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

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.

[1]  F. Clauss,et al.  Application of Symbolic Approach to the Bernstein Expansion for Program Analysis and Optimization , 2004, Programming and Computer Software.

[2]  Carla Piazza,et al.  Set-Based Analysis for Biological Modeling , 2019, Automated Reasoning for Systems Biology and Medicine.

[3]  Jürgen Garloff,et al.  Matrix methods for the simplicial Bernstein representation and for the evaluation of multivariate polynomials , 2017, Appl. Math. Comput..

[4]  Tommaso Dreossi,et al.  Sapo: Reachability Computation and Parameter Synthesis of Polynomial Dynamical Systems , 2016, HSCC.

[5]  Siegfried M. Rump,et al.  INTLAB - INTerval LABoratory , 1998, SCAN.

[6]  Pedro Alonso,et al.  Conditioning and accurate computations with Pascal matrices , 2013, J. Comput. Appl. Math..

[7]  Rida T. Farouki,et al.  The Bernstein polynomial basis: A centennial retrospective , 2012, Comput. Aided Geom. Des..

[8]  Jürgen Garloff,et al.  Fast Determination of the Tensorial and Simplicial Bernstein Forms of Multivariate Polynomials and Rational Functions , 2017 .

[9]  Jürgen Garloff,et al.  Solution of Systems of Polynomial Equation by Using Bernstein Expansion , 2001, Symbolic Algebraic Methods and Verification Methods.

[10]  Jürgen Garloff.  Convergent Bounds for the Range of Multivariate Polynomials , 1985, Interval Mathematics.

[11]  Jürgen Garloff,et al.  Matrix methods for the tensorial Bernstein form , 2019, Appl. Math. Comput..

[12]  Philippe Clauss,et al.  Symbolic Polynomial Maximization Over Convex Sets and Its Application to Memory Requirement Estimation , 2009, IEEE Transactions on Very Large Scale Integration (VLSI) Systems.

[13]  Rida T. Farouki,et al.  Algorithms for polynomials in Bernstein form , 1988, Comput. Aided Geom. Des..

[14]  T. J. Rivlin.  Bounds on a polynomial , 1970 .

[15]  Andrew Paul Smith,et al.  Fast construction of constant bound functions for sparse polynomials , 2009, J. Glob. Optim..

[16]  Jihad Titi.  Matrix Methods for the Tensorial and Simplicial Bernstein Forms with Application to Global Optimization , 2019 .