Bernstein polynomial

In the mathematical field of numerical analysis, a Bernstein polynomial, named after Sergei Natanovich Bernstein, is a polynomial in the Bernstein form, that is a linear combination of Bernstein basis polynomials. In the mathematical field of numerical analysis, a Bernstein polynomial, named after Sergei Natanovich Bernstein, is a polynomial in the Bernstein form, that is a linear combination of Bernstein basis polynomials. A numerically stable way to evaluate polynomials in Bernstein form is de Casteljau's algorithm. Polynomials in Bernstein form were first used by Bernstein in a constructive proof for the Stone–Weierstrass approximation theorem. With the advent of computer graphics, Bernstein polynomials, restricted to the interval , became important in the form of Bézier curves. The n + 1 Bernstein basis polynomials of degree n are defined as where ( n ν ) {displaystyle { binom {n}{ u }}} is a binomial coefficient. So, for example, b 2 , 5 ( x ) = ( 5 2 ) x 2 ( 1 − x ) 3 = 10 x 2 ( 1 − x ) 3 . {displaystyle b_{2,5}(x)={ binom {5}{2}}x^{2}(1-x)^{3}=10x^{2}(1-x)^{3}.} The Bernstein basis polynomials of degree n form a basis for the vector space Πn of polynomials of degree at most n with real coefficients. A linear combination of Bernstein basis polynomials is called a Bernstein polynomial or polynomial in Bernstein form of degree n. The coefficients β ν {displaystyle eta _{ u }} are called Bernstein coefficients or Bézier coefficients.