Topics: Interpolation

Let be a function whose exact definition we don’t know. All we know is various values of , like so:

Unlike the tables we use for Newton’s finite differences method, the spacing between every does not need to be constant.

We will be able to interpolate for any within the constraints of the table by using a method known as Newton’s finite divided interpolation method.


Just like Newton’s finite differences method, Newton’s divided differences method uses, well, differences.

For this method, the first differences are as follows:

The second differences are:

In general, the th difference is:

Of course, the amount of difference tiers will depend on the amount of points we know.

Interpolation Formulas

Given an contained within the constraints of the table (i.e. ), we can use these differences to interpolate its with one of two formulas that are used to obtain an interpolating polynomial.

For both of them, we choose a specific in our table as an anchor value; we’ll set this value to be .

Our choice between these two formulas will depend on the amount of differences that can be used with a given one (the greater, the better the approximation), as well as on how close is to (the closer, the better).

Forwards Interpolation

We can obtain an approximation to our with the forwards interpolation formula:

Backwards Interpolation

We can also obtain our with the backwards interpolation formula: