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Behavior of the Steepest Descent Method in Minimizing Rayleigh Quotient
Takashi OZEKI Taizo IIJIMA
Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences
Vol.E80A
No.1
pp.176182 Publication Date: 1997/01/25 Online ISSN:
DOI: Print ISSN: 09168508 Type of Manuscript: PAPER Category: Numerical Analysis and Optimization Keyword: numerical analysis, steepest descent method, Rayleigh quotient, eigenvalue problem, search direction,
Full Text: PDF>>
Summary:
In this paper we discuss the limiting behavior of the search direction of the steepest descent method in minimizing the Rayleigh quotient. This minimization problem is equivalent to finding the smallest eigenvalue of a matrix. It is shown that the search direction asymptotically alternates between two directions represented by linear combinations of two eigenvectors of the matrix. This is similar to the phenomenon in minimizing the quadratic form. We also show that these eigenvectors correspond to the largest and secondsmallest eigenvalues, unlike in the case of the quadratic form.

