For n = 2 ... 11 set up Vandermonde matrix A of size m × n and for each n given data set a. Matlab

Question

I am given set of 50 data points with values {a^(i),b^(i)} for i=1,...,50 stored in the arrays a and b. I know that the Vandermonde matrix A has size m x n, where n = 2 ... 11 and m is the size of the array a.

I want to to fit the data with a polynomial of degree (n − 1), for n = 2,...,11. To do that for each n I have to set up the Vandermonde matrix A of size m × n.

The Vandermonde matrix A solves the following equation: A^T*A*x = A^T*b

Where the A^T is the transpose matrix and I have b already given. Also we know that Aij = (a^(i))^(j−1) for j = 1,...,n, What confuses me is how to set the matrix for n = 2,..,11.

What my line of thought is: I have m = length(a); this will set up m = 50;

n = 11;

Then A=ones(m,n); This creates a matrix A filled with ones that has the correct size.

However I am not sure how to populate the matrix. I wrote the following for loop which I thought will populate the matrix:

for n = 2:11
    j=n;
    for i = 1:50
        A(i,n) = (a^(i))^(j-1);
    end
end

Could you help me please with setting up the matrix?

Answer 1

Try to avoid for-loops if you can in matlab. A hint is you can use

meshgrid(1:k,1:n)

for example

[rows,cols] = meshgrid(1:3,1:4)

returns $$rows = \left[\begin{array}{ccc}1&2&3\\1&2&3\\1&2&3\\1&2&3\end{array}\right], cols= \left[\begin{array}{ccc}1&1&1\\2&2&2\\3&3&3\\4&4&4\end{array}\right]$$

Then you can use point operators on matrices.

a.*b

calculates elementwise multiplication

a.^b

calculates elementwise power