IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v315y2017icp246-258.html
   My bibliography  Save this article

Matrix methods for the simplicial Bernstein representation and for the evaluation of multivariate polynomials

Author

Listed:
  • Titi, Jihad
  • Garloff, Jürgen

Abstract

In this paper, multivariate polynomials in the Bernstein basis over a simplex (simplicial Bernstein representation) are considered. Two matrix methods for the computation of the polynomial coefficients with respect to the Bernstein basis, the so-called Bernstein coefficients, are presented. Also matrix methods for the calculation of the Bernstein coefficients over subsimplices generated by subdivision of the standard simplex are proposed and compared with the use of the de Casteljau algorithm. The evaluation of a multivariate polynomial in the power and in the Bernstein basis is considered as well. 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. The latter one enables the use of the Fast Fourier Transform hereby reducing the amount of arithmetic operations.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:apmaco:v:315:y:2017:i:c:p:246-258
    DOI: 10.1016/j.amc.2017.07.026
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300317304770
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.amc.2017.07.026?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. de Klerk, E. & den Hertog, D. & Elabwabi, G., 2008. "On the complexity of optimization over the standard simplex," European Journal of Operational Research, Elsevier, vol. 191(3), pages 773-785, December.
    2. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Titi, Jihad & Garloff, Jürgen, 2019. "Matrix methods for the tensorial Bernstein form," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 254-271.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Titi, Jihad & Garloff, Jürgen, 2019. "Matrix methods for the tensorial Bernstein form," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 254-271.
    2. Evrim Dalkiran & Hanif Sherali, 2013. "Theoretical filtering of RLT bound-factor constraints for solving polynomial programming problems to global optimality," Journal of Global Optimization, Springer, vol. 57(4), pages 1147-1172, December.
    3. James Chok & Geoffrey M. Vasil, 2023. "Convex optimization over a probability simplex," Papers 2305.09046, arXiv.org.
    4. Tareq Hamadneh & Hassan Al-Zoubi & Saleh Ali Alomari, 2020. "Fast Computation of Polynomial Data Points Over Simplicial Face Values," Journal of Information & Knowledge Management (JIKM), World Scientific Publishing Co. Pte. Ltd., vol. 19(01), pages 1-13, March.
    5. P. S. Dhabe & P. S. V. Nataraj, 2017. "The Bernstein algorithm using the modified implicit Bernstein form and its GPU parallelization using CUDA," 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(4), pages 826-841, December.
    6. Sadek, Lakhlifa & Bataineh, Ahmad Sami & Isik, Osman Rasit & Alaoui, Hamad Talibi & Hashim, Ishak, 2023. "A numerical approach based on Bernstein collocation method: Application to differential Lyapunov and Sylvester matrix equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 212(C), pages 475-488.
    7. Immanuel Bomze & Stefan Gollowitzer & E. Yıldırım, 2014. "Rounding on the standard simplex: regular grids for global optimization," Journal of Global Optimization, Springer, vol. 59(2), pages 243-258, July.
    8. 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.
    9. Priyadarshan Dhabe & P. S. V. Nataraj, 2020. "A GPU parallel Bernstein algorithm for polynomial global optimization," 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. 11(1), pages 21-44, February.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:315:y:2017:i:c:p:246-258. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.