Analytical inversion of general periodic tridiagonal matrices

In this paper we present an analytical forms for the inversion of general periodic tridiagonal matrices, and provide some very simple analytical forms which immediately lead to closed formulae for some special cases such as symmetric or perturbed Toeplitz for both periodic and non-periodic tridiagonal matrices. An efficient computational algorithm for finding the inverse of any general periodic tridiagonal matrices from the analytical form is given, it is suited for implementation using Computer Algebra systems such as MAPLE, MATLAB, MACSYMA, and MATHEMATICA. An example is also given to illustrate the algorithm.