In mathematics divided differences is a recursive division process.
Definition
Given n data points
the divided differences are defined as
![[y_{\nu}] := y_{\nu} \qquad \mbox{ , } \nu = 0,\ldots,n-1](a/../upload/math/7e259a30c728c3a3a5d9be60a9adaa67.png)
![[y_{\nu},\ldots,y_{\nu+j}] := \frac{[y_{\nu+1},\ldots y_{\nu+j}] - [y_{\nu},\ldots y_{\nu+j-1}]}{x_{\nu+j}-x_{\nu}} \qquad \mbox{ , } \nu = 0,\ldots,n-j,j=1,\ldots,n-1](a/../upload/math/4ccb54fb99305e36f031c593668fa907.png)
Notes
If the data points are given as a function f(x)
we sometimes write
![f[x_{\nu}] := f(x_{\nu}) \qquad \mbox{ , } \nu = 0,\ldots,n-1](a/../upload/math/202caee0efb35876a1210b6470570559.png)
![f[x_{\nu},\ldots,x_{\nu+j}] := \frac{f[x_{\nu+1},\ldots x_{\nu+j}] - f[x_{\nu},\ldots x_{\nu+j-1}]}{x_{\nu+j}-x_{\nu}} \qquad \mbox{ , } \nu = 0,\ldots,n-j,j=1,\ldots,n-1](a/../upload/math/f37b3359dfc720e3fbaae735ed6f3455.png)
Example
For the first few [yν] this yields
- [y0] = y0
![[y_0,y_1] = \frac{y_1-y_0}{x_1-x_0}](a/../upload/math/8e22437cf714b7f746e56ecf702834a2.png)
![[y_0,y_1,y_2] = \frac{\frac{y_2-y_1}{x_2-x_1}-\frac{y_1-y_0}{x_1-x_0}}{x_2-x_0}](a/../upload/math/6904f3d542741d4cd61bc719c1262747.png)
To make the recursive process more clear the divided differences can be put in a tabular form
![\begin{matrix} x_0 & y_0 = [y_0] & & & \\ & & [y_0,y_1] & & \\ x_1 & y_1 = [y_1] & & [y_0,y_1,y_2] & \\ & & [y_1,y_2] & & [y_0,y_1,y_2,y_3]\\ x_2 & y_2 = [y_2] & & [y_1,y_2,y_3] & \\ & & [y_2,y_3] & & \\ x_3 & y_3 = [y_3] & & & \\ \end{matrix}](a/../upload/math/36962f47338444c859795fffde59efb4.png)
Peano form
The divided differences can be expressed as
![f[x_0,\ldots,x_n] = \frac{1}{n!} \int_{x_0}^{x_n} f^{(n)}(t)B_{n-1}(t) \, dt](a/../upload/math/d8d253d490142fed289507e92cbd1270.png)
where Bn-1 is a B-spline of degree n-1 for the data points x0,...,xn and f(n)(x) is the n derivative 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
Example
Application
The method of divided differences can be used to calculate the coefficients in the interpolation polynomial in the Newton form.