code: def _fixed_point(t, N, k_sq, a_sq):
      """Calculate t-zeta*gamma^[l](t).
  
      Implementation of the function t-zeta*gamma^[l](t) derived from equation (30) in [1].
  
      References
      ----------
      .. [1] Kernel density estimation via diffusion.
         Z. I. Botev, J. F. Grotowski, and D. P. Kroese.
         Ann. Statist. 38 (2010), no. 5, 2916--2957.
      """
      k_sq = np.asfarray(k_sq, dtype=np.float64)
      a_sq = np.asfarray(a_sq, dtype=np.float64)
  
      l = 7
      f = np.sum(
          np.power(k_sq, l) * a_sq * np.exp(-k_sq * np.pi**2 * t)
      )
      f *= 0.5 * np.pi ** (2.0 * l)
  
      for j in np.arange(l - 1, 2 - 1, -1):
          c1 = (1 + 0.5 ** (j + 0.5)) / 3
          c2 = np.prod(np.arange(1.0, 2 * j + 1, 2, dtype=np.float64))
          c2 /= (np.pi / 2) ** 0.5
          t_j = np.power((c1 * (c2 / (N * f))), (2.0 / (3.0 + 2.0 * j)))
          f = np.sum(
              k_sq**j * a_sq * np.exp(-k_sq * np.pi**2.0 * t_j)
          )
          f *= 0.5 * np.pi ** (2 * j)
  
      out = t - (2 * N * np.pi**0.5 * f) ** (-0.4)
      return out
  
imports: import warnings
  import numpy
  from scipy.fftpack import fft
  from scipy.optimize import brentq
  from scipy.signal import convolve, convolve2d, gaussian
  from scipy.sparse import coo_matrix
  from scipy.special import ive
  from ..utils import _cov, _dot, _stack, conditional_jit
label_desc: Use image node_id 11 for calling a global function with example usage: _fixed_point(t, N, k_sq, a_sq) and returns: out