The paper offers arguments and illustrative examples supporting the
replacement, near a defective eigenvalue, of the common inverse
iterations (for which the convergence slows down) by a slightly
modified implicit determinant method which remains quadratically
convergent. The paper may be read as motivated by the recent
perceivable growth of popularity of working with certain
non-selfadjoint (one could call them Dyson's) representations of the
operators of observables in quantum physics (cf. my compact review
``Three-Hilbert-space formulation of Quantum Mechanics'' in SIGMA 5
(2009), 001, arXiv:0901.0700 for more details). This re-attracted
attention to the related Jordan-block degeneracies and to the
numerical tractability of matrices with defective eigenvalues
emerging at parameters called, in this context, the Kato's
exceptional points.


MR3235880 (Sent 2014-10-30) Akinola, R. O.; Spence, A. A comparison
of the implicit determinant method and inverse iteration. J.
Nigerian Math. Soc. 33 (2014), 205--230. 65F15 (15A18)