A homogeneous Rayleigh quotient with applications in gradient methods

Giulia Ferrandi; Michiel E. Hochstenbach

doi:10.1016/j.cam.2023.115440

A homogeneous Rayleigh quotient with applications in gradient methods

Giulia Ferrandi (Corresponding author), Michiel E. Hochstenbach

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Abstract

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.

Original language	English
Article number	115440
Number of pages	15
Journal	Journal of Computational and Applied Mathematics
Volume	437
DOIs	https://doi.org/10.1016/j.cam.2023.115440
Publication status	Published - Feb 2024

Funding

We are grateful to the referees for their very useful suggestions which considerably improved the quality of the paper. This work has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 812912. We are grateful to the referees for their very useful suggestions which considerably improved the quality of the paper. This work has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 812912 .

Funders	Funder number
European Union's Horizon 2020 - Research and Innovation Framework Programme
Marie Skłodowska‐Curie	812912
European Union's Horizon 2020 - Research and Innovation Framework Programme

Keywords

Eigenvalue problem
Generalized eigenvalue problem
Homogeneous Rayleigh quotient
Projective coordinates
Secant condition
Unconstrained optimization

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AB - 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.

KW - Eigenvalue problem

KW - Generalized eigenvalue problem

KW - Homogeneous Rayleigh quotient

KW - Projective coordinates

KW - Secant condition

KW - Unconstrained optimization

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JF - Journal of Computational and Applied Mathematics

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ER -

A homogeneous Rayleigh quotient with applications in gradient methods

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