求解矩阵方程AXB+CXD=F参数迭代法的最优参数分析
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摘要
本文针对求矩阵方程AXB+CXD=F唯一解的参数迭代法,分析当矩阵A, B,C,D均是Hermite正(负)定矩阵时,迭代矩阵的特征值表达式,给出了最优参数的确定方法,并提出了相应的加速算法.
In this paper, we considered the parameter iterative method of matrix equation AXB+CXD = F when it has a unique solution. Through eigenvalues analysis of iterative matrix, the optimal parameter is given when A, B, C, D are Hermite positive-definite matrices.Moreover the acceleration of algorithm about the parameter iterative method is proposed.Some numerical results show that this method is effective for the given problem.
In this paper, we considered the parameter iterative method of matrix equation AXB+CXD = F when it has a unique solution. Through eigenvalues analysis of iterative matrix, the optimal parameter is given when A, B, C, D are Hermite positive-definite matrices.Moreover the acceleration of algorithm about the parameter iterative method is proposed.Some numerical results show that this method is effective for the given problem.
引文
[1] Roger A Horn, Charles R. Johnson. Topics in Matrix Analysis[M].北京:人民邮电出版社,2005,241-242.
[2]张凯院,王同军.矩阵方程AX+XB=F的参数迭代解法[J].工程数学学报,2004,21(8):6-10.
[3]张凯院,蔡元虎.矩阵方程AXB+CXD=F的参数迭代解法[J].西北大学学报,2006, 36(1):13-16.
[4]李海合,傅永洁.矩阵方程AXB+CXD=F的两种参数迭代解法的讨论[J].天水师范学院学报,2006, 26(5):24-25.
[5] Dehghan M, Hajarian M. Two interation algorithms for solving coupled matrix equations over reflexive and anti-reflexive matrices[J]. Comput. Appl. Math., 2012, 31:353-371.
[6] Ding F, Liu P X, Ding J. Iterative solutions of the generalized Sylvester matrix equation by using the hierarchical identification principle[J].Appl. Math. Comput., 2008, 197:41-50.
[7] Ding J, Liu Y J, Ding F. Iterative solutions to matrix equations of the form A_i×B_i=F_i[J].Comput. Appl. Math., 2010, 59:3500-3507.
[8] Li Z Y, Wang Y, Zhou B, Duan G R. Least squares solution with the minimumnorm to general matrix equatioons via iteration[J]. Appl. Math. Comput.,2010,215:3547-3562.
[9] Li Z Y, Wang Y. Interative algorithm for minimal norm least squares solution to general linear matrix equations[J]. Int. J. Comput. Math., 2010, 87:2552-2567.
[10] Cai J, Chen G L. An interative algorithms for the least squares bisymmetrix solutions of the matrices equations A_1XB_1=C_1,A_2XB_2=C_2[J]. Math. Comput. Modelling., 2009, 50:1237-1244.
[11] Cai J, Chen G L. An interative algorithms for solving a kind of constrained linear matrix equations system[J]. Comput. Appl. Math., 2009, 28:309-325.
[12] Cai J, Chen G L, Liu Q B. An interative method for the bisymmetrix solutions of the consistent matrix equations A_1XB_1=C_1,A_2XB_2=C_2[J]. Int. J. Comput. Math., 2010, 87:2706-2715.
[2]张凯院,王同军.矩阵方程AX+XB=F的参数迭代解法[J].工程数学学报,2004,21(8):6-10.
[3]张凯院,蔡元虎.矩阵方程AXB+CXD=F的参数迭代解法[J].西北大学学报,2006, 36(1):13-16.
[4]李海合,傅永洁.矩阵方程AXB+CXD=F的两种参数迭代解法的讨论[J].天水师范学院学报,2006, 26(5):24-25.
[5] Dehghan M, Hajarian M. Two interation algorithms for solving coupled matrix equations over reflexive and anti-reflexive matrices[J]. Comput. Appl. Math., 2012, 31:353-371.
[6] Ding F, Liu P X, Ding J. Iterative solutions of the generalized Sylvester matrix equation by using the hierarchical identification principle[J].Appl. Math. Comput., 2008, 197:41-50.
[7] Ding J, Liu Y J, Ding F. Iterative solutions to matrix equations of the form A_i×B_i=F_i[J].Comput. Appl. Math., 2010, 59:3500-3507.
[8] Li Z Y, Wang Y, Zhou B, Duan G R. Least squares solution with the minimumnorm to general matrix equatioons via iteration[J]. Appl. Math. Comput.,2010,215:3547-3562.
[9] Li Z Y, Wang Y. Interative algorithm for minimal norm least squares solution to general linear matrix equations[J]. Int. J. Comput. Math., 2010, 87:2552-2567.
[10] Cai J, Chen G L. An interative algorithms for the least squares bisymmetrix solutions of the matrices equations A_1XB_1=C_1,A_2XB_2=C_2[J]. Math. Comput. Modelling., 2009, 50:1237-1244.
[11] Cai J, Chen G L. An interative algorithms for solving a kind of constrained linear matrix equations system[J]. Comput. Appl. Math., 2009, 28:309-325.
[12] Cai J, Chen G L, Liu Q B. An interative method for the bisymmetrix solutions of the consistent matrix equations A_1XB_1=C_1,A_2XB_2=C_2[J]. Int. J. Comput. Math., 2010, 87:2706-2715.