FST - Finite Differences

Finite Differences

- Michel Paul

Here is a function that will find the differences between each pair of consecutive terms in a list:

Here is a list of random numbers:

The differences between random items will also be random.

See what happens with the differences between pairs of consecutive terms in a list of fibonaccis:

The differences between consecutive fibonaccis will themselves be fibonaccis!

See what happens with the differences between consecutive odd numbers:

Now see what happens with the differences between consecutive squares:

The differences between consecutive squares are consecutive odd numbers!

And so the differences between the differences are:

It took us 2 levels of differences to get a row of constants, and the formula n^2 has degree 2.

It only took us 1 level of differences with the odd numbers, and in that formula n had degree 1: 2n+1.

Here's a list of cubes:

Here we had to find 3 levels of differences before we got a list of constants.

That's the whole point of finding these differences -

• if the values in the original list come from a polynomial function, then at some point finding differences will terminate in a row of constants,
• and the degree of the original polynomial will equal the number of levels of differences.

The following cell will create a fourth degree polynomial function:

Here are the values of f(x) for x values between 0 and 10:

It took 4 levels of differences to reach a row of constants.

Here is the original function:

Again, a constant row of differences will only occur when f(x) is a polynomial function.

In this case the formula was 2^x.  It's not a polynomial.