eigenvalue problems. The convergence is analyzed in terms of the distance of the solution to the subspace K m {K_m} and some error bounds are established showing, in particular, a similarity with the conjugate gradient method (for symmetric matrices) when the eigenvalues are real. Several numerical experiments are described and discussed.