Numerical Analysis/Vandermonde exercise

Using a Vandermonde matrix, find the interpolating polynomial that passes through the points $(1,9)$, $(2,9)$, $(3,6)$, $(7,3)$. Give both the polynomial, and the augmented matrix you used.

Solution:

The augmented matrix depends on your choice of index for the given points. One possibility is:

$\left(\begin{array}{ccccc}1& 1& 1& 1& 9\\ 8& 4& 2& 1& 9\\ 27& 9& 3& 1& 6\\ 343& 49& 7& 1& 3\end{array}\right)$

The polynomial is:

$p(x)=\frac{13}{40}{x}^3-\frac{69}{20}{x}^2+\frac{323}{40}x+\frac{81}{20}$.