# Newton Backward Interpolation

**Newton Backward Interpolation:**

Let the function y = f (x) take the values y_{0}, y_{1}, y_{2}, ...,y_{n} corresponding to the values x_{0}, x_{1}, x_{2}, ...,x_{n}. Where x_{i} = x_{0} ih, i = 0,1,2, ...,n. Suppose it is required to evaluate f (x) for the x = x_{0} ph, where p is any real number.

We have for any real number p, we have defined E such that

That is,

If y = f (x) is a polynimial of nth degree, then ^{n 1} and higher differences will be zero. Hence

This formula is known as **Newton’s Backward Formula.**

**Example: **The table gives the distance in nautical miles of the visible horizon for the given height in feet above the earth’s surface:

Find the values of y when x = 410 ft.

**Solution**: The difference table is

Since x = 400 is near the end of the table, we use Newton’s backward interpolation formula.

Thus required value;