Classical special functions of matrix arguments
This article focuses on a few of the most commonly used special functions and their key properties and defines an analytical approach to building their matrix-variate counterparts. To achieve this, we refrain from using any numerical approximation algorithms and instead rely on properties of matrice...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | Ukrainian |
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Igor Sikorsky Kyiv Polytechnic Institute
2024-12-01
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| Series: | Sistemnì Doslìdženâ ta Informacìjnì Tehnologìï |
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| Online Access: | http://journal.iasa.kpi.ua/article/view/322530 |
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| _version_ | 1850037012807024640 |
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| author | Dmytro Shutiak Gleb Podkolzin Victor Bondarenko Yury Chapovsky |
| author_facet | Dmytro Shutiak Gleb Podkolzin Victor Bondarenko Yury Chapovsky |
| author_sort | Dmytro Shutiak |
| collection | DOAJ |
| description | This article focuses on a few of the most commonly used special functions and their key properties and defines an analytical approach to building their matrix-variate counterparts. To achieve this, we refrain from using any numerical approximation algorithms and instead rely on properties of matrices, the matrix exponential, and the Jordan normal form for matrix representation. We focus on the following functions: the Gamma function as an example of a univariate function with a large number of properties and applications; the Beta function to highlight the similarities and differences from adding a second variable to a matrix-variate function; and the Jacobi Theta function. We construct explicit function views and prove a few key properties for these functions. In the comparison section, we highlight and contrast other approaches that have been used in the past to tackle this problem. |
| format | Article |
| id | doaj-art-06c8b33ebf7e4bd7ba23e7618529f614 |
| institution | DOAJ |
| issn | 1681-6048 2308-8893 |
| language | Ukrainian |
| publishDate | 2024-12-01 |
| publisher | Igor Sikorsky Kyiv Polytechnic Institute |
| record_format | Article |
| series | Sistemnì Doslìdženâ ta Informacìjnì Tehnologìï |
| spelling | doaj-art-06c8b33ebf7e4bd7ba23e7618529f6142025-08-20T02:56:59ZukrIgor Sikorsky Kyiv Polytechnic InstituteSistemnì Doslìdženâ ta Informacìjnì Tehnologìï1681-60482308-88932024-12-01411713210.20535/SRIT.2308-8893.2024.4.10361245Classical special functions of matrix argumentsDmytro Shutiak0https://orcid.org/0009-0008-6480-3706Gleb Podkolzin1https://orcid.org/0000-0002-7120-2772Victor Bondarenko2https://orcid.org/0000-0003-1663-4799Yury Chapovsky3https://orcid.org/0009-0001-8981-4742World Data Center for Geoinformatics and Sustainable Development of the National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", KyivEducational and Research Institute for Applied System Analysis of the National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", KyivEducational and Research Institute for Applied System Analysis of the National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", KyivEducational and Research Institute for Applied System Analysis of the National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", KyivThis article focuses on a few of the most commonly used special functions and their key properties and defines an analytical approach to building their matrix-variate counterparts. To achieve this, we refrain from using any numerical approximation algorithms and instead rely on properties of matrices, the matrix exponential, and the Jordan normal form for matrix representation. We focus on the following functions: the Gamma function as an example of a univariate function with a large number of properties and applications; the Beta function to highlight the similarities and differences from adding a second variable to a matrix-variate function; and the Jacobi Theta function. We construct explicit function views and prove a few key properties for these functions. In the comparison section, we highlight and contrast other approaches that have been used in the past to tackle this problem.http://journal.iasa.kpi.ua/article/view/322530matrixspecial functionmatrix functiongamma functionbeta functionjacobi theta functionjordan normal form |
| spellingShingle | Dmytro Shutiak Gleb Podkolzin Victor Bondarenko Yury Chapovsky Classical special functions of matrix arguments Sistemnì Doslìdženâ ta Informacìjnì Tehnologìï matrix special function matrix function gamma function beta function jacobi theta function jordan normal form |
| title | Classical special functions of matrix arguments |
| title_full | Classical special functions of matrix arguments |
| title_fullStr | Classical special functions of matrix arguments |
| title_full_unstemmed | Classical special functions of matrix arguments |
| title_short | Classical special functions of matrix arguments |
| title_sort | classical special functions of matrix arguments |
| topic | matrix special function matrix function gamma function beta function jacobi theta function jordan normal form |
| url | http://journal.iasa.kpi.ua/article/view/322530 |
| work_keys_str_mv | AT dmytroshutiak classicalspecialfunctionsofmatrixarguments AT glebpodkolzin classicalspecialfunctionsofmatrixarguments AT victorbondarenko classicalspecialfunctionsofmatrixarguments AT yurychapovsky classicalspecialfunctionsofmatrixarguments |