Gradients of quotients and eigenvalue problems

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A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
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Date

2025-06

Department

Department of Mathematics and Systems Analysis

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Mcode

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Language

en

Pages

26

Series

BIT Numerical Mathematics, Volume 65, issue 2, pp. 1-26

Abstract

Intertwining analysis, optimization, numerical analysis and algebra, computing conjugate co-gradients of real-valued quotients gives rise to eigenvalue problems. In the linear Hermitian case, by inspecting optimal quotients in terms of taking the conjugate co-gradient for their critical points, a generalized folded spectrum eigenvalue problem arises. Replacing the Euclidean norm in optimal quotients with the p-norm, a matrix version of the so-called p-Laplacian eigenvalue problem arises. Such nonlinear eigenvalue problems seem to be naturally classified as being a special case of homogeneous problems. Being a quite general class, tools are developed for recovering whether a given homogeneous eigenvalue problem is a gradient eigenvalue problem. It turns out to be a delicate issue to come up with a valid quotient. A notion of nonlinear Hermitian eigenvalue problem is suggested. Cauchy–Schwarz quotients are introduced to a have a way to approach non-gradient eigenvalue problems.

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Publisher Copyright: © The Author(s) 2025.

Keywords

Conjugate co-gradient, Folded spectrum method, Gradient eigenvalue problem, Homogeneous eigenvalue problem, Nonlinear eigenvalue problem, p-Laplacian, Quotient

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DOI

10.1007/s10543-025-01064-x

Citation

Huhtanen, M & Nevanlinna, O 2025, 'Gradients of quotients and eigenvalue problems', BIT Numerical Mathematics, vol. 65, no. 2, 21, pp. 1-26. https://doi.org/10.1007/s10543-025-01064-x

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https://urn.fi/URN:NBN:fi:aalto-202506195080

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[article-cris] Perustieteiden korkeakoulu / SCI