docs: def kunc_k_num(v, v0, k0, k0p, order=5, precision=1.e-5):
      """
      calculate bulk modulus numerically from volume, not pressure
      according to test this differs from analytical result by 1.e-5
  
      :param v: unit-cell volume in A^3
      :param v0: unit-cell volume in A^3 at 1 bar
      :param k0: bulk modulus at reference conditions
      :param k0p: pressure derivative of bulk modulus at reference conditions
      :param precision: precision for numerical calc (default = 1.e-5 * v0)
      :return: bulk modulus
      """
      return -1. * v * kunc_dPdV(v, v0, k0, k0p, order=order,
                                 precision=precision)