The ApproximateValue command computes the value(s) of the approximated polynomial at specified point(s) pts or at the extrapolated point(s) from the POLYINTERP structure, depending on whether pts is specified or not.

The approximate values are returned in a list of the form [[${\mathrm{point}}_i$, ${\mathrm{approx}}_i$], [...], ...], $i$=$1..\mathrm{number}$ $\mathrm{of}$ $\mathrm{points}$.

The POLYINTERP structure is created using the PolynomialInterpolation command or the CubicSpline command.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{NumericalAnalysis}\right]\right)\:$

$\mathrm{xy}≔\left[\left[1.0\,0.7651977\right]\,\left[1.3\,0.6200860\right]\,\left[1.6\,0.4554022\right]\,\left[1.9\,0.2818186\right]\right]$

$\mathrm{xy}≔\left[\left[1.0\,0.7651977\right]\,\left[1.3\,0.6200860\right]\,\left[1.6\,0.4554022\right]\,\left[1.9\,0.2818186\right]\right]$

$\mathrm{p1}≔\mathrm{PolynomialInterpolation}\left(\mathrm{xy}\,\mathrm{method}=\mathrm{neville}\,\mathrm{extrapolate}=\left[1.5\right]\right)\:$

$\mathrm{ApproximateValue}\left(\mathrm{p1}\right)$

$\left[\left[1.5\,0.5118126939\right]\right]$

$\mathrm{ApproximateValue}\left(\mathrm{p1}\,\left[1.7\,1.8\right]\right)$

$\left[\left[1.7\,0.3980028398\right]\,\left[1.8\,0.3400098828\right]\right]$

$\mathrm{xyy}≔\left[\left[0\,1\right]\,\left[\frac{1}{2}\,1\right]\,\left[1\,\frac{11}{10}\right]\,\left[\frac{3}{2}\,\frac{3}{4}\right]\,\left[2\,\frac{7}{8}\right]\,\left[\frac{5}{2}\,\frac{9}{10}\right]\,\left[3\,\frac{11}{10}\right]\,\left[\frac{7}{2}\,1\right]\right]$

$\mathrm{xyy}≔\left[\left[0\,1\right]\,\left[\frac{1}{2}\,1\right]\,\left[1\,\frac{11}{10}\right]\,\left[\frac{3}{2}\,\frac{3}{4}\right]\,\left[2\,\frac{7}{8}\right]\,\left[\frac{5}{2}\,\frac{9}{10}\right]\,\left[3\,\frac{11}{10}\right]\,\left[\frac{7}{2}\,1\right]\right]$

$\mathrm{p2}≔\mathrm{CubicSpline}\left(\mathrm{xyy}\,\mathrm{independentvar}='x'\right)\:$

$\mathrm{ApproximateValue}\left(\mathrm{p2}\,1.3\right)$

$\left[\left[1.3\,0.8776545517\right]\right]$