errorboundvar = name

The name to assign to the independent variable in the remainder term.

The RemainderTerm command returns the remainder term from the POLYINTERP structure p.

The POLYINTERP structure is created using the PolynomialInterpolation command.

In order for the remainder term to exist, the POLYINTERP structure p must have an associated exact function, given through the PolynomialInterpolation command.

POLYINTERP structures that were created with the CubicSpline command cannot be used with the RemainderTerm command, since they do not have a remainder term.

A remainder term is also called an error term.

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

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

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

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

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

$\left(\frac{\left(-2^{2-\mathrm{\xi }}{\ln \left(2\right)}^7\cos \left(\mathrm{\pi }\mathrm{\xi }\right)-72^{2-\mathrm{\xi }}{\ln \left(2\right)}^6\mathrm{\pi }\sin \left(\mathrm{\pi }\mathrm{\xi }\right)+212^{2-\mathrm{\xi }}{\ln \left(2\right)}^5{\mathrm{\pi }}^2\cos \left(\mathrm{\pi }\mathrm{\xi }\right)+352^{2-\mathrm{\xi }}{\ln \left(2\right)}^4{\mathrm{\pi }}^3\sin \left(\mathrm{\pi }\mathrm{\xi }\right)-352^{2-\mathrm{\xi }}{\ln \left(2\right)}^3{\mathrm{\pi }}^4\cos \left(\mathrm{\pi }\mathrm{\xi }\right)-212^{2-\mathrm{\xi }}{\ln \left(2\right)}^2{\mathrm{\pi }}^5\sin \left(\mathrm{\pi }\mathrm{\xi }\right)+72^{2-\mathrm{\xi }}\ln \left(2\right){\mathrm{\pi }}^6\cos \left(\mathrm{\pi }\mathrm{\xi }\right)+2^{2-\mathrm{\xi }}{\mathrm{\pi }}^7\sin \left(\mathrm{\pi }\mathrm{\xi }\right)\right)x\left(x-0.5\right)\left(x-1.0\right)\left(x-1.5\right)\left(x-2.0\right)\left(x-2.5\right)\left(x-3.0\right)}{5040}\right)\&where\left\{0.\le \mathrm{\xi }\le 3.0\right\}$

$\mathrm{xyyp}≔\left[\left[1\,1.105170918\,0.2210341836\right]\,\left[1.5\,1.252322716\,0.3756968148\right]\,\left[2\,1.491824698\,0.5967298792\right]\right]$

$\mathrm{xyyp}≔\left[\left[1\,1.105170918\,0.2210341836\right]\,\left[1.5\,1.252322716\,0.3756968148\right]\,\left[2\,1.491824698\,0.5967298792\right]\right]$

$\mathrm{p2}≔\mathrm{PolynomialInterpolation}\left(\mathrm{xyyp}\,\mathrm{method}=\mathrm{hermite}\,\mathrm{function}=\exp \left(0.1x^2\right)\,\mathrm{independentvar}='x'\,\mathrm{errorboundvar}='\mathrm{\xi }'\,\mathrm{digits}=5\right)\:$

$\mathrm{RemainderTerm}\left(\mathrm{p2}\right)$

$\left(\frac{\left(0.120{\ⅇ}^{0.1{\mathrm{\xi }}^2}+0.0720{\mathrm{\xi }}^2{\ⅇ}^{0.1{\mathrm{\xi }}^2}+0.00480{\mathrm{\xi }}^4{\ⅇ}^{0.1{\mathrm{\xi }}^2}+0.000064{\mathrm{\xi }}^6{\ⅇ}^{0.1{\mathrm{\xi }}^2}\right){\left(x-1.\right)}^2{\left(x-1.5\right)}^2{\left(x-2.\right)}^2}{720}\right)\&where\left\{1.\le \mathrm{\xi }\le 2.\right\}$