On generalized symmetric SOR method for augmented systems

Abstract

In this paper, we establish the generalized symmetric SOR method (GSSOR) for solving the large sparse augmented systems of linear equations, which is the extension of the SSOR iteration method. The convergence of the GSSOR method for augmented systems is studied. Numerical resume shows that this method is effective.

The Semi-convergence of Generalized SSOR Method for Singular Augmented Systems

“…When A is singular, the following two lemmas give the semi-convergence property about the iteration method (2). Before the semi-convergence of the GSSOR method is discussed, now, we review the GSSOR method presented in [18]. In fact, the augmented systems (1) can be written as the following equivalent form…”

“…When the augmented systems (1) are nonsingular, the convergence of GSSOR method is studied in [18]. When r < n, the matrix A is singular.…”

“…By (8), µ max is also the maximum eigenvalue of Q −1 1 B T r A −1 B r . From Theorem 3.2 of [18], we know that ρ( H 1 (ω, τ )) < 1 if 0 < ω < 2 and τ satisfy (7). By the above analysis, the proof of the theorem is completed.…”

“…The preconditioned iterative methods [8,9], the inexact Uzawa methods [6,10], the SOR-like method [7], the general SOR (GSOR) method [13], the symmetric SOR (SSOR) method [15,16] and the modified SSOR (MSSOR) method [17] for solving the nonsingular augmented systems (1) were proposed and analyzed, respectively. Furthermore, the general symmetric SOR (GSSOR) method [18] was presented. In most cases, the matrix B is full column rank in scientific computing and engineering applications, but not always.…”

“…In this paper, the GSSOR method for solving singular linear augmented systems (1) is further investigated and the semi-convergence conditions are proposed, which generalize the result of Zhang and Lu [18] for the nonsingular augmented systems to the singular augmented systems.…”