Matrix Exponential Research Articles

Dirac cones in three-layered plates

The proper Dirac-like cones are observed in dispersion portraits of guided waves propagating in isotropic three-layered traction-free plates. It is revealed that if both layers have equal Poisson’s ratios and satisfy Wiechert condition, the Dirac-like cones appear regardless of the Poisson’s ratio value. This phenomenon opens a new approach for the creation of the layered plates possessing Dirac-like cones. The analysis is based on the Cauchy sextic formalism combined with the exponential fundamental matrix method.

Extending nonstandard finite difference scheme rules to systems of nonlinear ODEs with constant coefficients

In this paper, we present a reformulation of Mickens' rules for nonstandard finite difference (NSFD) schemes to adapt them to systems of ODEs. We want to compare several methods within the class of NSFD schemes. These methods are expressed as exponential integrators which treat exactly the linear part of the equations. While the classical methods address each equation of a system separately, we propose here to address them as a whole, which involves a reformulation of Mickens' rules. This leads to enhance the results compared to the classical case, even if approximations of the matrix exponentials are used. Precisely, using Cayley–Hamilton theorem we obtain two correction factors, one of which is linked to the size of the system and the other to the nonlinear term, we are thus able to use test cases which show the impact of one or another corrector. We prove the consistency and the zero-stability of the method in the nonlinear case to conclude to the convergence of the method.

Validity and error analysis of calculating matrix exponential function and vector product

Validity and error analysis of calculating matrix exponential function and vector product

A Unified Dispersive Media and PML FDTD Formulation Based on the Matrix Exponential Method

A Unified Dispersive Media and PML FDTD Formulation Based on the Matrix Exponential Method

Solving the Matrix Exponential Function for Special Orthogonal Groups SO(n) up to n = 9 and the Exceptional Lie Group G2

In this work the matrix exponential function is solved analytically for the special orthogonal groups SO(n) up to n=9. The number of occurring k-th matrix powers gets limited to 0≤k≤n−1 by exploiting the Cayley–Hamilton relation. The corresponding expansion coefficients can be expressed as cosine and sine functions of a vector-norm V and the roots of a polynomial equation that depends on a few specific invariants. Besides the well-known case of SO(3), a quadratic equation needs to be solved for n=4,5, a cubic equation for n=6,7, and a quartic equation for n=8,9. As an interesting subgroup of SO(7), the exceptional Lie group G2 of dimension 14 is constructed via the matrix exponential function through a remarkably simple constraint on an invariant, ξ=1. The traces of the SO(n)-matrices arising from the exponential function are sums of cosines of several angles. This feature confirms that the employed method is equivalent to exponentiation after diagonalization, but avoids complex eigenvalues and eigenvectors and operates only with real-valued quantities.

A time splitting Chebyshev-Fourier spectral method for the time-dependent rotating nonlocal Schrödinger equation in polar coordinates

In this article, we propose a time splitting Chebyshev-Fourier spectral method to compute the dynamics of rotating nonlocal Schrödinger equation in polar coordinates. Since the rotation term is diagonalizable with Fourier spectral method in azimuthal direction, we split the Hamiltonian into a linear part, i.e., the Laplacian and rotation terms, and the nonlinear part. The linear part is discretized by Chebyshev-Fourier spectral method in space and integrated by Crank-Nicolson or exact matrix exponential. The nonlinear part is solved exactly in physical space, and the nonlocal potential, which is defined as convolution φ:=U⁎|ψ|2 with a singular kernel U, is computed via Kernel Truncation method with spectral accuracy. Then we construct a high order time splitting spectral method. Our scheme is spectrally accurate in space and of second/fourth order in the temporal direction. Extensive numerical results are presented to confirm the accuracy and efficiency for both the nonlocal potential and the wave function, together with one application to a two-component rotating dipolar Bose-Einstein condensates. In addition, we investigate the rotational symmetry preserving performance by a comprehensive comparison with existing scheme.

Characteristic curves and the exponentiation in the Riordan Lie group: A connection through examples

We point out how to use the classical characteristic method, that is used to solve quasilinear PDE's, to obtain the matrix exponential of some lower triangle infinite matrices. We use the Lie Fréchet structure of the Riordan group described in [4]. After that, we describe some linear dynamical systems in K[[x]] with a concrete involution being a symmetry or a time-reversal symmetry for them. We take this opportunity to assign some dynamical properties to Pascal's triangle.

A fast and efficient computation method for reflective diffraction simulations

We present a new computation method for simulating reflection high-energy electron diffraction and the total-reflection high-energy positron diffraction experiments. The two experiments are used commonly for the structural analysis of material surface. The present paper improves the conventional numerical method, the multi-slice method, for faster computation, since the present method avoids the matrix-eigenvalue solver for the computation of matrix exponentials and can adopt higher-order ordinary differential equation solvers. Moreover, we propose a high-performance implementation based on multi-thread parallelization and cache-reusable subroutines. In our tests, this new method performs up to 2,000 times faster than the conventional method without compromising accuracy.

On infinite group velocity in composite plates

The analytical and geometrical conditions for the possible appearance of the infinite group velocity (IGV) points belonging to the dispersion curves of Lamb waves propagating in traction-free plates are studied by a combined method comprising Cauchy sextic formalism and the exponential fundamental matrix method. According to the obtained geometrical condition, the IGV corresponds to the coincidence of a tangent line to any of the dispersion curves with a straight line passing through the origin. The developed technique is demonstrated by applying to the analysis of Lamb wave dispersion in three-layered plates with a soft inner core.

High-order exponential integrators for the Riesz space-fractional telegraph equation

In this paper, we study the numerical solution of a class of Riesz space fractional telegraph equation. In the spatial direction, the equations are discretized using the fractional central difference scheme, and an equivalent semi-linear form is obtained. Then, a fourth-order exponential Runge–Kutta method is chosen in the temporal direction. Moreover, an efficient method for calculating the matrix exponent and matrix φ-function is proposed by performing a series of matrix transformations on the coefficient matrix in the semi-linear form, improving the efficiency of the matrix functions calculation. Several numerical experiments show that the convergence order of the scheme is O(h2+τ4), where h is the space step and τ is the time step. The effectiveness of the scheme is also verified.

A coupled hygro-elastic 3D model for steady-state analysis of functionally graded plates and shells

Abstract This 3D coupled hygro-elastic model proposes the three-dimensional (3D) equilibrium equations associated with the 3D Fick diffusion equation for spherical shells. The primary unknowns of the problem are the displacements and the moisture content. This coupled 3D exact shell model allows to understand the effects of the moisture field in relation with the elastic field on stresses and deformations in different plates and shells. This model is specifically developed for configurations including functionally graded material (FGM) layers. Four different geometries are analyzed using an orthogonal mixed curvilinear reference system. The main advantage of this reference system for spherical shells is the degeneration of the equations to those for simpler geometries. The solving method is the exponential matrix method in the thickness direction. The closed-form solution is possible because of simply supported sides and harmonic forms for displacements and moisture content. The moisture content amplitudes are directly applied at the top and bottom outer faces through steady-state hypotheses. The final system is based on a set of coupled homogeneous second-order differential equations. The moisture field effects are evaluated for the static analysis in terms of displacement, strain, and stress components. After preliminary validations, used to better understand how to properly define the calculation of the curvature-related terms and FGM properties, four new benchmarks are proposed for several thickness ratios, geometrical data, FGM configurations, and moisture values imposed at the external surfaces. From the results, it is clear the accordance between the uncoupled hygro-elastic model and this new coupled hygro-elastic model when the 3D Fick diffusion law is employed. Both effects connected with the thickness layer and the embedded material are included in the 3D hygro-elastic analyses proposed. The 3D coupled hygro-elastic model is simpler than the uncoupled one because the 3D Fick diffusion law does not have to be separately solved.

Order reduction of matrix exponentials by proper orthogonal decomposition

Many applications of the matrix exponential exp(At) of a real matrix A and a real parameter t require repeated evaluation of it for different values of t. Such evaluations are time consuming and encounter the curse of dimensionality, especially in the large scale cases. Therefore, reduced order versions of the matrix exponential will be of great use and ease in these situations. The proper orthogonal decomposition (POD) method has wide applications in the reduced order modeling of dynamical systems. The matrix exponential can be considered as the solution of an initial value problem (IVP) as a dynamical system and the reduced order approximations to it be obtained by utilizing the POD method. Following this approach, the matrix exponential exp(At) is viewed as a scalar to matrix function and the optimal subspace(s) of its range space for the order reduction is computed by the POD method. Next, the IVP defining the matrix exponential is projected on the obtained optimal subspace(s). Then the solution of projected IVP is adopted as the desired reduced order approximation of the matrix exponential. An upper estimate for the error of the reduced order approximation is obtained thorough an error analysis. Numerical experiments illustrating efficiency of the method and quality of the obtained approximations is provided.

A stronger form of Yamamoto's theorem on singular values

For a matrix T∈Mm(C), let |T|:=T⁎T. For A∈Mm(C), we show that the matrix sequence {|An|1n}n∈N converges to a positive-semidefinite matrix H whose jth-largest eigenvalue is equal to the jth-largest eigenvalue-modulus of A (for 1≤j≤m). In fact, we give an explicit description of the spectral projections of H in terms of the eigenspaces of the diagonalizable part of A in its Jordan-Chevalley decomposition. This gives us a stronger form of Yamamoto's theorem which asserts that limn→∞⁡sj(An)1n is equal to the jth-largest eigenvalue-modulus of A, where sj(An) denotes the jth-largest singular value of An. Moreover, we also discuss applications to the asymptotic behaviour of the matrix exponential function, t↦etA.

Optimal Active Vibration Control of Tensegrity Structures Using Fast Model Predictive Control Strategy

Active vibration control of tensegrity structures is often challenging due to the geometrical nonlinearity, assemblage uncertainties of connections, and actuator saturation of controllers. To tackle these technical difficulties, a fast model predictive control (FMPC) strategy is herein implemented to effectively mitigate the structural vibration. Specifically, based on the explicit expression form of the Newmark- β method, the computation of the matrix exponential is avoided and replaced by one online and two offline transient analyses at each sampling instant on the structure, and the optimal control input is attainted from the second-order dynamic equation without forming an expanded state-space equation. Meanwhile, the artificial fish swarm algorithm (AFSA) is embedded to automatically derive optimal arrangement of actuators with the selection of a reasonable objective function. Two illustrative examples, including two standard and clustered tensegrity beams and a clustered tensegrity tower, have been fully investigated. The outcomes from illustrative examples prove the effectiveness and feasibility of the proposed method in optimal active vibration control of tensegrity structures, implying a promising prospect of the investigated approach in analyzing and solving relevant engineering problems.

Greybody factors for higher-dimensional non-commutative geometry inspired black holes

Greybody factors are computed for massless fields of spin 0, 1/2, 1, and 2 emitted from higher-dimensional non-commutative geometry inspired black holes. Short-range potentials are used with path-ordered matrix exponentials to numerically calculate transmission coefficients. The resulting absorption cross sections and emission spectra are computed on the brane and compared with the higher-dimensional Schwarzschild–Tangherlini black hole. A non-commutative black hole at its maximum temperature in seven extra dimensions will radiate a particle flux and power of 0.72–0.81 and 0.75–0.81, respectively, times lower than a Schwarzschild–Tangherlini black hole of the same temperature. A non-commutative black hole at its maximum temperature in seven extra dimensions will radiate a particle flux and power of 0.64–0.72 and 0.60–0.64, respectively, times lower than a Schwarzschild–Tangherlini black hole of the same mass.

Solutions of linear systems of moment differential equations via generalized matrix exponentials

A generalized exponential matrix based on the construction of kernel operators for generalized summability is defined and analyzing its main properties, generalizing the classical exponential matrix and fractional exponential matrix.This object serves as a practical tool to express the solutions of linear systems of moment differential equations in a compact manner, in the spirit of the classical exponential matrix.

A scalable approach for continuous time Markov models with covariates.

Existing methods for fitting continuous time Markov models (CTMM) in the presence of covariates suffer from scalability issues due to high computational cost of matrix exponentials calculated for each observation. In this article, we propose an optimization technique for CTMM which uses a stochastic gradient descent algorithm combined with differentiation of the matrix exponential using a Padé approximation. This approach makes fitting large scale data feasible. We present two methods for computing standard errors, one novel approach using the Padé expansion and the other using power series expansion of the matrix exponential. Through simulations, we find improved performance relative to existing CTMM methods, and we demonstrate the method on the large-scale multiple sclerosis NO.MS data set.

A Numerical Approach to Modeling Electromagnetic Response of Dynamic Medium in Arbitrary Shape

In this letter, the electromagnetic (EM) response of a dynamic medium (moving or deforming) is modeled via a suggested method belonging to the class of matrix exponential (ME). Because of the dynamicity in addition to the inherent nonlinearity of the medium, the interaction between it and EM fields becomes much tangled compared to when the medium is stationary. Such a complication is accounted for by assigning the spacetime variation to EM parameters. Moreover, the possible explicit parameter nonlinearity is coped with in a forward time marching routine. Our method retains superiorities in simplicity, numerical stability, and generality. Two numerical tests pertaining, respectively, to the EM scattering of dynamic ferromagnetic block and cylinder are provided for confirming the validity and for exemplifying the application of the suggested method.

Exponential-type extended Riordan arrays and reciprocity law for generalized Stirling numbers

We propose an exponential-type extended Riordan array (or exponential recursive matrix) with permitted negative indices using the Roman factorial. In this context, we establish a fundamental theory of the generalized Stirling numbers introduced by Hsu and Shiue [11]. Subsequently, we provide a lucid explanation for the reciprocity law among these numbers found by Choi et al. [5] and Maltenfort [19]. Moreover, our method reveals that interesting numbers emerge in the fourth quadrant of the proposed exponential recursive matrix.

A Hybrid Full-Discretization Method of Multiple Interpolation Polynomials and Precise Integration for Milling Stability Prediction

As milling chatter can lead to poor machining quality and limit the efficiency of productivity, it is of great significance to learn about milling chatter stability and research the effective and fast prediction of milling stability. In this study, a hybrid full-discretization method of multiple interpolation polynomials and precise integration (HFDM) is proposed for milling stability prediction. Firstly, the third-order Newton interpolation polynomial, third-order Hermite interpolation polynomial and linear interpolation are applied to approximate the state term, delay term and periodic coefficient matrix, respectively. Meanwhile, the matrix exponentials can be calculated based on the precise integration algorithm, which can improve computational accuracy and efficiency. The numerical simulation results indicate that the proposed method can not only effectively generate a stability lobe diagram (SLD) but also obtain better prediction accuracy and computation efficiency. A milling experiment is offered to demonstrate the feasibility of the method.