$$ u_0, u_1, u_2 = 0, 1, 5$$
$$ p_0, p_1, p_2 = 1, 2, -6 $$
How can we derive a collocation matrix when the Newton's Polynomials are like this:
$$ P_i(u) = \prod^{i-1}_{j=0}(u-u_{i}) $$ $$ P_0(u) = 1 $$
The answer is supposed to be this:
$$ \begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 5 & 20 \\ \end{bmatrix} $$
But I don't understand how. I understand it for the first column because $$P_0(u) = 1$$ but I don't get which exact values of $u$ are being used for for the other columns $P_1$ and $P_2$. I am probably interpreting the product expression wrongly.
Each column corresponds to a function $P_0, P_1$, and $P_2$ and the rows corresponds to the points $u_0, u_1, u_2$.
$$P_1(u)=u-u_0=u-0=u$$
Hence the second column just takes the value of $u_0, u_1, u_2$.
$$P_2(u)=(u-u_0)(u-u_1)$$
Hence the first two entries of it is $0$ and the last entry is
$$(u_2-u_1)(u_2-u_0)=5(4)=20$$
$p_i$ doesn't play a role in the current computation.