Calculating the matrix exponential of a constant matrix on time scales

AbstractIn this paper we describe an elementary method for calculating the matrix exponential on an arbitrary time scale. An example is also given to illustrate the result.

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More Explicit Formulas for the Matrix Exponential

Linear Algebra and its Applications ◽

10.1016/s0024-3795(96)00478-8 ◽

1997 ◽

Vol 262 (1-3) ◽

pp. 131-163 ◽

Cited By ~ 13

Author(s):

H Cheng

Keyword(s):

Matrix Exponential ◽

Explicit Formulas ◽

The Matrix

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Krylov SSP Integrating Factor Runge–Kutta WENO Methods

Mathematics ◽

10.3390/math9131483 ◽

2021 ◽

Vol 9 (13) ◽

pp. 1483

Author(s):

Shanqin Chen

Keyword(s):

Large Time ◽

Hyperbolic Equations ◽

Strong Stability ◽

Matrix Exponential ◽

Linear Component ◽

Nonlinear Hyperbolic Equations ◽

Time Step ◽

Integrating Factor ◽

The Matrix ◽

Runge Kutta

Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.

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Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales

Open Mathematics ◽

10.1515/math-2020-0021 ◽

2020 ◽

Vol 18 (1) ◽

pp. 353-377 ◽

Cited By ~ 1

Author(s):

Zhien Li ◽

Chao Wang

Keyword(s):

Time Scales ◽

Quaternion Algebra ◽

Matrix Exponential ◽

Dynamic Equations ◽

Necessary Condition ◽

Quaternion Matrix ◽

Exponential Functions ◽

Cauchy Matrix ◽

Adjoint Matrix ◽

Sufficient And Necessary Condition

Abstract In this study, we obtain the scalar and matrix exponential functions through a series of quaternion-valued functions on time scales. A sufficient and necessary condition is established to guarantee that the induced matrix is real-valued for the complex adjoint matrix of a quaternion matrix. Moreover, the Cauchy matrices and Liouville formulas for the quaternion homogeneous and nonhomogeneous impulsive dynamic equations are given and proved. Based on it, the existence, uniqueness, and expressions of their solutions are also obtained, including their scalar and matrix forms. Since the quaternion algebra is noncommutative, many concepts and properties of the non-quaternion impulsive dynamic equations are ineffective, we provide several examples and counterexamples on various time scales to illustrate the effectiveness of our results.

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Optimal parameter selection in Weeks’ method for numerical Laplace transform inversion based on machine learning

Journal of Algorithms & Computational Technology ◽

10.1177/1748302621999621 ◽

2021 ◽

Vol 15 ◽

pp. 174830262199962

Author(s):

Patrick O Kano ◽

Moysey Brio ◽

Jacob Bailey

Keyword(s):

Laplace Transform ◽

Optimal Parameter ◽

Large Data ◽

Matrix Exponential ◽

Resolvent Matrix ◽

Data Set ◽

Network Training ◽

The Matrix ◽

The Laplace Transform ◽

Space Function

The Weeks method for the numerical inversion of the Laplace transform utilizes a Möbius transformation which is parameterized by two real quantities, σ and b. Proper selection of these parameters depends highly on the Laplace space function F( s) and is generally a nontrivial task. In this paper, a convolutional neural network is trained to determine optimal values for these parameters for the specific case of the matrix exponential. The matrix exponential eA is estimated by numerically inverting the corresponding resolvent matrix [Formula: see text] via the Weeks method at [Formula: see text] pairs provided by the network. For illustration, classes of square real matrices of size three to six are studied. For these small matrices, the Cayley-Hamilton theorem and rational approximations can be utilized to obtain values to compare with the results from the network derived estimates. The network learned by minimizing the error of the matrix exponentials from the Weeks method over a large data set spanning [Formula: see text] pairs. Network training using the Jacobi identity as a metric was found to yield a self-contained approach that does not require a truth matrix exponential for comparison.

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