errorboundvar = name

The name to assign to the independent variable in the remainder term. By default, the errorboundvar given when the POLYINTERP structure was created is used.

independentvar = name

The name to assign to the independent variable in the approximated polynomial. By default, the independentvar given when the POLYINTERP structure was created is used.

showapproximatepoly = true or false

Whether to return the approximated polynomial. By default this is set to true.

showremainder = true or false

Whether to return the remainder term. By default, this is set to true.

The InterpolantRemainderTerm command returns the approximate polynomial and remainder term from a POLYINTERP structure.

The interpolant and remainder term are returned in an expression sequence of the form $\mathrm{Pn}$, $\mathrm{Rn}$, where $\mathrm{Pn}$ is the interpolant and $\mathrm{Rn}$ is the remainder term.

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

If the POLYINTERP structure p was created using the CubicSpline command then the InterpolantRemainderTerm command can only return the approximate polynomial and therefore showremainder must be set to false.

In order for the remainder term to exist, the POLYINTERP structure p must have an associated exact function that has been given.

The remainder term is also called an error term.

The interpolant is also called the approximating polynomial or interpolating polynomial.

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

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

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

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

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

$0.3555555556\left(x-0.5\right)\left(x-1.0\right)\left(x-1.5\right)\left(x-2.0\right)\left(x-2.5\right)\left(x-3.0\right)-2.666666667x\left(x-0.5\right)\left(x-1.5\right)\left(x-2.0\right)\left(x-2.5\right)\left(x-3.0\right)+1.333333333x\left(x-0.5\right)\left(x-1.0\right)\left(x-1.5\right)\left(x-2.5\right)\left(x-3.0\right)-0.04444444444x\left(x-0.5\right)\left(x-1.0\right)\left(x-1.5\right)\left(x-2.0\right)\left(x-2.5\right),\left(\frac{\left(-2^{2-\mathrm{\xi }}{\ln \left(2\right)}^7\cos \left(\mathrm{\pi }\mathrm{\xi }\right)-72^{2-\mathrm{\xi }}{\ln \left(2\right)}^6\mathrm{\pi }\sin \left(\mathrm{\pi }\mathrm{\xi }\right)+212^{2-\mathrm{\xi }}{\ln \left(2\right)}^5{\mathrm{\pi }}^2\cos \left(\mathrm{\pi }\mathrm{\xi }\right)+352^{2-\mathrm{\xi }}{\ln \left(2\right)}^4{\mathrm{\pi }}^3\sin \left(\mathrm{\pi }\mathrm{\xi }\right)-352^{2-\mathrm{\xi }}{\ln \left(2\right)}^3{\mathrm{\pi }}^4\cos \left(\mathrm{\pi }\mathrm{\xi }\right)-212^{2-\mathrm{\xi }}{\ln \left(2\right)}^2{\mathrm{\pi }}^5\sin \left(\mathrm{\pi }\mathrm{\xi }\right)+72^{2-\mathrm{\xi }}\ln \left(2\right){\mathrm{\pi }}^6\cos \left(\mathrm{\pi }\mathrm{\xi }\right)+2^{2-\mathrm{\xi }}{\mathrm{\pi }}^7\sin \left(\mathrm{\pi }\mathrm{\xi }\right)\right)x\left(x-0.5\right)\left(x-1.0\right)\left(x-1.5\right)\left(x-2.0\right)\left(x-2.5\right)\left(x-3.0\right)}{5040}\right)\&where\left\{0.\le \mathrm{\xi }\le 3.0\right\}$