Nonlinear Matrix Equation Research Articles

Unified interpolative of a Reich-Rus-Ćirić-type contraction in relational metric space with an application

In this paper, we introduce the notion of unified interpolative contractions of the Reich–Rus–Ćirić type and give some results about the fixed points for these mappings in the framework of relational metric spaces. We present examples where the results of some previous research are not relevant. Also, we apply our results to solving problems related to nonlinear matrix equations, emphasizing their practical importance.

One step inversion free iterative algorithm for computing generalised bisymmetric solution for nonlinear matrix equations

In this paper, we investigate one step inversion free iterative algorithm for computing generalised bisymmetric solution to the nonlinear matrix equations (NLME) X + A T X − q A + B T X − q B = H ( q ≥ 2 ) , where A , B , H are given matrices, while X is a generalised bisymmetric matrix to be computed. The necessary conditions for the existence of a generalised bisymmetric solution and the convergence of the suggested iterative method are derived. Some lemmas and theorems are provided and proved where the iterative solutions are obtained. Four numerical examples are presented to exhibit the effectiveness of the suggested method and to verify the theoretical findings of this paper.

Multiple standing waves of matrix nonlinear Schrödinger equations with mixed growth nonlinearities in [formula omitted]

In the whole space RN, we study the existence of two standing waves for a class of matrix nonlinear Schrödinger equations with potentials by using variational methods, where the nonlinearities are sublinear or asymptotically linear at infinity. The novelties are as follows. (1) The matrix nonlinear equations are defined in the whole space RN. (2) The nonlinearities are composed of two mixed nonlinear terms with different growth conditions. (3) The weight function may be sign-changing. (4) Our results can be applied to many examples.

Nonlinear matrix equations involving Kubo–Ando means

Nonlinear matrix equations involving Kubo–Ando means

Orthogonality and unitary conditions for solutions to some consistent linear matrix equations

The nonlinear matrix equation (A1 + B1X1D1)(A2 + B2X2D2) = A, presents a matrix identity, where A, Ai, Bi, and Di are known matrices of suitable sizes and Xi are unknown matrices for i = 1,2, over the field of complex numbers ℂ. Several known simple methods in linear algebra can handle the orthogonality problem. In this paper, we select some linear matrix equations as illustrative examples to discuss some properties of pairs of identically dimensional consistent linear matrix equations, then apply specific matrix analytic tools, among them is the matrix rank method, the rank of a matrix is one of the most basic quantities and useful methods and tools that are widely used in linear algebra specifically, in matrix theory and its applications. We consider certain forms of linear matrix equations such as T1X1 = N1 and the system A3X3 = C3, X3B3 = D3, to derive novel conditions dictating the orthogonality of all solutions to two consistent matrix equations, as well as the necessary and sufficient conditions for all solutions to be unitary, further we give an illustrative example.

New multivariable mean from nonlinear matrix equation associated to the harmonic mean

New multivariable mean from nonlinear matrix equation associated to the harmonic mean

An analytical method for free vibrations of the fuel rod with non-uniform mass of small modular reactor

The intricate internal structure of fuel rods results in a non-uniform mass distribution, making it imperative to employ analytical methods for accurate assessment. The study utilizes Euler beam theory to derive the transverse vibration equation for beams with varying mass distribution. The approach involves transforming the non-uniform mass beam into a multi-segment beam with concentrated mass points. Modal function relationships between adjacent uniform segments are established based on continuous conditions at connection points. This transformation leads to the conversion of the variable coefficient differential equation into a nonlinear matrix equation. The Newton-Raphson method is then applied to calculate the characteristic equation and mode shapes, essential for determining natural frequencies. To validate precision, the results obtained are compared with those derived from the finite element method. Furthermore, the developed method is employed to assess the impact of gas plenum location and length on the natural frequency of fuel rods. The proposed methodology serves as a rapid design tool, particularly beneficial during the design phase of fuel rods with non-uniform mass distribution, aiding in configuring structural aspects effectively.

Fixed point results for generalized almost contractions and application to a nonlinear matrix equation

<abstract><p>The goal of this paper was to improve some known results of fixed points by using $ w $-distances and properties of locally symmetric $ \mathcal{H} $-transitivity of binary relations. Also, we gave the application of the obtained results for finding the solution of nonlinear matrix equations. Finally, we gave a numerical example to demonstrate the applicability of our results.</p></abstract>

Teoreme postojanja za jedinstvenu interpolativnu Kananovu kontrakciju sa primenama kod nelinearnih matričnih jednačina

Introduction/purpose: This paper established a new mathematical framework by uncovering the relationships between Kannan contractions and interpolative Kannan contractions. The concept of unified interpolative Kannan contractions was introduced in the framework of a relational metric space. Additionally, the study aimed to broaden the concept of alpha admissibility by incorporating specific relational metric ideas. Methods: A detailed exploration of the properties and characteristics of Kannan contractions and interpolative Kannan contractions was conducted. The research introduced the concept of unified interpolative Kannan contractions and formulated new fixed point results for these mappings. Result: The study successfully established fixed point results for unified interpolative Kannan contractions within the framework of relational metric spaces. Additionally, an application of these results to solve a problem concerning nonlinear matrix equations was provided, further emphasizing their significance. Conclusion: The findings of this study significantly advanced the understanding of Kannan contractions and interpolative Kannan contractions, offering a unified framework for their analysis. The introduction of unified interpolative Kannan contractions and the expansion of alpha admissibility have broad implications for the field of mathematics.

Solving nonlinear matrix and Riesz-Caputo fractional differential equations via fixed point theory in partial metric spaces

A modified implicit relation and ?-implicit contractive condition are introduced in the setting of relational partial metric spaces and some related fixed point results are derived. Two suitable examples are provided. As an application, sufficient conditions are derived for the existence of a unique positive definite solution of the non-linear matrix equation X = B + Pk i=1A* i T(X)Ai. An example is given, using matrices that are randomly generated, as well as convergence and error analysis and average CPU time analysis. Solving fractional differential equations of Riesz-Caputo type with anti-periodic boundary conditions is also discussed, followed by two illustrations.

Suzuki-type fixed point theorems in relational metric spaces with applications

In this paper, we establish a relation-theoretic version of the results presented by Kim et al. (Journal of Nonlinear and Convex Analysis, 16 (9) (2015), 1779-1786). To showcase the versatility of our results, we furnish some illustrative examples. Furthermore, we exhibit an application of our results to establish sufficient conditions for the existence of a positive definite common solution to a pair of nonlinear matrix equations.

On positive definite solutions of the matrix equation $ X-\sum_{i = 1}^{m}A_{i}^{\ast}X^{-p_{i}}A_{i} = Q $

<p>In this paper, we used the outstanding properties of the Thompson metric to conclusively demonstrate the existence of a unique positive definite solution for the nonlinear matrix equation $ X-\sum_{i = 1}^{m}A_{i}^{\ast}X^{-p_{i}}A_{i} = Q $ without any additional assumptions. Furthermore, we designed an iterative algorithm to compute this unique positive definite solution, and derive its corresponding error estimate formula. Additionally, we presented three refined existence intervals for positive definite solutions of this equation. Finally, numerical examples were employed to validate the practicability of our iterative algorithm.</p>

Controlled Extended Branciari Quasi-b-Metric Spaces, Results, and Applications to Riesz-Caputo Fractional Differential Equations and Nonlinear Matrix Equations

We introduce the concept of controlled extended Branciari quasi-b-metric spaces, as well as a Gq-implicit type mapping. Under this new space setting, we derive some new fixed points, periodic points, right and left Ulam–Hyers stability, right and left weak well-posed properties, and right and left weak limit shadowing results. Additionally, we use these findings to solve the fractional differential equations of a Riesz–Caputo type with integral anti-periodic boundary values, as well of nonlinear matrix equations. All ideas, results, and applications are properly illustrated with examples.

Analiz mnogotochechnoy kraevoy zadachi dlya nelineynogo matrichnogo differentsial'nogo uravneniya

For a nonlinear differential matrix equation, we study a multipoint boundary valueproblem by a constructive method of regularization over the linear part of the equation usingthe corresponding fundamental matrices. Based on the initial data of the problem, sufficientconditions for its unique solvability are obtained. Iterative algorithms containing relativelysimple computational procedures are proposed for constructing a solution. Effective estimatesare given that characterize the rate of convergence of the iteration sequence to the solution, aswell as estimates of the solution localization domain.

Fixed point in ℳvb M_{\rm{v}}^{\rm{b}} – metric space and applications

Abstract The aim is to utilize a new metric called an M v b M_{\rm{v}}^{\rm{b}} –metric which is an improvement and generalization of Mv−metric to revisit the celebrated Banach and Sehgal contractions in M v b M_{\rm{v}}^{\rm{b}} –metric space. We demonstrate that the collection of open balls forms a basis on M v b M_{\rm{v}}^{\rm{b}} -metric space. Further, we give some examples for the verification of established results. Towards the end, we solve a non-linear matrix equation and an equation of rotation of a hanging cable to substantiate the utility of these extensions.

Bauer’s spectral factorization method for low order multiwavelet filter design

Para-Hermitian polynomial matrices obtained by matrix spectral factorization lead to functions useful in control theory systems, basis functions in numerical methods or multiscaling functions used in signal processing. We introduce a fast algorithm for matrix spectral factorization based on Bauer’s method. We convert Bauer’s method into a nonlinear matrix equation (NME). The NME is solved by two different numerical algorithms (Fixed Point Iteration and Newton’s Method) which produce approximate scalar or matrix factors, as well as a symbolic algorithm which produces exact factors in closed form for some low-order scalar or matrix polynomial matrices, respectively. Convergence rates of the two numerical algorithms are investigated for a number of singular and nonsingular scalar and matrix polynomials taken from different areas. In particular, one of the singular examples leads to new orthogonal multiscaling and multiwavelet filters. Since the NME can also be solved as a Generalized Discrete Time Algebraic Riccati Equation (GDARE), numerical results using built-in routines in Maple 17.0 and six Matlab versions are presented.

Efficient iterative schemes based on Newton's method and fixed-point iteration for solving nonlinear matrix equation Xp = Q±A(X−1+B)−1AT

PurposeIn this paper, the authors study the nonlinear matrix equation Xp=Q±A(X-1+B)-1AT, that occurs in many applications such as in filtering, network systems, optimal control and control theory.Design/methodology/approachThe authors present some theoretical results for the existence of the solution of this nonlinear matrix equation. Then the authors propose two iterative schemes without inversion to find the solution to the nonlinear matrix equation based on Newton's method and fixed-point iteration. Also the authors show that the proposed iterative schemes converge to the solution of the nonlinear matrix equation, under situations.Findings The efficiency indices of the proposed schemes are presented, and since the initial guesses of the proposed iterative schemes have a high cost, the authors reduce their cost by changing them. Therefore, compared to the previous scheme, the proposed schemes have superior efficiency indices.Originality/value Finally, the accuracy and effectiveness of the proposed schemes in comparison to an existing scheme are demonstrated by various numerical examples. Moreover, as an application, by using the proposed schemes, the authors can get the optimal controller state feedback of $x(t+1) = A x(t) + C v(t)$.

On the Iterative Methods for the Solution of Three Types of Nonlinear Matrix Equations

In this paper, we investigate the iterative methods for the solution of different types of nonlinear matrix equations. More specifically, we consider iterative methods for the minimal nonnegative solution of a set of Riccati equations, a nonnegative solution of a quadratic matrix equation, and the maximal positive definite solution of the equation X+A∗X−1A=Q. We study the recent iterative methods for computing the solution to the above specific type of equations and propose more effective modifications of these iterative methods. In addition, we make comments and comparisons of the existing methods and show the effectiveness of our methods by illustration examples.

Convex (α, β)-Generalized Contraction and Its Applications in Matrix Equations

This paper investigates the existence and convergence of solutions for linear and nonlinear matrix equations. This study explores the potential of convex (α,β)-generalized contraction mappings in geodesic spaces, ensuring the existence of solutions for both linear and nonlinear matrix equations. This paper extends the concept to partially ordered geodesic spaces and establishes new existence and convergence results. Illustrative examples are provided to demonstrate the practical relevance of the findings. Overall, this research contributes a novel approach to the field of matrix equations.

Optimization of Stabilization of the Lateral Motion of an Aircraft Using the Decomposition Method of Modal Synthesis

For the fourth-order model of the lateral motion of an aircraft with two controls, analytical expressions for the laws of stabilization control are obtained, which ensure the optimal placement of the poles. The synthesis is based on a two-level decomposition of the control object and the method of modal control of MIMO systems developed earlier by the authors with the optimal placement of the poles of a closed control system. The method is based on the features of quadratic control obtained by solving the nonlinear Lurie-Riccati matrix equation. In this case, for the optimal controller, it is necessary that the closed control object be asymptotically stable, and the matrix obtained by the product of the matrix of feedback coefficients by the control matrix of the dynamic plant must be positive-definite symmetric. Using this approach, final analytical expressions for the matrix of feedback coefficients are obtained and, accordingly, they can be used for any aircraft that has the same structure of its own dynamics and control matrices. The results of modeling the stabilization of the lateral motion of an aircraft using the obtained analytical control laws that ensure the optimal placement of the poles and, accordingly, the control laws using the decomposition method of synthesis with the same dynamic properties in the form of the value of the poles of a closed control system are presented. These properties correspond, as in the first case, to the optimal values of the placed poles. A comparison of transient processes by components of the maximum deviation of the controls shows that with optimal control, the maximum deviation of the rudder is 1.5 times less than with control using the standard decomposition method. All other parameters of the transient process, both in terms of the components of the state vector and the control vector, are approximately the same.