Acceleration of Relaxation Methods for Non-Hermitian Linear Systems

Abstract

Let $A = I - B \in \mathbb{C}^{n,n} $, with diag$(B) = 0$, denote a nonsingular non-Hermitian matrix. To iteratively solve the linear system $A{\bf x} = {\bf b}$, two splittings of A, together with induced relaxation methods, have been recently investigated in [W. Niethammer and R. S. Varga, Results in Math., 16 (1989), pp. 308–320]. The Hermitian splitting of A is defined by $A = M^h - N^h $, where $M^h : = ( A + A^ * )/2$ is the Hermitian part of A. The skew-Hermitian splitting of A is similarly defined by $A = M^s - N^s $ with $M^s : = I + ( A - A^ *)/2$. This paper considers k-step iterative methods to accelerate the relaxation schemes (involving a relaxation factor $\omega $) that are generated by these two splittings. The primary interest is not to determine the optimal relaxation factor $\omega $ that minimizes the spectral radius of the associated iteration operator. Rather, a value of $\omega $ is sought such that the resulting relaxation method can be most efficiently accelerated by a k-step method. For the Hermitian splitting, the choice $\omega = 1$ (together with a suitable Chebyshev acceleration) turns out to be optimal in this sense. For the skew-Hermitian splitting, a hybrid scheme is proposed that is nearly optimal. As another application of this latter hybrid procedure, the block Jacobi method arising from a model equation for a convection-diffusion problem is analyzed.