NationMaster - Encyclopedia: Divided differences

In mathematics divided differences is a recursive division process. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... A Sierpinski triangle â€”a confined recursion of triangles to form a geometric lattice. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ...

The method can be used to calculate the coefficients in the interpolation polynomial in the Newton form. In the mathematical subfield of numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial. ... In the mathematical subfield of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is the interpolation polynomial for a given set of data points in the Newton form. ...

1 Definition
2 Notes
3 Example
4 Peano form
5 Forward differences
- 5.1 Definition
- 5.2 Example
6 See also

Definition

Given n data points

$(x_0, y_0),ldots,(x_{n-1}, y_{n-1})$

the divided differences are defined as

$[y_{nu}] := y_{nu} qquad mbox{ , } nu = 0,ldots,n-1$

$[y_{nu},ldots,y_{nu+j}] := frac{[y_{nu+1},ldots y_{nu+j}] - [y_{nu},ldots y_{nu+j-1}]}{x_{nu+j}-x_{nu}} qquad mbox{ , } nu = 0,ldots,n-j,j=1,ldots,n-1.$

Notes

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

$(x_0, f(x_0)),ldots,(x_{n-1}, f(x_{n-1}))$

we sometimes write

$f[x_{nu}] := f(x_{nu}) qquad mbox{ , } nu = 0,ldots,n-1$

$f[x_{nu},ldots,x_{nu+j}] := frac{f[x_{nu+1},ldots x_{nu+j}] - f[x_{nu},ldots x_{nu+j-1}]}{x_{nu+j}-x_{nu}} qquad mbox{ , } nu = 0,ldots,n-j,j=1,ldots,n-1.$

Example

For the first few [y_ν] this yields

[y 0] = y 0

$[y_0,y_1] = frac{y_1-y_0}{x_1-x_0}$

$[y_0,y_1,y_2] = frac{frac{y_2-y_1}{x_2-x_1}-frac{y_1-y_0}{x_1-x_0}}{x_2-x_0}.$

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

$begin{matrix} x_0 & y_0 = [y_0] & & & & & [y_0,y_1] & & x_1 & y_1 = [y_1] & & [y_0,y_1,y_2] & & & [y_1,y_2] & & [y_0,y_1,y_2,y_3] x_2 & y_2 = [y_2] & & [y_1,y_2,y_3] & & & [y_2,y_3] & & x_3 & y_3 = [y_3] & & & end{matrix}$

Peano form

The divided differences can be expressed as

$f[x_0,ldots,x_n] = frac{1}{n!} int_{x_0}^{x_n} f^{(n)}(t)B_{n-1}(t) , dt$

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). In the mathematical subfield of numerical analysis a B-spline is a spline function which has minimal support with respect to a given degree, smoothness, and domain partition. ... In mathematics, a derivative is the rate of change of a quantity. ...

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. Giuseppe Peano Giuseppe Peano (August 27, 1858 â€“ April 20, 1932) was an Italian mathematician and philosopher best known for his contributions to set theory. ...

Forward differences

For more details on this topic, see Finite difference.

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. In mathematics, a finite difference is like a differential quotient, except that it uses finite quantities instead of infinitesimal ones. ...

Definition

Given n data points

$(x_0, y_0),ldots,(x_{n-1}, y_{n-1})$

with

$x_{nu} = x_0 + nu h mbox{ , } h > 0 mbox{ , } nu=0,ldots,n-1$

the divided differences can be calculated via forward differences defined as

$triangle^{(0)}y_{i} := y_{i}$

$triangle^{(k)}y_{i} := triangle^{(k-1)}y_{i+1} - triangle^{(k-1)}y_{i} mbox{ , } k ge 1.$

Example

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

Contents

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Notes

Example

Peano form

Forward differences

Definition

Example

See also

Definition