The Matrix Sign Function Method and the Computation of Invariant Subspaces

Abstract

A perturbation analysis shows that if a numerically stable procedure is used to compute the matrix sign function, then it is competitive with conventional methods for computing invariant subspaces. Stability analysis of the Newton iteration improves an earlier result of Byers and con rms that ill-conditioned iterates may cause numerical instability. Numerical examples demonstrate the theoretical results. Short Title. The Matrix Sign Function and Invariant Subspaces.

Iterative and doubling algorithms for Riccati‐type matrix equations: A comparative introduction

“…] is a basis matrix for the invariant subspace associated to the other eigenvalues of , those outside the unit circle. Indeed, (30) gives…”

“…Thus  2 k (as well as its inverse) has some eigenvalues that converge to zero, and some that diverge to infinity, as k grows. This is one of the reasons why it is preferable to keep  in its factored form (30). On the other hand, the eigenvalues of  k converge to finite values c −1 (0) = − and c −1 (∞) = , so this computation suggests that the direct computation of  k is feasible.…”

“…Scaling also has an impact on stability; the stability of the sign iteration as a method to compute invariant subspaces (and hence ultimately Riccati solutions) has been studied by Bai and Demmel [3] and Byers et al [30]. The two interesting messages are that (expectedly) the sign function method suffers when is ill‐conditioned, but that (unexpectedly) the invariant subspaces extracted from has better stability properties than itself.…”

Iterative and doubling algorithms for Riccati-type matrix equations: a comparative introduction

“…More general versions of the factorization (30) and of the iteration (33), which guarantee existence and boundedness of the iterates under much weaker conditions, have been explored by Mehrmann and Poloni [73]. Kuo, Lin and Shieh [63] studied the theoretical properties of the factorization (30) for general powers  , ∈ ℝ, drawing a parallel with the so-called Toda flow for the QR algorithm.…”

“…Thus  2 (as well as its inverse) has some eigenvalues that converge to zero, and some that diverge to infinity, as grows. This is one of the reasons why it is preferable to keep  in its factored form (30). On the other hand, the eigenvalues of  converge to finite values −1 (0) = − and −1 (∞) = , so this computation suggests that the direct computation of  is feasible.…”