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Computing tensor generalized inverses via specialization and rationalization

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper, we introduce the notion of outer generalized inverses, with predefined range and null space, of tensors with rational function entries equipped with the Einstein product over an arbitrary field, of characteristic zero, with or without involution. We assume that the involved tensor entries are rational functions of unassigned variables or rational expressions of functional entries. The research investigates the replacements in two stages. The lower-stage replacements assume replacements of unknown variables by constant values from the field. The higher-order stage assumes replacements of functional entries by unknown variables. This approach enables the calculation on tensors over meromorphic functions to be simplified by analogous calculations on matrices whose elements are rational expressions of variables. In general, the derived algorithms permit symbolic computation of various generalized inverses which belong to the class of outer generalized inverses, with prescribed range and null space, over an arbitrary field of characteristic zero. More precisely, we focus on a few algorithms for symbolic computation of outer inverses of matrices whose entries are elements of a field of characteristic zero or a field of meromorphic functions in one complex variable over a connected open subset of \({\mathbb {C}}\). Illustrative numerical results validate the theoretical results.

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Acknowledgements

Predrag Stanimirović and Dijana Mosić are supported from the Ministry of Education and Science of Republic of Serbia, Grants No. 174013/451-03-68/2020-14/200124 and 174007/451-03-68/2020-14/200124. Ratikanta Behera is supported from the Mohapatra Family Foundation and the College of Graduate Studies, University of Central Florida, Orlando, FL, USA. J.R. Sendra and J. Sendra are are partially supported by the Spanish Ministerio de Economía y Competitividad, and by the European Regional Development Fund (ERDF), under the Project MTM2017-88796-P. A. Lastra and J.R. Sendra are members of the Research Group ASYNACS (Ref.CT-CE2019/683). Alberto Lastra is partially supported by the project PID2019-105621GB-I00 of Ministerio de Ciencia e Innovación. A. Lastra and J. R. Sendra also partially supported by Comunidad de Madrid and Universidad de Alcalá under grant CM/JIN/2019-010.

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Authors and Affiliations

  1. Faculty of Science and Mathematics, University of Niš, Višegradska 33, 18000, Niš, Serbia

    Predrag S. Stanimirović & Dijana Mosić

  2. Universidad de Alcalá, Dpto. Física y Matemáticas, Alcalá de Henares, Madrid, Spain

    J. Rafael Sendra & Alberto Lastra

  3. Department of Mathematics, University of Central Florida, Orlando, USA

    Ratikanta Behera

  4. Department of Mathematics, Birla Institute of Technology and Science Pilani-K.K. Birla Goa Campus, Zuarinagar, Goa, India

    Jajati Keshari Sahoo

  5. Dpto. Matemática Aplicada a las TIC, Universidad Politécnica de Madrid, Madrid, Spain

    Juana Sendra

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  1. Predrag S. Stanimirović
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Correspondence to Predrag S. Stanimirović.

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Stanimirović, P.S., Sendra, J.R., Behera, R. et al. Computing tensor generalized inverses via specialization and rationalization. RACSAM 115, 116 (2021). https://doi.org/10.1007/s13398-021-01057-9

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  • DOI: https://doi.org/10.1007/s13398-021-01057-9

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