The ApproximateExactUpperBound command computes three values: the approximating polynomial, the exact value of the function and the upper bound of the remainder term, for each point specified in pts.

If pts is not specified, the extrapolated points in the given POLYINTERP structure are used instead.

The computed values are returned in the form of a list: [pts, approximate values, exact values, upper bounds].

The POLYINTERP structure is created using the PolynomialInterpolation command.

POLYINTERP structures whose interpolation method is the cubic spline method may not be used with this command.

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

$\mathrm{xy}≔\left[\left[0\,4.0\right]\,\left[0.5\,0\right]\,\left[1.0\,-2.0\right]\,\left[1.5\,0\right]\,\left[2.0\,1.0\right]\,\left[2.5\,0\right]\,\left[3.0\,-0.5\right]\right]$

$\mathrm{xy}≔\left[\left[0\,4.0\right]\,\left[0.5\,0\right]\,\left[1.0\,\mathrm{-2.0}\right]\,\left[1.5\,0\right]\,\left[2.0\,1.0\right]\,\left[2.5\,0\right]\,\left[3.0\,\mathrm{-0.5}\right]\right]$

$\mathrm{p1}≔\mathrm{PolynomialInterpolation}\left(\mathrm{xy}\,\mathrm{function}=2^{2-x}\cos \left(\mathrm{\pi }x\right)\,\mathrm{method}=\mathrm{lagrange}\,\mathrm{extrapolate}=\left[0.25\,0.75\,1.25\right]\,\mathrm{errorboundvar}=\mathrm{\xi }\right)\:$

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

$\left[\left[0.25\,2.685058594\,2.378414230\,1.793805002\right]\,\left[0.75\,\mathrm{-1.746582031}\,\mathrm{-1.681792830}\,0.4892195460\right]\,\left[1.25\,\mathrm{-1.166503906}\,\mathrm{-1.189207114}\,0.2717886368\right]\right]$

$\mathrm{ApproximateExactUpperBound}\left(\mathrm{p1}\,\left[0.5\,1.5\right]\right)$

$\left[\left[0.5\,0.\,\mathrm{-5.801199657}\times {10}^{\mathrm{-10}}\,0.\right]\,\left[1.5\,0.\,8.701799482\times {10}^{\mathrm{-10}}\,0.\right]\right]$