recursion stability of the golden mean

[Jay Wang]

The downward recursion of the golden ratio (Project 1.3) may be done as follows:

y2 = 0.  # y_{n+1}
y1 = 1.  # y_n
nmax = 50
yn = []

for i in range(nmax):
    y0 = y2 - y1  # y0 is y_{n-1}
    yn.append(y0)
    y2 = y1         # reseed
    y1 = y0
# reverse, scale, using NumPy array
yn = yn[::-1]      # or yn.reverse() for Python 3.x
yn = np.array(yn)/yn[0]    # elementwise op (vector)

One can now plot the relative error.

Things to try: One should be able to obtain the positive phi series in upward recursion with arbitrary initial seeds also.
Try y0=0, y1 = 1., and y2=y1+y0 for nmax iterations (say nmax=50). Scale the series so the last element is correct, i.e., multiply it by phi**(nmax+1)/yn[-1]. Compare with the exact series and plot the error. What do you see? why?