Degree-Raising Matrix for Bernstein Polynomials

• September 24th 2012, 12:14 PM
zodiacbrave
Degree-Raising Matrix for Bernstein Polynomials
Question :
Let v = (a, b, c)T be a column vector which represents a coordinate vector of a polynomial in P2 with
respect to the Bernstein basis. Find the 4 × 3 matrix which transforms v to the standard basis of P3.
(Hint: First transform v to the standard basis of P2, then transform to the standard basis of P3 by just
adding an extra row of zeros.)

So, the Bernstein basis of P2 is { (t-1)^2, -2t(1-t), t^2}

so the change of basis matrix, from BB to standard is given by [1,0,0 ; -2,2,0 ; 0, 0, 1]

So, the question asks for a 4x3 matrix that will take a P2 Bernstein polynomial and transform it to p3 Bernstein polynomial.

So, the answer should be [1,0,0 ; -2,2,0 ; 0, 0, 1 ; 0, 0 ,0]

is this right?

Thank you
• September 24th 2012, 12:42 PM
Deveno
Re: Degree-Raising Matrix for Bernstein Polynomials
in the the standard basis {1,t,t2} (note: the order matters here), the coordinate form of (t-1)2 is (1,-2,1), and the coordinate form of (-2t)(1-t) = 2t(t-1) = 2t2-2t is (0,-2,2).

EDIT: according to wikipedia, your second basis vector should be 2t(1-t), which would be (0,2,-2).

Bernstein polynomial - Wikipedia, the free encyclopedia

what i get for the matrix is:

$P = \begin{bmatrix} 1&0&0\\-2&2&0\\1&-2&1\\0&0&0 \end{bmatrix}$