Symbolic algorithms for the inverses of general kk-tridiagonal matrices

Two symbolic algorithms for inverting kk-tridiagonal matrices have been recently found by El-Mikkawy and Atlan (2014, 2015). These two algorithms are mainly based on the Doolittle LU factorization of the kk-tridiagonal matrix. In the current paper, we present a new explicit analytic expression for the inverses of general tridiagonal matrices at first. By using a block diagonalization technique, we then relate kk-tridiagonal matrix inversion to tridiagonal matrix inversion. Meanwhile, an efficient algorithm is derived for computing the inverses of nonsingular kk-tridiagonal matrices with the help of any algorithm for computing the inverses of tridiagonal matrices. Three examples are given in order to illustrate the performance and efficiency of the proposed algorithms.