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Least squares problems involving generalized Kronecker products and application to bivariate polynomial regression

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Abstract

A method for solving least squares problems (A ⊗ Bi)x = b whose coefficient matrices have generalized Kronecker product structure is presented. It is based on the exploitation of the block structure of the Moore-Penrose inverse and the reflexive minimum norm g-inverse of the coefficient matrix, and on the QR method for solving least squares problems. Firstly, the general case where A is a rectangular matrix is considered, and then the special case where A is square is analyzed. This special case is applied to the problem of bivariate polynomial regression, in which the involved matrices are structured matrices (Vandermonde or Bernstein-Vandermonde matrices). In this context, the advantage of using the Bernstein basis instead of the monomial basis is shown. Numerical experiments illustrating the good behavior of the proposed algorithm are included.

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Acknowledgements

We are grateful to the anonymous reviewers for their suggestions, which have contributed to improve our paper.

Funding

This research has been partially supported by Spanish Research Grant MTM2015-65433-P (MINECO/FEDER) from the Spanish Ministerio de Economía y Competitividad. A. Marco, J. J. Martínez and R. Viaña are members of the Research Group asynacs (Ref. ccee2011/r34) of Universidad de Alcalá.

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Authors and Affiliations

  1. Departamento de Física y Matemáticas, Universidad de Alcalá, Campus Universitario, 28871, Alcalá de Henares (Madrid), Spain

    Ana Marco, José-Javier Martínez & Raquel Viaña

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  1. Ana Marco
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  2. José-Javier Martínez
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  3. Raquel Viaña
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Correspondence to José-Javier Martínez.

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Marco, A., Martínez, JJ. & Viaña, R. Least squares problems involving generalized Kronecker products and application to bivariate polynomial regression. Numer Algor 82, 21–39 (2019). https://doi.org/10.1007/s11075-018-0592-1

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  • DOI: https://doi.org/10.1007/s11075-018-0592-1

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