Convergence acceleration

We'll give an example of the application of generators for convergence acceleration. This presentation closely follows the example given in [9].

Note that a generator may take another 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, see [33]. It transforms a sequence  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 = sequence.__next__() # Si
    s1 = sequence.__next__() # Si+1
    s2 = sequence.__next__() # Si+2
    while True:
    ...