Alternates of polyval and polyfit?

I'm working on an assignment that has to do with interpolating polynomials where I create a picture reading in the data points from a file. However, I am not allowed to use the functions polyfit or polyval. How would I go about finding the coefficients of a polynomial and generating a curve based on those coefficients?

Answer 1

[Walter Roberson]

https://en.wikipedia.org/wiki/Vandermonde_matrix

Suppose you had a polynomial of degree 2, y = a2*x^2 + a1*x^1 + a0*x^0, a2, a1, a0 unknown then you can write that as

[x^2, x^1, x^0] * [a2; a1; a0] = y

Now fill in those values as columns for all known x, y

A = [x1^2, x1, 1
     x2^2, x2, 1
     x3^2, x3, 1
     ...]
b = [y1;
     y2
     y3
     ...]
     

and then it follows that

  A * [a2; a1; a0] = b

which is a (possibly overdetermined) system of simultaneous linear equations you can solve by your favorite method