Convergence on successive over-relaxed iterative methods for non-Hermitian positive definite linear systems

Cheng-Yi Zhang,Yan Zhu,Guangyan Miao

doi:10.1186/s13660-016-1100-6

Abstract

Some convergence conditions on successive over-relaxed (SOR) iterative method and symmetric SOR (SSOR) iterative method are proposed for non-Hermitian positive definite linear systems. Some examples are given to demonstrate the results obtained.

Highlights

We propose some convergence conditions on successive over-relaxed (SOR) and symmetric SOR (SSOR) iterative methods for non-Hermitian positive definite linear systems
Many problems in scientific computing give rise to a system of n linear equationsAx = b, A = ∈ Cn×n is nonsingular, and x, b ∈ Cn, ( )where A is a large sparse non-Hermitian positive definite matrix, that is, its Hermitian part H = (A + A∗)/ is Hermitian positive definite, where A∗ denotes the conjugate transpose of a matrix A
Considerable interest appears in the work on the Hermitian and skew-Hermitian splitting (HSS) method for this system introduced by Bai et al [ ] and some generalized HSS methods such as the NSS method [ ], PSS method [ ], PHSS method [, ], and LHSS method [ ], and several significant theoretical results are proposed

Summary

Introduction

We propose some convergence conditions on SOR and SSOR iterative methods for non-Hermitian positive definite linear systems. The SOR iterative method converges to the unique solution of ( ) for any choice of the initial guess x , that is, ρ(Lω) < if and only if < ω < /η if η > or /η < ω if η < or < ω if η = , where x∗[(I – U)∗(I – U) – L∗L]x η=

Results

Conclusion