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A homogeneous Rayleigh quotient with applications in gradient methods. (arXiv:2206.13457v3 [math.NA] UPDATED)

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Given an approximate eigenvector, its (standard) Rayleigh quotient and
harmonic Rayleigh quotient are two well-known approximations of the
corresponding eigenvalue. We propose a new type of Rayleigh quotient, the
homogeneous Rayleigh quotient, and analyze its sensitivity with respect to
perturbations in the eigenvector. Furthermore, we study the inverse of this
homogeneous Rayleigh quotient as stepsize for the gradient method for
unconstrained optimization. The notion and basic properties are also extended
to the generalized eigenvalue problem.

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