ACCELERATION OF RELAXATION METHODS FOR NON-HERMITIAN LINEAR-SYSTEMS

Let A = I - B is-an-element-of C(n,n), with diag(B) = 0, denote a nonsingular non-Hermitian matrix. To iteratively solve the linear system Ax = 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.