Polynomial Form of the Matrix Exponential

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.

2010

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.

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.

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.

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.

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.

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.

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...

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)).