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====== Polynomial Interpolation via Newton Divided Differences ====== Newton's Divided Difference algorithm is a slick way to compute an $N$th order polynomial interpolant to a set of $N+1$ data points $(x_0, y_0), (x_1, y_1), \ldots, (x_N, y_N)$ with distinct $x_i$'s. It produces a polynomial of in the form of [[gibson:teaching:fall-2016:math753:horner|Horner's method]] with base points, e.g. \begin{equation*} P(x) = c_0 + (x - x_0) \, [c_1 + (x - x_1) [c_2 + (x - x_2) [c_3 + (x - x_3) \, c_4]]] \end{equation*} If we set $y = P(x)$ to the above equation (or its $N$th-order generalization) and plug in the data points $(x_i, y_i)$ for $i=0,1,\ldots, N$, we get a series of $N+1$ equations in the $N+1$ unknowns $c_0, \ldots c_N$. \begin{eqnarray*} y_0 &=& c_0 \\ y_1 &=& c_0 + (x_1-x_0) \, c_1 \\ y_2 &=& c_0 + (x_2-x_0) \, c_1 + (x_2-x_0)(x_2-x_1) \, c_2 \\ &\vdots& \end{eqnarray*} Lo and behold this is lower-triangular system of equations, which can be written in matrix form like this \begin{equation*} \left[\begin{array}{ccccc} 1 & & \ldots & & 0 \\ 1 & x_1-x_0 & & & \\ 1 & x_2-x_0 & (x_2-x_0)(x_2-x_1) & & \vdots \\ \vdots & \vdots & & \ddots & \\ 1 & x_k-x_0 & \ldots & \ldots & \prod_{j=0}^{N-1}(x_N - x_j) \end{array}\right] \left[\begin{array}{c} c_0 \\ c_1 \\ c_2 \\ \vdots \\ \\ c_{N} \end{array}\right] = \left[\begin{array}{c} y_0 \\ y_1 \\ y_2 \\ \vdots \\ \\ y_{N} \end{array}\right] \end{equation*} Lower-triangular systems can be solved easily via forward substitution.
