Hybrid Ikebe–Newton’s iteration for inverting general nonsingular Hessenberg matrices

After a concise survey, the expanded Ikebe algorithm for inverting the lower half plus the superdiagonal of an n × n unreduced upper Hessenberg matrix H is extended to general nonsingular upper Hessenberg matrices by computing, in the reduced case, a block diagonal form of the factor matrix HL   in the inverse factorization H−1=HLU−1H−1=HLU−1. This factorization enables us to propose hybrid and accurate (nongaussian) procedures for computing H−1H−1. Thus, HL   is computed directly in the aim to be used as a fine initial guess for Newton’s iteration, which converges to H−1H−1 in a suitable number of iterations.