我的 Tetration（复数）函数必须更好地矢量化（Python）

Question

Inspired by 3blue1brown I'm trying to graph the escape (divergence) of the Tetration function with Python –– something akin to this beautiful graphic on Wikipedia.

def tetration_com(base, tol=10**-15, max_step=10**6):
  # returns t, the infinite tetration of base.
  # if t does not converge, the function returns an escape value, aka how fast it diverges..

  t = 1.0
  step = 0
  escape = None
  ln_base = cmath.log(base)

  t_last = 0
  try:
    while(abs(t - t_last) > tol):
      if(step > max_step):
        raise OverflowError
      t_last = t
      t = cmath.exp(ln_base*t)   # [ base^t == e^(ln(base)*t) ]
      step += 1

  except(OverflowError):
    t = None
    escape = 1000/step
    # the escape value is is inversely related to the number of steps it took
    # us to diverge to infinity

  return t, escape

我试图让它与网格一起工作，以便绘制 x-y 平面上的逃逸图。 Python 不喜欢输出是 2 个解压缩变量（限制或转义）——我绝对可以通过拆分成两个函数来解决这个问题。

但另一个问题是复杂的数学运算（cmath.log、cmath.exp）仅适用于标量...

我尝试将函数矢量化：

nx, ny = 700, 500
x, y = np.linspace(-3.5, 3.5, nx), np.linspace(-2.5, 2.5, ny)
xv, yv = np.meshgrid(x, y)

tetration_vec = np.vectorize(tetration_com)
t, escape = tetration_vec(xv + yv*1j, max_step=500)

但它永远运行。

关于如何处理复杂的数学运算和向量化有什么建议吗？

Answer 1

下面是我最后如何绘制逃生图：

def tetration_com(base, tol=10**-15, max_step=10**6, max_val=10**2):
  # returns t, the infinite tetration of base.
  # if t does not converge, the function returns an escape value
  # aka how fast it diverges..

  t = 1.0
  step = 0
  escape = None

  t_last = 0
  try:
    while(abs(t - t_last) > tol):
      if(step > max_step or abs(t) > max_val):
        raise OverflowError
      t_last = t
      t = pow(base, t)
      step += 1
  except(OverflowError):
    t = None
    escape = 1000/step
    # the escape value is is inversely related to the number of steps it took
    # us to diverge to infinity

  return t, escape

矢量化辅助函数：

def tetra_graph_escape(real, imag, tol=10**-15, max_step=10**3, max_val=10**2):
  return np.array([np.array([tetration_com(r + im*1j, tol=tol, max_step=max_step, max_val=max_val)[1]
                             for im in imag]) for r in real])

作图：

# graph our escape:
nx, ny = 700, 500
x, y = np.linspace(-3.5, 3.5, nx), np.linspace(-2.5, 2.5, ny)

val, escape = tetra_graph_conv(x, y), tetra_graph_escape(x, y)

import matplotlib.pyplot as plt

for r in range(len(escape)):
  for c in range(len(escape[0])):
    if escape[r][c] is None:
      escape[r][c] = -100

escape[460][250]

plt.contour(escape)

Tetration逃逸等高线图：

我的 Tetration（复数）函数必须更好地矢量化（Python）

My Tetration (complex-number) function has to be better vectorized (Python)

python

math

numpy

vectorization

complex-numbers