In mathematics divided differences is a recursive division process.

The method can be used to calculate the coefficients in the interpolation polynomial in the Newton form.

Definition

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Given n data points

$(x\_0,\; y\_0),\backslash ldots,(x\_\{n-1\},\; y\_\{n-1\})$

the divided differences are defined as

$*:=\; y\_\{\backslash nu\}\; \backslash qquad\; \backslash mbox\{\; ,\; \}\; \backslash nu\; =\; 0,\backslash ldots,n-1$

$:=\; \backslash frac\{[y\_\{\backslash nu+1\},\backslash ldots\; y\_\{\backslash nu+j\}-y\_\{\backslash nu+j-1\}\}\{x\_\{\backslash nu+j\}-x\_\{\backslash nu\}\}\; \backslash qquad\; \backslash mbox\{\; ,\; \}\; \backslash nu\; =\; 0,\backslash ldots,n-j,j=1,\backslash ldots,n-1.$

Notes

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If the data points are given as a function f(x)

$(x\_0,\; f(x\_0)),\backslash ldots,(x\_\{n-1\},\; f(x\_\{n-1\}))$

we sometimes write

$f*:=\; f(x\_\{\backslash nu\})\; \backslash qquad\; \backslash mbox\{\; ,\; \}\; \backslash nu\; =\; 0,\backslash ldots,n-1$

$f:=\; \backslash frac\{f[x\_\{\backslash nu+1\},\backslash ldots\; x\_\{\backslash nu+j\}-\; fx\_\{\backslash nu+j-1\}\}\{x\_\{\backslash nu+j\}-x\_\{\backslash nu\}\}\; \backslash qquad\; \backslash mbox\{\; ,\; \}\; \backslash nu\; =\; 0,\backslash ldots,n-j,j=1,\backslash ldots,n-1.$

Example

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For the first few ^* this yields

$*=\; y\_0$

$*=\; \backslash frac\{y\_1-y\_0\}\{x\_1-x\_0\}$

$*=\; \backslash frac\{\backslash frac\{y\_2-y\_1\}\{x\_2-x\_1\}-\backslash frac\{y\_1-y\_0\}\{x\_1-x\_0\}\}\{x\_2-x\_0\}.$

To make the recursive process more clear the divided differences can be put in a tabular form

$$

\begin{matrix} x_0 & y_0 = ^* & & & \\ & & ^* & & \\ x_1 & y_1 = & & [y_0,y_1,y_2 & \\ & & & & [y_0,y_1,y_2,y_3\\ x_2 & y_2 = & & [y_1,y_2,y_3 & \\ & & ^* & & \\ x_3 & y_3 = ^* & & & \\ \end{matrix}

Peano form

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The divided differences can be expressed as

$f*=\; \backslash frac\{1\}\{n!\}\; \backslash int\_\{x\_0\}^\{x\_n\}\; f^\{(n)\}(t)B\_\{n-1\}(t)\; \backslash ,\; dt$

where B_n-1 is a B-spline of degree n-1 for the data points x₀,...,x_n and f⁽ⁿ⁾(x) is the n derivative of the function f(x).

This is called the Peano form of the divided differences and B_n-1 is called the Peano kernel for the divided differences, both named after Giuseppe Peano.

A related theorem states that there exists p in the interval x_n such that

$f*=\; \backslash frac\{f^\{(n)\}(p)\}\{n!\}\; .\; \backslash ,$

This is a generalization of the mean value theorem, which can be recovered by setting n=1 above.

Forward differences

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When the data points are equidistantly distributed we get the special case called forward differences. They are easier to calculate than the more general divided differences.

Definition

Given n data points

$(x\_0,\; y\_0),\backslash ldots,(x\_\{n-1\},\; y\_\{n-1\})$

with

$x\_\{\backslash nu\}\; =\; x\_0\; +\; \backslash nu\; h\; \backslash mbox\{\; ,\; \}\; h\; >\; 0\; \backslash mbox\{\; ,\; \}\; \backslash nu=0,\backslash ldots,n-1$

the divided differences can be calculated via forward differences defined as

$\backslash triangle^\{(0)\}y\_\{i\}\; :=\; y\_\{i\}$

$\backslash triangle^\{(k)\}y\_\{i\}\; :=\; \backslash triangle^\{(k-1)\}y\_\{i+1\}\; -\; \backslash triangle^\{(k-1)\}y\_\{i\}\; \backslash mbox\{\; ,\; \}\; k\; \backslash ge\; 1.$

Example

$$

\begin{matrix} y_0 & & & \\ & \triangle y_0 & & \\ y_1 & & \triangle^{2} y_0 & \\ & \triangle y_1 & & \triangle^{3} y_0\\ y_2 & & \triangle^{2} y_1 & \\ & \triangle y_2 & & \\ y_3 & & & \\ \end{matrix}

Finite differences

Différences divisées | Różnica dzielona | Diferença dividida