compute_fixed_point and approximate fixed points

[oyamad]

I think compute_fixed_point should return v, instead of new_v, when np.max(np.abs(new_v - v)) <= error_tol becomes true, where new_v = T(v), provided that this routine is designed to return an approximate fixed point (for given error_tol).

There are two issues:

1. What is the definition of an approximate fixed point?
   My definition is: given error_tol, x_star is an approximate fixed point of f if |f(x_star) - x_star| <= error_tol (let's suppose | | is the max-norm).

2. Should the routine return an approximate fixed point whenever it stops before max_iter is reached?

If the answer to the question in 2 is yes according to the definition in 1, then the current implementation of compute_fixed_point is not correct.

def f(x):
    return 2 * x if x <= 0.5 else 2 * (1 - x)

x0 = 0.955
error_tol = 0.1
x_star = qe.compute_fixed_point(f, x0, error_tol, verbose=0)

abs(f(x_star) - x_star) <= error_tol

The outcome is False.

(Of course this is totally irrelevant if use of compute_fixed_point is restricted to contraction mappings. I am opening this issue because I am now implementing an algorithm that computes a fixed point of a function that satisfies the assumptions of Brouwer's fixed point theorem.)