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西北工业大学应用数学系
纸质出版:2012
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刘晓敏, 张凯院, 谢培月. 求双变量LME一种异类约束最小二乘解的MCG算法[J]. 中北大学学报(自然科学版), 2012,33(4):358-362.
刘晓敏, 张凯院, 谢培月. Modified Conjugate Gradient Method for a Different Constrained Least Square Solution of Two-Variables Linear Matrix Equation[J]. 2012, 33(4): 358-362.
借鉴求线性矩阵方程(LME)同类约束最小二乘解的修正共轭梯度法
建立了求双变量LME的一种异类约束最小二乘解的修正共轭梯度法
并证明了该算法的收敛性.在不考虑舍入误差的情况下
利用该算法不仅可在有限步计算后得到LME的一组异类约束最小二乘解
而且选取特殊初始矩阵时
可求得LME的极小范数异类约束最小二乘解.另外
还可求得指定矩阵在该LME的异类约束最小二乘解集合中的最佳逼近.算例表明
该算法是有效的.
Based on the modified conjugate gradient method
which gets same constrained least square solution of the linear matrix equation
a modified conjugate gradient method is constructed for different constrained least square solution of the linear matrix equation with two variables.And the convergence of this method is proved.By this method
a different constrained least square solution can be obtained within finite iterative steps in absence of round-off errors
and the different constrained least square solution with least-norm can be got by choosing special initial matrices.In addition
the optimal approximation matrix to any given matrix can be obtained in the set of the different constrained least square solutions.Examples show the efficiency of the method.
彭卓华,胡锡炎,张磊.矩阵方程A1X1B1+…+AlXlBl=C的中心对称解及其最佳逼近[J].数学物理学报,2009,29A(1):193-207.Peng Zhuohua,Hu Xiyan,Zhang Lei.The centrosymmetric solutions of matrix equationA1X1B1+…+AlXlBl=Cand its optimal approximation[J].ACTA Mathmatica Scientia,2009,29A(1):193-207.(in Chinese)
袁仕芳,廖安平,雷渊.矩阵方程AXB+CYD=F的对称极小范数最小二乘解[J].计算数学,2007,29(2):203-216.Yuan Shifang,Liao Anping,Lei Yuan.Least squares symmetric solution of the matrix equationAXB+CYD=Fwith the least norm[J].Mathematica NumericaSinica,2007,29(2):203-216.(in Chinese)
Liao Anping.Best approximate solution of matrix equationAXB+CYD=E[J].SIAM J.Matrix Anal.Appl,2006,27(3):675-688.
郑凤芹,张凯院.求多变量线性矩阵方程组自反解的迭代算法[J].数值计算与计算机应用,2010,31(1):39-54.Zheng Fengqin,Zhang Kaiyuan.An iterative method for the reflexive solutions of the linear matrix equations with several variables[J].Journal on Numerical Methods and Computer Applications,2010,31(1):39-54.(in Chinese)
郑凤芹,张凯院.一类矩阵方程组双对称解的修正共轭梯度法[J].中北大学学报(自然科学版),2011,32(2):128-134.Zheng Fengqin,Zhang Kaiyuan.The modified conjugate gradient method for the bisymmetric solutions of a class matrix equations[J].Journal of North University of China(Natural Science Edition),2011,32(2):128-134.(in Chinese)
Zhang Kaiyuan,Wu Jian,Xie Peiyue.An iterative method for solving linear matrix equation with several variables over different constrained matrices[C].In:Jiang Erxiong,Proceedings of9th ICMTA,Vol.2,World Academic Press,2010:152-155.
张凯院,田小红,郑凤芹.求线性矩阵方程自反最小二乘解的迭代方法[J].中北大学学报(自然科学版),2010,31(6):548-553.Zhang Kaiyuan,Tian Xiaohong,Zheng Fengqin.Iterative method for least squares reflexive solution of general linear matrix equation[J].Journal of North University of China(Natural Science Edition),2010,31(6):548-553.(in Chinese)
Dehghan M,Hajarian M.Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equationA1X1B1+A2X2B2=C[J].Mathematical andComputer Modelling,2009,49:1937-1959.
袁飞,张凯院.矩阵方程AXB+CXTD=F自反最小二乘解的迭代算法[J].数值计算与计算机应用,2009,30(3):195-201.Yuan Fei,Zhang Kaiyuan.An iterative method for the least squares reflexive matrix solution of matrix equationAXB+CXTD=F[J].Journal on Numerical Methods and Computer Applications[J],2009,30(3):195-201.(in Chinese)
尚丽娜,张凯院.一类Lyapunov矩阵方程对称最小二乘解的迭代算法[J].中北大学学报(自然科学报),2008,29(4):301-307.Shang Lina,Zhang Kaiyuan.An iterative method for the least squares symmetric solution of the Lyapunov matrix equation[J].Journal of North University of China(Natural Science Edition),2008,29(4):301307.(in Chinese)
陈梅枝,张凯院,尚丽娜.矩阵方程AXB=C的中心对称最小二乘解及其最佳逼近的迭代算法[J].工程数学学报,2008,25(6):1125-1128.Chen Meizhi,Zhang Kaiyuan,Shang Lina.An iterative method for the least squares central symmetric solutions of the matrix equationAXB=C and its optimal approximation[J].Chinese Journal of Engineering Mathematics,2008,25(6):1125-1128.(in Chinese)
林宏程.双对称线性矩阵方程的最佳逼近解[J].数学杂志,2011,31(2):233-244.Lin Hongcheng.The anti-symmetric optimal approximation solution for a linear matrix equationpair[J].Journal of Mathematics,2011,31(2):233-244.(in Chinese)
袁晖坪.行(列)对称矩阵的奇异值分解[J].中北大学学报(自然科学版),2009,30(2):100-104.Yuan Huiping.Singular value decomposition of row(column)symmetric matrix[J].Journal of North University of China(Natural Science Edition),2009,30(2):100-104.(in Chinese)
刘畔畔,李庆春.矩阵方程AX+XB=C的对称解及其最佳逼近[J].江西师范大学学报(自然科学版),2009,33(1):17-21.Liu Panpan,Li Qingchun.The symmetric solution of matrix equation and its optimal approximation[J].Journal of Jiangxi Normal University(Natural Sciences Edition),2009,33(1):17-21.(in Chinese)
张凯院,徐仲.数值代数[M].第2版.北京:科学出版社,2010.
张贤达.矩阵分析与应用[M].北京:清华大学出版社,2006.
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