Matrix Exponential Research Articles

Euler polynomials for the matrix exponential approximation

In this work, a new method to compute the matrix exponential function by using an approximation based on Euler polynomials is proposed. These polynomials are used in combination with the scaling and squaring technique, considering an absolute forward-type theoretical error. Its numerical and computational properties have been evaluated and compared with the most current and competitive codes dedicated to the computation of the matrix exponential. Under a heterogeneous test battery and a set of exhaustive experiments, it has been demonstrated that the new method offers performance in terms of accuracy and stability which is as good as or even better than those of the considered methods, with an intermediate computational cost among all of them. All of the above makes this a very competitive alternative that should be considered in the growing list of available numerical methods and implementations dedicated to the approximation of the matrix exponential.

Accurate Approximation of the Matrix Hyperbolic Cosine Using Bernoulli Polynomials

This paper presents three different alternatives to evaluate the matrix hyperbolic cosine using Bernoulli matrix polynomials, comparing them from the point of view of accuracy and computational complexity. The first two alternatives are derived from two different Bernoulli series expansions of the matrix hyperbolic cosine, while the third one is based on the approximation of the matrix exponential by means of Bernoulli matrix polynomials. We carry out an analysis of the absolute and relative forward errors incurred in the approximations, deriving corresponding suitable values for the matrix polynomial degree and the scaling factor to be used. Finally, we use a comprehensive matrix testbed to perform a thorough comparison of the alternative approximations, also taking into account other current state-of-the-art approaches. The most accurate and efficient options are identified as results.

Hitting times, number of jumps, and occupation times for continuous-time finite state Markov chains

In this paper we discuss three random variables associated with a finite state continuous-time Markov chain having a constant transition rate matrix: the number of jumps between two different states, the amount of time spent in a given state, known as an occupation time, and the time it takes for the Markov chain to first reach a particular state, known as a hitting time or first passage time. First, we calculate expected values for each of these random variables. We then derive the moment generating functions for these variables. Our approach is to use the vector/matrix representations of the various processes. Each solution is written in terms of an integral of the exponential of a matrix.

Jordan–Schur Algorithms for Computing the Matrix Exponential

In this paper, two new versions of the Schur algorithm for computing the matrix exponential of an n × n complex matrix A are presented. Instead of the Schur form, these algorithms use the Jordan–Schur form of A. The Jordan–Schur form is found by less computation and it is determined more reliable than the reduction to Jordan form since it is obtained using only unitary similarity transformations. In contrast to the known methods, the diagonal blocks of the matrix exponential are obtained by using finite Taylor series. This improves the accuracy and avoids the decisions made about the termination of the series expansion. The off‐diagonal blocks of the exponential are determined by modifications of the Schur–Parlett or Schur–Fréchet method, which takes advantage of the Jordan–Schur form of the matrix. The numerical features of the new algorithms are discussed, revealing their advantages and disadvantages in comparison with the other methods for computing the matrix exponential. Computational experiments show that using the new algorithms, the matrix exponential is determined in certain cases with higher accuracy than some widely used methods, however, at the price of an increase in the computational cost which is of order n4. It is shown that the Jordan–Schur algorithms for computing the matrix exponential are appropriate for matrices with multiple eigenvalues and are especially efficient in cases of large Weyr characteristics.

Исследование решений линейной однородной системы дифференциальных уравнений

The paper shows that the fundamental zero-normalized solution of a linear homogeneous differ-ential equations system can be represented as an exponential matrices products formal series. If the system satisfies the equations system triangulation Perron theorem conditions, then the system solution can be represented as an exponential matrices finite product. In addition, an exponential matrix function differentiating formula is derived. Also, the transformation constructing problem is considered. Such, a homogeneous differential equations system allows to reduce to a triangular form.

New digital signature scheme based on MDLP and multinacci matrices

A new digital signature scheme based on Matrices Discrete Logarithm Problem (MDLP) and generalized Fibonacci or Multinacci matrices is proposed. The security of the scheme is based on the difficulty of solving the Discrete Logarithm Problem (DLP) in matrices. MDLP is a new one-way function based on matrices that provides the same security as the DLP. The use of matrices increases the complexity of the scheme, as it involves matrix exponentiation rather than integers. In the proposed scheme, the signer uses a Multinacci matrix Fkn to generate the signature and an inverse Multinacci matrix Fn-k to verify it. The computational complexity and security of the scheme are also discussed.

Three Dimensional Coupling between Elastic and Thermal Fields in the Static Analysis of Multilayered Composite Shells

This new work aims to develop a full coupled thermomechanical method including both the temperature profile and displacements as primary unknowns of the model. This generic full coupled 3D exact shell model permits the thermal stress investigation of laminated isotropic, composite and sandwich structures. Cylindrical and spherical panels, cylinders and plates are analyzed in orthogonal mixed curved reference coordinates. The 3D equilibrium relations and the 3D Fourier heat conduction equation for spherical shells are coupled and they trivially can be simplified in those for plates and cylindrical panels. The exponential matrix methodology is used to find the solutions of a full coupled model based on coupled differential relations with respect to the thickness coordinate. The analytical solution is based on theories of simply supported edges and harmonic relations for displacement components and sovra-temperature. The sovra-temperature magnitudes are directly applied at the outer faces through static state hypotheses. As a consequence, the sovra-temperature description is assumed to be an unknown variable of the model and it is calculated in the same way as the three displacements. The final system is based on a set of coupled homogeneous differential relations of second order in the thickness coordinate. This system is reduced in a first order differential relation system by redoubling the number of unknowns. Therefore, the exponential matrix methodology is applied to calculate the solution. The temperature field effects are evaluated in the static investigation of shells and plates in terms of displacement and stress components. After an appropriate preliminary validation, new benchmarks are discussed for several thickness ratios, geometrical data, lamination sequences, materials and sovra-temperature values imposed at the outer faces. Results make evident the accordance between the uncoupled thermo-mechanical model and this new full coupled thermo-mechanical model without the need to separately solve the Fourier heat conduction relation. Both effects connected with the thickness layer and the related embedded materials are included in the conducted thermal stress analysis.

Incomplete exponential type of $ R $-matrix functions and their properties

<abstract><p>In the present paper, we establish the incomplete exponential type (IEF) of $ R $-matrix functions and identify some properties of the incomplete exponential matrix functions including integral representation, some derivative formula and generating functions of the incomplete exponential of $ R $-matrix functions. Finally, special cases of the presented results are pointed out.</p></abstract>

Spectral properties of the exponential distance matrix

Given a graph $G$, the exponential distance matrix is defined entry-wise by letting the $(u,v)$-entry be $q^{\text{dist}(u,v)}$, where $\text{dist}(u,v)$ is the distance between the vertices $u$ and $v$ with the convention that if vertices are in different components, then $q^{\text{dist}(u,v)}=0$. In this paper, we will establish several properties of the characteristic polynomial (spectrum) for this matrix, give some families of graphs which are uniquely determined by their spectrum, and produce cospectral constructions.

Exact solutions and Hyers-Ulam stability of fractional equations with double delays

In this paper, we discuss the exact solutions of linear homogeneous and nonhomogeneous fractional differential equations with double delays. Firstly, a new concept of double-delayed Mittag-Leffler type matrix function is introduced, which is the promotion of the double-delayed matrix exponential. Secondly, we apply the double-delayed Mittag-Leffler type matrix function and Laplace transform approach to obtain the exact solutions of fractional differential equations with double delays. Furthermore, the solution is used to investigate the Hyers-Ulam stability of the system. Lastly, we illustrate our techniques by an example.

Precise integration solutions for the static and dynamic responses of axially graded solid beams

By virtue of the scaled boundary finite element method (SBFEM), the precise integration solutions (PIS) to deflections, stresses and transverse vibration parameters of the functionally graded solid beams are provided. It is worth to note that the mechanical parameters of the graded beams are altered across the x-axis according to power-law, exponential, trigonometric or other arbitrary theories. In SBFEM, the in-homogeneous beam is simplified into the one-dimensional model and just two high-order linear spectral elements are exploited to discretize the axially non-homogeneous beam. Nevertheless, the high accuracy can be ensured. It is essential to remind that the displacement-based beam spectral element is characterized by owning only two degrees of freedom. Additionally, the changing patterns of the displacement and stress field through the transverse direction can be expressed as an analytical matrix exponent. Aided by the two-dimensional fundamental equations of elastic materials and the employed scaled boundary coordinate, the governing equation of the axially graded solid beam is derived in the form of second order ordinary differential equation. Furthermore, the stiffness matrix and PIS of displacements and stresses are acquired from evaluating the governing equation by means of the dual vector technology and precise integration methodology (PIM). Applying the kinetic energy theory constructs the mass matrix of the axially heterogeneous beam. Afterwards the PIS to natural frequencies are obtained according to solving the eigenvalue equation. Finally, numerical exercises are presented to portray the excellent agreement and fast convergence ratio of the introduced PIS and discuss influences of supported conditions, gradient exponents, length-to-thickness ratios and formulae of gradation functions on the flexural behaviors and free vibration responses of the axially graded elastic beams.

Relative controllability of nonlinear delayed multi-agent systems

Controllability of nonlinear delayed leader–follower multi-agent system with fixed communication topology is introduced. A solution of a semilinear time-delay control system and Gramian matrix for the linear part which has Kronecker delay perturbation of exponential matrix function are given. The assumptions and conditions under which the nonlinear delay multi-agent system can be controlled are determined thanks to control problem into fixed point problem. A comprehensive example to illustrate given theoretical contents is presented.

The exponential distance matrix of block graphs

Let G be a connected graph with vertex set {v1,v2,⋯,vn}. As a variant of distance matrix, the exponential distance matrix was proposed by Yan and Yeh, and Bapat et al. independently. Given a nonzero indeterminate q, the exponential distance matrix F=(Fij)n×n of G is defined as Fij=qDij, where Dij is the distance between vertices vi and vj in G (i.e., the (i,j)-entry of the distance matrix of G).A graph is called a block graph, if each block is a clique (of possibly varying orders). In this paper, the determinant, inverse and cofactor sum of the exponential distance matrix of block graphs are obtained. Some known results about the exponential distance matrix and q-Laplacian matrix are generalized.

Method of Lines for Valuation and Sensitivities of Bermudan Options

In this paper, we present a computationally efficient technique based on the Method of Lines for the approximation of the Bermudan option values via the associated partial differential equations. The method of lines converts the Black Scholes partial differential equation to a system of ordinary differential equations. The solution of the system of ordinary differential equations so obtained only requires spatial discretization and avoids discretization in time. Additionally, the exact solution of the ordinary differential equations can be obtained efficiently using the exponential matrix operation, making the method computationally attractive and straightforward to implement. An essential advantage of the proposed approach is that the associated Greeks can be computed with minimal additional computations. We illustrate, through numerical experiments, the efficacy of the proposed method in pricing and computation of the sensitivities for a European call, cash-or-nothing, powered option, and Bermudan put option.

Numerical simulation of temperature distribution of heat flow on reservoir tanks connected in a series

The flow of temperature distribution through a medium in thermodynamic studies plays an important role in understanding physical phenomena in chemical science and petroleum engineering, while temperature distribution indicates the degree of reaction that must be undergone to obtain the final product. Therefore, this paper aims to present and apply the exponential matrix algorithm (EMA), differential transformation algorithm (DTA), and Runge-Kutta (RK5) to simulate the temperature distribution in five heating tanks in series. successive preheating of multicomponent oil solutions. A mathematical model of the energy balance equations of the reservoir is considered. Two computer experiments were performed to test and investigate the relationship between two constant parameters appearing in the model. Numerical simulation of saturated steam Tsteam temperature of 500 °C and 1000 °C used to heat the tanks and initial temperature T035°Cand100°C of the first tank feed oil are considered. The fluids in the reservoirs were considered homogeneous throughout the experiment and changes in the cell configuration at two constant parameters were presented in the 2D plot control with the use of the MAPLE 18 software package. The study revealed the nature of the temperature distribution that the higher temperature distribution is obtained when heat is transferred from the first tank to the fifth tank and the reverse reaction occurs in all five reservoirs when ψ = 0.0025 and ω = 0.0025 respectively. Numerical results obtained are prototypes of oil temperature distribution performed under laboratory conditions in a thermodynamic experiment.

Pseudo-Marginal Inference for CTMCs on Infinite Spaces via Monotonic Likelihood Approximations

Bayesian inference for Continuous-Time Markov chains (CTMCs) on countably infinite spaces is notoriously difficult because evaluating the likelihood exactly is intractable. One way to address this challenge is to first build a nonnegative and unbiased estimate of the likelihood—involving the matrix exponential of finite truncations of the true rate matrix—and then to use the estimates in a pseudo-marginal inference method. In this work, we show that we can dramatically increase the efficiency of this approach by avoiding the computation of exact matrix exponentials. In particular, we develop a general methodology for constructing an unbiased, nonnegative estimate of the likelihood using doubly-monotone matrix exponential approximations. We further develop a novel approximation in this family—the skeletoid—as well as theory regarding its approximation error and how that relates to the variance of the estimates used in pseudo-marginal inference. Experimental results show that our approach yields more efficient posterior inference for a wide variety of CTMCs. Supplementary materials for this article are available online.

A Fourth-Order Time-Stepping Method for Two-Dimensional, Distributed-Order, Space-Fractional, Inhomogeneous Parabolic Equations

Distributed-order, space-fractional diffusion equations are used to describe physical processes that lack power-law scaling. A fourth-order-accurate, A-stable time-stepping method was developed, analyzed, and implemented to solve inhomogeneous parabolic problems having Riesz-space-fractional, distributed-order derivatives. The considered problem was transformed into a multi-term, space-fractional problem using Simpson’s three-eighths rule. The method is based on an approximation of matrix exponential functions using fourth-order diagonal Padé approximation. The Gaussian quadrature approach is used to approximate the integral matrix exponential function, along with the inhomogeneous term. Partial fraction splitting is used to address the issues regarding stability and computational efficiency. Convergence of the method was proved analytically and demonstrated through numerical experiments. CPU time was recorded in these experiments to show the computational efficiency of the method.

Detecting “invisible” layers in layered composites

Lamb waves propagating in a layered structure consisting of layers with equal acoustic impedances, is studied by a variant of Cauchy sextic formalism and the exponential fundamental matrix method. It is shown that the dispersion portraits provide enough information to distinguish individual layers, even if their acoustic impedances are identical, making such layers indistinguishable by the pulse – echo method. It is revealed that the developed methodology is adequate for determining geometrical and physical properties of the individual layers with equal acoustic impedances.

Taming numerical errors in simulations of continuous variable non-Gaussian state preparation

Numerical simulation of continuous variable quantum state preparation is a necessary tool for optimization of existing quantum information processing protocols. A powerful instrument for such simulation is the numerical computation in the Fock state representation. It unavoidably uses an approximation of the infinite-dimensional Fock space by finite complex vector spaces implementable with classical digital computers. In this approximation we analyze the accuracy of several currently available methods for computation of the truncated coherent displacement operator. To overcome their limitations we propose an alternative with improved accuracy based on the standard matrix exponential. We then employ the method in analysis of non-Gaussian state preparation scheme based on coherent displacement of a two mode squeezed vacuum with subsequent photon counting measurement. We compare different detection mechanisms, including avalanche photodiodes, their cascades, and photon number resolving detectors in the context of engineering non-linearly squeezed cubic states and construction of qubit-like superpositions between vacuum and single photon states.

Fractional Laplace transform for matrix valued functions with applications

We extend several fractional Laplace transform results to matrix-valued functions in this paper, which we will utilize to obtain some useful and valuable theorems, like results on the set of the piecewise continuous functions with conformable exponential order and the low and conditions to obtain the fractional Laplace transform of matrix-valued functions. As an application, we apply the obtained theorems to solve certain fractional initial value problems for vector-valued and matrix-valued functions and the solution will be victor or matrix and it will exact. Also, we establish a fractional system transfer matrix and the conformable fractional Laplace transform of the conformable exponential matrix function, which will give us directly the exact solution for a precisely type of fractional initial value problems.