A note on “A fast numerical algorithm for the determinant of a pentadiagonal matrix”

A fast numerical algorithm for evaluating the determinant of a pentadiagonal matrix has been recently proposed [T. Sogabe, A fast numerical algorithm for the determinant of a pentadiagonal matrix, Appl. Math. Comput. 196 (2008) 835–841]. The algorithm whose cost is 14n − 28, where n(⩾3) denotes the size of the matrix, is composed of two steps: first, transform a pentadiagonal matrix into sparse Hessenberg form; second, run a numerical algorithm for computing the determinant of the sparse Hessenberg matrix. In this note, it is shown that we have an algorithm with the cost of 13n − 24 by applying twice the idea of the sparse Hessenberg transformation to a pentadiagonal matrix.