Convergence acceleration

We give an example of application of generators for convergence acceleration. This presentation follows closely the example given by Pramode C.E in Python Generator Tricks (refer [9] for more information).

Note that a generator may take an other generator as an input parameter. For instance, suppose that we have defined a generator that generates the elements of a converging sequence. It is then possible to improve the convergence by an acceleration technique due to Euler and Aitken, often called Aitken’s Δ²-method (Refer [33]). It transforms a sequence s_i into another by defining 

Both sequences have the same limit, but the sequence   converges significantly faster. One possible implementation is as follows:

def Euler_accelerate(sequence):
    """
    Accelerate the iterator in the variable `sequence`.
    """
    s0 = next(sequence) # Si
    s1 = next(sequence) # Si+1
    s2 = next(sequence) # Si+2
    while True:
      ...