Divided Differences - Polynomials and Power Series

Polynomials and Power Series

Divided differences of polynomials are particularly interesting, because they can benefit from the Leibniz rule. The matrix with

J= \begin{pmatrix} x_0 & 1 & 0 & 0 & \cdots & 0 \\ 0 & x_1 & 1 & 0 & \cdots & 0 \\ 0 & 0 & x_2 & 1 & & 0 \\ \vdots & \vdots & & \ddots & \ddots & \\ 0 & 0 & 0 & 0 & & x_n \end{pmatrix}

contains the divided difference scheme for the identity function with respect to the nodes, thus contains the divided differences for the power function with exponent . Consequently you can obtain the divided differences for a polynomial function with respect to the polynomial by applying (more precisely: its corresponding matrix polynomial function ) to the matrix .

= \begin{pmatrix} \varphi(p) & \varphi(p) & \varphi(p) & \ldots & \varphi(p) \\ 0 & \varphi(p) & \varphi(p) & \ldots & \varphi(p) \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & \ldots & 0 & 0 & \varphi(p) \end{pmatrix}

This is known as Opitz' formula.

Now consider increasing the degree of to infinity, i.e. turn the Taylor polynomial to a Taylor series. Let be a function which corresponds to a power series. You can compute a divided difference scheme by computing the according matrix series applied to . If the nodes are all equal, then is a Jordan block and computation boils down to generalizing a scalar function to a matrix function using Jordan decomposition.

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