Binary Splitting

The Fast Fourier Transform (FFT) is an algorithm for computing based on binary splitting.  Here is how it works.  Start with the polynomial

    

Collect the terms with even powers of x together and the other terms are the ones with an odd power of x.

Let
       
and
     
     
Notice that both   and are polynomials of degree and that

             

For ,  to compute both  and notice that

    
    
    
    
Therefore, to compute both  and   we need only and and then do one additional multiplication:  .

If we use this binary splitting technique, we can evaluate our polynomial of degree N-1 at N points by

(1)  Evaluating two polynomials (  and ) at points each.   This takes    multiplications.
(2)  Combine these evaluations to get pairs of  values of and   as above with more multiplications.

Therefore we could cut the number of multiplications in half to by using binary splitting once.