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1.
研究了Sylvester矩阵方程最小二乘解以及极小范数最小二乘解的迭代解法,首先利用递阶辨识原理,得到了求解矩阵方程AX+YB=C的极小范数最小二乘解的一种迭代算法,进而,将这种算法推广到一般线性矩阵方程A_iX_iB_i=C的情形,最后,数值例子验证了算法的有效性.  相似文献   

2.
彭雪梅  张爱华  张志强 《数学杂志》2014,34(6):1163-1169
本文研究了矩阵方程AXB+CY D=E的三对角中心对称极小范数最小二乘解问题.利用矩阵的Kronecker积和Moore-Penrose广义逆方法,得到了矩阵方程AXB+CY D=E的三对角中心对称极小范数最小二乘解的表达式.  相似文献   

3.
周海林 《计算数学》2023,45(1):93-108
应用共轭梯度方法和线性投影算子,给出迭代算法求解了线性矩阵方程AX=B在任意线性子空间上的最小二乘解问题.在不考虑舍入误差的情况下,可以证明,所给迭代算法经过有限步迭代可得到矩阵方程AX=B的最小二乘解、极小范数最小二乘解及其最佳逼近.文中的数值例子证实了该算法的有效性.  相似文献   

4.
给定矩阵Y, X和B,得到了矩阵方程YAX=B的反中心对称最小二乘解.利用矩阵的标准相关分解给出解存在的充要条件及其解的一般表达式.  相似文献   

5.
张凯院  王娇 《数学杂志》2015,35(2):469-476
本文研究了一类Riccati矩阵方程广义自反解的数值计算问题.利用牛顿算法将Riccati矩阵方程的广义自反解问题转化为线性矩阵方程的广义自反解或者广义自反最小二乘解问题,再利用修正共轭梯度法计算后一问题,获得了求Riccati矩阵方程的广义自反解的双迭代算法.拓宽了求解非线性矩阵方程的迭代算法.数值算例表明双迭代算法是有效的.  相似文献   

6.
利用逆矩阵的Neumann级数形式,将在Schur插值问题中遇到的含未知矩阵二次项之逆的非线性矩阵方程转化为高次多项式矩阵方程,然后采用牛顿算法求高次多项式矩阵方程的对称解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的对称解或者对称最小二乘解,建立求非线性矩阵方程的对称解的双迭代算法.双迭代算法仅要求非线性矩阵方程有对称解,不要求它的对称解唯一,也不对它的系数矩阵做附加限定.数值算例表明,双迭代算法是有效的.  相似文献   

7.
研究一类双矩阵变量Riccati矩阵方程(R-ME)对称解的数值计算问题.运用牛顿算法求R-ME的对称解时,会导出求双矩阵变量线性矩阵方程的对称解或者对称最小二乘解的问题,采用修正共轭梯度法解决导出的线性矩阵方程约束解问题,可建立求R-ME的对称解的迭代算法.数值算例表明,迭代算法是有效的.  相似文献   

8.
肖庆丰  胡锡炎  张磊 《数学杂志》2015,35(3):505-512
本文研究了矩阵方程AX=B的中心对称解.利用矩阵对的广义奇异值分解和广义逆矩阵,获得了该方程有中心对称解的充要条件以及有解时,最大秩解、最小秩解的一般表达式,并讨论了中心对称最小秩解集合中与给定矩阵的最佳逼近解.  相似文献   

9.
本文研究了在控制理论和随机滤波等领域中遇到的一类含高次逆幂的矩阵方程的等价矩阵方程对称解的数值计算问题.采用牛顿算法求等价矩阵方程的对称解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的对称解或者对称最小二乘解,建立了求这类矩阵方程对称解的双迭代算法,数值算例验证了双迭代算法是有效的.  相似文献   

10.
利用J-中心对称矩阵的结构和约化性质,本文研究了J-中心对称矩阵方程的通解、最小二乘解,然后考虑了方程解集合中对给定矩阵的最佳逼近问题,并给出了唯一最佳逼近解的表达式。  相似文献   

11.
This paper presents an iterative method for solving the matrix equation AXB + CYD = E with real matrices X and Y. By this iterative method, the solvability of the matrix equation can be determined automatically. And when the matrix equation is consistent, then, for any initial matrix pair [X0, Y0], a solution pair can be obtained within finite iteration steps in the absence of round‐off errors, and the least norm solution pair can be obtained by choosing a special kind of initial matrix pair. Furthermore, the optimal approximation solution pair to a given matrix pair [X?, ?] in a Frobenius norm can be obtained by finding the least norm solution pair of a new matrix equation AX?B + C?D = ?, where ? = E ? AX?B ? C?D. The given numerical examples show that the iterative method is efficient. Copyright © 2005 John Wiley & Sons, Ltd.  相似文献   

12.
刘莉  王伟 《工科数学》2012,(6):67-73
基于共轭梯度法的思想,通过特殊的变形,建立了一类求矩阵方程AXA^T+BYB^T=C的双对称最小二乘解的迭代算法.对任意的初始双对称矩阵.在没有舍人误差的情况下,经过有限步迭代得到它的双对称最小二乘解;在选取特殊的初始双对称矩阵时,能得到它的的极小范数双对称最小二乘解.另外,给定任意矩阵,利用此方法可得到它的最佳逼近双对称解,数值例子表明,这种方法是有效的.  相似文献   

13.
This paper is concerned with solutions to the so-called coupled Sylvester-transpose matrix equations, which include the generalized Sylvester matrix equation and Lyapunov matrix equation as special cases. By extending the idea of conjugate gradient method, an iterative algorithm is constructed to solve this kind of coupled matrix equations. When the considered matrix equations are consistent, for any initial matrix group, a solution group can be obtained within finite iteration steps in the absence of roundoff errors. The least Frobenius norm solution group of the coupled Sylvester-transpose matrix equations can be derived when a suitable initial matrix group is chosen. By applying the proposed algorithm, the optimal approximation solution group to a given matrix group can be obtained by finding the least Frobenius norm solution group of new general coupled matrix equations. Finally, a numerical example is given to illustrate that the algorithm is effective.  相似文献   

14.
设矩阵X=(xij) ∈Rn×n, 如果xij=xn+1-i, n+1-j (i,j=1,2, …,n), 则称X是中心对称矩阵. 该文构造了一种迭代法求矩阵方程A1X1B1+A2X2B2+…+AlXlBl=C的中心对称解组(其中[X1, X2, …, Xl]是实矩阵组). 当矩阵方程相容时, 对任意初始的中心对称矩阵组[X1(0), X2(0), …, Xl(0)], 在没有舍入误差的情况下,经过有限步迭代,得到它的一个中心对称解组, 并且, 通过选择一种特殊的中心对称矩阵组, 得到它的最小范数中心对称解组. 另外, 给定中心对称矩阵组[X1, X2, …, Xl], 通过求矩阵方程A1X1B1+A2X2B2+…+AlXlBl=C(其中C=C-A1X1B1-A2X2B2-…-AlXlBl)的中心对称解组, 得到它的最佳逼近中心对称解组. 实例表明这种方法是有效的.  相似文献   

15.
A new matrix based iterative method is presented to compute common symmetric solution or common symmetric least-squares solution of the pair of matrix equations AXB = E and CXD = F. By this iterative method, for any initial matrix X0, a solution X can be obtained within finite iteration steps if exact arithmetic was used, and the solution X with the minimum Frobenius norm can be obtained by choosing a special kind of initial matrix. In addition, the unique nearest common symmetric solution or common symmetric least-squares solution to given matrix in Frobenius norm can be obtained by first finding the minimum Frobenius norm common symmetric solution or common symmetric least-squares solution of the new pair of matrix equations. The given numerical examples show that the matrix based iterative method proposed in this paper has faster convergence than the iterative methods proposed in [1] and [2] to solve the same problems.  相似文献   

16.
In the present paper, we propose an iterative algorithm for solving the generalized (P,Q)-reflexive solution to the quaternion matrix equation $\sum^{u}_{l=1}A_{l}XB_{l}+\sum^{v}_{s=1} C_{s}\overline{X}D_{s}=F$ . By this iterative algorithm, the solvability of the problem can be determined automatically. When the matrix equation is consistent over generalized (P,Q)-reflexive matrix X, a generalized (P,Q)-reflexive solution can be obtained within finite iteration steps in the absence of roundoff errors, and the least Frobenius norm generalized (P,Q)-reflexive solution can be obtained by choosing an appropriate initial iterative matrix. Furthermore, the optimal approximate generalized (P,Q)-reflexive solution to a given matrix X 0 can be derived by finding the least Frobenius norm generalized (P,Q)-reflexive solution of a new corresponding quaternion matrix equation. Finally, two numerical examples are given to illustrate the efficiency of the proposed methods.  相似文献   

17.
This paper an iterative method is presented to solve the minimum Frobenius norm residual problem: with unknown symmetric matrix . By the iterative method, for any initial symmetric matrix , a solution can be obtained within finite iteration steps in the absence of roundoff errors, and the solution with least Frobenius norm can be obtained by choosing a special kind of initial symmetric matrix. In addition, in the solution set of the minimum Frobenius norm residual problem, the unique optimal approximation solution to a given matrix in Frobenius norm can be expressed as , where is the least norm symmetric solution of the new minimum residual problem: with . Given numerical examples are show that the iterative method is quite efficient.Research supported by Scientific Research Fund of Hunan Provincial Education Department of China (05C797), by China Postdoctoral Science Foundation (2004035645) and by National Natural Science Foundation of China (10571047).  相似文献   

18.
An iterative method is proposed to solve generalized coupled Sylvester matrix equations, based on a matrix form of the least-squares QR-factorization (LSQR) algorithm. By this iterative method on the selection of special initial matrices, we can obtain the minimum Frobenius norm solutions or the minimum Frobenius norm least-squares solutions over some constrained matrices, such as symmetric, generalized bisymmetric and (RS)-symmetric matrices. Meanwhile, the optimal approximate solutions to the given matrices can be derived by solving the corresponding new generalized coupled Sylvester matrix equations. Finally, numerical examples are given to illustrate the effectiveness of the present method.  相似文献   

19.
In this paper, an iterative algorithm is constructed to solve the minimum Frobenius norm residual problem: min over bisymmetric matrices. By this algorithm, for any initial bisymmetric matrix X0, a solution X* can be obtained in finite iteration steps in the absence of roundoff errors, and the solution with least norm can be obtained by choosing a special kind of initial matrix. Furthermore, in the solution set of the above problem, the unique optimal approximation solution to a given matrix in the Frobenius norm can be derived by finding the least norm bisymmetric solution of a new corresponding minimum Frobenius norm problem. Given numerical examples show that the iterative algorithm is quite effective in actual computation.  相似文献   

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