The linear system $V p = f$ can be solved to find coefficients of the polynomial $P(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 $ that interpolates the points $(-1,f_1), (2,f_2), (1,f_3), (0,f_4)$, where $V$ is the Vandermonde matrix, $p$ contains the coefficients, and $f$ contains the function outputs. See below:
$$\begin{bmatrix} 1&-1&1&-1\\ 1&2&4&8\\ 1&1&1&1\\ 1&0&0&0 \end{bmatrix} \begin{bmatrix} a_0\\ a_1\\ a_2\\ a_3\\ \end{bmatrix} = \begin{bmatrix} f_1\\ f_2\\ f_3\\ f_4\\ \end{bmatrix}$$
One can use $LU$-decomposition of the Vandermonde matrix to solve this system for any given coefficients of $a_i$ for $i = 1,\dots, 4$ whereby: $$ L = \begin{bmatrix} 1&0&0&0\\ 1&1&0&0\\ 1&2/3&1&0\\ 1&1/3&1&1 \end{bmatrix}, \, \, \, \, U = \begin{bmatrix} 1&-1&1&-1\\ 0&3&3&9\\ 0&0&-2&-4\\ 0&0&0&2 \end{bmatrix} $$
My question is: how could one calculate the Lagrange polynomials to interpolate this polynomial $\textbf{without}$ explicitly using the formula for the Lagrange Polynomials? I am told this is possible using the results from the $LU$-decomposition, however I am not sure where the link between the two methods are? Might anyone have any suggestions how this would be possible?
The Lagrange polynomials, $L_i(x)$ are in this case polynomials of degree $\leq 3$ satisfying $L_i(x_j)=\delta_{ij}$. You can use the $LU$ decomposition to solve the systems $V p = e_i$, yielding the coefficients of $L_i(x)$.