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:

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