RemainderTerm - Maple Help

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

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{NumericalAnalysis}\right]\right):$
 > $\mathrm{xy}≔\left[\left[0,4.0\right],\left[0.5,0\right],\left[1.0,-2.0\right],\left[1.5,0\right],\left[2.0,1.0\right],\left[2.5,0\right],\left[3.0,-0.5\right]\right]$
 ${\mathrm{xy}}{≔}\left[\left[{0}{,}{4.0}\right]{,}\left[{0.5}{,}{0}\right]{,}\left[{1.0}{,}{-2.0}\right]{,}\left[{1.5}{,}{0}\right]{,}\left[{2.0}{,}{1.0}\right]{,}\left[{2.5}{,}{0}\right]{,}\left[{3.0}{,}{-0.5}\right]\right]$ (1)
 > $\mathrm{p1}≔\mathrm{PolynomialInterpolation}\left(\mathrm{xy},\mathrm{function}={2}^{2-x}\mathrm{cos}\left(\mathrm{\pi }x\right),\mathrm{method}=\mathrm{lagrange},\mathrm{extrapolate}=\left[0.25,0.75,1.25\right],\mathrm{errorboundvar}='\mathrm{\xi }'\right):$
 > $\mathrm{RemainderTerm}\left(\mathrm{p1}\right)$
 $\left(\frac{\left({-}{{2}}^{{2}{-}{\mathrm{\xi }}}{}{{\mathrm{ln}}{}\left({2}\right)}^{{7}}{}{\mathrm{cos}}{}\left({\mathrm{\pi }}{}{\mathrm{\xi }}\right){-}{7}{}{{2}}^{{2}{-}{\mathrm{\xi }}}{}{{\mathrm{ln}}{}\left({2}\right)}^{{6}}{}{\mathrm{\pi }}{}{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{\mathrm{\xi }}\right){+}{21}{}{{2}}^{{2}{-}{\mathrm{\xi }}}{}{{\mathrm{ln}}{}\left({2}\right)}^{{5}}{}{{\mathrm{\pi }}}^{{2}}{}{\mathrm{cos}}{}\left({\mathrm{\pi }}{}{\mathrm{\xi }}\right){+}{35}{}{{2}}^{{2}{-}{\mathrm{\xi }}}{}{{\mathrm{ln}}{}\left({2}\right)}^{{4}}{}{{\mathrm{\pi }}}^{{3}}{}{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{\mathrm{\xi }}\right){-}{35}{}{{2}}^{{2}{-}{\mathrm{\xi }}}{}{{\mathrm{ln}}{}\left({2}\right)}^{{3}}{}{{\mathrm{\pi }}}^{{4}}{}{\mathrm{cos}}{}\left({\mathrm{\pi }}{}{\mathrm{\xi }}\right){-}{21}{}{{2}}^{{2}{-}{\mathrm{\xi }}}{}{{\mathrm{ln}}{}\left({2}\right)}^{{2}}{}{{\mathrm{\pi }}}^{{5}}{}{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{\mathrm{\xi }}\right){+}{7}{}{{2}}^{{2}{-}{\mathrm{\xi }}}{}{\mathrm{ln}}{}\left({2}\right){}{{\mathrm{\pi }}}^{{6}}{}{\mathrm{cos}}{}\left({\mathrm{\pi }}{}{\mathrm{\xi }}\right){+}{{2}}^{{2}{-}{\mathrm{\xi }}}{}{{\mathrm{\pi }}}^{{7}}{}{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{\mathrm{\xi }}\right)\right){}{x}{}\left({x}{-}{0.5}\right){}\left({x}{-}{1.0}\right){}\left({x}{-}{1.5}\right){}\left({x}{-}{2.0}\right){}\left({x}{-}{2.5}\right){}\left({x}{-}{3.0}\right)}{{5040}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{{0.}{\le }{\mathrm{\xi }}{\le }{3.0}\right\}$ (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)
 > $\mathrm{p2}≔\mathrm{PolynomialInterpolation}\left(\mathrm{xyyp},\mathrm{method}=\mathrm{hermite},\mathrm{function}=\mathrm{exp}\left(0.1{x}^{2}\right),\mathrm{independentvar}='x',\mathrm{errorboundvar}='\mathrm{\xi }',\mathrm{digits}=5\right):$
 > $\mathrm{RemainderTerm}\left(\mathrm{p2}\right)$
 $\left(\frac{\left({0.120}{}{{ⅇ}}^{{0.1}{}{{\mathrm{\xi }}}^{{2}}}{+}{0.0720}{}{{\mathrm{\xi }}}^{{2}}{}{{ⅇ}}^{{0.1}{}{{\mathrm{\xi }}}^{{2}}}{+}{0.00480}{}{{\mathrm{\xi }}}^{{4}}{}{{ⅇ}}^{{0.1}{}{{\mathrm{\xi }}}^{{2}}}{+}{0.000064}{}{{\mathrm{\xi }}}^{{6}}{}{{ⅇ}}^{{0.1}{}{{\mathrm{\xi }}}^{{2}}}\right){}{\left({x}{-}{1.}\right)}^{{2}}{}{\left({x}{-}{1.5}\right)}^{{2}}{}{\left({x}{-}{2.}\right)}^{{2}}}{{720}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{{1.}{\le }{\mathrm{\xi }}{\le }{2.}\right\}$ (4)