divided_diff: Divided differencing

In dspline: Tools for Computations with Discrete Splines

Divided differencing

Description

Computes the divided difference of a function (or vector of function evaluations) with respect to given centers.

Usage

divided_diff(f, z)

Arguments

f	Function, or vector of function evaluations at the centers.
z	Centers for the divided difference calculation.

Details

The divided difference of a function f with respect to centers z_1, \ldots, z_{k+1} is defined recursively as:

f[z_1,\ldots,z_{k+1}] = \displaystyle \frac{f[z_2,\ldots,z_{k+1}] - f[z_1,\ldots,z_k]}{z_{k+1}-z_1},

with base case f[z_1] = f(z_1) (that is, divided differencing with respect to a single point reduces to function evaluation).

A notable special case is when the centers are evenly-spaced, say, z_i = z+ih, i=0,\ldots,k for some spacing h>0, in which case the divided difference becomes a (scaled) forward difference, or equivalently a (scaled) backward difference,

k! \cdot f[z,\ldots,z+kh] = \displaystyle \frac{1}{h^k} (F^k_h f)(z) = \frac{1}{h^k} (B^k_h f)(z+kh),

where we use F^k_h and B^k_v to denote the forward and backward difference operators, respectively, of order k and with spacing h.

Value

Divided difference of f with respect to centers z.

Examples

f = runif(4)
z = runif(4)
divided_diff(f[1], z[1])
f[1]
divided_diff(f[1:2], z[1:2])
(f[1]-f[2])/(z[1]-z[2])
divided_diff(f[1:3], z[1:3])
((f[1]-f[2])/(z[1]-z[2]) - (f[2]-f[3])/(z[2]-z[3])) / (z[1]-z[3]) 
divided_diff(f, 1:4)
diff(f, diff = 3) / factorial(3)

dspline documentation built on June 8, 2025, 9:40 p.m.