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Well-posedness and regularity properties of the Grassmann-Rayleigh quotient iteration

Summary: Well-posedness and regularity properties of the
Grassmann-Rayleigh quotient iteration
P.-A. Absil
U. Helmke
K. H¨uper
Technical Report FSU-CSIT-04-26
December 9, 2004
Abstract. A generalization of the Rayleigh quotient iteration has recently been proposed
on the Grassmann manifold. This iteration has been shown to converge locally cubically to
the invariant subspaces of symmetric matrices. The present paper studies global properties
of the iteration mapping. Results are obtained e.g. concerning fixed points, smoothness, and
singularities of the iteration mapping.
Key words. Grassmann-Rayleigh quotient iteration, Block-Rayleigh quotient iteration,
Grassmann manifold, singularities, continuous extension, fixed points.
AMS subject classification. 65F15.
1 Introduction
The Rayleigh quotient iteration (RQI) is a well-known method for computing an eigenvector
of a symmetric matrix. It consists in a shifted inverse iteration that maps y Rn\{0} to y+
defined by
(A - (y)I)z = y, y+ = µz (1)


Source: Absil, Pierre-Antoine - Département d'ingénierie Mathématique, Université Catholique de Louvain


Collections: Mathematics