Numerical Methods

Newtons Divided Difference Interpolation

The Newton's divided difference interpolation is used to find higher order divided difference.

 

We write the interpolating polynomial as

              

                                                         f(x) = Pn(x)   =    c0 (x – x0) c1 (x – x0)(x – x1) c2 ... (x – x0)(x – x1)...(x – xn-1) cn.  ....................1.1

The polynomial fits the data                                                                                 Pn(xi) = f(xi) = fi

Setting                                                                                                                      Pn(x0) = f0,

we obtain

Pn(x0) = f0 = c0

since all the remaining terms vanish.

Setting Pn(x1) = f1, we obtain

Setting Pn(x2) = f2, we obtain

or

By induction, we can prove that

cn = f [x0, x1, x2, ..., xn].

Hence, we can write the interpolating polynomial as

....................1.2

 

This polynomial is called the Newton’s divided difference interpolating polynomial.