四元数亚正定系统混参分裂迭代方法

Mixed Parameter Splitting Iteration Method for Quaternion Subpositive Definite System

随着四元数的广泛应用,大型四元数结构矩阵方程的求解成为科学计算的重要课题。针对四元数亚正定系统AX=B,在新自共轭正定和斜自共轭分裂迭代(new positive definite and skew-self-conjugate splitting,NPSS)基础上,通过引入双参数和松弛加速技术,构建出两种新的混参分裂迭代格式,即非对称新自共轭正定和斜自共轭分裂迭代(asymmetric new positive definite and skew-self-conjugate splitting,ANPSS),以及超松弛非对称新自共轭正定和斜自共轭分裂迭代(successive over relaxation asymmetric new positive definite and skew-self-conjugate splitting,SANPSS),同时运用四元数矩阵特征值理论,证明了这两种迭代的收敛性,并给出相关参数的取值范围。采用四元数矩阵的复表示方法,在MATLAB环境下实现该系统的数值求解。数值算例表明,多参数的灵活选取,显示出所提混参分裂迭代相比NPSS迭代具有更高的收敛效率。

With the wide application of quaternion,the solution of large quaternion structured matrix equations has become an important topic in scientific computing.Based on the new positive definite and skew-self-conjugate splitting iteration(NPSS),two new mixed-parameter splitting iteration schemes were constructed for the quaternion sub-positive definite system AX=B by introducing the two-parameter and relaxation acceleration techniques.That is,asymmetric new positive definite and skew-self-conjugate splitting(ANPSS),successive over relaxation asymmetric new positive definite and skew-self-conjugate splitting(SANPSS),and the convergence of these two iterations was proved by using the eigenvalue theory of quaternion matrix,and the range of relevant parameters was given.In addition,the complex representation method of quaternion matrix was used to realize the numerical solution of the system in MATLAB environmen t.Numerical examples show that the flexible selection of multiple parameters shows that the proposed mixed parameter splitting iteration has higher convergence efficiency than NPSS iteration.