The LinearSystem command retrieves the matrix and vector in the linear system of equations that were solved when computing the cubic spline interpolants.

The LinearSystem command returns an expression sequence of the form: $A$, $b$, where $A$ is a matrix and $b$ is a vector and $A·x=b$ is the linear system.

The POLYINTERP structure is created using the CubicSpline command.

The LinearSystem command only accepts interpolation structures that were created using the CubicSpline command, since the cubic spline interpolation method is the only method that has an associated linear system.

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

$\mathrm{xy}≔\left[\left[0\,1\right]\,\left[\frac{1}{2}\,1\right]\,\left[1\,\frac{11}{10}\right]\,\left[\frac{3}{2}\,\frac{3}{4}\right]\,\left[2\,\frac{7}{8}\right]\,\left[\frac{5}{2}\,\frac{9}{10}\right]\,\left[3\,\frac{11}{10}\right]\,\left[\frac{7}{2}\,1\right]\right]$

$\mathrm{xy}≔\left[\left[0\,1\right]\,\left[\frac{1}{2}\,1\right]\,\left[1\,\frac{11}{10}\right]\,\left[\frac{3}{2}\,\frac{3}{4}\right]\,\left[2\,\frac{7}{8}\right]\,\left[\frac{5}{2}\,\frac{9}{10}\right]\,\left[3\,\frac{11}{10}\right]\,\left[\frac{7}{2}\,1\right]\right]$

$\mathrm{p1}≔\mathrm{CubicSpline}\left(\mathrm{xy}\,\mathrm{independentvar}='x'\right)\:$

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

$\left[\begin{array}{cccccccc}1.& 0.& 0.& 0.& 0.& 0.& 0.& 0.\\ 0.5000000000& 2.& 0.5000000000& 0.& 0.& 0.& 0.& 0.\\ 0.& 0.5000000000& 2.& 0.5000000000& 0.& 0.& 0.& 0.\\ 0.& 0.& 0.5000000000& 2.& 0.5000000000& 0.& 0.& 0.\\ 0.& 0.& 0.& 0.5000000000& 2.& 0.5000000000& 0.& 0.\\ 0.& 0.& 0.& 0.& 0.5000000000& 2.& 0.5000000000& 0.\\ 0.& 0.& 0.& 0.& 0.& 0.5000000000& 2.& 0.5000000000\\ 0.& 0.& 0.& 0.& 0.& 0.& 0.& 1.\end{array}\right],\left[\begin{array}{c}0.\\ 0.8291995878\\ \mathrm{-2.116798351}\\ 2.237993817\\ \mathrm{-1.135176915}\\ 1.102713844\\ \mathrm{-1.175678461}\\ 0.\end{array}\right]$