On successive‐overrelaxation acceleration of the Hermitian and skew‐Hermitian splitting iterations
On successive‐overrelaxation acceleration of the Hermitian and skew‐Hermitian splitting iterations
复制标题
DOI：
10.1002/nla.517
复制
发表时间：
2007-05
影响因子：
4.3
通讯作者：
Z. Bai;G. Golub;M. Ng
中科院分区：
文献类型：
--
作者：
Z. Bai;G. Golub;M. Ng
我们进一步将求解大型稀疏非厄米正定线性方程组的厄米/斜厄米分裂（HSS）迭代方法推广到正态/斜厄米分裂（NS）迭代，得到了一类正态/斜厄米分裂（NSS）迭代方法。理论分析表明，NSS方法无条件收敛于线性方程组的精确解。此外，我们还导出了NSS迭代的收缩因子的上界，该上界仅依赖于正分裂矩阵的谱，而与所涉及的矩阵的特征向量无关。我们提出了NSS迭代的连续-超松弛（SOR）加速方案，具体地说，这导致了HSS迭代的加速方案。在相应的块Jacobi迭代矩阵的特征值位于复平面的某些区域的假设下，导出了该SOR方案的收敛条件。数值算例表明，SOR技术可以显著加快NSS或HSS迭代方法的收敛速度。版权所有©2006约翰威利父子有限公司
We further generalize the technique for constructing the Hermitian/skew‐Hermitian splitting (HSS) iteration method for solving large sparse non‐Hermitian positive definite system of linear equations to the normal/skew‐Hermitian (NS) splitting obtaining a class of normal/skew‐Hermitian splitting (NSS) iteration methods. Theoretical analyses show that the NSS method converges unconditionally to the exact solution of the system of linear equations. Moreover, we derive an upper bound of the contraction factor of the NSS iteration which is dependent solely on the spectrum of the normal splitting matrix, and is independent of the eigenvectors of the matrices involved. We present a successive‐overrelaxation (SOR) acceleration scheme for the NSS iteration, which specifically results in an acceleration scheme for the HSS iteration. Convergence conditions for this SOR scheme are derived under the assumption that the eigenvalues of the corresponding block Jacobi iteration matrix lie in certain regions in the complex plane. A numerical example is used to show that the SOR technique can significantly accelerate the convergence rate of the NSS or the HSS iteration method. Copyright © 2006 John Wiley & Sons, Ltd.