On the theory of operator interpolation in spaces of rectangular matrixes

Authors

  • Marina V. Ignatenko Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus
  • Leonid A. Yanovich Institute of Mathematics, National Academy of Sciences of Belarus, 11 Surhanava Street, Minsk 220072, Belarus

Keywords:

pseudo-inverse matrix, skeletal decomposition of a matrix, function of a matrix, matrix polynomial, operator interpolation
Supporting Agencies
This work was supported by the state program of scientific research «Convergence-2025» (subprogram «Mathematical models and methods», task 1.3.01)

Abstract

The problem of constructing and studying interpolation operator polynomials of an arbitrary fixed degree, defined in spaces of rectangular matrices, which would be generalisations of the corresponding interpolation formulas in the case of square matrices, is considered. Linear interpolation formulas of various structures are constructed for rectangular matrices. Matrix polynomials, with respect to which the resulting interpolation formulas are invariant, are indicated. As a generalisation of linear formulas, formulas for quadratic interpolation and interpolation by polynomials of arbitrary fixed degree in the space of rectangular matrices are constructed. Particular cases of the obtained formulas are considered: when square matrices are chosen as nodes or when the values of the interpolated function are square matrices, as well as the case when both of these conditions are satisfied. For the last variant, the possibilities of different and identical matrix orders for nodes and function values are explored. The obtained results are based on the application of some well-known provisions of the theory of matrices and the theory of interpolation of scalar functions. The presentation of the material is illustrated by a number of examples.

Author Biographies

  • Marina V. Ignatenko, Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus

    PhD (physics and mathematics), docent; associate professor at the department of web-technologies and computer simulation, faculty of mechanics and mathematics

  • Leonid A. Yanovich, Institute of Mathematics, National Academy of Sciences of Belarus, 11 Surhanava Street, Minsk 220072, Belarus

    corresponding member of the National Academy of Sciences, doctor of science (physics and mathematics), full professor; chief researcher

References

  1. Gantmakher FR. Teoriya matrits [Matrix theory]. Moscow: Nauka; 1967. 575 p. Russian.
  2. Magnus YaR, Neidekker Kh. Matrichnoe differentsial’noe ischislenie s prilozheniyami k statistike i ekonometrike [Matrix differential calculus with applications to statistics and econometrics]. Moscow: Fizmatlit; 2002. 496 p. Russian.
  3. Makarov VL, Khlobystov VV, Yanovich LA. Methods of operator interpolation. Kyiv: Institute of Mathematics of the National Academy of Sciences of Ukraine; 2010. 517 p. (Praci instytutu matematyky NAN Ukrai’ny; volume 83).
  4. Yanovich LA, Ignatenko MV. Osnovy teorii interpolirovaniya funktsii matrichnykh peremennykh [Bases of the theory of interpolation of functions of matrix variables]. Minsk: Belaruskaja navuka; 2016. 281 р. Russian.
  5. Yanovich LA, Ignatenko MV. Interpolyatsionnye metody approksimatsii operatorov, zadannykh na funktsional’nykh prostranstvakh i mnozhestvakh matrits [Interpolation methods for approximation of operators defined on function spaces and sets of matrices]. Minsk: Belaruskaja navuka; 2020. 476 р. Russian.

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Published

2022-12-19

Issue

No. 3 (2022): Journal of the Belarusian State University. Mathematics and Informatics

Section

Computational Mathematics

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How to Cite

[1]
Ignatenko, M.V. and Yanovich, L.A. 2022. On the theory of operator interpolation in spaces of rectangular matrixes. Journal of the Belarusian State University. Mathematics and Informatics. 3 (Dec. 2022), 91–106. DOI:https://doi.org/10.33581/2520-6508-2022-3-91-106.