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On the Kronecker Structure of linearization of Cubic Two-Parameter Eigenvalue Problems

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Title: On the Kronecker Structure of linearization of Cubic Two-Parameter Eigenvalue Problems
Authors: Bora, Niranjan, Borgohain, Bharati
Source: Malaysian Journal of Science (MJS); Vol 44 No 1 (2025): March 2025; 70-86 ; 2600-8688 ; 1394-3065
Publisher Information: Faculty of Science, Universiti Malaya
Publication Year: 2025
Subject Terms: Cubic two-parameter eigenvalue problem, Linear two-parameter eigenvalueproblem, linearization, matrix polynomial
Description: Linearization is a classical approach to study matrix polynomial of the form P(lambda)=Sum lambdaj Aj, where Aj Cnxn . It converts into a matrix pencil of the form L(lambda)=A+lambda B of high dimension, where A and B are matrices over C , and lambda is the spectral parameter. In this paper we consider Cubic two-parameter eigenvalue problems ( CTEP) and study three different linearization process of the problem. Usinglinearization techniques, CTEP is first converted into a linear two-parameter eigenvalue problem(L2EP) of coefficient matrices of different sizes. The main advantage of these linearizations lies in the fact that, after transforming them into suitable linearized forms, existing numerical techniques for linear multiparameter eigenvalue problems (LMEP) can be applied to CTEP without solving the original problem. While solving CTEP by formulating suitable linearizations, several transformations are generally used. In the current paper, it is also intended to report on these transformations, which have not been studied completely due to the complexity of their Kronecker structures.
Document Type: article in journal/newspaper
File Description: application/pdf
Language: English
Relation: https://ejournal.um.edu.my/index.php/index/article/view/45991/18200; https://ejournal.um.edu.my/index.php/index/article/view/45991
DOI: 10.22452/mjs.vol44no1.9
Availability: https://ejournal.um.edu.my/index.php/index/article/view/45991
https://doi.org/10.22452/mjs.vol44no1.9
Rights: Copyright (c) 2025 Malaysian Journal of Science ; https://creativecommons.org/licenses/by-nc/4.0
Accession Number: edsbas.9275F745
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  Data: On the Kronecker Structure of linearization of Cubic Two-Parameter Eigenvalue Problems
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  Data: <searchLink fieldCode="AR" term="%22Bora%2C+Niranjan%22">Bora, Niranjan</searchLink><br /><searchLink fieldCode="AR" term="%22Borgohain%2C+Bharati%22">Borgohain, Bharati</searchLink>
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  Data: Malaysian Journal of Science (MJS); Vol 44 No 1 (2025): March 2025; 70-86 ; 2600-8688 ; 1394-3065
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  Data: Faculty of Science, Universiti Malaya
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  Data: 2025
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  Data: <searchLink fieldCode="DE" term="%22Cubic+two-parameter+eigenvalue+problem%22">Cubic two-parameter eigenvalue problem</searchLink><br /><searchLink fieldCode="DE" term="%22Linear+two-parameter+eigenvalueproblem%22">Linear two-parameter eigenvalueproblem</searchLink><br /><searchLink fieldCode="DE" term="%22linearization%22">linearization</searchLink><br /><searchLink fieldCode="DE" term="%22matrix+polynomial%22">matrix polynomial</searchLink>
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  Data: Linearization is a classical approach to study matrix polynomial of the form P(lambda)=Sum lambdaj Aj, where Aj Cnxn . It converts into a matrix pencil of the form L(lambda)=A+lambda B of high dimension, where A and B are matrices over C , and lambda is the spectral parameter. In this paper we consider Cubic two-parameter eigenvalue problems ( CTEP) and study three different linearization process of the problem. Usinglinearization techniques, CTEP is first converted into a linear two-parameter eigenvalue problem(L2EP) of coefficient matrices of different sizes. The main advantage of these linearizations lies in the fact that, after transforming them into suitable linearized forms, existing numerical techniques for linear multiparameter eigenvalue problems (LMEP) can be applied to CTEP without solving the original problem. While solving CTEP by formulating suitable linearizations, several transformations are generally used. In the current paper, it is also intended to report on these transformations, which have not been studied completely due to the complexity of their Kronecker structures.
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  Data: https://ejournal.um.edu.my/index.php/index/article/view/45991/18200; https://ejournal.um.edu.my/index.php/index/article/view/45991
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  Data: Copyright (c) 2025 Malaysian Journal of Science ; https://creativecommons.org/licenses/by-nc/4.0
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      – SubjectFull: Linear two-parameter eigenvalueproblem
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