Graphing Polynomials in MATLAB

Question

I need to create a polynomial of the form:

P(x) = q(1,1) + q(2,2)(x-z(1)) + q(3,3)(x-z(1))(x-z(2)) + --- + q(2n, 2n)(x-z(1))(x-z(2))...(x-z(2n)) NOTE: The indices of the equation have been shifted to accomodate MATLAB.

in MATLAB. Consult this link here specifically slides 15 and 16.

I have the matrix Q filled, so I have the diagonal, and I also have z(1:2n) filled.

I'm having a hard time figuring out a way to create a polynomial that I can graph this polynomial. I've tried to use a for loop to append each term to P(x), but it doesn't operate the way I thought it would.

So far, my code will calculate the coefficients (presented as Q(0,0) -> Q(2n+1, 2n+1) in the problem above) without a problem.

I'm having an issue with the construction of a degree n polynomial of the form described above. Plotting makes more sense now, create a vector x with evaluative values, and then run them through the polynomial "function" and plot the x vector against the resulting vector.

So I just need to create this polynomial.

Answer 1

I would use diag and cumprod to help you accomplish this. First use diag to extract the diagonals of your matrix Q. After, use cumprod to generate a vector of cumulative products.

How cumprod works on a vector is that for each element in the vector, the i'th element collects products from 1 up to the i'th element. As an example, if we had a vector V = [1 2 3 4 5], cumprod(V) would produce [1 2 6 24 120]. The 4th element (as an example) would be 1*2*3*4, representing the products from the 1st to the 4th element.

As such, this is the code that I would do:

qdiag = diag(Q);
xMinusZ = x - z; % Takes z and does x - z for every element in z
cumProdRes = cumprod(xMinusZ);
P = sum(qdiag .* [1;cumProdRes(1:end-1)]);

P should give you P(x) that you desired. Make sure that z is a column vector to make it compatible with the diagonals extracted from Q.

NB: I believe there is a typo in your equation. The last term of your equation (going with your convention) should have (x-z(2n-1)) and not (x-z(2n)). This is because the first term in your equation does not have z.

Here's an example. Let's suppose Q is defined

Q = [1 2 3 4; 5 6 7 8; 9 10 11 12; 13 14 15 16];

The vector z is:

z = [4;3;2;1];

Let's evaluate the function at x = 2

Extracting the diagonals of Q should give us Q = [1;6;11;16]. Subtract x from every element of z should give us:

xMinusZ = [-2;-1;0;1];

Using the equation that you have above, we have:

P = 1 + 6*(-2) + 11*(-2)*(-1) + 16*(-2)*(-1)*(0) = 11

This is what the code should give.

What if we want to do this for more than one value of x?

As you have stated in your post, you want to evaluate this for a series of x values. As such, you need to modify the code so that it looks like this (make sure that x is a column vector):

qdiag = diag(Q);
xMinusZ = repmat(x,1,length(z)) - repmat(z',length(z),1);
cumProdRes = cumprod(xMinusZ,2);
P = sum(repmat(qdiag',length(z),1).*[ones(length(z),1) cumProdRes(:,1:end-1)],2);

P should now give you a vector of outputs, and so if you want to plot this, simply do plot(x,P);