Newton's Divided Difference Method:

In this method, the n^th order polynomial is taken as

y = a₀ + a₁ (x - x₀) + a₂ (x - x₀) (x - x₁) + . . . + an (x - x₀) (x - x₁) . . . (x - x_n - 1)

A recursive formula is written down to determine the coefficients. The higher order coefficients are found out from the lower order ones. The coefficients are evaluated beginning with a₁, a₂ and a₃ and so on up to an.

The general form of the nth order Newton's polynomial may be written as following

y = c₀ + c₁ (x - x₀) + c₂ (x - x₀) (x - x₁) + . . . + c_n (x - x₀) (x - x₁) . . . (x - x_n - 1)

To find out the (n + 1) coefficients, we need n + 1 data points such as (x₀, y₀), (x₁, y₁) . . .  (x_n, y_n).

The coefficients are given by

c₀ = y₀

c₁  =   y₁ - y₀/ ( x₁ - x₀ )  =  f ( x₁ , x₀ )

c₂  = [ f ( x₂ , x₁ ) - f ( x₁, x₀ )] /( x ₂ - x₀ )= f ( x₂ , x₁ , x ₀)

:

:

c_n  = f ( x_n , x_{n -1}, . . . , x₁, x₀ )