The BasisFunctions routine retrieves the basis functions from a POLYINTERP structure.

The POLYINTERP structure is created using the PolynomialInterpolation command.

Only POLYINTERP structures that contain data points that have been interpolated using the Lagrange, Newton or Hermite methods can use the BasisFunctions command.

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

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

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

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

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

$\left[0.08888888889\left(x-0.5\right)\left(x-1.0\right)\left(x-1.5\right)\left(x-2.0\right)\left(x-2.5\right)\left(x-3.0\right)\,-0.5333333333x\left(x-1.0\right)\left(x-1.5\right)\left(x-2.0\right)\left(x-2.5\right)\left(x-3.0\right)\,1.333333333x\left(x-0.5\right)\left(x-1.5\right)\left(x-2.0\right)\left(x-2.5\right)\left(x-3.0\right)\,-1.777777778x\left(x-0.5\right)\left(x-1.0\right)\left(x-2.0\right)\left(x-2.5\right)\left(x-3.0\right)\,1.333333333x\left(x-0.5\right)\left(x-1.0\right)\left(x-1.5\right)\left(x-2.5\right)\left(x-3.0\right)\,-0.5333333333x\left(x-0.5\right)\left(x-1.0\right)\left(x-1.5\right)\left(x-2.0\right)\left(x-3.0\right)\,0.08888888889x\left(x-0.5\right)\left(x-1.0\right)\left(x-1.5\right)\left(x-2.0\right)\left(x-2.5\right)\right]$

$\mathrm{Draw}\left(\mathrm{p1}\,\mathrm{objects}=\left[\mathrm{BasisFunctions}\right]\right)$

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

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

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

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

$\left[1\,x-1.\,{\left(x-1.\right)}^2\,{\left(x-1.\right)}^2\left(x-1.5\right)\,{\left(x-1.\right)}^2{\left(x-1.5\right)}^2\,{\left(x-1.\right)}^2{\left(x-1.5\right)}^2\left(x-2.\right)\right]$