author: Kagstrom, Bo; Wiberg, Petter
shorttitle: Extracting partial canonical structure
source: Numer. Algorithms 24, No. 3, 195-237 (2000)
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).