正则化HSS预处理鞍点矩阵的特征值估计

最近，Bai和Benzi针对鞍点问题提出了一类正则化HSS（Regularized Hermitian and skew-Hermitian splitting，RHSS）预处理子（BIT Numer.Math.，57（2017）287-311）.为了进一步分析RHSS预处理子的效果，本文重点研究了RHSS预处理鞍点矩阵特征值的估计，分析了复特征值实部和模的上下界、实特征值的上下界，还给出了特征值均为实数的充分条件.当正则化矩阵取为零矩阵时，RHSS预处理子退化为HSS预处理子，分析表明本文给出的复特征值实部的界比已有的结果更精确.数值算例验证了本文给出的理论结果.

EIGENVALUE ESTIMATES OF THE REGULARIZED HSS PRECONDITIONED SADDLE POINT MATRIX

Recently, Bai and Benzi proposed a class of regularized HSS (RHSS) preconditioners for saddle point problems (BIT Numer. Math., 57 (2017) 287–311). To further analyze the effectiveness of the RHSS preconditioners, in this paper the eigenvalue estimates of the RHSS preconditioned saddle point matrix will be studied. The upper and the lower bounds for the real part and the modulus of the complex eigenvalues are analyzed. The upper and the lower bounds for the real eigenvalues are obtained. In addition, a sufficient condition for the eigenvalues to be real is derived. If the regularization matrix is set to be the zero matrix, then the RHSS preconditioner is reduced to the HSS preconditioner. Theoretical results show that the bounds for the real part of the complex eigenvalues are more accurate than the existing results. Numerical experiments are used to verify the theoretical results.