fractals.py

from typing import Tuple
import numpy as np
from numpy.polynomial import Polynomial

def mandelbrot(width: int, height: int, *,
      x: float = -0.5, y: float = 0, zoom: int = 1, max_iterations: int = 100) -> np.array:
    """
    From https://www.learnpythonwithrune.org/numpy-compute-mandelbrot-set-by-vectorization/.
    """
    # To make navigation easier we calculate these values
    x_width, y_height = 1.5, 1.5*height/width
    x_from, x_to = x - x_width/zoom, x + x_width/zoom
    y_from, y_to = y - y_height/zoom, y + y_height/zoom

    # Here the actual algorithm starts
    x = np.linspace(x_from, x_to, width).reshape((1, width))
    y = np.linspace(y_from, y_to, height).reshape((height, 1))
    c = x + 1j*y

    # Initialize z to all zero
    z = np.zeros(c.shape, dtype=np.complex128)

    # To keep track in which iteration the point diverged
    div_time = np.zeros(z.shape, dtype=int)

    # To keep track on which points did not converge so far
    m = np.full(c.shape, True, dtype=bool)
    for i in range(max_iterations):
        z[m] = z[m]**2 + c[m]
        diverged = np.greater(np.abs(z), 2, out=np.full(c.shape, False), where=m) # Find diverging
        div_time[diverged] = i      # set the value of the diverged iteration number
        m[np.abs(z) > 2] = False    # to remember which have diverged

    return div_time

def julia(width: int, height: int, *,
        c: complex = -0.4 + 0.6j, x: float = 0, y: float = 0, zoom: int = 1, max_iterations: int = 100) -> np.array:
    """
    From https://www.learnpythonwithrune.org/numpy-calculate-the-julia-set-with-vectorization/.
    """
    # To make navigation easier we calculate these values
    x_width, y_height = 1.5, 1.5*height/width
    x_from, x_to = x - x_width/zoom, x + x_width/zoom
    y_from, y_to = y - y_height/zoom, y + y_height/zoom

    # Here the actual algorithm starts
    x = np.linspace(x_from, x_to, width).reshape((1, width))
    y = np.linspace(y_from, y_to, height).reshape((height, 1))
    z = x + 1j*y

    # Initialize z to all zero
    c = np.full(z.shape, c)

    # To keep track in which iteration the point diverged
    div_time = np.zeros(z.shape, dtype=int)

    # To keep track on which points did not converge so far
    m = np.full(c.shape, True, dtype=bool)
    for i in range(max_iterations):
        z[m] = z[m]**2 + c[m]
        m[np.abs(z) > 2] = False
        div_time[m] = i

    return div_time

Range = Tuple[float, float]

def newton(width: int, height: int, *,
        p: Polynomial, a: complex, xr: Range = (-2.5, 1), yr: Range = (-1, 1), max_iterations: int = 100) -> (np.array, np.array):
    """ """
    # To make navigation easier we calculate these values
    x_from, x_to = xr
    y_from, y_to = yr

    # Here the actual algorithm starts
    x = np.linspace(x_from, x_to, width).reshape((1, width))
    y = np.linspace(y_from, y_to, height).reshape((height, 1))
    z = x + 1j*y

    # Compute the derivative
    dp = p.deriv()

    # Compute roots
    roots = p.roots()
    epsilon = 1e-5

    # Set the initial conditions
    a = np.full(z.shape, a)

    # To keep track in which iteration the point diverged
    div_time = np.zeros(z.shape, dtype=int)

    # To keep track on which points did not converge so far
    m = np.full(a.shape, True, dtype=bool)

    # To keep track which root each point converged to
    r = np.full(a.shape, 0, dtype=int)

    for i in range(max_iterations):
        z[m] = z[m] - a[m]*p(z[m])/dp(z[m])

        for j, root in enumerate(roots):
            converged = (np.abs(z.real - root.real) < epsilon) & (np.abs(z.imag - root.imag) < epsilon)
            m[converged] = False
            r[converged] = j + 1

        div_time[m] = i

    return div_time, r