Relaxation Methods for Non-Hermitian Linear Systems

Abstract

For the matrix equation Ax = b, we consider here two splittings A = M1 − N1 = M2 − N2 of the matrix A, where M1 ≔ (A + A*)/2 is the Hermitian part of A, and M2 ≔ I + (A − A*)/2 is the identity plus the skew-Hermitian part of A. To these two splittings of A, we apply an extrapolation, with extrapolation factor ω, and we find associated regions for ω, in the complex plane, for which these extrapolated splittings yield convergent iterative methods. From this, further applications to semiiterative methods are indicated.