Degree-Raising Formulas for Bernstein Polynomials

[zodiacbrave]

Two part question...

Homework Statement

Question 1:
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.)

Question 2:
Multiply the matrix in the previous part (on the left) by the conversion matrix from the standard basis of
P3 to the Bernstein basis of P3. The resulting matrix should convert from degree 2 to degree 3 Bernstein
representation (degree-raising). Test your matrix on the Bernstein polynomials of degree 2. Check by
expanding everything back to the standard basis.

Homework Equations

the Bernstein basis of P2 is { (t-1)^2, -2t(t-1), t^2}
the Bernstein basis of P3 is { -(t-1)^3, 3t(t-1)^2, -3t^2(t-1), t^3}

The Attempt at a Solution

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]


For question 2, I am to multiple the the matrix above with the matrix that goes from P3 (standard) to P3 (BB)

The matrix that goes from BB->S in p3 is [1,0,0,0; -3,3,0,0 ; 3,-6,3,0; -1,3,-3,1]^-1

so I multiple both matrices together and that should be the answer to question 2, right?


Thank you

 

[mighty2000]

for your question! I would approach this problem by first understanding the concepts involved and then solving the problem step by step.

For question 1, you are correct in your approach. To transform v to the standard basis of P3, you first need to transform it to the standard basis of P2 by using the change of basis matrix given. Then, you can simply add an extra row of zeros to get a 4x3 matrix that will transform v to the standard basis of P3.

For question 2, you are on the right track. However, when multiplying the two matrices, remember that the order matters. The matrix that goes from P3 (standard) to P3 (BB) should be multiplied on the left of the matrix you found in question 1. So, the resulting matrix would be [1,0,0,0; -3,3,0,0 ; 3,-6,3,0; -1,3,-3,1]^-1 [1,0,0 ; -2,2,0 ; 0, 0, 1 ; 0, 0 ,0]. This will give you the matrix that converts from degree 2 to degree 3 Bernstein representation.

To test this matrix, you can plug in the Bernstein polynomials of degree 2 and see if they are correctly transformed to the Bernstein polynomials of degree 3. You can also expand everything back to the standard basis to verify the results.

I hope this helps and good luck with your problem!