Gradients of quotients and eigenvalue problems
Huhtanen, Marko; Nevanlinna, Olavi (2025-04-23)
Springer
23.04.2025
Huhtanen, M., Nevanlinna, O. Gradients of quotients and eigenvalue problems. Bit Numer Math 65, 21 (2025). https://doi.org/10.1007/s10543-025-01064-x
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© The Author(s) 2025. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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https://urn.fi/URN:NBN:fi:oulu-202505023044
https://urn.fi/URN:NBN:fi:oulu-202505023044
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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.
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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