Numerical Analysis/Vandermonde exercise

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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:

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

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