Scarfs

A library to find an approximate fixed point for a for a bounded vector valued function.

Installation

pip install scarfs

Usage

Define the function you want to find a fixed point of using numba:

from numba import njit

@njit
def roll(simp: np.ndarray) -> np.ndarray:
    return np.roll(simp, 1)

For performance reasons, this function must be compiled by numba as a cfunc or in nopython mode. Jitclass functions are currently not supported. The function must also lie in a bounded space, three default spaces are provided: the simplex, the simplotope, and the unit hypercube. If your bounded space is not one of these, you'll need to first compute a homeomorphism between your space and one of these. The main algorithm runs on the simplex, so you may find it faster if you can project there directly.

Once your function is defined, simply call one of the fixed point functions with an initial position and a discretization:

from scarfs import simplex_fixed_point

sol = simplex_fixed_point(roll, np.array([1, 0, 0, 0], float), 100)

The result is guaranteed to be within 1 / discretization of a true fixed point (or a little larger for the other bounded spaces).

Note that fixed points are difficult to approximate generally, so this may run for a very long time.

Also note that this library "trusts" you, so if you pass in invalid inputs, you may get arcane errors.