The UpperBoundOfRemainderTerm command returns the value(s) of upper bound of the remainder term of the approximated polynomial at the specified point(s) pts or at the extrapolated point(s) from the POLYINTERP structure, depending on whether pts is specified or not.

The pts must be within the range of the approximating polynomial.

The upper bounds are returned in a list of the form: [[${\mathrm{point}}_i$, ${\mathrm{upperbound}}_i$, [...], ...], $i$=$1..\mathrm{number}$ $\mathrm{of}$ $\mathrm{points}$.

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

In order for the upper bound to be computed, the POLYINTERP structure p must have an associated function, given by the PolynomialInterpolation command.

If the POLYINTERP structure was created with the CubicSpline command, the boundary conditions must be clamped.

A remainder term is sometimes called an error term.

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

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

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

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

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

$\left[\left[1.25\,0.2717886368\right]\right]$

$\mathrm{UpperBoundOfRemainderTerm}\left(\mathrm{p1}\,1.7\right)$

$\left[\left[1.7\,0.2519320807\right]\right]$