docstr: Test optimized iterated integrability tensor derivatives.
  
  The optimized version of the iterated integrability tensor
  :math:`A_X A_Y A_X Y`, computed with the horizontal lift of
  quotient-parallel vector fields extending the tangent vectors
  :math:`X,Y` of Kendall shape spaces (identified to horizontal vectors
  of the pre-shape space), is the recursive application of two general
  integrability tensor derivatives with proper derivatives.
  Intermediate computations returned are also verified.
code: def test_iterated_integrability_tensor_derivative_parallel_optimized(
      self, n_points, atol
  ):
      """Test optimized iterated integrability tensor derivatives.
  
      The optimized version of the iterated integrability tensor
      :math:`A_X A_Y A_X Y`, computed with the horizontal lift of
      quotient-parallel vector fields extending the tangent vectors
      :math:`X,Y` of Kendall shape spaces (identified to horizontal vectors
      of the pre-shape space), is the recursive application of two general
      integrability tensor derivatives with proper derivatives.
      Intermediate computations returned are also verified.
      """
      base_point = self.data_generator.random_point(n_points)
      horizontal_vec_x = self._get_horizontal_vec(base_point)
      horizontal_vec_y = self._get_horizontal_vec(base_point)
  
      a_x_y = self.bundle.integrability_tensor(
          horizontal_vec_x, horizontal_vec_y, base_point
      )
      nabla_x_v, a_x_y = self.bundle.integrability_tensor_derivative(
          horizontal_vec_x,
          horizontal_vec_x,
          gs.zeros_like(horizontal_vec_x),
          horizontal_vec_y,
          a_x_y,
          base_point,
      )
  
      (
          nabla_x_a_y_a_x_y,
          a_y_a_x_y,
      ) = self.bundle.integrability_tensor_derivative(
          horizontal_vec_x,
          horizontal_vec_y,
          a_x_y,
          a_x_y,
          nabla_x_v,
          base_point,
      )
  
      a_x_a_y_a_x_y = self.bundle.integrability_tensor(
          horizontal_vec_x, a_y_a_x_y, base_point
      )
  
      (
          nabla_x_a_y_a_x_y_qp,
          a_x_a_y_a_x_y_qp,
          nabla_x_v_qp,
          a_y_a_x_y_qp,
          ver_v_qp,
      ) = self.bundle.iterated_integrability_tensor_derivative_parallel(
          horizontal_vec_x, horizontal_vec_y, base_point
      )
      self.assertAllClose(a_x_y, ver_v_qp, atol=atol)
      self.assertAllClose(a_y_a_x_y, a_y_a_x_y_qp, atol=atol)
      self.assertAllClose(nabla_x_v, nabla_x_v_qp, atol=atol)
      self.assertAllClose(a_x_a_y_a_x_y, a_x_a_y_a_x_y_qp, atol=atol)
      self.assertAllClose(
          nabla_x_a_y_a_x_y, nabla_x_a_y_a_x_y_qp, atol=atol
      )
  
imports: import pytest
  import geomstats.backend
  from geomstats.geometry.matrices import Matrices
  from geomstats.test.random import KendalShapeRandomDataGenerator, get_random_times
  from geomstats.test.vectorization import generate_vectorization_data
  from geomstats.test_cases.geometry.base import LevelSetTestCase
  from geomstats.test_cases.geometry.fiber_bundle import FiberBundleTestCase
  from geomstats.test_cases.geometry.quotient_metric import QuotientMetricTestCase
  from geomstats.vectorization import get_batch_shape
label_desc: Use image node_id 22 for calling the PreShapeSpaceBundleTestCase obj's underlying member method code with example usage: obj.test_iterated_integrability_tensor_derivative_parallel_optimized(n_points, atol) without return types