code: def _eq_10_42(lam_1, lam_2, t_12):
      """
      Equation (10.42) of Functions of Matrices: Theory and Computation.
  
      Notes
      -----
      This is a helper function for _fragment_2_1 of expm_2009.
      Equation (10.42) is on page 251 in the section on Schur algorithms.
      In particular, section 10.4.3 explains the Schur-Parlett algorithm.
      expm([[lam_1, t_12], [0, lam_1])
      =
      [[exp(lam_1), t_12*exp((lam_1 + lam_2)/2)*sinch((lam_1 - lam_2)/2)],
      [0, exp(lam_2)]
      """
  
      # The plain formula t_12 * (exp(lam_2) - exp(lam_2)) / (lam_2 - lam_1)
      # apparently suffers from cancellation, according to Higham's textbook.
      # A nice implementation of sinch, defined as sinh(x)/x,
      # will apparently work around the cancellation.
      a = 0.5 * (lam_1 + lam_2)
      b = 0.5 * (lam_1 - lam_2)
      return t_12 * _exp_sinch(a, b)
  
imports: import numpy
  from scipy.linalg._basic import solve, solve_triangular
  from scipy.sparse._base import issparse
  from scipy.sparse.linalg import spsolve
  from scipy.sparse._sputils import is_pydata_spmatrix, isintlike
  import scipy.sparse
  import scipy.sparse.linalg
  from scipy.sparse.linalg._interface import LinearOperator
  from scipy.sparse._construct import eye
  from ._expm_multiply import _ident_like, _exact_1_norm
label_desc: Use image node_id 11 for calling a global function with example usage: _eq_10_42(lam_1, lam_2, t_12) and returns: unknown