It is often useful to compute a lot of matrix exponentials in computer graphics (CG). The exponential of a matrix is used for the smooth deformation of 2D or 3D meshed CG objects. Hence, we need to compute a large number of the exponentials of 3 × 3 rotational matrices and 3 × 3 real symmetric matrices. For rotational matrices, Rodrigues’ formula is known to compute their exponentials. We inves...

Auxiliary matrix exponential method is used to derive simple and numerically efficient general expressions for the following, historically rather cumbersome, and hard to compute, theoretical methods: (1) average Hamiltonian theory following interaction representation transformations; (2) Bloch-Redfield-Wangsness theory of nuclear and electron relaxation; (3) gradient ascent pulse engineering ve...

In this paper, considering fractional matrix power definition, we define powers of matrixusing exponential and logarithm. Finally present a guide for computingthe matrix.

Abstract This paper presents new exponential stability conditions for a class of linear systems with time-varying delay. Unlike the previous works, the delay function considered in this paper is time-varying without any requirement on the derivative, which means that a fast time-varying delay is allowed. Based on an improved Lyapunov-Krasovskii functional combined with Newton-Leibniz formula an...

In this note we present a theoretical analysis of some Krylov subspace approximations to the matrix exponential operation exp(A)v and establish a priori and a posteriori error estimates. Several such approximations are considered. The main idea of these techniques is to approximately project the exponential operator onto a small Krylov subspace and carry out the resulting small exponential matr...

A real square matrix is said to be essentially non-negative if all of its off-diagonal entries are non-negative. It has recently been shown that the exponential of an essentially non-negative matrix is determined entrywise to high relative accuracy by its entries up to a condition number intrinsic to the exponential function (Numer. Math., 110 (2008), pp. 393-403). Thus the smaller entries of t...

The problems of delay-dependent exponential passivity analysis and exponential passification of uncertain Markovian jump systems MJSs with partially known transition rates are investigated. In the deterministic model, the time-varying delay is in a given range and the uncertainties are assumed to be norm bounded. With constructing appropriate Lyapunov-Krasovskii functional LKF combining with Je...

The implementation of exponential integrators requires the action of the matrix exponential and related functions of a large matrix. There are various methods in the literature for carrying out this task. In this paper we describe a new implementation of a method based on interpolation at Leja points. We numerically compare this method with others from the literature. As we are interested in ex...

By using the matrix exponential operator, solution of the system can be written as y(t) = exp(−tA)v. Numerical algorithms, which are based on this approach, are called exponential time integration methods. The essential point is that not the matrix exponential itself but rather its action on the vector v is computed. An attractive feature of the formula y(t) = exp(−tA)v is that it provides solu...

In this paper, the problem of exponential stability for a class of polytopic neutral systems with nondifferentiable interval time-varying delays is studied. By using an improved Lyapunov–Krasovskii parameter-dependent functional and linear matrix inequality (LMI) technique, new delay-dependent sufficient conditions for the exponential stability of the systems are first established in terms of l...