Properties and Applications of Hermite Matrix Exponential Polynomials

Author(s):  
Subuhi Khan ◽  
Shahid Ahmad Wani
2009 ◽  
Vol 42 (4) ◽  
Author(s):  
M. S. Metwally ◽  
M. T. Mohamed ◽  
A. Shehata

AbstractIn this paper, we introduce a new generalization of the Hermite matrix polynomials expansions of some relevant matrix functions appearing in the solution of differential systems. An explicit representation and an expansion of the matrix exponential in a series of these matrix polynomials is obtained. Properties of Hermite matrix polynomials such as the recurrence formula permit an efficient computations of matrix functions are established. A new expansions of the matrix exponential for a wide class of matrices in terms of Hermite matrix polynomials is proposed.


Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1483
Author(s):  
Shanqin Chen

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.


2020 ◽  
Vol 18 (1) ◽  
pp. 353-377 ◽  
Author(s):  
Zhien Li ◽  
Chao Wang

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.


2021 ◽  
Vol 15 ◽  
pp. 174830262199962
Author(s):  
Patrick O Kano ◽  
Moysey Brio ◽  
Jacob Bailey

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.


2008 ◽  
Vol 190 (2) ◽  
pp. 459-477 ◽  
Author(s):  
L. Bodrog ◽  
A. Heindl ◽  
G. Horváth ◽  
M. Telek

Analysis ◽  
1994 ◽  
Vol 14 (2-3) ◽  
pp. 103-112 ◽  
Author(s):  
Eberhard U. Stichel

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