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In this paper, we establish some efficient iteration methods for solving the general coupled discrete-time periodic matrix equation. More concretely, we propose some structure preserving matrix versions of subspace methods, such as bi-conjugate gradient, bi-conjugate residual, conjugate gradient squared, bi-conjugate gradient stabilized, generalized minimal residual and restarted generalized minimal residual methods. We demonstrate experimentally that the proposed iteration methods are feasible and efficient for the periodic matrix equation. [ABSTRACT FROM AUTHOR]

In this paper, some accelerated iterative algorithms are developed to find the minimal positive solution of the nonsymmetric algebraic Riccati equation arising in transport theory. The convergence analysis shows that the sequences of vectors generated by iterative algorithms with the initial vector (e , e) are monotonically increasing and converge to the minimal positive solution of the vector equations. Numerical examples are provided to illustrate the efficiency of the proposed algorithms and testify the conclusions suggested in this paper. [ABSTRACT FROM AUTHOR]

This paper describes iterative methods for solving the general linear matrix equation including the well-known Lyapunov matrix equation, Sylvester matrix equation and some related matrix equations encountered in control system theory, as special cases. We develop the methods from the optimization point of view in the sense that the iterative algorithms are constructed to solve some optimization problems whose solutions are closely related to the unique solution to the linear matrix equation. Actually, two optimization problems are considered and, therefore, two iterative algorithms are proposed to solve the linear matrix equation. To solve the two optimization problems, the steepest descent method is adopted. By means of the so-called weighted inner product that is defined and studied in this paper, the convergence properties of the algorithms are analysed. It is shown that the algorithms converge at least linearly for arbitrary initial conditions. The proposed approaches are expected to be numerically reliable as only matrix manipulation is required. Numerical examples show the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate the quaternion matrix equation X−AXF=C and X−A[Xtilde] F=C. For convenience, we named the quaternion matrix equations X−AXF=C and X−A[Xtilde] F=C as quaternion Stein matrix equation and quaternion Stein-conjugate matrix equation. Based on the Kronecker map and complex representation of a quaternion matrix, we give the solution expressions of the quaternion Stein matrix equation and quaternion Stein-conjugate matrix equation. Through these expressions, we can easily obtain the solution of the above two equations. In order to compare the direct algorithm with the indirect algorithm, we propose an example to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is concerned with numerical solutions to general linear matrix equations including the well-known Lyapunov matrix equation and Sylvester matrix equation as special cases. Gradient based iterative algorithm is proposed to approximate the exact solution. A necessary and sufficient condition guaranteeing the convergence of the algorithm is presented. A sufficient condition that is easy to compute is also given. The optimal convergence factor such that the convergence rate of the algorithm is maximized is established. The proposed approach not only gives a complete understanding on gradient based iterative algorithm for solving linear matrix equations, but can also be served as a bridge between linear system theory and numerical computing. Numerical example shows the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider the positive-definite solution to a system of nonlinear matrix equations,and, whereare two positive constants andare given matrices. We proposed two inversion-free iterative algorithms for obtaining the maximal positive-definite solution of the system. And we prove that the sequences generated by the iterative schemes are monotonically increasing and bounded above. Finally, some numerical examples are given to show the behaviour of the considered algorithms. [ABSTRACT FROM AUTHOR]

In this paper, the Hermitian positive-definite solutions of the matrix equationXs+A*X−tA=Qare considered. New necessary and sufficient conditions for the equation to have a Hermitian positive-definite solution are derived. In particular, whenAis singular, a new estimate of Hermitian positive-definite solutions is obtained. In the end, based on the fixed point theorem, an iterative algorithm for obtaining the positive-definite solutions of the equation withQ=Iis discussed. The error estimations are found. [ABSTRACT FROM AUTHOR]

LetAbe a nilpotent matrix of index three and consider the Yang–Baxter-like matrix equationAXA=XAX. We first obtain a simplified matrix equation of the same type withAreplaced by its Jordan form, whose Jordan blocks are at most. Then we obtain a system of matrix equations of smaller sizes to find all the commuting solutions of the original matrix equation. [ABSTRACT FROM PUBLISHER]

In this paper, we present an iterative algorithm for solving the following coupled Sylvester-transpose matrix equationsover the generalized centro-symmetric matrix group (X1,X2, …,Xq). The solvability of the problem can be determined by the proposed algorithm, automatically. If the coupled Sylvester-transpose matrix equations are consistent over the generalized centro-symmetric matrices, then a generalized centro-symmetric solution group can be obtained within finite iterative steps for any initial generalized centro-symmetric matrix group in the exact arithmetic. Furthermore, it is shown that the least-norm generalized centro-symmetric solution group of the coupled Sylvester-transpose matrix equations can be computed by choosing an appropriate initial iterative matrix group. Moreover, the optimal approximate generalized centro-symmetric solution group to a given arbitrary matrix group (V1,V2, …,Vq) can be derived by finding the least-norm generalized centro-symmetric solution group of a new coupled Sylvester-transpose matrix equations. Finally, some numerical results are given to illustrate the validity and practicability of the theoretical results established in this work. [ABSTRACT FROM PUBLISHER]

In this paper, we present an iterative method to compute the symmetric solutions of the matrix inequalityAX≥Bin least-squares sense. The presented iteration method can be also used to compute the symmetric solutions of the matrix equationAX=Band the consistent matrix inequalityAX≥B. Numerical experiments are given to illustrate the usefulness of the proposed approach. [ABSTRACT FROM PUBLISHER]

In this paper, combining the equivalent form of the unified coupled algebraic Riccati equation (UCARE) with the eigenvalue inequalities of a matrix's sum and product, using the properties of an M-matrix and its inverse matrix, we offer new lower and upper matrix bounds for the solution of the UCARE. Furthermore, applying the derived lower and upper matrix bounds and a fixed-point theorem, an existence uniqueness condition of the solution of the UCARE is proposed. Then, we propose a new fixed-point iterative algorithm for the solution of the UCARE. Finally, we present a corresponding numerical example to demonstrate the effectiveness of our results. [ABSTRACT FROM PUBLISHER]

In this paper, an iterative method is presented to solve the following constrained minimum Frobenius norm residual problem: [image omitted] where [image omitted] is a linear operator from Rm×n onto [image omitted] , [image omitted] , [image omitted] is a linear self-conjugate involution operator. By this method, for any initial matrix [image omitted] , a solution can be obtained in finite iteration steps in the absence of roundoff errors. The least norm solution can be derived when an appropriate initial matrix is chosen. In addition, the optimal approximation solution in the solution set of the above problem to a given matrix can also be derived by this method. Several numerical examples are given to show the efficiency of the proposed iterative method. [ABSTRACT FROM AUTHOR]

This paper is devoted to the perturbation analysis for the generalized Sylvester equation and the generalized Lyapunov equation. The explicit expressions and upper bounds of normwise, mixed and componentwise condition numbers for these equations are presented. The results are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, an iterative method is presented for finding the bisymmetric solutions of a pair of consistent matrix equations A1XB1=C1, A2XB2=C2, by which a bisymmetric solution can be obtained in finite iteration steps in the absence of round-off errors. Moreover, the solution with least Frobenius norm can be obtained by choosing a special kind of initial matrix. In the solution set of the matrix equations, the optimal approximation bisymmetric solution to a given matrix can also be derived by this iterative method. The efficiency of the proposed algorithm is shown by some numerical examples. [ABSTRACT FROM AUTHOR]

This paper is concerned with minimal norm least squares solution to general linear matrix equations including the well-known Lyapunov matrix equation and Sylvester matrix equation as special cases. Two iterative algorithms are proposed to solve this problem. The first method is based on the gradient search principle for solving optimization problem and the second one can be regarded as its dual form. For both algorithms, necessary and sufficient conditions guaranteeing the convergence of the algorithms are presented. The optimal step sizes such that the convergence rates of the algorithms are maximized are established in terms of the singular values of some coefficient matrix. It is believed that the proposed methods can perform important functions in many analysis and design problems in systems theory. [ABSTRACT FROM AUTHOR]

Given two polynomials, this paper is devoted to describing the natural relation between the Euclidean algorithm and the block LU factorization of the Hankel and Bezout matrices associated to such polynomials. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show how to use the Gauss map to produce efficient triangulations of parametric surfaces. The central idea is to invert canonical triangulations of the sphere using the Gauss map. The main problem is that the inverse of the Gauss map is not in general well defined. We show how to overcome this problem and formulate an algorithm to generate the desired triangulation. [ABSTRACT FROM PUBLISHER]

In this paper, we present the explicit expressions of the perturbation of the Drazin inverse under different conditions. Also, we give the upper bounds of ‖(A+E)D−A D‖ P /‖A D‖ P for these cases. [ABSTRACT FROM PUBLISHER]

The recursive solution to the Procrustes problem -with or without constraints- is thoroughly investigated. Given known matrices A and B , the proposed solution minimizes the square of the Frobenius norm of the difference A X − B when rows or columns are added to A and B. The proposed method is based on efficient strategies which reduce the computational cost by utilizing previous computations when new data are acquired. This is particularly useful in the iterative solution of an unbalanced orthogonal Procrustes problem. The results show that the computational efficiency of the proposed recursive algorithms is more significant when the dimensions of the matrices are large. This demonstrates the usefulness of the proposed algorithms in the presence of high-dimensional data sets. The practicality of the new method is demonstrated through an application in machine learning, namely feature extraction for image processing. [ABSTRACT FROM AUTHOR]

We perform extensive theoretical analysis on an inversion-free algorithm to obtain a maximal positive definite solution (PDS) of a general nonlinear matrix equation (NME) with suitable square matrices. We illustrate some estimations for obtaining the solution, rate of convergence and convergence analysis of the presented algorithm. To ensure its validity and usefulness, several different types of test problems, including randomly generated ones, are given. Comparisons with the existing iterative methods are done through error and computational time. [ABSTRACT FROM AUTHOR]

We consider a second-order damped-vibrational system described by the equation M ẍ+C(v) x˙+K x=0, where M, C(v), K are real, symmetric matrices of order n. We assume that the undamped eigenfrequencies (eigenvalues of (λ2 M+K) x=0) ω1, ω2, ..., ωn, are multiple in the sense that ω1=ω2, ω3=ω4, ..., ωn-1=ωn, or are given in close pairs ω1 ≈ ω2, ω3 ≈ ω4, ..., ωn-1 ≈ ωn. We present a formula which gives the solution of the corresponding phase space Lyapunov equation, which then allows us to calculate the first and second derivatives of the trace of the solution, with no extra cost. It can serve for the efficient trace minimization. [ABSTRACT FROM AUTHOR]

By applying the canonical correlation decomposition of matrix pairs, the general fixed rank least square solutions of matrix equation Xβ=Y are derived. As statistical applications, an algorithm for computing the least square estimator of the multivariate reduced rank regression model Y=Xβ+ε, r(β)=t is given. [ABSTRACT FROM AUTHOR]

We present an iterative method based on the Hermitian and skew-Hermitian splitting (HSS) for solving the continuous Sylvester equation. By using the HSS of the coefficient matricesAandB, we establish a method which is practically inner/outer iterations, by employing a conjugate gradient on the normal equations (CGNR)-like method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergent splitting of the coefficient matrices. Via this method, a Sylvester equation with coefficient matricesand(which are the skew-Hermitian part ofAandB, respectively) is solved iteratively by a CGNR-like method. Convergence conditions of this method are studied and numerical examples show the efficiency of this method. In addition, we show that the quasi-Hermitian splitting can induce accurate, robust and effective preconditioned Krylov subspace methods. [ABSTRACT FROM PUBLISHER]

In this work, we propose a class of linearized energy-conserved finite difference schemes for nonlinear space-fractional Schrödinger equations. We prove the energy conservation, stability, and convergence of our schemes. In the proposed schemes, we only need to solve linear algebraic systems to obtain the numerical solutions. Numerical examples are presented to verify the accuracy, energy conservation, and stability of these schemes. [ABSTRACT FROM PUBLISHER]

In this article, we propose an iterative algorithm to compute the minimum norm least-squares solution of AXB+CYD=E, based on a matrix form of the algorithm LSQR for solving the least squares problem. We then apply this algorithm to compute the minimum norm least-squares centrosymmetric solution of min X|AXB-E|F. Numerical results are provided to verify the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]