, Volume 36, Issue 3, pp 494-508

On conjugate gradient-like methods for eigen-like problems

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Abstract

Numerical analysts, physicists, and signal processing engineers have proposed algorithms that might be called conjugate gradient for problems associated with the computation of eigenvalues. There are many variations, mostly one eigenvalue at a time though sometimes block algorithms are proposed. Is there a correct “conjugate gradient” algorithm for the eigenvalue problem? How are the algorithms related amongst themselves and with other related algorithms such as Lanczos, the Newton method, and the Rayleigh quotient?

This paper will also appear in the Proceedings of the AMS/IMS/SIAM Joint Summer Research Conference on Linear and Nonlinear Conjugate Gradient-Related Methods held in Seattle, 9–13 July 1995.
Supported by a fellowship from the Alfred P. Sloan Foundation and NSF Grant 9404326-CCR.
This work was sponsored by DARPA under Air Force contract F19628-95-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the author and are not necessarily endorsed by the United States Air Force.