Matrix Inequalities Research Articles

The relaxed gradient based iterative algorithm for solving the generalized coupled complex conjugate and transpose Sylvester matrix equations

Inspired by the idea of Ma et al. (Journal of the Franklin Institute, 2018), we adopt relaxation technique and introduce relaxation factors into the gradient based iterative (GI) algorithm, and the relaxed based iterative (RGI) algorithm is established to solve the generalized coupled complex conjugate and transpose Sylvester matrix equations. By applying the real representation and straighten operation, we contain the sufficient and necessary condition for convergence of the RGI method. In order to effectively utilize this algorithm, we further derive the optimal convergence parameter and some related conclusions. Moreover, to overcome the high dimension calculation problem, a sufficient condition for convergence with less computational complexity is determined. Finally, numerical examples are reported to demonstrate the availability and superiority of the constructed iterative algorithm.

Matrix Pencil Optimal Iterative Algorithms and Restarted Versions for Linear Matrix Equation and Pseudoinverse

We derive a double-optimal iterative algorithm (DOIA) in an m-degree matrix pencil Krylov subspace to solve a rectangular linear matrix equation. Expressing the iterative solution in a matrix pencil and using two optimization techniques, we determine the expansion coefficients explicitly, by inverting an m×m positive definite matrix. The DOIA is a fast, convergent, iterative algorithm. Some properties and the estimation of residual error of the DOIA are given to prove the absolute convergence. Numerical tests demonstrate the usefulness of the double-optimal solution (DOS) and DOIA in solving square or nonsquare linear matrix equations and in inverting nonsingular square matrices. To speed up the convergence, a restarted technique with frequency m is proposed, namely, DOIA(m); it outperforms the DOIA. The pseudoinverse of a rectangular matrix can be sought using the DOIA and DOIA(m). The Moore–Penrose iterative algorithm (MPIA) and MPIA(m) based on the polynomial-type matrix pencil and the optimized hyperpower iterative algorithm OHPIA(m) are developed. They are efficient and accurate iterative methods for finding the pseudoinverse, especially the MPIA(m) and OHPIA(m).

Unified method for typical gear failure modeling and stiffness calculation based on the matrix equation

Unified method for typical gear failure modeling and stiffness calculation based on the matrix equation

A Dynamic Parameter Noise-Tolerant Zeroing Neural Network for Time-Varying Quaternion Matrix Equation With Applications.

As a common and significant problem in the field of industrial information, the time-varying quaternion matrix equation (TV-QME) is considered in this article and addressed by an improved zeroing neural network (ZNN) method based on the real representation of the quaternion. In the light of an improved dynamic parameter (IDP) and an innovative activation function (IAF), a dynamic parameter noise-tolerant ZNN (DPNTZNN) model is put forward for solving the TV-QME. The presented IDP with the character of changing with the residual error and the proposed IAF with the remarkable performance can strongly enhance the convergence and robustness of the DPNTZNN model. Therefore, the DPNTZNN model possesses fast predefined-time convergence and superior robustness under different noise environments, which are theoretically analyzed in detail. Besides, the provided simulative experiments verify the advantages of the DPNTZNN model for solving the TV-QME, especially compared with other ZNN models. Finally, the DPNTZNN model is applied to image restoration, which further illustrates the practicality of the DPNTZNN model.

Solution to Several Split Quaternion Matrix Equations

Split quaternions have various applications in mathematics, computer graphics, robotics, physics, and so on. In this paper, two useful, real representations of a split quaternion matrix are proposed. Based on this, we derive their fundamental properties. Then, via the real representation method, we obtain the necessary and sufficient conditions for the existence of solutions to two split quaternion matrix equations. In addition, two experimental examples are provided to show their feasibility.

The (anti‐)η$$ \eta $$‐Hermitian solution to a novel system of matrix equations over the split quaternion algebra

In comparison to quaternions, split quaternions exhibit a more intricate algebraic structure, characterized by the presence of nontrivial zero factors. Furthermore, in various fields such as geometry and electromagnetism, split quaternions serve as more convenient research tools than quaternions. This paper aims to establish the necessary and sufficient conditions for the existence of (anti‐) ‐Hermitian solutions to a novel system of matrix equations formulated over split quaternions. We also provide the general expression of such solutions when the system is solvable, along with the least squares (anti‐) ‐Hermitian solution in the case of inconsistency. To illustrate the key findings of this paper, numerical examples are presented.

Aerodynamic modeling and performance analysis of model predictive controller for fixed wing vertical takeoff and landing unmanned aerial vehicle

This research focuses mainly on the aerodynamic modelling and performance analysis of a model predictive controller for a hybrid fixed-wing vertical takeoff and landing of unmanned aerial vehicle. The aerodynamics, which comprises several aerodynamic characteristics including the lift, drag, and thrust coefficients, is modelled using Newton’s second law of motion. The force and moment equations were obtained, and they were then converted into matrix and state equation form, together with the transformation matrices. The essential equations with six degrees of freedom (6 DoF) and 12 state matrices including the control input and manipulating variables were obtained. The proposed model predictive controller (MPC) was designed using the optimal model predictive controller design parameters, such as a sampling time of 0.1, a prediction horizon of 15, a control horizon of 3 and additional controller settings. With a settling period of 3 s, an overshoot of 0.5865, and a steady state inaccuracy of 0.00785, the MATLAB simulation demonstrates that the system variables, including roll, pitch, and yaw, are stabilized. This MPC control is more effective in anticipating and optimizing the UAV than other control strategies. Eventually, the controller’s simulation on MATLAB Simulink demonstrates the controller’s ability to stabilize and control the system in a real-time application. Autonomous vertical takeoff and landing operations depend heavily on the mathematical model and architecture of the flight controller. Among the uses are inspection, monitoring, and rescue.

Low-rank generalized alternating direction implicit iteration method for solving matrix equations

Low-rank generalized alternating direction implicit iteration method for solving matrix equations

Ribbed elastic metasurface with lateral scalability for flexural wave manipulation

With the goal of engineering applications, the scalability and structural portability of elastic metasurfaces have become significant and attractive. However, most existing elastic metasurfaces cannot be arbitrarily scalable, which will inevitably limit their applications. Here, a novel conceptual design of ribbed elastic metasurfaces (REMs) is proposed for anomalous wavefront manipulation of flexural waves in thin plates. Ribbed subunits, as functional subunits of REMs, are characterized by simple constructions. The phase shift mechanism for the flexural waves across the ribbed subunits is revealed by analyzing the lowest dispersion bands. The analytical model for ribbed subunits is also established based on the equivalent orthotropic plate theory and transfer matrix method (TMM) to accurately predict the phase shift and amplitude of the transmitted waves. In addition, lateral scalability can be observed when the ratio of rib width to subunit width is constant, owing to the constant equivalent bending stiffness. By considering the lateral scalability of subunits, the continuous adjustment of the refraction angle can be perfectly achieved. The deflecting and focusing functionalities of the designed REMs are numerically and experimentally demonstrated. Both simulated and experimental results are consistent with theoretical results. Since ribbed plates are typical load-bearing structures in engineering, our design has broad application prospects, including but not limited to vibration control, energy harvesting, and structural health monitoring.

Tests for Large-Dimensional Shape Matrices via Tyler’s M Estimators

Tyler’s M estimator, as a robust alternative to the sample covariance matrix, has been widely applied in robust statistics. However, classical theory on Tyler’s M estimator is mainly developed in the low-dimensional regime for elliptical populations. It remains largely unknown when the parameter of dimension p grows proportionally to the sample size n for general populations. By using the eigenvalues of Tyler’s M estimator, this article develops tests for the identity and equality of shape matrices in a large-dimensional framework where the dimension-to-sample size ratio p/n has a limit in (0, 1). The proposed tests can be applied to a broad class of multivariate distributions including the family of elliptical distributions (see model (2.1) for details). To analyze both the null and alternative distributions of the proposed tests, we provide a unified theory on the spectrum of a large-dimensional Tyler’s M estimator when the underlying population is general. Simulation results demonstrate good performance and robustness of our tests. An empirical analysis of the Fama-French 49 industrial portfolios is carried out to demonstrate the shape of the portfolios varying. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

A Low-Rank Global Krylov Squared Smith Method for Solving Large-Scale Stein Matrix Equation

A Low-Rank Global Krylov Squared Smith Method for Solving Large-Scale Stein Matrix Equation

Programmable Wave‐Based Meta‐Computer

AbstractIn light of the escalating demand for computing powers, wave‐based analog computers have emerged as a compelling study area due to their remarkable capacity for ultra‐fast operation. Here, a programmable wave‐based computer, based on programmable meta‐structures is proposed, which is also called as meta‐computer. The meta‐computer can perform various types of analog computations in real‐time in the electromagnetic (EM) wave space, including matrix‐vector multiplication, discrete Fourier transform, filtering, and solving complex matrix equations. Owing to the powerful capability of the programmable meta‐structures in manipulating the EM fields and waves in real‐time, the proposed meta‐computer exhibits the advantages of flexible re‐programmability, low complexity, and direct computation in the EM space. Two prototypes of the meta‐computer in the microwave frequency are designed and fabricated, and the computing capability is validated through both numerical simulations and experiments. This work contributes a prominent approach for fully programmable wave‐based computers with noteworthy implications for expanding the applications of analog computing.

Optimization of some types of Rényi divergences between unitary orbits

In this article, we investigate explicitly the minimal and maximal values of the α-z-Rényi divergences and some other types of Rényi divergences between unitary orbits. Our main tools are the majorization theory of eigenvalues of matrices, and some matrix inequalities such as Golden–Thompson's and Araki–Lieb–Thirring's inequalities.

The Shear Effect of Large-Diameter Piles under Different Lateral Loading Levels: The Transfer Matrix Method

In various analytical models, modeling the behavior of large-diameter monopiles and piles can be challenging due to these foundations with huge body sizes carrying mechanisms of lateral loads to the surrounding soils. In this paper, the transfer matrix method with the Timoshenko beam theory was used to modify the shear rotation of pile sections under different loading stages, including serviceability limit stages and the ultimate loading stage. In this transfer matrix method, a large-diameter pile is considered according to the Timoshenko beam theory, and the recurring variables in the matrix equation are replaced with constants to simplify the calculation steps. Two model test cases were used to verify the accuracy of the method. Then, a series of comparisons between the Timoshenko beam and the Euler–Bernoulli beam theories, with the relative pile–soil stiffness being equal to 0.15, 0.45, and 0.75, was conducted to investigate the differences in pile response after considering the shear deformation. The results show that the effect of shear deformation of large-diameter piles changes with different loading levels. The values of the pile deformation based on the Timoshenko beam theory divided by those of that based on the Euler–Bernoulli beam theory were in the range of 1.0 to 1.10, and they increased slightly with increasing loads, reaching their maximum value, and then rapidly decreased to 1.0 when close to the ultimate lateral load; the maximum value was influenced by the relative pile–soil stiffness. Furthermore, the ratio of the shear rotation of the pile section to the slope of the deflection curve was in the range of 1.0 to 1.10; these also showed similar but more moderate trends compared with the values of pile deformation based on the Timoshenko beam theory divided by those of that based on the Euler–Bernoulli beam theory.

Thermal Vibration of Thick FGM Conical Shells by Using Third-Order Shear Deformation Theory.

A time-dependent third-order shear deformation theory (TSDT) approach on the displacements of thick functionally graded material (FGM) conical shells under dynamic thermal vibration is studied. Dynamic equations of motion with TSDT for thick FGM conical shells are applied directly with the partial derivative of variable R*θ in the curve coordinates (x, θ, z) instead of y in the Cartesian coordinates (x, y, z) for thick FGM plates, where R* is the middle-surface radius at any point on conical shells. The generalized differential quadrature (GDQ) numerical method is used to solve the dynamic differential equations in equilibrium matrix forms under thermal loads. It is the novelty of the current study to identify the parametric effects of shear correction coefficient, environment temperature, TSDT model, and FGM power law index on the displacements and stresses in the thick conical shells only subjected to sinusoidal heating loads. The physical parts with values on the length-to-thickness ratio equals 5, and 10 FGMs can be used in an area of an airplane engine that usually operates near more than 1000 K of temperatures when the thermal stress is considered and affected. The important findings of the presented study are listed as follows. The values of normal stress are in decreasing tendencies with time in cases when the coefficient c1 equals 0.925925/mm2 in TSDT and length-to-thickness ratio equals 5. The shear stress values in x plane z direction on the minor middle-surface radius (r) equals the major middle-surface radius (R) over 8 and length-to-thickness ratio equals to 5 can withstand T = 1000 K of pressure.

Iterative methods based on low-rank matrix for solving the Yang–Baxter-like matrix equation

Iterative methods based on low-rank matrix for solving the Yang–Baxter-like matrix equation

SOME COUNTING FORMULAE FOR -QUIDDITIES OVER THE RINGS

Abstract The $\lambda $ -quiddities of size n are n-tuples of elements of a fixed set, solutions of a matrix equation appearing in the study of Coxeter’s friezes. Their number and properties are closely linked to the structure and the cardinality of the chosen set. Our main objective is an explicit formula giving the number of $\lambda $ -quiddities of odd size, and a lower and upper bound for the number of $\lambda $ -quiddities of even size, over the rings ${\mathbb {Z}}/2^{m}{\mathbb {Z}}$ ( $m \geq 2$ ). We also give explicit formulae for the number of $\lambda $ -quiddities of size n over ${\mathbb {Z}}/8{\mathbb {Z}}$ .

Simulation of Sloped-Bed Tuned Liquid Dampers Using a Nonlinear Shallow Water Model

Simulation of Sloped-Bed Tuned Liquid Dampers Using a Nonlinear Shallow Water Model

Constrained Wave-telescope Technique

Constrained Wave-telescope Technique

Stability and nonlinear vibration characteristics of cantilevered fluid-conveying pipe with nonlinear energy sink

Considering the midline non-expansion theory and nonlinear coordinate method, this paper mainly investigates the stability and nonlinear vibration characteristics of cantilevered fluid-conveying pipe with energy sink. Based on Hamilton 's variational principle, the dynamic equation of a cantilevered pipe conveying fluid with nonlinear energy sink (NES) under pulsating flow is established. The system equation is processed by Galerkin method, a novel matrix equations of single-degree-of-freedom and multi-degree-of-freedom coupling system is established. The results show that the physical factors in the flow pipe system have a great influence on the critical flow velocity of the whole system, comparing with the case of adding NES. Increasing the proportion of the mass of the fluid will have a greater impact on the real part frequency, thereby increasing the critical velocity of the fluid-conveying pipe system. Addition of nonlinear energy sink can significantly suppress the amplitude and decrease the unstable vibration of the pipe.