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 = H L U - 1 . This factorization enables us to propose hybrid and accurate (nongaussian) procedures for computing H - 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 - 1 in a suitable number of iterations.

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