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2016
in this paper we present several kinds of methods that allow us to compute the exponential matrix tA e exactly. These methods include calculating eigenvalues and Laplace transforms are well known, and are mentioned here for completeness. Other method, not well known is mentioned in the literature, that don't including the calculation of eigenvectors, and which provide general formulas applicable to any matrix.
SIAM review, 2001
The exponential of a matrix and the spectral decomposition of a matrix can be computed knowing nothing more than the eigenvalues of the matrix and the Cayley-Hamilton theorem. The arrangement of the ideas in this paper is simple enough to be taught to beginning students of ODEs.
SIAM Review, 1978
In principle, the exponential of a matrix could be computed in many ways. Methods involving approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polynomial have been proposed. In practice, consideration of computational stability and efficiency indicates that some of the methods are preferable to others but that none are completely satisfactory. Most of this paper was originally published in 1978. An update, with a separate bibliography, describes a few recent developments.
Matrix exponential is widely used in science area especially in matrix analysis. We pay particular attention to the matrix exponential. The matrix exponential is a very important subclass of control theory. In control theory it is needed to evaluate matrix exponential. In classical methods we calculate the eigenvalues of the matrix, but that the problem can be complicated if the eigenvalues are not easy to calculate. In this paper we use same methods and same procedure, but the eigenvalues of A are not needed for the construction of tA e , since most of our results use only the coefficients of the polynomial w , we explain some examples how the procedure works in the method of Dr. Luis Verde Star, in his article, where it develops, he gave in his article only theory without any applied. Finally, we developed the method for evaluate the characteristic polynomial.
In principle, the exponential of a matrix could be computed in many ways. Methods involving approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polynomial have been proposed. In practice, consideration of computational stability and efficiency indicates that some of the methods are preferable to others, but that none are completely satisfactory. Most of this paper was originally published in 1978. An update, with a separate bibliography, describes a few recent developments.
2000
Abstract. In this note, we consider the problem of computing the exponential of a real matrix. It is shown that if A is a real n�� n matrix and A can be diagonalized over C, then there is a formula for computing eA involving only real matrices. When A is a skew symmetric matrix, the formula reduces to the generalization of Rodrigues's formula given in Gallier and Xu [1].
The matrix exponential has many applications in the fields of mathematics, physics and economics. There are many explicit formulas that have been developed for compute the matrix exponential. In this paper we give some explicit formulas for the exponentials of some special matrices. The main results are the extension of Beibei Wu\'s work.
IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 2001
We give a simple condition on a matrix for which if the exponential matrix is diagonal, lower or upper triangular, then so is . It is also shown that for diagonalizable and any matrix , and commute if and only if and commute. These results are useful in problems in which knowledge about has to be extracted from structural information about its exponential, such as in large-scale sampled-data systems. They also find application in the area of blind system identification.
Computing, 1989
A Self-Validating Numerical Method fur the Matrix Exponential. An algorithm is presented, which produces highly accurate and automatically verified bounds for the matrix exponential function. Our computational approach involves iterative defect correction, interval analysis and advanced computer arithmetic. The algorithm presented is based on the "scaling and squaring" scheme, utilizing Pad6 approximations and safe error monitoring. A PASCAL-SC program is reported and numerical results are discussed.
2009
The matrix exponential plays a fundamental role in linear systems arising in engineering, mechanics and control theory. In this paper, an efficient Taylor method for computing matrix exponentials is presented. Taylor series truncation together with a modification of the PatersonStockmeyer method avoiding factorial evaluations, and the scaling-squaring technique, allow efficient computation of the matrix exponential approximation. A careful backward-error analysis of the approximation is given and a theoretical estimate for the optimal scaling of matrices is obtained. The modified Paterson-Stockmeyer implementation was compared with the classical implementation and other efficient state of the art methods on dense matrices for different dimensions from 2× 2 to 100× 100. Numerical tests show that it obtains higher precision than all compared methods in the majority of cases. We show that it presents lower computational cost in terms of matrix products than efficient Padé methods for g...
Journal of Computational and Applied Mathematics, 2016
This work presents a new algorithm for matrix exponential computation that significantly simplifies a Taylor scaling and squaring algorithm presented previously by the authors, preserving accuracy. A Matlab version of the new simplified algorithm has been compared with the original algorithm, providing similar results in terms of accuracy, but reducing processing time. It has also been compared with two state-of-the-art implementations based on Padé approximations, one commercial and the other implemented in Matlab, getting better accuracy and processing time results in the majority of cases.
Linear Algebra and its Applications, 1996
We analyze the Pad& method for computing the exponential of a real matrix.
The matrix exponential e At forms the basis for the homogeneous (unforced) and the forced response of LTI systems. We consider here a method of determining e At based on the the Cayley-Hamiton theorem. Consider a square matrix A with dimension n and with a characteristic polynomial ∆(s) = |sI − A| = s n + c n−1 s n−1 +. .. + c 0 , and define a corresponding matrix polynomial, formed by substituting A for s above ∆(A) = A n + c n−1 A n−1 +. .. + c 0 I where I is the identity matrix. The Cayley-Hamilton theorem states that every matrix satisfies its own characteristic equation, that is ∆(A) ≡ [0] where [0] is the null matrix. (Note that the normal characteristic equation ∆(s) = 0 is satisfied only at the eigenvalues (λ 1 ,. .. , λ n)).
arXiv: Numerical Analysis, 2016
An algorithm for numerically computing the exponential of a matrix is presented. We have derived a polynomial expansion of $e^x$ by computing it as an initial value problem using a symbolic programming language. This algorithm is shown to be comparable in operation count and convergence with the state--of--the--art method which is based on a Pade approximation of the exponential matrix function. The present polynomial form, however, is more reliable because the evaluation requires only linear combinations of the input matrix. We also show that the technique used to solve the differential equation, when implemented symbolically, leads to a rational as well as a polynomial form of the solution function. The rational form is the well-known diagonal Pade approximation of $e^x$. The polynomial form, after some rearranging to minimize operation count, will be used to evaluate the exponential of a matrix so as to illustrate its advantages as compared with the Pade form.
Symmetry, 2014
We discuss a method to obtain closed-form expressions of f (A), where f is an analytic function and A a square, diagonalizable matrix. The method exploits the Cayley-Hamilton theorem and has been previously reported using tools that are perhaps not sufficiently appealing to physicists. Here, we derive the results on which the method is based by using tools most commonly employed by physicists. We show the advantages of the method in comparison with standard approaches, especially when dealing with the exponential of low-dimensional matrices. In contrast to other approaches that require, e.g., solving differential equations, the present method only requires the construction of the inverse of the Vandermonde matrix. We show the advantages of the method by applying it to different cases, mostly restricting the calculational effort to the handling of two-by-two matrices.
Our main purpose in this project is to help reader find a clear and glaring relationship between linear algebra and differential equations, such that the applications of the former may solve the system of the latter using exponential of a matrix. Applications to linear differential equations on account of eigen values and eigenvectors, diagonalization of n-square matrix using computation of an exponential of a matrix using results and ideas from elementary studies form the core study of our project.
Linear Algebra and its Applications
This paper is devoted to the study of some formulas for polynomial decomposition of the exponential of a square matrix A. More precisely, we suppose that the minimal polynomial M A (X) of A is known and has degree m. Therefore, e tA is given in terms of P 0 (A),. .. , P m−1 (A), where the P j (A) are polynomials in A of degree less than m, and some explicit analytic functions. Examples and applications are given. In particular, the two cases m = 5 and m = 6 are considered.
2018
In this paper we give a new technique for the calculation of the matrix exponential function e as well as the matrix trigonometric functions sinA and cosA, where A ∈ M2 (C). We also determine the real logarithm of the matrix xI2, when x ∈ R∗, as well as the real logarithm of scalar multiple of rotation and reflection matrices. The real logarithm of a real circulant matrix and a symmetric matrix are also determined.
SIAM Journal on Scientific Computing, 2015
The matrix exponential plays a fundamental role in linear differential equations arising in engineering, mechanics, and control theory. The most widely used, and the most generally efficient, technique for calculating the matrix exponential is a combination of "scaling and squaring" with a Padé approximation. For alternative scaling and squaring methods based on Taylor series, we present two modifications that provably reduce the number of matrix multiplications needed to satisfy the required accuracy bounds, and a detailed comparison of the several algorithmic variants is provided.
International Journal of Computer Mathematics, 2014
This work gives a new formula for the forward relative error of matrix exponential Taylor approximation and proposes new bounds for it depending on the matrix size and the Taylor approximation order, providing a new efficient scaling and squaring Taylor algorithm for the matrix exponential. A Matlab version of the new algorithm is provided and compared with Padé state-of-the-art algorithms obtaining higher accuracy in the majority of tests at similar or even lower cost.
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