In this paper, by the transformation form of the discrete algebraic Riccati equation (DARE), we propose a new inverse-free iterative algorithm to obtain the positive definite solution of the DARE. Furthermore, the mon...
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In this paper, by the transformation form of the discrete algebraic Riccati equation (DARE), we propose a new inverse-free iterative algorithm to obtain the positive definite solution of the DARE. Furthermore, the monotone convergence is proved and convergence rate analysis is presented for the derived algorithm. Compared with some existing algorithms, numerical examples demonstrate the feasibility and effectiveness of our algorithm.
Linear matrix equations play an important role in many areas, such as control theory, system theory, stability theory and some other fields of pure and applied mathematics. In the present paper, we consider the genera...
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Linear matrix equations play an important role in many areas, such as control theory, system theory, stability theory and some other fields of pure and applied mathematics. In the present paper, we consider the generalized coupled Sylvester-transpose and conjugate matrix equations T-v(X) = F-v, v = 1, 2, ... , N, where X = (X-1,X-2, ... , X-p) is a group of unknown matrices and for v = 1,2, ... , N, T-v(X) = (p)Sigma(i=1) (s1)Sigma(mu=1) A(vi mu)X(i)B(vi mu) + (s2)Sigma(mu=1) (Cvi mu XiDvi mu)-D-T + (s3)Sigma(mu=1) M-vi mu(X) over bar N-i(vi mu) + (s4)Sigma(mu=1) H(vi mu)X(i)(H)G(vi mu,) in which A(vi mu), B-vi mu, C-vi mu, D-vi mu, M-vi mu, N-vi mu, H-vi mu, G(vi mu) and F-v are given matrices with suitable dimensions defined over complex number field. By using the hierarchical identification principle, an iterative algorithm is proposed for solving the above coupled linear matrix equations over the group of reflexive (anti-reflexive) matrices. Meanwhile, sufficient conditions are established which guarantee the convergence of the presented algorithm. Finally, some numerical examples are given to demonstrate the validity of our theoretical results and the efficiency of the algorithm for solving the mentioned coupled linear matrix equations.
This paper is concerned with a new composite iteration approximating to common fixed points for a finite family of nonexpansive mappings in Banach spaces which have a uniformly Gateaux differentiable norm. Utilizing t...
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This paper is concerned with a new composite iteration approximating to common fixed points for a finite family of nonexpansive mappings in Banach spaces which have a uniformly Gateaux differentiable norm. Utilizing the iterative algorithm, we obtain the strong convergence theorems for a finite family of nonexpansive mappings. Furthermore, the problem of image recovery is considered in the above result. Our results extend and improve the corresponding results. (C) 2009 Elsevier B.V. All rights reserved.
In this article, a finite difference parallel iterative (FDPI) algorithm for solving 2D Poisson equation was presented. Based on the domain decomposition, the domain was divided into four sub-domains and the four iter...
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In this article, a finite difference parallel iterative (FDPI) algorithm for solving 2D Poisson equation was presented. Based on the domain decomposition, the domain was divided into four sub-domains and the four iterative schemes were constructed from the classical five-point difference scheme to implement the algorithm differently with the number of iterations of odd or even. Although the iterative schemes are semi-implicit, they can be computed explicitly and in parallel in combining with the boundary conditions. The convergence of the presented algorithm was proved. Particularly, a relaxation factor. was added into the iterative schemes to improve the convergence rate and decrease the number of iterations. Finally, several numerical experiments were presented to examine the efficiency and accuracy of the iterative algorithm. Also, the comparison between the numerical results that were derived from Jacobi, Gauss-Seidel iterative algorithms, Mathematics Stencil (Fen et al., China Sci 35 (2005), 901-909) method, and the presented algorithm with the optimal relaxation *** demonstrated that the presented algorithm has smaller number of iterations, shorter computational time, and faster convergence rate. Furthermore, the presented algorithm is also applicable to 2D variable coefficient elliptic problems. (C) 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 829-853, 2011
A proximal iterative algorithm for the mulitivalue operator equation 0 ∈ T(x) is presented, where T is a maximal monotone operator. It is an improvement of the proximal point algorithm as well know. The convergence o...
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A proximal iterative algorithm for the mulitivalue operator equation 0 ∈ T(x) is presented, where T is a maximal monotone operator. It is an improvement of the proximal point algorithm as well know. The convergence of the algorithm is discussed and all example is given.
This note presents an iterative algorithm to solve the coupled Sylvester-transpose matrix equations (including the generalized coupled Sylvester matrix equations and Lyapunov matrix equations as special cases) over ge...
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This note presents an iterative algorithm to solve the coupled Sylvester-transpose matrix equations (including the generalized coupled Sylvester matrix equations and Lyapunov matrix equations as special cases) over generalized centro-symmetric matrices. When the considered matrix equations are consistent, for any initial generalized centro-symmetric matrix group, a generalized centro-symmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors. The least Frobenius norm generalized centro-symmetric solution group of the coupled Sylvester-transpose matrix equations can be derived when a suitable initial generalized centro-symmetric matrix group is chosen. In addition, for a given generalized centro-symmetric matrix group, the optimal approximation generalized centro-symmetric solution group can be obtained by finding the least Frobenius norm generalized centro-symmetric solution group of new coupled Sylvester-transpose matrix equations. Finally, a numerical example is given to demonstrate the efficiency of the introduced iterative algorithm.
The nonlinear Poisson problems in heat diffusion governed by elliptic type partial differential equations are solved by a modified globally optimal iterative algorithm (MGOIA). The MGOIA is a purely iterative method f...
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The nonlinear Poisson problems in heat diffusion governed by elliptic type partial differential equations are solved by a modified globally optimal iterative algorithm (MGOIA). The MGOIA is a purely iterative method for searching the solution vector x without using the invert of the Jacobian matrix D. Moreover, we reveal the weighting parameter alpha(c) in the best descent vector w = alpha E-c + (DE)-E-T and derive the convergence rate and find a criterion of the parameter gamma. When utilizing alpha(c) and gamma, we can farther accelerate the convergence speed several times. Several numerical experiments are carefully discussed and validated the proposed method.
Twin support vector machine (TSVM) is a practical machine learning algorithm, whereas traditional TSVM can be limited for data with outliers or noises. To address this problem, we propose a novel TSVM with the symmetr...
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Twin support vector machine (TSVM) is a practical machine learning algorithm, whereas traditional TSVM can be limited for data with outliers or noises. To address this problem, we propose a novel TSVM with the symmetric LINEX loss function (SLTSVM) for robust classification. There are several advantages of our method: (1) The performance of the proposed SLTSVM for data with outliers or noise can be improved by using the symmetric LINEX loss function. (2) The introduction of regularization term can effectively improve the generalization ability of our model. (3) An efficient iterative algorithm is developed to solve the optimization problems of our SLTSVM. (4) The convergence and time complexity of the iterative algorithm are analyzed in detail. Furthermore, our model does not involve loss function parameter, which makes our method more competitive. Experimental results on synthetic, benchmark and image datasets with label noises and feature noises demonstrate that our proposed method slightly outperforms other state-of-the-art methods on most datasets.(c) 2023 Elsevier Ltd. All rights reserved.
In this paper, we discuss some iterated method for solving the saddle point problem. We propose some new schemes and prove its convergence. The method has weaker convergence condition than the classic Uzawa method. Th...
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In this paper, we discuss some iterated method for solving the saddle point problem. We propose some new schemes and prove its convergence. The method has weaker convergence condition than the classic Uzawa method. The analysis is supported by numerical experiments. (c) 2005 Elsevier Inc. All rights reserved.
In this paper, an iterative algorithm is constructed to solve the minimum Frobenius norm residual problem: min parallel to((A1XB1)(A2XB2)) - ((C1)(C2))parallel to over bisymmetric matrices. By this algorithm, for any ...
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In this paper, an iterative algorithm is constructed to solve the minimum Frobenius norm residual problem: min parallel to((A1XB1)(A2XB2)) - ((C1)(C2))parallel to over bisymmetric matrices. By this algorithm, for any initial bisymmetric matrix X-0, a solution X* can be obtained infinite 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 (X) over cap to a given matrix (X) over bar 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. (C) 2009 Elsevier Ltd. All rights reserved.
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