In this paper we present algorithms that approximate invariant subspaces of a linear operator on a finite (but very high) dimensional space. First we will show, following Stewart and Sun (1990), that a small-enough error in an approximation for such a subspace, is the solution of a generalized Riccati equation. The solution of this Riccati equation will be approximated by Picard iterations, and we comment on convergence speed, costs and interrelations. As a by-product, we give an overview of iterative methods to solve a Sylvester equation with one large and sparse and one small and dense matrix.
Next, we will accelerate the Picard iterations in the same way as the Raleigh Quotient Iteration accelerates Shift and Invert. This results in Newton-like methods for the generalized algebraic Riccati equation. Additionally, so-called subspace acceleration will be applied, in the same way as the Arnoldi method is a subspace acceleration of the power method. Finally, forced by efficiency considerations, we consider the effects of inexact solution of equations at each level of the nested algorithms.
For invariant subspaces of dimension one, one of the resulting algorithms is the Jacobi-Davidson by Sleijpen and Van der Vorst (1996).
@article{1709,
title = {Computation of Invariant Subspaces of Large and Sparse Matrices},
journal = {CUBO, A Mathematical Journal},
volume = {5},
year = {2003},
language = {en},
url = {http://dml.mathdoc.fr/item/1709}
}
Brandts, Jan. Computation of Invariant Subspaces of Large and Sparse Matrices. CUBO, A Mathematical Journal, Tome 5 (2003) . http://gdmltest.u-ga.fr/item/1709/