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Miloslav Znojil
Reviewer number:
Item's zbl-Number:
DE 016 589 168
Ovtchinnikov, E. E.; Xanthis, L. S.:
Successive eigenvalue relaxation: a new method for the generalized eigenvalue problem and convergence estimates
Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 457, No. 2006, 441-451 (2001).
65F15Eigenvalues, eigenvectors
Primary Classification:
Secondary Classification:
eigensolvers with preconditioning; subspace iteration; multiple and clustered eigenvalues; large scale problems

The slightly more than 10 pages of this paper form an archetype
presentation (of a method) with a marvelously composed title
(which is openly and almost exhaustively descriptive) and using a
high-brow style (which is not quite usual, using citation in Greek
and framing the text between strongly philosophical ``prolegomena"
and exceptionally enthusiastic ``epilegomena"). In this spirit the
authors combine the extensive historical review with the Draconian
selection of the material itself (referring, e.g., to the proofs
published separately in two reports, or omitting the explicit
formulae from theorems in order to maximize the simplicity of the
message). Somebody may like the maximal digestibility of the
resulting text. I would be more sceptical - the idea itself
(iterations in subspaces plus a convergence estimate) is nice and
verbally ambitious (we read that it does not require any
information about the preconditioner used) but the ``proof of the
pudding" may look like missing (the promised ``effective dealing
with large-scale eigenvalue problems" is consummated via the mere
single two-dimensional Laplacean-like numerical illustration).
Hence, I would personally wait with the evaluation of the
recommended SER method until the above-mentioned two internal
reports of the University of Westminster appear in print (it is
really difficult not to accept and mimic the style, so let me add,
in this spirit, a small fraction from a poem by Pablo Neruda, in
Spanish: ``Primero vivo, ... despues video que pueda SER").
Remarks to the editors:

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