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Concept# Divided differences

Summary

In mathematics, divided differences is an algorithm, historically used for computing tables of logarithms and trigonometric functions. Charles Babbage's difference engine, an early mechanical calculator, was designed to use this algorithm in its operation.
Divided differences is a recursive division process. Given a sequence of data points (x_0, y_0), \ldots, (x_{n}, y_{n}), the method calculates the coefficients of the interpolation polynomial of these points in the Newton form.
Definition
Given n + 1 data points
(x_0, y_0),\ldots,(x_{n}, y_{n})
where the x_k are assumed to be pairwise distinct, the forward divided differences are defined as:
\begin{align}
\mathopen[y_k] &:= y_k, && k \in { 0,\ldots,n} \
\mathopen[y_k,\ldots,y_{k+j}] &:= \frac{[y_{k+1},\ldots , y_{k+j}] - [y_{k},\ldots , y_{k+j-1}]}{x_{k+j}-x_k}, && k\in{0,\ldots,n-j},\ j\in{1,\ldots,n}.
\end{align}
To ma

Official source

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