求解连续Sylvester方程的预处理非对称HSS分裂迭代法

Preconditioned asymmetric HSS iteration method for continuous Sylvester equations

针对系数矩阵为大型非Hermitian正定/半正定稀疏矩阵的连续Sylvester方程组,提出了预处理不对称的埃尔米特和反埃尔米特分裂(PAHSS)迭代方法,并对所提算法进行了收敛性分析,讨论了PAHSS方法的准最优参数.为了进一步减少计算量,在内迭代求解子线性方程组时,基于该子线性系统具有特殊结构,采用某种有效的迭代方法去求解,得到了不精确的PAHSS迭代方法,并分析了其收敛性.数值实验验证了所提算法的有效性.

A preconditioned asymmetric Hermitian and skew-Hermitian splitting （PAHSS） method was present-ed for solving large sparse continuous Sylvester equations with non-Hermitian and positive definite/semi-definite matrices. The convergent property of the PAHSS method was also discussed. The choice of the quasi-optimal parameter of the PAHSS method was studied. To further reduce the computing cost, an inexact preconditioned asymmetric HSS splitting （IPAHSS） iteration method was used, together with a certain e℃ient iterative method to approximately solve the two specially preconditioned structured continuous Sylvester equations involved in each step of the AHSS iteration. The convergence of the IPAHSS iteration method was also studied in detail. Numerical experiments demonstrated the e℃iency of the PAHSS method with different choices of the parame-ters for the continuous Sylvester equations.