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In mathematicsdivided differences is a recursive division process.
Definition
Given n data points
the divided differences are defined as
Notes
If the data points are given as a function f(x)
we sometimes write
Example
For the first few [yν] this yields
To make the recursive process more clear the divided differences can be put in a tabular form
Peano form
The divided differences can be expressed as
where Bn-1 is a B-spline of degree n-1 for the data points x0,...,xn and f(n)(x) is the nderivative of the function f(x)
This is called the Peano form of the divided differences and Bn-1 is called the Peano kernel for the divided differences.
Forward differences
When the data points are equidistantly distributed we get the special case called forward differences. They are easier to calculate then the more general divided differences.
Definition
Given n data points
with
the divided differences can be calculated via forward differences defined as