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Divided DifferencesIn mathematics divided differences is a recursive division process. Definition Given n data points -
the divided differences are defined as -
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Notes If the data points are given as a function f(x) -
we sometimes write -
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Example For the first few yν this yields -
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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 -
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 -
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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} Application The method of divided differences can be used to calculate the coefficients in the interpolation polynomial in the Newton form.
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