Conjugate gradient algorithm for consistent generalized Sylvester-transpose matrix equations

<abstract><p>We develop an effective algorithm to find a well-approximate solution of a generalized Sylvester-transpose matrix equation where all coefficient matrices and an unknown matrix are rectangular. The algorithm aims to construct a finite sequence of approximated solutions from any given initial matrix. It turns out that the associated residual matrices are orthogonal, and thus, the desire solution comes out in the final step with a satisfactory error. We provide numerical experiments to show the capability and performance of the algorithm.</p></abstract>

Stationary multi-kinks in the discrete sine-Gordon equation

We consider the existence and spectral stability of static multi-kink structures in the discrete sine-Gordon equation, as a representative example of the family of discrete Klein–Gordon models. The multi-kinks are constructed using Lin’s method from an alternating sequence of well-separated kink and antikink solutions. We then locate the point spectrum associated with these multi-kink solutions by reducing the spectral problem to a matrix equation. For an m-structure multi-kink, there will be m eigenvalues in the point spectrum near each eigenvalue of the primary kink, and, as long as the spectrum of the primary kink is imaginary, the spectrum of the multi-kink will be as well. We obtain analytic expressions for the eigenvalues of a multi-kink in terms of the eigenvalues and corresponding eigenfunctions of the primary kink, and these are in very good agreement with numerical results. We also perform numerical time-stepping experiments on perturbations of multi-kinks, and the outcomes of these simulations are interpreted using the spectral results.

On Hermitian Solutions of the Generalized Quaternion Matrix Equation A X B + C X D = E

The paper deals with the matrix equation A X B + C X D = E over the generalized quaternions. By the tools of the real representation of a generalized quaternion matrix, Kronecker product as well as vec-operator, the paper derives the necessary and sufficient conditions for the existence of a Hermitian solution and gives the explicit general expression of the solution when it is solvable and provides a numerical example to test our results. The paper proposes a unificated algebraic technique for finding Hermitian solutions to the mentioned matrix equation over the generalized quaternions, which includes many important quaternion algebras, such as the Hamilton quaternions and the split quaternions.

Controlling Stochastic Sensitivity by Feedback Regulators in Nonlinear Dynamical Systems with Incomplete Information

The problem of synthesis of stochastic sensitivity for equilibrium modes in nonlinear randomly forced dynamical systems with incomplete information is considered. We construct a feedback regulator that uses noisy data on some system state coordinates. For parameters of the regulator providing assigned stochastic sensitivity, a quadratic matrix equation is derived. Attainability of the assigned stochastic sensitivity is reduced to the solvability of this equation. We suggest a constructive algorithm for solving this quadratic matrix equation. These general theoretical results are used to solve the problem of stabilizing equilibrium modes of nonlinear stochastic oscillators under conditions of incomplete information. Details of our approach are illustrated on the example of a van der Pol oscillator.

Two-stage Algorithm for solving Arbitrary Trapezoidal Fully Fuzzy Sylvester Matrix Equations

Many authors proposed analytical methods for solving fully fuzzy Sylvester matrix equation (FFSME) based on Vec-operator and Kronecker product. However, these methods are restricted to nonnegative fuzzy numbers and cannot be extended to FFSME with near-zero fuzzy numbers. The main intention of this paper is to develop a new numerical method for solving FFSME with near-zero trapezoidal fuzzy numbers that provides a wider scope of trapezoidal fully fuzzy Sylvester matrix equation (TrFFSME) in scientific applications. This numerical method can solve the trapezoidal fully fuzzy Sylvester matrix equation with arbitrary coefficients and find all possible finite arbitrary solutions for the system. In order to obtain all possible fuzzy solutions, the TrFFSME is transferred to a system of non-linear equations based on newly developed arithmetic fuzzy multiplication between trapezoidal fuzzy numbers. The fuzzy solutions to the TrFFSME are obtained by developing a new two-stage algorithm. To illustrate the proposed method numerical example is solved.

Some Rank Formulas for the Yang-Baxter Matrix Equation AXA = XAX

Let A be an arbitrary square matrix, then equation AXA =XAX with unknown X is called Yang-Baxter matrix equation. It is difficult to find all the solutions of this equation for any given A . In this paper, the relations between the matrices A and B are established via solving the associated rank optimization problem, where B = AXA = XAX , and some analytical formulas are derived, which may be useful for finding all the solutions and analyzing the structures of the solutions of the above Yang-Baxter matrix equation.