# 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) = P_{n}(x) = c_{0} (x – x_{0}) c_{1} (x – x_{0})(x – x_{1}) c_{2} ... (x – x_{0})(x – x_{1})...(x – x_{n-1}) c_{n}. ....................1.1

The polynomial fits the data Pn(xi) = f(x_{i}) = f_{i}

Setting P_{n}(x_{0}) = f_{0},

we obtain

Pn(x_{0}) = f_{0} = c_{0}

since all the remaining terms vanish.

Setting P_{n}(x_{1}) = f_{1}, we obtain

Setting P_{n}(x_{2}) = f_{2}, we obtain

or

By induction, we can prove that

cn = f [x_{0}, x_{1}, x_{2}, ..., x_{n}].

Hence, we can write the interpolating polynomial as

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

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