On new algorithms for inverting Hessenberg matrices

A modification of the Ikebe algorithm for computing the lower half of the inverse of an (unreduced) upper Hessenberg matrix, extended to compute the entries of the superdiagonal, is considered in this paper. It enables us to compute the inverse of a quasiseparable Hessenberg matrix in O(n2) times. A new factorization expressing the inverse of a nonsingular Hessenberg matrix as a product of two suitable matrices is obtained. Because this allows us the use of back substitution for the inversion of triangular matrices, the inverse is computed with complexity O(n3). Some comparisons with results obtained using other recent inversion algorithms are also provided.