Nonlinear Matrix Equation Research Articles

Extended Branciari quasi-b-distance spaces, implicit relations and application to nonlinear matrix equations

This study introduces extended Branciari quasi-b-distance spaces, a novel implicit contractive condition in the underlying space, and basic fixed-point results, a weak well-posed property, a weak limit shadowing property and generalized Ulam–Hyers stability. The given notions and results are exemplified by suitable models. We apply these results to obtain a sufficient condition ensuring the existence of a unique positive-definite solution of a nonlinear matrix equation (NME) mathcal{X}=mathcal{Q} + sum_{i=1}^{k}mathcal{A}_{i}^{*} mathcal{G(X)}mathcal{A}_{i}, where mathcal{Q} is an ntimes n Hermitian positive-definite matrix, mathcal{A}_{1}, mathcal{A}_{2}, …, mathcal{A}_{m} are n times n matrices, and mathcal{G} is a nonlinear self-mapping of the set of all Hermitian matrices that are continuous in the trace norm. We demonstrate this sufficient condition for the NME mathcal{X}= mathcal{Q} +mathcal{A}_{1}^{*}mathcal{X}^{1/3} mathcal{A}_{1}+mathcal{A}_{2}^{*}mathcal{X}^{1/3} mathcal{A}_{2}+ mathcal{A}_{3}^{*}mathcal{X}^{1/3}mathcal{A}_{3}, and visualize this through convergence analysis and a solution graph.

Approximate solutions of a SIR epidemiological model of computer viruses

In the present study, a collocation procedure based upon two different polynomials (Bessel and Legendre) is presented to solve a modified nonlinear epidemiological model of computer viruses, which is a three-dimensional system of ordinary differential equations (ODEs) with quadratic nonlinearities. Representing the unknown solutions and their derivatives in the matrix forms along with the collocation points, the presented approximation algorithm transforms the given system of equations into a nonlinear matrix equation. In addition to direct Bessel or Legendre-collocation method, a combination of the idea of quasi-linearization and the Bessel/Legendre-collocation is applied to the original nonlinear system. The main benefit of the combined approach is the efficiency while keeping the accuracy. To assess the accuracy of the results, an error estimation based upon residual is performed. We evaluate the accuracy and performance of the proposed algorithm through some numerical experiments and comparison with different available alternative algorithms are also carried out in order to show the validity of the scheme. Overall, it was found that the combined approach indicated highly satisfactory performance and is more efficient compared with other numerical results.

A New Inversion-Free Iterative Scheme to Compute Maximal and Minimal Solutions of a Nonlinear Matrix Equation

The goal of this article is to investigate a new solver in the form of an iterative method to solve X+A∗X−1A=I as an important nonlinear matrix equation (NME), where A,X,I are appropriate matrices. The minimal and maximal solutions of this NME are discussed as Hermitian positive definite (HPD) matrices. The convergence of the scheme is given. Several numerical tests are also provided to support the theoretical discussions.

Common positive solutions for two non-linear matrix equations using fixed point results in b-metric-like spaces

Common positive solutions for two non-linear matrix equations using fixed point results in b-metric-like spaces

Inversion free Iterative Method for Finding Symmetric Solution of the Nonlinear Matrix Equation 𝑿 − 𝑨∗𝑿𝒒𝑨 = 𝑰 (𝒒 ≥ 𝟐)

In this paper, we propose the inversion free iterative method to find symmetric solution of thenonlinear matrix equation 𝑿 − 𝑨∗𝑿𝒒𝑨 = 𝑰 (𝒒 ≥ 𝟐), where 𝑋 is an unknown symmetricsolution, 𝐴 is a given Hermitian matrix and 𝑞 is a positive integer. The convergence of theproposed method is derived. Numerical examples demonstrate that the proposed iterative methodis quite efficient and converges well when the initial guess is sufficiently close to the approximatesolution.
 Keywords: Symmetric solution, nonlinear matrix equation, inversion free, iterative method

Generalized Bessel Quasilinearization Technique Applied to Bratu and Lane–Emden-Type Equations of Arbitrary Order

The ultimate goal of this study is to develop a numerically effective approximation technique to acquire numerical solutions of the integer and fractional-order Bratu and the singular Lane–Emden-type problems especially with exponential nonlinearity. Both the initial and boundary conditions were considered and the fractional derivative being considered in the Liouville–Caputo sense. In the direct approach, the generalized Bessel matrix method based on collocation points was utilized to convert the model problems into a nonlinear fundamental matrix equation. Then, the technique of quasilinearization was employed to tackle the nonlinearity that arose in our considered model problems. Consequently, the quasilinearization method was utilized to transform the original nonlinear problems into a sequence of linear equations, while the generalized Bessel collocation scheme was employed to solve the resulting linear equations iteratively. In particular, to convert the Neumann initial or boundary condition into a matrix form, a fast algorithm for computing the derivative of the basis functions is presented. The error analysis of the quasilinear approach is also discussed. The effectiveness of the present linearized approach is illustrated through several simulations with some test examples. Comparisons with existing well-known schemes revealed that the presented technique is an easy-to-implement method while being very effective and convenient for the nonlinear Bratu and Lane–Emden equations.

An implicit relation, relational theoretic approach under w-distance and application to nonlinear matrix equations

We propose a new class of implicit relations and an implicit type contractive condition based on it in the relational metric spaces under w-distance functional. Further we derive fixed points results based on them. Useful examples illustrate the applicability and effectiveness of the presented results. We apply these results to discuss sufficient conditions ensuring the existence of a unique positive definite solution of the nonlinear matrix equation (NME) of the form mathcal{U}=mathcal{Q} + sum_{i=1}^{k}mathcal{A}_{i}^{*} mathcal{G}mathcal{(U)}mathcal{A}_{i}, where mathcal{Q} is an ntimes n Hermitian positive definite matrix, mathcal{A}_{1}, mathcal{A}_{2}, …, mathcal{A}_{m} are n times n matrices and mathcal{G} is a nonlinear self-mapping of the set of all Hermitian matrices which are continuous in the trace norm. In order to demonstrate the obtained conditions, we consider an example together with convergence and error analysis and visualisation of solutions in a surface plot.

On Iterative Algorithm and Perturbation Analysis for the Nonlinear Matrix Equation

On Iterative Algorithm and Perturbation Analysis for the Nonlinear Matrix Equation

Common Positive Solution of Two Nonlinear Matrix Equations Using Fixed Point Results

We discuss a pair of nonlinear matrix equations (NMEs) of the form X=R1+∑i=1kAi*F(X)Ai, X=R2+∑i=1kBi*G(X)Bi, where R1,R2∈P(n), Ai,Bi∈M(n), i=1,⋯,k, and the operators F,G:P(n)→P(n) are continuous in the trace norm. We go through the necessary criteria for a common positive definite solution of the given NME to exist. We develop the concept of a joint Suzuki-implicit type pair of mappings to meet the requirement and achieve certain existence findings under weaker assumptions. Some concrete instances are provided to show the validity of our findings. An example is provided that contains a randomly generated matrix as well as convergence and error analysis. Furthermore, we offer graphical representations of average CPU time analysis for various initializations.

θ*-Weak Contractions and Discontinuity at the Fixed Point with Applications to Matrix and Integral Equations

In this paper, the notion of θ*-weak contraction is introduced, which is utilized to prove some fixed point results. These results are helpful to give a positive response to certain open question raised by Kannan and Rhoades on the existence of contractive definition which does not force the mapping to be continuous at the fixed point. Some illustrative examples are also given to support our results. As applications of our result, we investigate the existence and uniqueness of a solution of non-linear matrix equations and integral equations of Volterra type as well.

Latest Inversion-Free Iterative Scheme for Solving a Pair of Nonlinear Matrix Equations

In this work, the following system of nonlinear matrix equations is considered, X 1 + A ∗ X 1 − 1 A + B ∗ X 2 − 1 B = I and X 2 + C ∗ X 2 − 1 C + D ∗ X 1 − 1 D = I , where A , B , C , and D are arbitrary n × n matrices and I is the identity matrix of order n . Some conditions for the existence of a positive-definite solution as well as the convergence analysis of the newly developed algorithm for finding the maximal positive-definite solution and its convergence rate are discussed. Four examples are also provided herein to support our results.

Elegant Iterative Methods for Solving a Nonlinear Matrix Equation X-A* eX A=I

The nonlinear matrix equation was solved by Gao (2016) via standard fixed point method. In this paper, three more elegant iterative methods are proposed to find the approximate solution of the nonlinear matrix equation namely: Newton’s method; modified fixed point method and a combination of Newton’s method and fixed point method. The convergence of Newton’s method and modified fixed point method are derived. Comparative numerical experimental results indicate that the new developed algorithms have both less computational time and good convergence properties when compared to their respective standard algorithms.
 Keywords: Hermitian positive definite solution; nonlinear matrix equation; modified fixed point method; iterative method

Approximating Solutions of Non-Linear Troesch’s Problem via an Efficient Quasi-Linearization Bessel Approach

Two collocation-based methods utilizing the novel Bessel polynomials (with positive coefficients) are developed for solving the non-linear Troesch’s problem. In the first approach, by expressing the unknown solution and its second derivative in terms of the Bessel matrix form along with some collocation points, the governing equation transforms into a non-linear algebraic matrix equation. In the second approach, the technique of quasi-linearization is first employed to linearize the model problem and, then, the first collocation method is applied to the sequence of linearized equations iteratively. In the latter approach, we require to solve a linear algebraic matrix equation in each iteration. Moreover, the error analysis of the Bessel series solution is established. In the end, numerical simulations and computational results are provided to illustrate the utility and applicability of the presented collocation approaches. Numerical comparisons with some existing available methods are performed to validate our results.

Some Fixed Point Results on Relational Quasi Partial Metric Spaces and Application to Non-Linear Matrix Equations

We introduce a qϱ-implicit contractive condition by an implicit relation on relational quasi partial metric spaces and establish new (unique) fixed point results and periodic point results based on it. We justify the results by two suitable examples and compare with them related work. We discuss sufficient conditions ensuring the existence of a unique positive definite solution of the non-linear matrix equation U=B+∑i=1mAi*G(U)Ai, where B is an n×n Hermitian positive definite matrix, A1, A2, … Am are n×n matrices, and G is a non-linear self-mapping of the set of all Hermitian matrices which is continuous in the trace norm. Two examples (with randomly generated matrices and complex matrices, respectively) are given, together with convergence and error analysis, as well as average CPU time analysis and visualization of solution in surface plot.

On Hermitian positive definite solutions of a nonlinear matrix equation

In this paper, the Hermitian positive definite solutions of the matrix equation $$X^s +A^* X^{ - t}A = Q$$ , where A is an $$n \times n$$ nonsingular complex matrix, Q is an $$n \times n$$ Hermitian positive definite matrix and $$s, t> 0$$ , are discussed. Some conditions for the existence of Hermitian positive definite solutions of this equation are derived. In addition, two iterative methods to obtaining the maximum or minimum Hermitian positive definite solutions of this equation are proposed. In addition, a necessary and sufficient condition for the existence of these solutions is presented. Theoretical results are illustrated by some numerical examples.

Iterative Method for Positive Definite Solution of a Class of Nonlinear Matrix Equation

In this paper, we study the matrix equation X+A * (R+B * XB) -t A=Q (0<t⩽1), where A, B, R, Q are matrices of appropriate size and R, Q are both positive definite matrices. Based on the fixed point theorem, we suggest four iterative algorithms for solving this equation and prove that the suggested iterative algorithms are convergent with proper conditions. What’s more, the conditions for the existence of the positive definite solution are given. The convergence analysis of the suggested algorithms is established. Some numerical examples are presented to illustrate the convergence behaviour of the various algorithms.

New algorithm for finding the solution of nonlinear matrix equations based on the weak condition with relation-theoretic F-contractions

The aim of this work is to introduce the concept of an $$(F,\gamma )_{\mathfrak {R}}$$ -contraction which is a weak idea of an F-contraction in metric spaces endowed with a binary relation. Fixed point results for such new contractions are investigated, and its benefit is showing from an example. As applications, we apply theoretical results together with the Thompson metric for solving nonlinear matrix equations. Finally, we give a numerical example showing the usage of the proposed algorithm to solve nonlinear matrix equations.

Analysis of the electromagnetic wave illumination on the transmission lines with nonlinear terminations

AbstractIn this paper, an unconditionally stable time-domain method is proposed to investigate the external electromagnetic (EM) field illumination on the nonlinearly loaded transmission lines. In the proposed algorithm, the field to line coupling equations and linear/nonlinear boundary conditions are incorporated and expressed in a matrix form. The derived differential matrix equations are then discretized and solved using the proposed finite difference time domain (FDTD) method approach. The discretized nonlinear matrix equations are also solved by applying a globally convergent iterative method to ensure the stability of the method. Moreover, the experimental investigations are conducted using a transverse EM (TEM) cell for a single microstrip line and a power detector circuit as a nonlinearly loaded transmission line to confirm the accuracy and stability of the proposed method. With the merit of satisfactory accuracy and unconditional stability for the entire range of time steps, the presented approach would be applicable for analyzing the microwave linear/nonlinear circuits subjected to the external incident EM wave.

Investigation of positive definite solution of nonlinear matrix equation $$X^{p}=Q +\sum \nolimits _{i=1}^m A_i^*X^{\delta }A_i$$

In this paper, we consider the nonlinear matrix equation $$X^{p}=Q +\sum \nolimits _{i=1}^m A_i^*X^{\delta }A_i,$$ where $$A_i (i=1,2,\ldots ,m)$$ are $$n\times n$$ nonsingular complex matrices, Q is a $$n\times n$$ Hermitian positive definite (HPD) matrix, $$p\ge 1, m\ge 1$$ are positive integers, and $$\delta \in (0,1)$$ . We discuss the solution of this equation via properties of Thompson metric and two fixed point theorems in ordered Banach spaces and estimate the bounds of the HPD solution. Furthermore, perturbation analysis is investigated.

An efficient approximation technique applied to a non-linear Lane–Emden pantograph delay differential model

The primary focus of our work is to propose a computationally effective approximation algorithm to find the numerical solution of the so-called a new design of second-order Lane–Emden pantograph delayed problem with singularity and non-linearity. Our approach based upon the novel Bessel matrix representation together with the collocation points which transforms the newly designed model problem into a non-linear fundamental matrix equation. To testify the validity and applicability of the proposed method, three test examples with non-linearity are given. The computational results are accurate as compared with the exact solutions as well as with those of numerical values reported in the literature.