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.