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Symbolic-Numeric Computation of the Bernstein Coefficients of a Polynomial from Those of One of Its Partial Derivatives and of the Product of Two Polynomials

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Computer Algebra in Scientific Computing (CASC 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12291))

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Abstract

The expansion of a given multivariate polynomial into Bernstein polynomials is considered. Matrix methods for the calculation of the Bernstein expansion of the product of two polynomials and of the Bernstein expansion of a polynomial from the expansion of one of its partial derivatives are provided which allow also a symbolic computation.

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Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Physics, Palestine Polytechnic University, Hebron, Palestine

    Jihad Titi

  2. Department of Mathematics and Statistics, University of Konstanz, 78464, Konstanz, Germany

    Jürgen Garloff

  3. University of Applied Sciences/HTWG Konstanz, Institute for Applied Research, Alfred-Wachtel-Str. 8, 78462, Konstanz, Germany

    Jürgen Garloff

Authors
  1. Jihad Titi
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  2. Jürgen Garloff
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Corresponding author

Correspondence to Jürgen Garloff .

Editor information

Editors and Affiliations

  1. University of Lille, Villeneuve d’Ascq, France

    François Boulier

  2. Coventry University, Coventry, UK

    Matthew England

  3. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  4. Institute of Theoretical and Applied Mechanics, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

Appendices

Appendix A. Example for the Performance of Algorithm 1

Let p and q be bivariate polynomials of degree (4, 2) and (3, 2), respectively, with Bernstein matrices over u

$$\begin{aligned} \mathcal {B}(p, {\textit{\textbf{u}}}) = \begin{bmatrix} 2 &{} 0 &{} -1 ~\\ -2 &{} 3 &{} 0 ~\\ 1 &{} 3 &{} -3 ~\\ 1 &{} 0 &{} 1 ~\\ 0 &{} -1 &{} 0 ~ \end{bmatrix} \text{ and } \mathcal {B}(q, {\textit{\textbf{u}}}) = \begin{bmatrix} -1 &{} -4 &{} -1 ~\\ 2 &{} 0 &{} 5 ~\\ -2 &{} 3 &{} 0 ~\\ 0 &{} 1 &{} 1~ \end{bmatrix}. \end{aligned}$$

Then

$$\begin{aligned} D_1= & {} \mathrm {diag}(1, 4, 6, 4, 1) \text{ and } D_2 = \mathrm {diag}(1, 2, 1),\\ C_1(p)= & {} (D_1 C_0(p))^c = \begin{bmatrix} 2 &{} -8 &{} 6 &{} 4 &{} 0~\\ 0 &{} 12 &{} 18 &{} 0 &{} -1~\\ -1 &{} 0 &{} -18 &{} 4 &{} 0~ \end{bmatrix}, \\ C_2(p)= & {} (D_2 C_1(p))^c = \begin{bmatrix} 2 &{} 0 &{} -1~\\ -8 &{} 24 &{} 0~\\ 6 &{} 36 &{} -18~\\ 4&{} 0 &{} 4~\\ 0 &{} -2 &{} 0~ \end{bmatrix}; \end{aligned}$$
$$M_{0}^{(1,1)} = M_{0}^{(2,2)} =M_{0}^{(3,0)} = O_{5, 3},$$
$$\begin{aligned} M_{0}^{(0,0)} = \begin{bmatrix} -2 &{} 0 &{} 1~\\ 8 &{} -24 &{} 0~\\ -6 &{} -36 &{} 18~\\ -4&{} 0 &{} -4~\\ 0 &{} 2 &{} 0~ \end{bmatrix}, \;&M_{0}^{(0,1)} = \begin{bmatrix} -16 &{} 0 &{} 8~\\ 64 &{} -192 &{} 0~\\ -48 &{} -288 &{} 144~\\ -32 &{} 0 &{} -32~\\ 0 &{} 16 &{} 0~ \end{bmatrix}, \; M_{0}^{(0,2)} = \begin{bmatrix} -2 &{} 0 &{} 1~\\ 8 &{} -24 &{} 0~\\ -6 &{} -36 &{} 18~\\ -4&{} 0 &{} -4~\\ 0 &{} 2 &{} 0~ \end{bmatrix}\!, \end{aligned}$$
$$\begin{aligned} M_{0}^{(1,0)}&= \begin{bmatrix} 12 &{} 0 &{} -6~\\ -48 &{} 144 &{} 0~\\ 36 &{} 216 &{} -108~\\ 24 &{} 0 &{} 24~\\ 0 &{} -12 &{} 0~ \end{bmatrix}, \;&M_{0}^{(1,2)}&= \begin{bmatrix} 30 &{} 0 &{} -15~\\ -120 &{} 360 &{} 0~\\ 90 &{} 540 &{} -270~\\ 60 &{} 0 &{} 60~\\ 0 &{} -30 &{} 0~ \end{bmatrix}, \; \\ M_{0}^{(2,0)}&= \begin{bmatrix} -12 &{} 0 &{} 6~\\ 48 &{} -144 &{} 0~\\ -36 &{} -216 &{} 108~\\ -24&{} 0 &{} -24~\\ 0 &{} 12 &{} 0~ \end{bmatrix},&M_{0}^{(2,1)}&= \begin{bmatrix} 36 &{} 0 &{} -18~\\ -144 &{} 432 &{} 0~\\ 108 &{} 648 &{} -324~\\ 72&{} 0 &{} 72~\\ 0 &{} -36 &{} 0~ \end{bmatrix}, \; \\ M_{0}^{(3,1)}&= \begin{bmatrix} 4 &{} 0 &{} -2~\\ -16 &{} 48 &{} 0~\\ 12 &{} 72 &{} -36~\\ 8&{} 0 &{} 8~\\ 0 &{} -4 &{} 0~ \end{bmatrix}, \;&M_{0}^{(3,2)}&= \begin{bmatrix} 2 &{} 0 &{} -1~\\ -8 &{} 24 &{} 0~\\ 6 &{} 36 &{} -18~\\ 4&{} 0 &{} 4~\\ 0 &{} -2 &{} 0~ \end{bmatrix}; \end{aligned}$$
$$\begin{aligned} M_{1}^{(0,0)}&= \begin{bmatrix} -2 &{} 8 &{} -6 &{} -4 &{} 0 &{} 0 &{} 0 &{} 0~\\ 0 &{} -24 &{} -36 &{} 0 &{} 2 &{}0 &{} 0 &{} 0~\\ 1 &{} 0 &{} 18 &{} -4 &{} 0 &{} 0 &{} 0 &{} 0~ \end{bmatrix},&M_{2}^{(0,0)}&= \begin{bmatrix} -2 &{} 0 &{} 1 &{} 0 &{} 0~\\ 8 &{} -24 &{} 0 &{} 0 &{} 0~\\ -6 &{} -36 &{} 18 &{} 0 &{} 0~\\ -4 &{} 0 &{} -4 &{} 0 &{} 0~\\ 0 &{} 2 &{} 0 &{} 0 &{} 0~\\ 0 &{} 0 &{} 0 &{} 0 &{} 0~\\ 0 &{} 0 &{} 0 &{} 0 &{} 0~\\ 0 &{} 0 &{} 0 &{} 0 &{} 0~ \end{bmatrix}, \\ M_{1}^{(0,1)}&= \begin{bmatrix} -16 &{} 64 &{} -48 &{} -32 &{} 0 &{} 0 &{} 0 &{} 0~\\ 0 &{} -192 &{} -288 &{} 0 &{} 16 &{}0 &{} 0 &{} 0~\\ 8 &{} 0 &{} 144 &{} -32 &{} 0 &{} 0 &{} 0 &{} 0~ \end{bmatrix},&M_{2}^{(0,1)}&=\begin{bmatrix} ~0 &{}-16 &{} 0 &{} 8 &{} 0~ \\ ~0 &{} 64 &{} -192 &{} 0 &{} 0~ \\ ~0 &{} -48 &{} -288 &{} 144 &{} 0~\\ ~0 &{} -32 &{} 0 &{} -32 &{} 0~ \\ ~0 &{} 0 &{} 16 &{} 0 &{} 0~ \\ ~0 &{} 0 &{} 0 &{} 0 &{} 0~ \\ ~0 &{} 0 &{} 0 &{} 0 &{} 0~ \\ ~0 &{} 0 &{} 0 &{} 0 &{} 0~ \end{bmatrix}, \end{aligned}$$
$$\begin{aligned} M_{1}^{(0,2)}= & {} \begin{bmatrix} -2 &{} 8 &{} -6 &{} -4 &{} 0 &{} 0 &{} 0 &{} 0~\\ 0 &{} -24 &{} -36 &{} 0 &{} 2 &{}0 &{} 0 &{} 0~\\ 1 &{} 0 &{} 18 &{} -4 &{} 0 &{} 0 &{} 0 &{} 0~ \end{bmatrix}, \, M_{2}^{(0,2)} = \begin{bmatrix} ~0 &{} 0 &{} -2 &{} 0 &{} 1 ~\\ ~0 &{} 0 &{} 8 &{} -24 &{} 0 ~\\ ~0 &{} 0 &{} -6 &{} -36 &{} 18~\\ ~0 &{} 0 &{} -4 &{} 0 &{} -4 ~\\ ~0 &{} 0 &{} 0 &{} 2 &{} 0 ~\\ ~0 &{} 0 &{} 0 &{} 0 &{} 0 ~\\ ~0 &{} 0 &{} 0 &{} 0 &{} 0 ~\\ ~0 &{} 0 &{} 0 &{} 0 &{} 0 ~ \end{bmatrix}. \end{aligned}$$

The remaining 18 matrices \(M_s^{(i)}\), \(s=1, 2\), are formed analogously.

$$\begin{aligned} F_0 = C_0(pq) = \begin{bmatrix} -2 &{} -16 &{} -1 &{} 8 &{} 1~\\ 20 &{} 40 &{} -160 &{}-24 &{} -15~\\ -66 &{} 96 &{} -390 &{} 450 &{} 18~\\ 80 &{} -100 &{} 408 &{} 506 &{} -275~\\ -12 &{} -122 &{} 896 &{} -298 &{} 60~\\ -24 &{} 72 &{} 54 &{} 42 &{} -18~\\ 0 &{} 20 &{} -32 &{} 8 &{} 4~\\ 0 &{} 0 &{} 4 &{} -2 &{} 0~ \end{bmatrix}; \end{aligned}$$
$$\begin{aligned} D_{1}^{\prime } = \mathrm {diag}(1, \frac{1}{7}, \frac{1}{21}, \frac{1}{35}, \frac{1}{35}, \frac{1}{21}, \frac{1}{7}, 1), \; D_{2}^{\prime } = \mathrm {diag}(1, \frac{1}{4}, \frac{1}{6}, \frac{1}{4}, 1). \end{aligned}$$

The Bernstein matrix of the product of the polynomials p and q is given by

$$\begin{aligned} \mathcal {B}(pq, {\textit{\textbf{u}}}) = F_2 =(D_{2}^{\prime }(D_{1}^{\prime }F_0)^c)^c = \begin{bmatrix} -2 &{} -4 &{} \frac{-1}{6} &{} 2 &{} 1 \\ \frac{20}{7} &{} \frac{10}{7} &{} \frac{-80}{21} &{}\frac{-6}{7} &{} \frac{-15}{7} \\ \frac{-22}{7} &{} \frac{8}{7} &{} \frac{-65}{21} &{} \frac{75}{14} &{} \frac{6}{7} \\ \frac{16}{7} &{} \frac{-5}{7} &{} \frac{68}{35} &{} \frac{253}{70} &{} \frac{-55}{7} \\ \frac{-12}{35} &{} \frac{-61}{70} &{} \frac{64}{15} &{} \frac{-149}{70} &{} \frac{12}{7} \\ \frac{-8}{7} &{} \frac{6}{7} &{} \frac{3}{7} &{} \frac{1}{2} &{} \frac{-6}{7} \\ 0 &{} \frac{5}{7} &{} \frac{-16}{21} &{} \frac{2}{7} &{} \frac{4}{7} \\ 0 &{} 0 &{} \frac{2}{3} &{} \frac{-1}{2} &{} 0 \end{bmatrix}. \end{aligned}$$

Appendix B. Example for the Performance of the Method Presented in Sect. 5

Let \(p(x_1, x_2) = -504 x_1^4 x_2^2 - 84 x_1^4 x_2 + 288 x_1^4 + 6 x_1^3 x_2^2 + 30 x_1^3 x_2 - 60x_1^3 +36 x_1^2 x_2^2 - 20 x_1^2 x_2 + 28 x_1^2 -54 x_1 x_2^2 +21x_1 x_2 -24 x_1 + 24x_2^2 -24 x_2 +48\). Then

$$\begin{aligned} \mathcal {B}(\frac{\partial p}{\partial x_1}, {\textit{\textbf{u}}})= \begin{bmatrix} -24 &{}\frac{-27}{2} &{} -57 \\ -4 &{} \frac{-9}{6} &{} \frac{-83}{3} \\ \frac{-140}{3} &{} \frac{-207}{6} &{} \frac{-67}{6} \\ 1004 &{} \frac{-1743}{2} &{} -1241 \end{bmatrix} \text{ and } C_1^{\prime } = \mathcal {B}(p(0,x_2), [0, 1])= \begin{bmatrix} 48&36&48 \end{bmatrix}; \end{aligned}$$
$$\begin{aligned} B_1^{\prime } = \begin{bmatrix} 48 &{} 36 &{} 48\\ -6 &{}\frac{-27}{8} &{} \frac{-57}{4} \\ \frac{-4}{3} &{} \frac{-9}{24} &{} \frac{-83}{12} \\ \frac{-140}{12} &{} \frac{-207}{24} &{} \frac{-67}{12} \\ 251 &{} \frac{-1743}{8} &{} \frac{-1241}{4} \end{bmatrix}. \end{aligned}$$

The Bernstein coefficients of p over u are obtained from

$$\begin{aligned} \mathcal {B}(p, {\textit{\textbf{u}}})= & {} H_1 H_2 H_3 H_4 B_1^{\prime } \\= & {} \begin{bmatrix} 1 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 1 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 &{} 1 &{} 1 \end{bmatrix} \begin{bmatrix} 1 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 1 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 1 &{} 0 &{} 0\\ 0 &{} 0 &{} 1 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \end{bmatrix} \begin{bmatrix} 1 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 1 &{} 0 &{} 0 &{} 0\\ 0 &{} 1 &{} 1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \end{bmatrix} \begin{bmatrix} 1 &{} 0 &{} 0 &{} 0 &{} 0\\ 1 &{} 1 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \end{bmatrix} \begin{bmatrix} 48 &{} 36 &{} 48 \\ -6 &{}\frac{-27}{8} &{} \frac{-57}{4} \\ \frac{-4}{3} &{} \frac{-9}{24} &{} \frac{-83}{12} \\ \frac{-140}{12} &{} \frac{-207}{24} &{} \frac{-67}{12} \\ 251 &{} \frac{-1743}{8} &{} \frac{-1241}{4} \end{bmatrix}\\= & {} \begin{bmatrix} 48 &{} 36 &{} 48 \\ 42 &{}\frac{261}{8} &{} \frac{135}{4} \\ \frac{122}{3} &{} \frac{774}{24} &{} \frac{322}{12} \\ 29 &{} \frac{567}{24} &{} \frac{255}{12} \\ 280 &{} \frac{483}{2} &{} -289 \end{bmatrix}. \end{aligned}$$

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Titi, J., Garloff, J. (2020). Symbolic-Numeric Computation of the Bernstein Coefficients of a Polynomial from Those of One of Its Partial Derivatives and of the Product of Two Polynomials. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2020. Lecture Notes in Computer Science(), vol 12291. Springer, Cham. https://doi.org/10.1007/978-3-030-60026-6_34

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