Hermitian and skew-Hermitian splitting methods for non-hermitian positive definite linear systems
Hermitian and skew-Hermitian splitting methods for non-hermitian positive definite linear systems
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DOI：
10.1137/s0895479801395458
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发表时间：
2003-01-01
影响因子：
1.5
通讯作者：
Ng, MK
中科院分区：
文献类型：
--
作者：
Bai, ZZ;Golub, GH;Ng, MK
基于系数矩阵的Hermitian分裂和斜Hermitian分裂，研究了大型稀疏非Hermitian正定线性方程组的高效迭代算法.这些方法包括Hermitian/skew-Hermitian分裂（HSS）迭代和它的不精确变体--不精确Hermitian/skew-Hermitian分裂（IHSS）迭代，IHSS迭代在外HSS迭代的每一步中采用Krylov子空间方法作为内迭代过程。理论分析表明，HSS方法无条件收敛于线性方程组的唯一解。此外，我们推导出一个上界的HSS迭代的收缩因子，这是完全依赖于埃尔米特部分的频谱和所涉及的矩阵的特征向量无关。数值算例说明了HSS和IHSS迭代的有效性。最后，通过一个三维对流扩散方程的模型问题，说明了本文方法的优越性.
We study efficient iterative methods for the large sparse non-Hermitian positive definite system of linear equations based on the Hermitian and skew-Hermitian splitting of the coefficient matrix. These methods include a Hermitian/skew-Hermitian splitting (HSS) iteration and its inexact variant, the inexact Hermitian/skew-Hermitian splitting (IHSS) iteration, which employs some Krylov subspace methods as its inner iteration processes at each step of the outer HSS iteration. Theoretical analyses show that the HSS method converges unconditionally to the unique solution of the system of linear equations. Moreover, we derive an upper bound of the contraction factor of the HSS iteration which is dependent solely on the spectrum of the Hermitian part and is independent of the eigenvectors of the matrices involved. Numerical examples are presented to illustrate the effectiveness of both HSS and IHSS iterations. In addition, a model problem of a three-dimensional convection-diffusion equation is used to illustrate the advantages of our methods.