code: def test_linear_ode_dense(self, solver):
      initial_time = 0.0
      jacobian = -np.float64(
          [[1.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]]
      )
      num_odes = jacobian.shape[0]
      initial_state_value = np.float64([1.0] * num_odes)
      initial_state = tf.constant(initial_state_value, dtype=tf.float64)
  
      def ode_fn(_, state):
          return tf.squeeze(tf.matmul(jacobian, state[:, tf.newaxis]))
  
      intermediate_time = 1.0
      final_time = 2.0
      solver_instance = solver(rtol=_RTOL, atol=_ATOL)
  
      def grad_fn(initial_state):
          results = solver_instance.solve(
              ode_fn,
              initial_time,
              initial_state,
              solution_times=[intermediate_time, final_time],
          )
          intermediate_state = results.states[0]
          return intermediate_state
  
      grad = self.evaluate(
          tfp_gradient.value_and_gradient(grad_fn, initial_state)[1]
      )
      matrix_exponential_of_jacobian = np.float64(
          [
              [
                  +2.3703878775011322,
                  +0.2645063729368097,
                  -0.8413751316275110,
              ],
              [
                  -0.0900545996427410,
                  +0.7326649140674798,
                  -0.4446155722222950,
              ],
              [
                  -1.5504970767866180,
                  -0.7991765448018465,
                  +0.9521439871829201,
              ],
          ]
      )
      grad_exact = np.dot(
          np.ones([num_odes]), matrix_exponential_of_jacobian
      )
      self.assertAllClose(grad, grad_exact)
  
imports: from absl.testing import parameterized
  import numpy
  import tensorflow.compat.v1
  import tensorflow.compat.v2
  from tensorflow_probability.python.internal import test_util
  from tensorflow_probability.python.math import gradient
  from tensorflow_probability.python.math.ode import base
  from tensorflow_probability.python.math.ode import bdf
  from tensorflow_probability.python.math.ode import dormand_prince
label_desc: Use image node_id 1 for calling the GradientTestPforJacobian obj's underlying member method code with example usage: obj.test_linear_ode_dense(solver) and returns: tf, intermediate_state