Bernstein polynomials

Bernstein polynomials#

Without loss of generality, we restrict to the case of the unit interval, namely \(a=0\) and \(b=1\). In figure (Fig. \ref{fig:bernstein-polynomials}), we plot the first sixth Bernstein polynomials.

Definition 1

Fro \(n \in \mathbb{N}\), the \(n\)-th degree Bernstein polynomials \(B_{j,n}\,:\, [0,1] \longrightarrow \mathbb{R}\) are defined by

\[ B_{j,n}(x)=\frac{n!}{(n-j)!j!} x^j (1-x)^{n-j}, \quad x \in [0,1], \quad j=0, \ldots,n. \]

Bernstein polynomials exhibit interesting properties highlighted in the following proposition,

# needed imports
import numpy as np
import matplotlib.pyplot as plt

We first consider the evaluation of the Bernstein polynomials.

Evaluation of Bernstein polynomials#

The following function evaluates all Bernstein polynomials of degree \(n\) at \(x\)

def all_bernstein(n, x):
    b = np.zeros(n+1)
    b[0] = 1.
    x1 = 1.-x
    for j in range(1, n+1):
        saved = 0.
        for i in range(0, j):
            tmp = b[i]
            b[i] = saved + x1*tmp
            saved = x*tmp
        b[j] = saved
    return b

Bernstein polynomials properties#

Proposition 1

We have the following properties of Bernstein Polynomials,

  • Positivity: \(B_{j,n}(x) \ge 0\), for all \(x \in \left[ 0, 1 \right]\) and \(0 \leq j \leq n\).

  • Partition of unity: \(\sum_{j=0}^n B_{j,n}(x) = 1\), for all \(x \in \left[ 0, 1 \right]\).

  • \(B_{0,n}(0) = B_{n,n}(1) = 1\).

  • \(B_{j,n}\) has exactly one maximum on the interval \(\left[ 0, 1 \right]\), at \(\frac{j}{n}\).

  • Symmetry: \(B_{j,n}\) is symmetric with respect to \(x = \frac{1}{2}\) for \(j=0,\ldots,n\).

  • Bernstein polynomials can be defined recursively using the formula

\[ \begin{align} B_{j,n}(x) = (1-x) B_{j,n-1}(x) + x B_{k-1,n-1}(x), \end{align} \]

where we assume \(B_{j,n}(x) = 0\) if \(j < 0\) or \(j > n\).

  • Bernstein derivatives can be computed using the formulae

\[ \begin{align} {B_{j,n}}^\prime(x) = n \left(B_{j-1,n-1}(x) - B_{j,n-1}(x) \right). \end{align} \]

using the same assumption as before.

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