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# Example z^3 - 1 & z^2 - 2z - 2 fractal
# Written by Austin Walters
from PIL import Image
# -1 <= x <= 1
x_min = -1.0
x_max = 1.0
# -i <= y <= i
y_min = -1.0
y_max = 1.0
colors = [\
(180, 0, 30), (0, 180, 30), (0, 30, 180), \
(0, 190, 180), (180, 0, 175), (180, 255, 0), \
(155, 170, 180), (70, 50, 0), (255, 255, 255)]
def cubed(z): return z * z * z - 1.0
def dcubed(z): return 3.0 * (z*z)
def squared(z): return (z*z) - (2.0 * z) + 2.0
def dsquared(z): return 2.0 * z - 2.0
def cubedplus(z): return z * z * z - 2.0 * z + 2.0
def dcubedplus(z): return 3.0 * z * z - 2.0
def fun(z): return (z*z*z*z*z*z) - (z*z*z) + 2.0
def dfun(z): return 6.0 * (z*z*z*z*z) - 3.0 * (z*z)
def fifth(z): return (z*z*z*z*z) - (z)
def dfifth(z): return 5.0 * (z*z*z*z) - 1
def newtons_method(f, f_prime, z):
# Fourty iterations for safe measure
for i in range(40):
zplus = z - f(z)/f_prime(z)
# Checks for Overflow
if abs(f(z)) > 10e10:
return None
# Checks for underflow
if abs(f(z)) < 10e-14:
return None
# Checks for convergence
if abs(zplus - z) < 10e-4:
return z
z = zplus
return None
# Draws the functions
def draw(f, f_prime, img, size, img_name):
roots = []
for y in range(size):
z_y = y * (y_max - y_min)/(size - 1) + y_min
for x in range(size):
z_x = x * (x_max - x_min)/(size - 1) + x_min
root = newtons_method(f, f_prime, complex(z_x, z_y))
if root:
flag = False
for test_root in roots:
if abs(test_root - root) < 10e-3:
root = test_root
flag = True
break
if not flag:
roots.append(root)
if root:
img.putpixel((x, y), colors[roots.index(root)])
print roots
img.save(img_name, "PNG")
size = 1024
img = Image.new("RGB", (size, size), (255, 255, 255))
draw(lambda z: cubed(z), lambda z: dcubed(z), img, size, "fig1.png");
draw(lambda z: squared(z), lambda z: dsquared(z), img, size, "fig2.png");
draw(lambda z: fun(z), lambda z: dfun(z), img, size, "fig3.png");
draw(lambda z: cubedplus(z), lambda z: dcubedplus(z), img, size, "fig4.png");
draw(lambda z: fifth(z), lambda z: dfifth(z), img, size, "fig5.png");
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