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Student[NumericalAnalysis]

 ApproximateExactUpperBound
 compute approximate, exact and upper bound remainder term values

 Calling Sequence ApproximateExactUpperBound(p) ApproximateExactUpperBound(p, pts)

Parameters

 p - a POLYINTERP structure pts - (optional) algebraic, list(algebraic); the point(s) at which the approximating polynomial, the exact value(s) of the function, and the upper bound(s) of the remainder term are computed

Description

 • The ApproximateExactUpperBound command computes three values: the approximating polynomial, the exact value of the function and the upper bound of the remainder term, for each point specified in pts.
 • If pts is not specified, the extrapolated points in the given POLYINTERP structure are used instead.
 • The computed values are returned in the form of a list: [pts, approximate values, exact values, upper bounds].
 • The POLYINTERP structure is created using the PolynomialInterpolation command.

Notes

 • POLYINTERP structures whose interpolation method is the cubic spline method may not be used with this command.

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)
 > $\mathrm{p1}≔\mathrm{PolynomialInterpolation}\left(\mathrm{xy},\mathrm{function}={2}^{2-x}\mathrm{cos}\left(\mathrm{\pi }x\right),\mathrm{method}=\mathrm{lagrange},\mathrm{extrapolate}=\left[0.25,0.75,1.25\right],\mathrm{errorboundvar}=\mathrm{\xi }\right):$
 > $\mathrm{ApproximateExactUpperBound}\left(\mathrm{p1}\right)$
 $\left[\left[{0.25}{,}{2.685058594}{,}{2.378414230}{,}{1.793805002}\right]{,}\left[{0.75}{,}{-1.746582031}{,}{-1.681792830}{,}{0.4892195461}\right]{,}\left[{1.25}{,}{-1.166503906}{,}{-1.189207114}{,}{0.2717886368}\right]\right]$ (2)
 > $\mathrm{ApproximateExactUpperBound}\left(\mathrm{p1},\left[0.5,1.5\right]\right)$
 $\left[\left[{0.5}{,}{0.}{,}{-5.801199657}{}{{10}}^{{-10}}{,}{0.}\right]{,}\left[{1.5}{,}{0.}{,}{8.701799482}{}{{10}}^{{-10}}{,}{0.}\right]\right]$ (3)