Re-ordering and partitioning algorithms enable the authors to
split a sparse matrix into submatrices and to extract spectral
components (modes) forming approximate large-scale solutions.
Critical conditions of success of the ``component-mode synthesis"
are identified. A connection of accuracy is observed with the
canonical matrix pencil congruent to the original problem (viz.,
with the distribution of the components of its eigenvectors).
Error bars are given leading to the effective choice of modes. A
finite-element analysis of the cavity resonances leads to the
generalized eigenvalue problem chosen as an ultimate illustrative
example.



MR2199912 Yang, Chao; Gao, Weiguo; Bai, Zhaojun; Li, Xiaoye S.; Lee,
Lie-Quan; Husbands, Parry; Ng, Esmond
An algebraic substructuring method for large-scale eigenvalue calculation.
SIAM J. Sci. Comput. 27 (2005), no. 3, 873--892 (electronic). 65F15