## Dept. of Mathematics and Computer Science (Berlin)

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**reviewer:** Znojil, Miloslav
**reviewernum:** 9689

**revieweremail:** znojil@ujf.cas.cz

**zblno:** DE015347716

**author:** Kagstrom, Bo; Wiberg, Petter

**shorttitle:** Extracting partial canonical structure

**source:** Numer. Algorithms 24, No. 3, 195-237 (2000)

**rpclass:** 15A21

**rsclass:** 15A22; 15A42; 65G40; 65K05; 65F10; 65J20

**keywords:** large scale eigenvalue problems, degenerate eigenvalues, eigenvlaue clustering, determination of the canonical structures, Jordan form, Weierstrass form, implicitly restarted Arnoldi, staircase algorithm, Schur versions

**revtext:**
The paper is motivated by the current large scale computations of
eigenvalues which only pay a marginal attention to possible
degeneracy of computed eigenvalues. The problem is extremely
challenging since the corresponding Jordan canonical form of a
matrix A (or the Weierstrass canonical form of a matrix pencil A -
$\lambda$B in general) is unstable with respect to the
perturbations and/or influence of the input errors. A convenient
regularization is needed, using some suitable deflation criteria.
The use of the so called Jordan- or Weierstrass-Schur forms for
nearby matrices is also recommended in place of the bundles of the
strict canonical representations. The paper describes the eligible
numerical machinery which, in essence, combines the implicitly
restarted Arnoldi method (giving the partial Schur forms within
invariant subspace(s)) and a subsequent (final) staircase
algorithm. Emphasis is given to a new Gershgorin clustering
heuristics (with a reliable control of errors). The reliability
and robustness are demonstrated via numerical experiments. The
reading of the paper is easy (presenting the encountered
difficulties with ill-conditioning etc at an appropriate speed)
and rewarding (due to its steady progress towards the clarity and
completeness of the discussion).

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Michael Jost* (jo@zblmath.FIZ-Karlsruhe.DE).