Computing the exponential of matrices

The following differential equation plays a fundamental role in the study of dynamical systems and linear systems: x˙=Ax; x(0)=x₀; 0≤t<∞, where x and x₀ are n-vectors and A is an nxn matrix of complex constants. The theoretical solution to this equation is given by x(t)=e^tAx₀. In this paper, we discuss an algebraic method for computing e^tA and give explicit formular for n=3.