Divided difference

For distinct nodes,

f [x_{0}, \dots, x_{k}] = \sum_{j = 0}^{k} \frac{f ( x _{j} )}{\prod _{i \neq = j} ( x _{j} - x _{i} )} .

It is the leading coefficient of the interpolating polynomial through those nodes and obeys

f [x_{0}, \dots, x_{k}] = \frac{f [ x _{1} , \dots , x _{k} ] - f [ x _{0} , \dots , x _{k - 1} ]}{x _{k} - x _{0}} .

Table of contents

Exponential divided difference ( $D_{t} (β, γ) = \int_{0}^{t} e^{(t - x) γ + x β} d x$ )

The displayed integral is

(e^{tβ} - e^{t γ}) / (β - γ)

for distinct exponents and

t e^{t γ}

when they coincide. The latter is the continuous limit, so a zero denominator is a removable singularity. This quantity appears in variation-of-constants formula estimates of matrix evolution and avoids omitting an equal-exponent case.

Leibniz rule for divided differences

For

h = f g

,

h [t_{0}, \dots, t_{k}] = \sum_{m = 0}^{k} f [t_{0}, \dots, t_{m}] g [t_{m}, \dots, t_{k}] .

Newton interpolation polynomial

Newton's nested interpolation form is

p_{n} (x) = f (x_{0}) + \sum_{k = 1}^{n} f [x_{0}, \dots, x_{k}] \prod_{i = 0}^{k - 1} (x - x_{i}) .

A triangular divided-difference table computes all its coefficients using exactly

n (n + 1) /2

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