The Study For The Iteration Methods Of Two Classes Of Non-Hermitian Linear Systems

The iteration methods of the linear system Ax=b are studied in this paper,where A=W+tT∈C^n×n is a large sparse non-Hermitian matrix, with W，T∈R^n×n，and x，b∈Cⁿ. Complex linear systems of this kind arise in many problems inscientific computing and eigineering applications. It’s valuable to study the numericalsolution of them. Two classes of this kind linear systems are studied in this paper. Andwe give different iteration methods for either class as following.When the real part W and the imaginary part T of the cofficient matrix Aare nonsymmetric positive semidefinite and, at least, one of them is positive definite,we use the modified Hermitian and skew-Hermitian splitting （MHSS） iterationmethod to solve the non-Hermitian linear system Ax b. The MHSS iterationmethod converges unconditionally to the unique solution of the linear system. At eachstep, the MHSS iteration method requires to solve two linear sub-systems with realnonsymmetric positive definite coefficient matrices. Inner iteration methods are usedto compute the approximate solutions of these linear sub-systems. Two numericalexamples are utilized to illustrate the performance of the MHSS method and itsinexact variant.When the real part W and the imaginary part T of the cofficient matrix Aare symmetric positive semidefinite and, at least, one of them is positive definite, weintroduce and analyze the lopsided Hermitian and skew-Hermitian splitting （LHSS）method and modify it to solve the complex symmetric linear systems Ax b. Themodified LHSS （MLHSS） iteration method converges to the unique solution of thelinear system for a loose restriction on the parameter. We illustrate theperformance of our method through one numerical experiment.