ELA, Volume 15, pp. 178-190, May 2006, abstract. 

Fibonacci-Horner Decomposition of the Matrix Exponential 
and the Fundamental System of Solutions

R. Ben Taher, M. Mouline, and M. Rachidi

This paper concerns the Fibonacci-Horner decomposition of 
the matrix powers A^n and the matrix exponential exp(tA) 
(A rxr complex matrix and t real), which is derived 
from the combinatorial properties of the generalized Fibonacci 
sequences in the algebra of square matrices. More precisely, 
exp(tA) is expressed in a natural way in the so-called 
Fibonacci-Horner basis with the aid of the dynamical solution 
of the associated ordinary differential equation. Two simple 
processes for computing the dynamical solution and the fundamental 
system of solutions are given. The connection to Verde-Star's 
approach is discussed. Moreover, an extension to the computation 
of f(A), where f is an analytic function is initiated. Finally, 
some illustative examples are presented.