Bernstein polynomial

In computational mathematics, Bernstein polynomials are algebraic polynomials , which are a linear combination of Bernstein basis polynomials . ^[1] ^[2]

A stable algorithm for calculating polynomials in the form of Bernstein is de Castelgio's algorithm .

Bernstein polynomials were described by Sergei Nathanovich Bernstein in 1912 and used by him in constructive proof of Weierstrass's approximation theorem . With the development of computer graphics , the Bernstein polynomials on the interval x ∈ [0, 1] began to play an important role in the construction of Bezier curves .

Content

Definition

( n + 1) Bernstein basis polynomials of degree n are given by

b_{k,n}(x)={\binom {n}{k}}x^{k}(1-x)^{n-k},\qquad k=0,\ldots ,n.

{\ displaystyle b_ {k, n} (x) = {\ binom {n} {k}} x ^ {k} (1-x) ^ {nk}, \ qquad k = 0, \ ldots, n.}

Where ${\binom {n}{k}}$ ${\ displaystyle {\ binom {n} {k}}}$ - Binomial coefficient .

Bernstein basis polynomials of degree n form a basis for linear space $\Pi _{n}$ ${\ displaystyle \ Pi _ {n}}$ polynomials of degree n .

Linear combination of Bernstein basis polynomials

B_{n}(f;x)=B_{n}(x)=\sum _{k=0}^{n}f\left({\frac {k}{n}}\right)b_{k,n}(x)

{\ displaystyle B_ {n} (f; x) = B_ {n} (x) = \ sum _ {k = 0} ^ {n} f \ left ({\ frac {k} {n}} \ right) b_ {k, n} (x)}

is called a Bernstein polynomial or, more precisely , a Bernstein-shaped polynomial of degree n . Coefficients $f\left({\frac {k}{n}}\right)$ ${\ displaystyle f \ left ({\ frac {k} {n}} \ right)}$ are called Bernstein coefficients or Bezier coefficients .

Examples

Here are some basic Bernstein polynomials:

b_{0,0}(x)=1

{\ displaystyle b_ {0,0} (x) = 1}

b_{0,1}(x)=1-x

{\ displaystyle b_ {0,1} (x) = 1-x}

b_{1,1}(x)=x

{\ displaystyle b_ {1,1} (x) = x}

b_{0,2}(x)=(1-x)^{2}

{\ displaystyle b_ {0,2} (x) = (1-x) ^ {2}}

b_{1,2}(x)=2x(1-x)

{\ displaystyle b_ {1,2} (x) = 2x (1-x)}

b_{2,2}(x)=x^{2}\ .

{\ displaystyle b_ {2,2} (x) = x ^ {2} \.}

Properties

Differentiation

$b'_{k,n}(x)=n\,b_{k,n-1}(x)+n\,b_{k-1,n-1}(x)$ ${\ displaystyle b '_ {k, n} (x) = n \, b_ {k, n-1} (x) + n \, b_ {k-1, n-1} (x)}$

$b_{k,n}^{(l)}(x)={\frac {n!}{(n-l)!}}\sum _{j=0}^{l}{\binom {l}{j}}b_{k-j,n-l}(x)$ ${\ displaystyle b_ {k, n} ^ {(l)} (x) = {\ frac {n!} {(nl)!}} \ sum _ {j = 0} ^ {l} {\ binom {l } {j}} b_ {kj, nl} (x)}$

Lemmas about moments

$\sum _{k=0}^{n}b_{k,n}(x)=1$ ${\ displaystyle \ sum _ {k = 0} ^ {n} b_ {k, n} (x) = 1}$ for any n and x , since $\sum _{k=0}^{n}b_{k,n}(x)=(x+1-x)^{n}=1^{n}$ ${\ displaystyle \ sum _ {k = 0} ^ {n} b_ {k, n} (x) = (x + 1-x) ^ {n} = 1 ^ {n}}$

$\sum _{k=0}^{n}b_{k,n}(x)(x-k/n)=0$ ${\ displaystyle \ sum _ {k = 0} ^ {n} b_ {k, n} (x) (xk / n) = 0}$ for any n and x

$\sum _{k=0}^{n}b_{k,n}(x)(x-k/n)^{2}=x(1-x)/n$ ${\ displaystyle \ sum _ {k = 0} ^ {n} b_ {k, n} (x) (xk / n) ^ {2} = x (1-x) / n}$ for any n and x

Approximation of continuous functions

Notes

↑ Bernstein, S.N. Collected Works. - M. , 1952. - T. 1. - p. 105-106.
↑ Bernstein, S.N. Collected Works. - M. , 1954. - T. 3. - S. 310-348.

[1] Bernstein, S.N. Collected Works. - M. , 1952. - T. 1. - p. 105-106.

[2] Bernstein, S.N. Collected Works. - M. , 1954. - T. 3. - S. 310-348.