The $k$th Newton polynomial of degree $k$ is defined by

By convention, the nodes are indexed from $0$, so $\mathrm{nodes}=\[\mathrm{x0},\mathrm{x1},...,\mathrm{xn}\]$.

At present, this can only be evaluated in Maple by prior use of the object-oriented representation obtained by P:=convert(p,MatrixPolynomialObject,x) and subsequent call to P:-Value(<x-value>), which uses Horner's method to evaluate the polynomial $p$.

$\mathrm{nodes}≔\left[-1\&comma;-\frac{1}{3}\&comma;\frac{1}{3}\&comma;1\right]$

$\mathrm{nodes}≔\left[\mathrm{-1}\&comma;-\frac{1}{3}\&comma;\frac{1}{3}\&comma;1\right]$

$p≔3\mathrm{NewtonBasis}\left(0\&comma;\mathrm{nodes}\&comma;x\right)+5\mathrm{NewtonBasis}\left(2\&comma;\mathrm{nodes}\&comma;x\right)+7\mathrm{NewtonBasis}\left(3\&comma;\mathrm{nodes}\&comma;x\right)$

$p≔3\mathrm{NewtonBasis}\left(0\&comma;\left[\mathrm{-1}\&comma;-\frac{1}{3}\&comma;\frac{1}{3}\&comma;1\right]\&comma;x\right)+5\mathrm{NewtonBasis}\left(2\&comma;\left[\mathrm{-1}\&comma;-\frac{1}{3}\&comma;\frac{1}{3}\&comma;1\right]\&comma;x\right)+7\mathrm{NewtonBasis}\left(3\&comma;\left[\mathrm{-1}\&comma;-\frac{1}{3}\&comma;\frac{1}{3}\&comma;1\right]\&comma;x\right)$

$P≔\mathrm{convert}\left(p\&comma;\mathrm{MatrixPolynomialObject}\&comma;x\right)$

$P≔\mathrm{Record}\left(\mathrm{Value}={\mathrm{Default}}_{\mathrm{value}}\&comma;\mathrm{Variable}=x\&comma;\mathrm{Degree}=3\&comma;\mathrm{Coefficient}=\mathrm{coe}\&comma;\mathrm{Dimension}=\left[1\&comma;1\right]\&comma;\mathrm{Basis}=\mathrm{NewtonBasis}\&comma;\mathrm{BasisParameters}=\left[\left[\mathrm{-1}\&comma;-\frac{1}{3}\&comma;\frac{1}{3}\&comma;1\right]\right]\&comma;\mathrm{IsMonic}=\mathrm{mon}\&comma;\mathrm{OutputOptions}=\left[\mathrm{shape}=\left[\right]\&comma;\mathrm{storage}=\mathrm{rectangular}\&comma;\mathrm{order}=\mathrm{Fortran\_order}\&comma;\mathrm{fill}=0\&comma;\mathrm{attributes}=\left[\right]\right]\right)$

$P:-\mathrm{Degree}\left(\right)$

$3$

$\mathrm{seq}\left(P:-\mathrm{Value}\left(\mathrm{nodes}\left[k\right]\right)\left[1,1\right]\&comma;k=1..\mathrm{nops}\left(\mathrm{nodes}\right)\right)$

$3,3,\frac{67}{9},\frac{259}{9}$

$P:-\mathrm{Value}\left(0.3\right)$

$\left[\begin{array}{c}6.924555556\end{array}\right]$

$\mathrm{factor}\left(P:-\mathrm{Value}\left(t\right)\left[1,1\right]\right)$

$7t^3+12t^2+\frac{53}{9}t+\frac{35}{9}$