解一般矩阵方程的GPBiCG(m,l)法

The GPBiCG($m,l$) Method for Solving General Matrix Equations

中文关键词:  GPBiCG$(m,l)$法  Krylov子空间法  矩阵方程  Kronecker积  向量化算子

英文关键词:GPBiCG($m,l$) method  Krylov Subspace method  matrix equations  Kronecker product  vectorization operator

中文摘要:

      最近提出的用于解非对称线性方程组$Ax=b$的广义乘积型双共轭梯度法GPBiCG$(m,l)$是一种基于GPBiCG和BiCGSTAB法的混合型算法,该算法在许多数值试验中都有不错的收敛表现.借助Kronecker积和向量化算子,本文将GPBiCG$(m,l)$法做了推广并用之解一般矩阵方程Equation 1和一般离散时间周期矩阵方程组Equation 2, 其中包括出现在许多应用领域的Lyapunov, Stein 和Sylvester 等矩阵方程.通过数值试验与一些现有算法对比,检验了所提GPBiCG$(m,l)$法的准确性和有效性.

英文摘要:

      The generalized product bi-conjugate gradient (GPBiCG($m,l$)) method has been recently proposed as a hybrid variant of the GPBiCG and the BiCGSTAB methods to solve the linear system $Ax = b$ with non-symmetric coefficient matrix, and its attractive convergence behavior has been authenticated in many numerical experiments. By means of the Kronecker product and the vectorization operator, this paper aims to develop the GPBiCG($m,l$) method to solve the general matrix equation $$\sum^{p}_{i=1}{\sum^{s_{i}}_{j=1} A_{ij}X_{i}B_{ij}} = C,$$ and the general discrete-time periodic matrix equations $$\sum^{p}_{i=1}{\sum^{s_{i}}_{j=1} (A_{i,j,k}X_{i,k}B_{i,j,k}+C_{i,j, k}X_{i,k+1}D_{i,j,k})} = M_{k},~~k = 1, 2, \ldots,t,$$ which include the well-known Lyapunov, Stein, and Sylvester matrix equations that arise in a wide variety of applications in engineering, communications and scientific computations. The accuracy and efficiency of the extended GPBiCG($m,l$) method assessed against some existing iterative methods are illustrated by several numerical experiments.