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Matrix methods for the tensorial Bernstein form

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  • Titi, Jihad
  • Garloff, Jürgen

Abstract

In this paper, multivariate polynomials in the Bernstein basis over a box (tensorial Bernstein representation) are considered. A new matrix method for the computation of the polynomial coefficients with respect to the Bernstein basis, the so-called Bernstein coefficients, are presented and compared with existing methods. Also matrix methods for the calculation of the Bernstein coefficients over subboxes generated by subdivision of the original box are proposed. All the methods solely use matrix operations such as multiplication, transposition, and reshaping; some of them rely also on the bidiagonal factorization of the lower triangular Pascal matrix or the factorization of this matrix by a Toeplitz matrix. In the case that the coefficients of the polynomial are due to uncertainties and can be represented in the form of intervals it is shown that the developed methods can be extended to compute the set of the Bernstein coefficients of all members of the polynomial family.

Suggested Citation

  • Titi, Jihad & Garloff, Jürgen, 2019. "Matrix methods for the tensorial Bernstein form," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 254-271.
    • Handle: RePEc:eee:apmaco:v:346:y:2019:i:c:p:254-271
    DOI: 10.1016/j.amc.2018.08.049
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    References listed on IDEAS

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    1. P. Nataraj & M. Arounassalame, 2011. "Constrained global optimization of multivariate polynomials using Bernstein branch and prune algorithm," Journal of Global Optimization, Springer, vol. 49(2), pages 185-212, February.
    2. P. S. Dhabe & P. S. V. Nataraj, 2017. "A parallel Bernstein algorithm for global optimization based on the implicit Bernstein form," International Journal of System Assurance Engineering and Management, Springer;The Society for Reliability, Engineering Quality and Operations Management (SREQOM),India, and Division of Operation and Maintenance, Lulea University of Technology, Sweden, vol. 8(2), pages 1654-1671, November.
    3. Titi, Jihad & Garloff, Jürgen, 2017. "Matrix methods for the simplicial Bernstein representation and for the evaluation of multivariate polynomials," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 246-258.
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