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.

Read more about this topic:  Divided Differences

Famous quotes containing the words power and/or series:

    The spider-mind acquires a faculty of memory, and, with it, a singular skill of analysis and synthesis, taking apart and putting together in different relations the meshes of its trap. Man had in the beginning no power of analysis or synthesis approaching that of the spider, or even of the honey-bee; but he had acute sensibility to the higher forces.
    Henry Brooks Adams (1838–1918)

    Autobiography is only to be trusted when it reveals something disgraceful. A man who gives a good account of himself is probably lying, since any life when viewed from the inside is simply a series of defeats.
    George Orwell (1903–1950)