In the mathematical field of numerical analysis, a Bernstein polynomial is a polynomial that is a linear combination of Bernstein basis polynomials.

A numerically stable way to evaluate polynomials in Bernstein form is de Casteljau's algorithm.

## Definition

The $n+1$​Bernstein basis polynomials of degree $n$​ are defined as
$$b_{v,n}(x)=\binom{n}{v}x^v(1-x)^{n-v},\quad v=0,\cdots,n,$$
where $\binom{n}{v}$​​ is a binomial coefficient. So, for example, $b_{2,5}(x)=\binom{5}{2}x^2(1-x)^3=10x^2(1-x)^3$.

The first few Bernstein basis polynomial for blending 1, 2, 3 or 4 value together are:
\begin{aligned}b_{0,0}(x) =1, \\ b_{0,1}(x) =1-x,\quad b_{1,1}(x)=x,\\ b_{0,2}(x) =(1-x)^2,\quad b_{1,2}=2x(1-x),\quad b_{2,2}(x)=x^2\\ b_{0,3}(x) =(1-x)^3,\quad b_{1,3}=3x(1-x),\quad b_{2,3}(x)=3x^2(1-x),\quad b_{3,3}=x^3,\end{aligned}
The Bernstein polynomial of degree $n$ from a basis for the vector space $\Pi_n$ of polynomials of degree at most $n$ with real coefficients. A linear combination of Bernstein basis polynomials
$$B_n=\sum_{v=0}^{n}\beta_vb_{v,n}(x)$$
is called a Bernstein polynomial or polynomial in Bernstein form of degree n. The coefficients $\beta_v$​​ are called Bernstein coefficients or Bezier coefficients. The first few Bernstein basis polynomials from above in monomial form are:
\begin{aligned}b_{0,0}(x) =1, \\ b_{0,1}(x) =1-1x,\quad b_{1,1}(x)=0+1x,\\ b_{0,2}(x) =1-2x+1x^2,\quad b_{1,2}=0+2x-2x^2,\quad b_{2,2}(x)=0+0x+1x^2\\ b_{0,3}(x) =1-3x+3x^2-x^3,\quad b_{1,3}=0+3x-6x^2+3x^3,\quad b_{2,3}(x)=0+0x+3x^2-3x^3,\quad b_{3,3}=0+0x+0x^2+x^3,\end{aligned}

## Properties

The Bernstein basis polynomials have the following properties:

• $b_{v,n}=0$, if $v<0$ or $v>n$.

• $b_{v,n}(x)\geq 0$ for $x\in\left[0,1\right]$.

• $b_{v,n}(1-x)=b_{n-v,n}(x)$.

• $b_{v,n}(0)=\delta_{v,0}$ and $b_{v,n}(1)=\delta_{v,n}$​.​

• $b_{v,n}(x)$ has a root with multiplicity $v$ at point $x=0$ (note: if $v=0$​, there is no root at 0).

• $b_{v,n}(x)$ has a root with multiplicity $(n-v)$ at point $x=1$ (note: if $v=n$, there is no root at 1).

• The derivative can be written as a combination of two polynomials of lower degree:
$$b'_{v,n}(x)=n(b_{v-1,n-1}(x)-b_{v,n-1}(x))$$

• The k:th derivative at 0:
$$b_{v,n}^{(k)}(0)=\sum_{v=0}^{n}\frac{n!}{(n-k)!}\binom{k}{v}(-1)^{v+k}.$$

• The k:th derivative at 1:
$$b_{v,n}^{(k)}(0)=(-1)^kb_{n-v,n}^{(k)}(0).$$

• The transformation of the Bernstein polynomial to monomials is
$$b_{v,n}(x)=\binom{n}{v}\sum_{k=0}^{n-v}\binom{n-v}{k}(-1)^{n-v-k}x^{v+k}=\sum_{\mathcal{l}=v}^n\binom{n}{\mathcal{l}}\binom{\mathcal{l}}{v}(-1)^{\mathcal{l}-v}x^\mathcal{l}.$$
and by the inverse binomial transformation, the reverse transformation is
$$x^k=\sum_{i=0}^{n-k}\binom{n-k}{i}\frac{1}{\binom{n}{i}}b_{n-i,n}(x)=\frac{1}{\binom{n}{k}}\sum_{j=k}^{n}\binom{j}{k}b_{j,n}(x).$$

• The indefinite integral is given by
$$\int b_{v,n}(x)dx=\frac{1}{n+1}\sum_{j=v+1}^{n+1}b_{j,n+1}(x).$$

• The definite integral is constant for a given $n$:
$$\int_0^1b_{v,n}(x)dx=\frac{1}{n+1},\quad \forall v=0,1,\cdots,n.$$

• if $n\neq 0$, then $b_{v,n}(x)$​ has a unique local maximum on the interval $[0,1]$ at $x=\frac{v}{n}$. This maximum takes the value
$$v^vn^{-n}(n-v)^{n-v}\binom{n}{v}.$$

• The Bernstein basis polynomials of degree $n$​ form a partition of unity:
$$\sum_{v=0}^nb_{v,n}(x)=\sum_{v=0}^{n}\binom{n}{v}x^v(1-x)^{n-v}=(x+(1-x))^n=1$$

• By taking the first $x$-derivative of $(x+y)^n$, treating $y$ as constant, then substituting the value $y=1-x$, it can be shown that
$$\sum_{v=1}^nvb_{v,n}(x)=n(n-1)x^2.$$

• A Bernstein polynomial can always be written as a linear combination of polynomials of higher degree:
$$b_{v,n-1}(x)=\frac{n-v}{n}b_{v,n}(x)+\frac{v+1}{n}b_{v+1,n}(x).$$

• The expansion of the Chebyshev Polynomials of the First Kind into the Bernstein basis is
$$T_n(u)=(2n-1)!!\sum_{k=0}^n\frac{(-1)^{n-k}}{(2k-1)!!(2n-2k-1)!!}b_{k,n}(u).$$