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Masoud Hajarian - One of the best experts on this subject based on the ideXlab platform.
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analysis of an Iterative Algorithm to solve the generalized coupled sylvester matrix equations
Applied Mathematical Modelling, 2011Co-Authors: Mehdi Dehghan, Masoud HajarianAbstract:Abstract A complex matrix P ∈ C n × n is said to be a generalized reflection if P = P H = P −1 . Let P ∈ C n × n and Q ∈ C n × n be two generalized reflection matrices. A complex matrix A ∈ C n × n is called a generalized centro-symmetric with respect to ( P ; Q ), if A = PAQ . It is obvious that any n × n complex matrix is also a generalized centro-symmetric matrix with respect to ( I ; I ). In this work, we consider the problem of finding a simple way to compute a generalized centro-symmetric solution pair of the generalized coupled Sylvester matrix equations (GCSY) ∑ i = 1 l A i XB i + ∑ i = 1 l C i YD i = M , ∑ i = 1 l E i XF i + ∑ i = 1 l G i YH i = N , (including Sylvester and Lyapunov matrix equations as special cases) and to determine solvability of these matrix equations over generalized centro-symmetric matrices. By extending the idea of conjugate gradient (CG) method, we propose an Iterative Algorithm for solving the generalized coupled Sylvester matrix equations over generalized centro-symmetric matrices. With the Iterative Algorithm, the solvability of these matrix equations over generalized centro-symmetric matrices can be determined automatically. When the matrix equations are consistent over generalized centro-symmetric matrices, for any (special) initial generalized centro-symmetric matrix pair [ X (1), Y (1)], a generalized centro-symmetric solution pair (the least Frobenius norm generalized centro-symmetric solution pair) can be obtained within finite number of iterations in the absence of roundoff errors. Also, the optimal approximation generalized centro-symmetric solution pair to a given generalized centro-symmetric matrix pair [ X ∼ , Y ∼ ] can be derived by finding the least Frobenius norm generalized centro-symmetric solution pair of new matrix equations. Moreover, the application of the proposed method to find a generalized centro-symmetric solution to the quadratic matrix equation Q ( X ) = AX 2 + BX + C = 0 is highlighted. Finally, two numerical examples are presented to support the theoretical results of this paper.
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an Iterative Algorithm for solving a pair of matrix equations ayb e cyd f over generalized centro symmetric matrices
Computers & Mathematics With Applications, 2008Co-Authors: Mehdi Dehghan, Masoud HajarianAbstract:A matrix [email protected]?R^n^x^n is said to be a symmetric orthogonal matrix if P=P^T=P^-^1. A matrix [email protected]?R^n^x^n is said to be generalized centro-symmetric (generalized central anti-symmetric) with respect to P, if A=PAP (A=-PAP). The generalized centro-symmetric matrices have wide applications in information theory, linear estimate theory and numerical analysis. In this paper, we propose a new Iterative Algorithm to compute a generalized centro-symmetric solution of the linear matrix equations AYB=E,CYD=F. We show, when the matrix equations are consistent over generalized centro-symmetric matrix Y, for any initial generalized centro-symmetric matrix Y"1, the sequence {Y"k} generated by the introduced Algorithm converges to a generalized centro-symmetric solution of matrix equations AYB=E,CYD=F. The least Frobenius norm generalized centro-symmetric solution can be derived when a special initial generalized centro-symmetric matrix is chosen. Furthermore, the optimal approximation generalized centro-symmetric solution to a given generalized centro-symmetric matrix can be derived. Several numerical examples are given to show the efficiency of the presented method.
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an Iterative Algorithm for the reflexive solutions of the generalized coupled sylvester matrix equations and its optimal approximation
Applied Mathematics and Computation, 2008Co-Authors: Mehdi Dehghan, Masoud HajarianAbstract:Abstract The generalized coupled Sylvester matrix equations ( AY - ZB , CY - ZD ) = ( E , F ) with unknown matrices Y , Z are encountered in many systems and control applications. Also these matrix equations have several applications relating to the problem of computing stable eigendecompositions of matrix pencils. In this work, we construct an Iterative Algorithm to solve the generalized coupled Sylvester matrix equations over reflexive matrices Y , Z . And when the matrix equations are consistent, for any initial matrix pair [ Y 0 , Z 0 ] , a reflexive solution pair can be obtained within finite iteration steps in the absence of roundoff errors, and the least Frobenius norm reflexive solution pair can be obtained by choosing a special kind of initial matrix pair. Also we obtain the optimal approximation reflexive solution pair to a given matrix pair [ Y ¯ , Z ¯ ] in the reflexive solution pair set of the generalized coupled Sylvester matrix equations ( AY - ZB , CY - ZD ) = ( E , F ) . Moreover, several numerical examples are given to show the efficiency of the presented Iterative Algorithm.
Huamin Zhang - One of the best experts on this subject based on the ideXlab platform.
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new proof of the gradient based Iterative Algorithm for a complex conjugate and transpose matrix equation
Journal of The Franklin Institute-engineering and Applied Mathematics, 2017Co-Authors: Huamin ZhangAbstract:Abstract By using the hierarchical identification principle, the gradient-based Iterative Algorithm is suggested for solving a complex conjugate and transpose matrix equation. With the tools of the real representation of a complex matrix and the vec-operator, a new convergence proof is offered. The necessary and sufficient conditions for the convergence factor is determined to guarantee the convergence of the Algorithm for any initial Iterative matrix. Also a conjecture proposed by Wu et al. (2011) is solved. Two numerical examples are offered to illustrate the effectiveness of the suggested Algorithm and conclusions.
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A property of the eigenvalues of the symmetric positive definite matrix and the Iterative Algorithm for coupled Sylvester matrix equations
Journal of the Franklin Institute, 2014Co-Authors: Huamin Zhang, Feng DingAbstract:Abstract In this paper, we discuss the properties of the eigenvalues related to the symmetric positive definite matrices. Several new results are established to express the structures and bounds of the eigenvalues. Using these results, a family of Iterative Algorithms are presented for the matrix equation AX=F and the coupled Sylvester matrix equations. The analysis shows that the Iterative solutions given by the least squares based Iterative Algorithms converge to their true values for any initial conditions. The effectiveness of the proposed Iterative Algorithm is illustrated by a numerical example.
Feng Ding - One of the best experts on this subject based on the ideXlab platform.
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extended gradient based Iterative Algorithm for bilinear state space systems with moving average noises by using the filtering technique
International Journal of Control Automation and Systems, 2021Co-Authors: Siyu Liu, Yanliang Zhang, Feng Ding, Ahmed Alsaedi, Tasawar HayatAbstract:This paper develops a filtering-based Iterative Algorithm for the combined parameter and state estimation problems of bilinear state-space systems, taking account of the moving average noise. In order to deal with the correlated noise and unknown states in the parameter estimation, a filter is chosen to filter the input-output data disturbed by colored noise and a Kalman state observer (KSO) is designed to estimate the states by minimizing the trace of the error covariance matrix. Then, a KSO extended gradient-based Iterative (KSO-EGI) Algorithm and a filtering based KSO-EGI Algorithm are presented to estimate the unknown states and unknown parameters jointly by the Iterative estimation idea. The simulation results demonstrate the effectiveness of the proposed Algorithms.
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two stage gradient based Iterative Algorithm for bilinear stochastic systems over the moving data window
Journal of The Franklin Institute-engineering and Applied Mathematics, 2020Co-Authors: Siyu Liu, Feng Ding, Ahmed Alsaedi, Li Xie, Tasawar HayatAbstract:Abstract For the bilinear stochastic system, the difficulty of identification lies in the product of the state vector and input in the system. This paper studies the Iterative estimation of the parameters and states for the bilinear state-space systems in the observer canonical form. The standard Kalman filter is recognized as the best state estimator for linear systems, but it is not applicable for bilinear systems. Therefore, this paper proposes a state filter (SF) for the bilinear systems based on the extremum principle. By means of the hierarchical principle, we decompose the identification model into two sub-identification models by introducing two fictitious output variables. Then an SF two-stage gradient-based Iterative Algorithm is proposed to achieve the combined parameter and state estimation according to the gradient search. For the purpose of improving the identification performance, an SF two-stage moving data window gradient-based Iterative Algorithm is derived by increasing the data utilization. The numerical example demonstrates the validity of the proposed Algorithms.
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multi step length gradient Iterative Algorithm for equation error type models
Systems & Control Letters, 2018Co-Authors: Jing Chen, Feng Ding, Yanjun Liu, Quanmin ZhuAbstract:Abstract This letter develops a multi-step-length gradient Iterative Algorithm for equation-error type (EET) models. The Algorithm analysis is based upon the gradient search principle. By applying the Gram–Schmidt procedure, the EET model can be turned into an orthogonal model, in which each parameter in the parameter vector is independent of the other parameters. Then a multi-step-length gradient Iterative Algorithm is proposed for the orthogonal model, and can estimate the parameters in one iteration. Finally, based on the estimated parameters, the parameter estimates of the EET model can be computed. Different from the traditional gradient Iterative Algorithm which has slower convergence rates and is sensitive to the initial values of the unknown variables, the method in this letter has quicker convergence rates and is robust to the initial values of the unknown variables. The simulation studies demonstrate the feasibility and effectiveness of the proposed Algorithm.
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A property of the eigenvalues of the symmetric positive definite matrix and the Iterative Algorithm for coupled Sylvester matrix equations
Journal of the Franklin Institute, 2014Co-Authors: Huamin Zhang, Feng DingAbstract:Abstract In this paper, we discuss the properties of the eigenvalues related to the symmetric positive definite matrices. Several new results are established to express the structures and bounds of the eigenvalues. Using these results, a family of Iterative Algorithms are presented for the matrix equation AX=F and the coupled Sylvester matrix equations. The analysis shows that the Iterative solutions given by the least squares based Iterative Algorithms converge to their true values for any initial conditions. The effectiveness of the proposed Iterative Algorithm is illustrated by a numerical example.
Ahmed M. E. Bayoumi - One of the best experts on this subject based on the ideXlab platform.
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Iterative Algorithm for solving a class of general sylvester conjugate matrix equation sum_ i 1 s a_ i v sum_ j 1 t b_ j w sum_ l 1 m e_ l overline v f_ l c
Journal of Applied Mathematics and Computing, 2014Co-Authors: Mohamed A Ramadan, Mokhtar Abdel A Naby, Ahmed M. E. BayoumiAbstract:This paper is concerned with Iterative solution to general Sylvester-conjugate matrix equation of the form \(\sum_{i = 1}^{s} A_{i}V + \sum_{j = 1}^{t} B_{j}W = \sum_{l = 1}^{m} E_{l}\overline{V}F_{l} + C\). An Iterative Algorithm is established to solve this matrix equation. When this matrix equation is consistent, for any initial matrices, the solutions can be obtained within finite Iterative steps in the absence of round off errors. Some lemmas and theorems are stated and proved where the Iterative solutions are obtained. Finally, a numerical example is given to verify the effectiveness of the proposed Algorithm.
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finite Iterative Algorithm for solving a complex of conjugate and transpose matrix equation
Journal of Discrete Mathematics, 2013Co-Authors: Mohamed A Ramadan, Talaat S Eldanaf, Ahmed M. E. BayoumiAbstract:We consider an Iterative Algorithm for solving a complex matrix equation with conjugate and transpose of two unknowns of the form:
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Finite Iterative Algorithm for Solving a Complex of Conjugate and Transpose Matrix Equation
Hindawi Publishing Corporation, 2013Co-Authors: Mohamed A Ramadan, Talaat S. El-danaf, Ahmed M. E. BayoumiAbstract:We consider an Iterative Algorithm for solving a complex matrix equation with conjugate and transpose of two unknowns of the form: A1VB1+C1WD1+A2V¯B2+C2W¯D2+A3VHB3+C3WHD3+A4VTB4 + C4WTD4=E. With the Iterative Algorithm, the existence of a solution of this matrix equation can be determined automatically. When this matrix equation is consistent, for any initial matrices V1, W1 the solutions can be obtained by Iterative Algorithm within finite Iterative steps in the absence of round-off errors. Some lemmas and theorems are stated and proved where the Iterative solutions are obtained. A numerical example is given to illustrate the effectiveness of the proposed method and to support the theoretical results of this paper
Mehdi Dehghan - One of the best experts on this subject based on the ideXlab platform.
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analysis of an Iterative Algorithm to solve the generalized coupled sylvester matrix equations
Applied Mathematical Modelling, 2011Co-Authors: Mehdi Dehghan, Masoud HajarianAbstract:Abstract A complex matrix P ∈ C n × n is said to be a generalized reflection if P = P H = P −1 . Let P ∈ C n × n and Q ∈ C n × n be two generalized reflection matrices. A complex matrix A ∈ C n × n is called a generalized centro-symmetric with respect to ( P ; Q ), if A = PAQ . It is obvious that any n × n complex matrix is also a generalized centro-symmetric matrix with respect to ( I ; I ). In this work, we consider the problem of finding a simple way to compute a generalized centro-symmetric solution pair of the generalized coupled Sylvester matrix equations (GCSY) ∑ i = 1 l A i XB i + ∑ i = 1 l C i YD i = M , ∑ i = 1 l E i XF i + ∑ i = 1 l G i YH i = N , (including Sylvester and Lyapunov matrix equations as special cases) and to determine solvability of these matrix equations over generalized centro-symmetric matrices. By extending the idea of conjugate gradient (CG) method, we propose an Iterative Algorithm for solving the generalized coupled Sylvester matrix equations over generalized centro-symmetric matrices. With the Iterative Algorithm, the solvability of these matrix equations over generalized centro-symmetric matrices can be determined automatically. When the matrix equations are consistent over generalized centro-symmetric matrices, for any (special) initial generalized centro-symmetric matrix pair [ X (1), Y (1)], a generalized centro-symmetric solution pair (the least Frobenius norm generalized centro-symmetric solution pair) can be obtained within finite number of iterations in the absence of roundoff errors. Also, the optimal approximation generalized centro-symmetric solution pair to a given generalized centro-symmetric matrix pair [ X ∼ , Y ∼ ] can be derived by finding the least Frobenius norm generalized centro-symmetric solution pair of new matrix equations. Moreover, the application of the proposed method to find a generalized centro-symmetric solution to the quadratic matrix equation Q ( X ) = AX 2 + BX + C = 0 is highlighted. Finally, two numerical examples are presented to support the theoretical results of this paper.
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an Iterative Algorithm for solving a pair of matrix equations ayb e cyd f over generalized centro symmetric matrices
Computers & Mathematics With Applications, 2008Co-Authors: Mehdi Dehghan, Masoud HajarianAbstract:A matrix [email protected]?R^n^x^n is said to be a symmetric orthogonal matrix if P=P^T=P^-^1. A matrix [email protected]?R^n^x^n is said to be generalized centro-symmetric (generalized central anti-symmetric) with respect to P, if A=PAP (A=-PAP). The generalized centro-symmetric matrices have wide applications in information theory, linear estimate theory and numerical analysis. In this paper, we propose a new Iterative Algorithm to compute a generalized centro-symmetric solution of the linear matrix equations AYB=E,CYD=F. We show, when the matrix equations are consistent over generalized centro-symmetric matrix Y, for any initial generalized centro-symmetric matrix Y"1, the sequence {Y"k} generated by the introduced Algorithm converges to a generalized centro-symmetric solution of matrix equations AYB=E,CYD=F. The least Frobenius norm generalized centro-symmetric solution can be derived when a special initial generalized centro-symmetric matrix is chosen. Furthermore, the optimal approximation generalized centro-symmetric solution to a given generalized centro-symmetric matrix can be derived. Several numerical examples are given to show the efficiency of the presented method.
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an Iterative Algorithm for the reflexive solutions of the generalized coupled sylvester matrix equations and its optimal approximation
Applied Mathematics and Computation, 2008Co-Authors: Mehdi Dehghan, Masoud HajarianAbstract:Abstract The generalized coupled Sylvester matrix equations ( AY - ZB , CY - ZD ) = ( E , F ) with unknown matrices Y , Z are encountered in many systems and control applications. Also these matrix equations have several applications relating to the problem of computing stable eigendecompositions of matrix pencils. In this work, we construct an Iterative Algorithm to solve the generalized coupled Sylvester matrix equations over reflexive matrices Y , Z . And when the matrix equations are consistent, for any initial matrix pair [ Y 0 , Z 0 ] , a reflexive solution pair can be obtained within finite iteration steps in the absence of roundoff errors, and the least Frobenius norm reflexive solution pair can be obtained by choosing a special kind of initial matrix pair. Also we obtain the optimal approximation reflexive solution pair to a given matrix pair [ Y ¯ , Z ¯ ] in the reflexive solution pair set of the generalized coupled Sylvester matrix equations ( AY - ZB , CY - ZD ) = ( E , F ) . Moreover, several numerical examples are given to show the efficiency of the presented Iterative Algorithm.
