code: def _ell(A, m):
      """
      A helper function for expm_2009.
  
      Parameters
      ----------
      A : linear operator
          A linear operator whose norm of power we care about.
      m : int
          The power of the linear operator
  
      Returns
      -------
      value : int
          A value related to a bound.
  
      """
      if len(A.shape) != 2 or A.shape[0] != A.shape[1]:
          raise ValueError("expected A to be like a square matrix")
  
      p = 2 * m + 1
  
      # The c_i are explained in (2.2) and (2.6) of the 2005 expm paper.
      # They are coefficients of terms of a generating function series expansion.
      choose_2p_p = scipy.special.comb(2 * p, p, exact=True)
      abs_c_recip = float(choose_2p_p * math.factorial(2 * p + 1))
  
      # This is explained after Eq. (1.2) of the 2009 expm paper.
      # It is the "unit roundoff" of IEEE double precision arithmetic.
      u = 2**-53
  
      # Compute the one-norm of matrix power p of abs(A).
      A_abs_onenorm = _onenorm_matrix_power_nnm(abs(A), p)
  
      # Treat zero norm as a special case.
      if not A_abs_onenorm:
          return 0
  
      alpha = A_abs_onenorm / (_onenorm(A) * abs_c_recip)
      log2_alpha_div_u = np.log2(alpha / u)
      value = int(np.ceil(log2_alpha_div_u / (2 * m)))
      return max(value, 0)
  
imports: from __future__ import division, print_function, absolute_import
  import math
  import numpy
  import scipy.special
  from scipy.linalg.basic import solve, solve_triangular
  from scipy.sparse.base import isspmatrix
  from scipy.sparse.construct import eye
  from scipy.sparse.linalg import spsolve
  import scipy.sparse
  import scipy.sparse.linalg
  from scipy.sparse.linalg.interface import LinearOperator
label_desc: Use image node_id 15 for calling a global function with example usage: _ell(A, m) and returns: max, int