NEWTON´S DIVIDED DIFFERENCES

Interpolation is an estimation of a value within two known values in a sequence of values.

Newton’s divided difference interpolation formula is a interpolation technique used when the interval difference is not same for all sequence of values.

Suppose f(x0), f(x1), f(x2)………f(xn) be the (n+1) values of the function y=f(x) corresponding to the arguments x=x0, x1, x2…xn, where interval differences are not same
Then the first divided difference is given by

 

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The second divided difference is given by

 

 

 

and so on…
Divided differences are symmetric with respect to the arguments i.e independent of the order of arguments.
so, 
f[x0, x1]=f[x1, x0]
f[x0, x1, x2]=f[x2, x1, x0]=f[x1, x2, x0]
By using first divided difference, second divided difference as so on .A table is formed which is called the divided difference table.