What are the explicit expression for this interpolation problem

Question

We want to fit

$f(x) = a_0 + a_1 *x + a_2 * x^2 + ... + a_n * x^n$

to the data $(x_i,f(x_i))$ for $i = 0 ... n.$

It will give rise to the following system

$ A a = b $

Here $ a = [ a_1 a_2 a_3 ... a_n]$

$ A = [ 1 x_0 x_0^2 ... x_0^n ; 1 x_1 x_1^2 ... x_1^n ; 1 x_2 x_2^2 ... x_2^n ; ... ; 1 x_n x_n^2 ... x_n^n]$

and

$ b = [f(x_0) f(x_1) f(x_2) ... f(x_n)].$

What are explict expression for $a_i$?

Answer 1

You are essentially asking for an explicit formula for the inverse of a Vandermonde matrix. See the formula on ProofWiki, for instance.