Introduction

• Introduction
• • Bernstein Expansion
  • Example
  • References

Bernstein Expansion

Consider a polynomial in $n$ variables, $x_1, \ldots, x_n$ expressed in its power form,

\[ p(x) = \\sum_{i=0}^l a_i x^i,\qquad x = (x_1, \ldots, x_n),\]

where we use the multi-index notation. The degree of $p$ is $l = (l_1, \ldots, l_n)$, $0 ≤ l_i < \infty$ for all $i = 1, \ldots, n$.

Box

\[X = [\bar{x}_1, \overline{x}_1]\]

Example

using BernsteinExpansions, DynamicPolynomials

@polyvar x

univariate(x^3, 3, 1..2)

4-element Vector{Float64}:
 1.0
 2.0
 4.0
 8.0

univariate(2x^3, 3, 1..2)

4-element Vector{Float64}:
  2.0
  4.0
  8.0
 16.0

References

[1] Smith, A. P. Fast construction of constant bound functions for sparse polynomials. Journal of Global Optimization 43.2 (2009): 445-458.

[2] Smith, A. P. Enclosure methods for systems of polynomial equations and inequalities. Doctoral dissertation, Universität Konstanz (2012).

[3] Titi, J., & Garloff, J. (2017). Fast determination of the tensorial and simplicial Bernstein forms of multivariate polynomials and rational functions. Konstanzer Schriften in Mathematik; 361.