The PolynomialInterpolation routine returns the polynomial of degree less than or equal to $n-1$ in variable v that interpolates the points $\{\left(\mathrm{x1},\mathrm{y1}\right),\left(\mathrm{x2},\mathrm{y2}\right),...,\left(\mathrm{xn},\mathrm{yn}\right)\}$.  If v is a numerical value, the value of the polynomial at this point is returned.

You can call the PolynomialInterpolation routine in two ways.

The first, PolynomialInterpolation(xydata, v, opts), accepts a list, Array, or Matrix, $\[\[\mathrm{x1},\mathrm{y1}\],\[\mathrm{x2},\mathrm{y2}\],...,\[\mathrm{xn},\mathrm{yn}\]\]$, of data points.

The second, PolynomialInterpolation(xdata, ydata, v, opts), accepts two lists, two Arrays, or two Vectors. In this form, the first set of data contains the independent values, $\[\mathrm{x1},\mathrm{x2},...,\mathrm{xn}\]$, and the second set contains the dependent values, $\[\mathrm{y1},\mathrm{y2},...,\mathrm{yn}\]$.  Each element must be of type algebraic and all of the independent values must be distinct.

$\mathrm{with}\left(\mathrm{CurveFitting}\right)\:$

$\mathrm{PolynomialInterpolation}\left(\left[\left[0\,0\right]\,\left[1\,3\right]\,\left[2\,1\right]\,\left[3\,3\right]\right]\,z\right)$

$\frac{3}{2}{z}^3-7z^2+\frac{17}{2}z$

$\mathrm{PolynomialInterpolation}\left(\left[0\,2\,4\,7\right]\,\left[2\,a\,1\,3\right]\,3\right)$

$\frac{19}{70}+\frac{3a}{5}$

$\mathrm{PolynomialInterpolation}\left(\left[0\,2\,4\,7\right]\,\left[2\,a\,1\,3\right]\,z\,\mathrm{form}=\mathrm{Lagrange}\right)$

$-\frac{\left(z-2\right)\left(z-4\right)\left(z-7\right)}{28}+\frac{az\left(z-4\right)\left(z-7\right)}{20}-\frac{z\left(z-2\right)\left(z-7\right)}{24}+\frac{z\left(z-2\right)\left(z-4\right)}{35}$