How to render Mandelbrot Set faster?

Question

I\'m currently drawing the Mandelbrot set pixel by pixel with PhotoImage and tkinter. I\'m using basically the algorithm directly with no modifications. Are there methods to

Answer 1

Complex numbers in python can be slow, especially if you call abs(x) every iteration. Representing the complex number with c_r and c_i, for real and imaginary parts, reduces the number of calculations you do every iteration.

def mandel(c):
    z = 0
    for i in range(ITERATIONS):
        z = z**2 + c
        if abs(z) > 2:
            return i     
    return ITERATIONS

instead of z = 0, replace it with z_r,z_i=0,0 we also have to change c in the parameters. Now we have:

def mandel(c_r,c_i):
    z_r = 0
    z_i = 0
    for i in range(ITERATIONS):
        z = z**2 + c
        if abs(z) > 2:
            return i     
    return ITERATIONS

Instead of using abs(z) > 2, we can now use z_r * z_r + z_i + z_i > 4 Also, we replace z**2 + c with a new version using our new variables (Know that (a+bi)^2 = a^2 - b^2 + 2abi

def mandel(c_r,c_i):
    z_r = 0
    z_i = 0
    z_r_squared = 0
    z_i_squared = 0
    for i in range(ITERATIONS):
        z_r_squared = z_r * z_r
        z_i_squared = z_i * z_i
        z_r = z_r_squared - z_i_squared + c_r
        z_i = 2 * z_r * z_i + c_i

        if z_r_squared + z_r_squared > 4:
            return i     
    return ITERATIONS

Finally, you have to change where the mandelbrot function is called, so

i = mandel(complex(real, imag))

becomes

i = mandel(real, imag)