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Student[NumericalAnalysis]

 RemainderTerm
 return the remainder term from an interpolation structure

 Calling Sequence RemainderTerm(p, opts)

Parameters

 p - a POLYINTERP structure opts - (optional) equation(s) of the form keyword=value, where keyword is: errorboundvar; options for returning the remainder term

Options

 • errorboundvar = name
 The name to assign to the independent variable in the remainder term.

Description

 • 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.

Notes

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{NumericalAnalysis}]\right):$
 > $\mathrm{xy}≔\left[\left[0,4.0\right],\left[0.5,0\right],\left[1.0,-2.0\right],\left[1.5,0\right],\left[2.0,1.0\right],\left[2.5,0\right],\left[3.0,-0.5\right]\right]$
 ${\mathrm{xy}}{:=}\left[\left[{0}{,}{4.0}\right]{,}\left[{0.5}{,}{0}\right]{,}\left[{1.0}{,}{-}{2.0}\right]{,}\left[{1.5}{,}{0}\right]{,}\left[{2.0}{,}{1.0}\right]{,}\left[{2.5}{,}{0}\right]{,}\left[{3.0}{,}{-}{0.5}\right]\right]$ (1)
 > $\mathrm{p1}≔\mathrm{PolynomialInterpolation}\left(\mathrm{xy},\mathrm{function}={2}^{2-x}\mathrm{cos}\left(\mathrm{π}x\right),\mathrm{method}=\mathrm{lagrange},\mathrm{extrapolate}=\left[0.25,0.75,1.25\right],\mathrm{errorboundvar}='\mathrm{ξ}'\right):$
 > $\mathrm{RemainderTerm}\left(\mathrm{p1}\right)$
 $\left(\frac{{1}}{{5040}}{}\left({-}{{2}}^{{2}{-}{\mathrm{ξ}}}{}{{\mathrm{ln}}{}\left({2}\right)}^{{7}}{}{\mathrm{cos}}{}\left({\mathrm{π}}{}{\mathrm{ξ}}\right){-}{7}{}{{2}}^{{2}{-}{\mathrm{ξ}}}{}{{\mathrm{ln}}{}\left({2}\right)}^{{6}}{}{\mathrm{π}}{}{\mathrm{sin}}{}\left({\mathrm{π}}{}{\mathrm{ξ}}\right){+}{21}{}{{2}}^{{2}{-}{\mathrm{ξ}}}{}{{\mathrm{ln}}{}\left({2}\right)}^{{5}}{}{{\mathrm{π}}}^{{2}}{}{\mathrm{cos}}{}\left({\mathrm{π}}{}{\mathrm{ξ}}\right){+}{35}{}{{2}}^{{2}{-}{\mathrm{ξ}}}{}{{\mathrm{ln}}{}\left({2}\right)}^{{4}}{}{{\mathrm{π}}}^{{3}}{}{\mathrm{sin}}{}\left({\mathrm{π}}{}{\mathrm{ξ}}\right){-}{35}{}{{2}}^{{2}{-}{\mathrm{ξ}}}{}{{\mathrm{ln}}{}\left({2}\right)}^{{3}}{}{{\mathrm{π}}}^{{4}}{}{\mathrm{cos}}{}\left({\mathrm{π}}{}{\mathrm{ξ}}\right){-}{21}{}{{2}}^{{2}{-}{\mathrm{ξ}}}{}{{\mathrm{ln}}{}\left({2}\right)}^{{2}}{}{{\mathrm{π}}}^{{5}}{}{\mathrm{sin}}{}\left({\mathrm{π}}{}{\mathrm{ξ}}\right){+}{7}{}{{2}}^{{2}{-}{\mathrm{ξ}}}{}{\mathrm{ln}}{}\left({2}\right){}{{\mathrm{π}}}^{{6}}{}{\mathrm{cos}}{}\left({\mathrm{π}}{}{\mathrm{ξ}}\right){+}{{2}}^{{2}{-}{\mathrm{ξ}}}{}{{\mathrm{π}}}^{{7}}{}{\mathrm{sin}}{}\left({\mathrm{π}}{}{\mathrm{ξ}}\right)\right){}{x}{}\left({x}{-}{0.5}\right){}\left({x}{-}{1.0}\right){}\left({x}{-}{1.5}\right){}\left({x}{-}{2.0}\right){}\left({x}{-}{2.5}\right){}\left({x}{-}{3.0}\right)\right){&where}\left\{{0.}{\le }{\mathrm{ξ}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathrm{ξ}}{\le }{3.0}\right\}$ (2)
 > $\mathrm{xyyp}≔\left[\left[1,1.105170918,0.2210341836\right],\left[1.5,1.252322716,0.3756968148\right],\left[2,1.491824698,0.5967298792\right]\right]$
 ${\mathrm{xyyp}}{:=}\left[\left[{1}{,}{1.105170918}{,}{0.2210341836}\right]{,}\left[{1.5}{,}{1.252322716}{,}{0.3756968148}\right]{,}\left[{2}{,}{1.491824698}{,}{0.5967298792}\right]\right]$ (3)
 > $\mathrm{p2}≔\mathrm{PolynomialInterpolation}\left(\mathrm{xyyp},\mathrm{method}=\mathrm{hermite},\mathrm{function}={ⅇ}^{0.1{x}^{2}},\mathrm{independentvar}='x',\mathrm{errorboundvar}='\mathrm{ξ}',\mathrm{digits}=5\right):$
 > $\mathrm{RemainderTerm}\left(\mathrm{p2}\right)$
 $\left(\frac{{1}}{{720}}{}\left({0.120}{}{{ⅇ}}^{{0.1}{}{{\mathrm{ξ}}}^{{2}}}{+}{0.0720}{}{{\mathrm{ξ}}}^{{2}}{}{{ⅇ}}^{{0.1}{}{{\mathrm{ξ}}}^{{2}}}{+}{0.00480}{}{{\mathrm{ξ}}}^{{4}}{}{{ⅇ}}^{{0.1}{}{{\mathrm{ξ}}}^{{2}}}{+}{0.000064}{}{{\mathrm{ξ}}}^{{6}}{}{{ⅇ}}^{{0.1}{}{{\mathrm{ξ}}}^{{2}}}\right){}{\left({x}{-}{1.}\right)}^{{2}}{}{\left({x}{-}{1.5}\right)}^{{2}}{}{\left({x}{-}{2.}\right)}^{{2}}\right){&where}\left\{{1.}{\le }{\mathrm{ξ}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathrm{ξ}}{\le }{2.}\right\}$ (4)