An iterative algorithm for the best approximate (P, Q)-orthogonal symmetric and skew-symmetric solution pair of coupled matrix equations

2015 ◽  
Vol 39 (4) ◽  
pp. 537-554 ◽  
Author(s):  
Fatemeh Panjeh Ali Beik ◽  
Davod Khojasteh Salkuyeh
Keyword(s):  
Least Squares ◽  
Iterative Algorithm ◽  
Symmetric Solution ◽  
Numerical Experiments ◽  
Matrix Equations ◽  
Arbitrary Matrix ◽  
Solution Sets ◽  
Solution Pair ◽  
Matrix Pair ◽  
Theoretical Results

This paper deals with developing a robust iterative algorithm to find the least-squares ( P, Q)-orthogonal symmetric and skew-symmetric solution sets of the generalized coupled matrix equations. To this end, first, some properties of these type of matrices are established. Furthermore, an approach is offered to determine the optimal approximate ( P, Q)-orthogonal (skew-)symmetric solution pair corresponding to a given arbitrary matrix pair. Some numerical experiments are reported to confirm the validity of the theoretical results and to illustrate the effectiveness of the proposed algorithm.

scholarly journals An Iterative Algorithm for the Generalized Reflexive Solutions of the Generalized Coupled Sylvester Matrix Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-28 ◽  
Author(s):  
Feng Yin ◽  
Guang-Xin Huang
Keyword(s):  
Iterative Algorithm ◽  
Special Kind ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Sylvester Matrix ◽  
Special Cases ◽  
The Matrix ◽  
Solution Pair ◽  
Sylvester Matrix Equations ◽  
Matrix Pair

An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations(AXB-CYD,EXF-GYH)=(M,N), which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matricesXandY. When the matrix equations are consistent, for any initial generalized reflexive matrix pair[X1,Y1], the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors, and the least Frobenius norm generalized reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair[X̂,Ŷ]to a given matrix pair[X0,Y0]in Frobenius norm can be derived by finding the least-norm generalized reflexive solution pair[X̃*,Ỹ*]of a new corresponding generalized coupled Sylvester matrix equation pair(AX̃B-CỸD,EX̃F-GỸH)=(M̃,Ñ), whereM̃=M-AX0B+CY0D,Ñ=N-EX0F+GY0H. Several numerical examples are given to show the effectiveness of the presented iterative algorithm.

An Iterative algorithm for $\eta$-(anti)-Hermitian least-squares solutions of quaternion matrix equations

2015 ◽  
Vol 30 ◽  
Author(s):  
Fatemeh Beik ◽  
Salman Ahmadi-Asl
Keyword(s):  
Least Squares ◽  
Iterative Algorithm ◽  
Numerical Experiments ◽  
Iterative Algorithms ◽  
Iterative Approach ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Hermitian Matrices ◽  
Convergence Properties ◽  
Least Squares Problems

Recently, some research has been devoted to finding the explicit forms of the η-Hermitian and η-anti-Hermitian solutions of several kinds of quaternion matrix equations and their associated least-squares problems in the literature. Although exploiting iterative algorithms is superior than utilizing the explicit forms in application, hitherto, an iterative approach has not been offered for finding η-(anti)-Hermitian solutions of quaternion matrix equations. The current paper deals with applying an efficient iterative manner for determining η-Hermitian and η-anti-Hermitian least-squares solutions corresponding to the quaternion matrix equation AXB + CY D = E. More precisely, first, this paper establishes some properties of the η-Hermitian and η-anti-Hermitian matrices. These properties allow for the demonstration of how the well-known conjugate gradient least- squares (CGLS) method can be developed for solving the mentioned problem over the η-Hermitian and η-anti-Hermitian matrices. In addition, the convergence properties of the proposed algorithm are discussed with details. In the circumstance that the coefficient matrices are ill-conditioned, it is suggested to use a preconditioner for accelerating the convergence behavior of the algorithm. Numerical experiments are reported to reveal the validity of the elaborated results and feasibility of the proposed iterative algorithm and its preconditioned version.

scholarly journals The optimal convergence factor of the gradient based iterative algorithm for linear matrix equations

Filomat ◽  
2012 ◽  
Vol 26 (3) ◽  
pp. 607-613 ◽  
Author(s):  
Xiang Wang ◽  
Dan Liao
Keyword(s):  
Iterative Algorithm ◽  
Numerical Experiments ◽  
Linear Matrix ◽  
Matrix Equations ◽  
Convergence Factor ◽  
Gradient Based ◽  
Gradient Type ◽  
And Mathematics ◽  
Computers And Mathematics ◽  
Linear Matrix Equations

A hierarchical gradient based iterative algorithm of [L. Xie et al., Computers and Mathematics with Applications 58 (2009) 1441-1448] has been presented for finding the numerical solution for general linear matrix equations, and the convergent factor has been discussed by numerical experiments. However, they pointed out that how to choose a best convergence factor is still a project to be studied. In this paper, we discussed the optimal convergent factor for the gradient based iterative algorithm and obtained the optimal convergent factor. Moreover, the theoretical results of this paper can be extended to other methods of gradient-type based. Results of numerical experiments are consistent with the theoretical findings.

scholarly journals Least Squares Based Iterative Algorithm for the Coupled Sylvester Matrix Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Hongcai Yin ◽  
Huamin Zhang
Keyword(s):  
Exact Solution ◽  
Least Squares ◽  
Iterative Algorithm ◽  
Iterative Solution ◽  
Sylvester Equation ◽  
Matrix Equations ◽  
Convergence Factor ◽  
The Matrix ◽  
Generalized Sylvester Equation ◽  
Sylvester Matrix Equations

By analyzing the eigenvalues of the related matrices, the convergence analysis of the least squares based iteration is given for solving the coupled Sylvester equationsAX+YB=CandDX+YE=Fin this paper. The analysis shows that the optimal convergence factor of this iterative algorithm is 1. In addition, the proposed iterative algorithm can solve the generalized Sylvester equationAXB+CXD=F. The analysis demonstrates that if the matrix equation has a unique solution then the least squares based iterative solution converges to the exact solution for any initial values. A numerical example illustrates the effectiveness of the proposed algorithm.

Improving the Incoherence of a Learned Dictionary via Rank Shrinkage

2017 ◽  
Vol 29 (1) ◽  
pp. 263-285 ◽  
Author(s):  
Shashanka Ubaru ◽  
Abd-Krim Seghouane ◽  
Yousef Saad
Keyword(s):  
Least Squares ◽  
Dictionary Learning ◽  
Numerical Experiments ◽  
Learning Algorithms ◽  
Sparse Signal ◽  
Mutual Coherence ◽  
Least Squares Estimate ◽  
Learned Dictionary ◽  
Sparse Signal Representation ◽  
Theoretical Results

This letter considers the problem of dictionary learning for sparse signal representation whose atoms have low mutual coherence. To learn such dictionaries, at each step, we first update the dictionary using the method of optimal directions (MOD) and then apply a dictionary rank shrinkage step to decrease its mutual coherence. In the rank shrinkage step, we first compute a rank 1 decomposition of the column-normalized least squares estimate of the dictionary obtained from the MOD step. We then shrink the rank of this learned dictionary by transforming the problem of reducing the rank to a nonnegative garrotte estimation problem and solving it using a path-wise coordinate descent approach. We establish theoretical results that show that the rank shrinkage step included will reduce the coherence of the dictionary, which is further validated by experimental results. Numerical experiments illustrating the performance of the proposed algorithm in comparison to various other well-known dictionary learning algorithms are also presented.

scholarly journals The Generalized Bisymmetric (Bi-Skew-Symmetric) Solutions of a Class of Matrix Equations and Its Least Squares Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Yifen Ke ◽  
Changfeng Ma
Keyword(s):  
Least Squares ◽  
General Expression ◽  
Numerical Experiments ◽  
Matrix Equations ◽  
Solvability Conditions ◽  
Least Squares Problem ◽  
The Matrix

The solvability conditions and the general expression of the generalized bisymmetric and bi-skew-symmetric solutions of a class of matrix equations(AX=B,XC=D)are established, respectively. If the solvability conditions are not satisfied, the generalized bisymmetric and bi-skew-symmetric least squares solutions of the matrix equations are considered. In addition, two algorithms are provided to compute the generalized bisymmetric and bi-skew-symmetric least squares solutions. Numerical experiments illustrate that the results are reasonable.

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