text: import jax
  import jax.numpy as jnp
  import numpy as np
  import pytest
  from scipy.integrate import solve_ivp
  
  import tornadox
  
  
  @pytest.fixture
  def ivp():
      return tornadox.ivp.vanderpol(t0=0.0, tmax=0.25, stiffness_constant=1.0)
  
  
  @pytest.fixture
  def d(ivp):
      return ivp.y0.shape[0]
  
  
  @pytest.fixture
  def steps():
      return tornadox.step.AdaptiveSteps(abstol=1e-3, reltol=1e-3)
  
  
  @pytest.fixture
  def num_derivatives():
      return 2
  
  
  @pytest.fixture
  def ek0_solution(ek0_version, num_derivatives, ivp, steps):
      ek0 = ek0_version(num_derivatives=num_derivatives, steprule=steps)
      state, _ = ek0.simulate_final_state(ivp=ivp)
  
      final_t_ek0 = state.t
      final_y_ek0 = state.y.mean[0]
      return final_t_ek0, final_y_ek0
  
  
  @pytest.fixture
  def scipy_solution(ivp):
      scipy_sol = solve_ivp(ivp.f, t_span=(ivp.t0, ivp.tmax), y0=ivp.y0)
      final_t_scipy = scipy_sol.t[-1]
      final_y_scipy = scipy_sol.y[:, -1]
      return final_t_scipy, final_y_scipy
  
  
  # Tests for full solves.
  
  
  # Handy abbreviation for the long parametrize decorator
  EK0_VERSIONS = [
      tornadox.ek0.ReferenceEK0,
      tornadox.ek0.KroneckerEK0,
      tornadox.ek0.DiagonalEK0,
  ]
  all_ek0_versions = pytest.mark.parametrize("ek0_version", EK0_VERSIONS)
  
  
  @all_ek0_versions
  def test_full_solve_compare_scipy(ek0_solution, scipy_solution):
      """Assert the ODEFilter solves an ODE appropriately."""
      final_t_scipy, final_y_scipy = scipy_solution
      final_t_ek0, final_y_ek0 = ek0_solution
  
      assert jnp.allclose(final_t_scipy, final_t_ek0)
      assert jnp.allclose(final_y_scipy, final_y_ek0, rtol=1e-3, atol=1e-3)
  
  
  @all_ek0_versions
  def test_info_dict(ek0_version, ivp, num_derivatives):
      """Assert the ODEFilter solves an ODE appropriately."""
      num_steps = 5
      steprule = tornadox.step.ConstantSteps((ivp.tmax - ivp.t0) / num_steps)
      ek0 = ek0_version(num_derivatives=num_derivatives, steprule=steprule)
      _, info = ek0.simulate_final_state(ivp=ivp)
      assert info["num_f_evaluations"] == num_steps
      assert info["num_steps"] == num_steps
      assert info["num_attempted_steps"] == num_steps
      assert info["num_f_evaluations"] == num_steps
      assert info["num_df_evaluations"] == 0
      assert info["num_df_diagonal_evaluations"] == 0
  
  
  # Test fixtures for attempt_step and initialize
  
  
  @pytest.fixture
  def solver_tuple(steps, num_derivatives, d):
      reference_ek0 = tornadox.ek0.ReferenceEK0(
          num_derivatives=num_derivatives, steprule=steps
      )
      kronecker_ek0 = tornadox.ek0.KroneckerEK0(
          num_derivatives=num_derivatives, steprule=steps
      )
  
      return kronecker_ek0, reference_ek0
  
  
  @pytest.fixture
  def initialized_both(solver_tuple, ivp):
      kronecker_ek0, reference_ek0 = solver_tuple
  
      kronecker_init = kronecker_ek0.initialize(*ivp)
      reference_init = reference_ek0.initialize(*ivp)
  
      return kronecker_init, reference_init
  
  
  @pytest.fixture
  def stepped_both(solver_tuple, ivp, initialized_both):
  
      kronecker_ek0, reference_ek0 = solver_tuple
      kronecker_init, reference_init = initialized_both
  
      kronecker_stepped, _ = kronecker_ek0.attempt_step(kronecker_init, 0.12345, *ivp)
      reference_stepped, _ = reference_ek0.attempt_step(reference_init, 0.12345, *ivp)
  
      return kronecker_stepped, reference_stepped
  
  
  # Tests for initialize
  
  
  def test_init_type(initialized_both):
      kronecker_init, _ = initialized_both
      assert isinstance(kronecker_init.y, tornadox.rv.LeftIsotropicMatrixNormal)
  
  
  def test_init_values(initialized_both, d):
      kronecker_init, reference_init = initialized_both
  
      kron_mean = kronecker_init.y.mean
      kron_cov_sqrtm = kronecker_init.y.dense_cov_sqrtm()
      kron_cov = kronecker_init.y.dense_cov()
      assert jnp.allclose(kronecker_init.t, reference_init.t)
      assert jnp.allclose(kron_mean, reference_init.y.mean)
      assert jnp.allclose(kron_cov_sqrtm, reference_init.y.cov_sqrtm)
      assert jnp.allclose(kron_cov, reference_init.y.cov)
  
  
  def test_init_shape_kronecker(initialized_both, d, num_derivatives):
      kronecker_init, _ = initialized_both
  
      # shorthand
      n = num_derivatives + 1
      y = kronecker_init.y
      m = y.mean
      sc1, sc2 = y.cov_sqrtm_1, y.cov_sqrtm_2
      c1, c2 = y.cov_1, y.cov_2
  
      assert m.shape == (n, d)
      assert sc2.shape == (n, n)
      assert sc1.shape == (d, d)
      assert c1.shape == (d, d)
      assert c2.shape == (n, n)
  
  
  def test_init_shape_reference(initialized_both, d, num_derivatives):
      _, reference_init = initialized_both
  
      # shorthand
      n = num_derivatives + 1
      y = reference_init.y
      m, sc, c = y.mean, y.cov_sqrtm, y.cov
  
      assert m.shape == (n, d)
      assert sc.shape == (d * n, d * n)
      assert c.shape == (d * n, d * n)
  
  
  # Tests for each attempt step
  
  
  @pytest.fixture
  def stepped_kronecker(stepped_both):
      stepped_kron, _ = stepped_both
      return stepped_kron
  
  
  @pytest.fixture
  def stepped_reference(stepped_both):
      _, stepped_reference = stepped_both
      return stepped_reference
  
  
  # Test for shapes of output
  
  
  def test_attempt_step_y_type(stepped_kronecker):
      assert isinstance(stepped_kronecker.y, tornadox.rv.LeftIsotropicMatrixNormal)
  
  
  def test_attempt_step_y_shapes_kronecker(stepped_kronecker, d, num_derivatives):
      n = num_derivatives + 1
      y = stepped_kronecker.y
      m = y.mean
      sc1, sc2 = y.cov_sqrtm_1, y.cov_sqrtm_2
      c1, c2 = y.cov_1, y.cov_2
  
      assert m.shape == (n, d)
      assert sc1.shape == (d, d)
      assert sc2.shape == (n, n)
      assert c1.shape == (d, d)
      assert c2.shape == (n, n)
  
  
  def test_attempt_step_y_shapes_reference(stepped_reference, d, num_derivatives):
      n = num_derivatives + 1
      assert stepped_reference.y.mean.shape == (n, d)
      assert stepped_reference.y.cov_sqrtm.shape == (d * n, d * n)
      assert stepped_reference.y.cov.shape == (d * n, d * n)
  
  
  def test_attempt_step_error_estimate_kronecker(stepped_kronecker, d):
  
      assert isinstance(stepped_kronecker.error_estimate, jnp.ndarray)
      assert stepped_kronecker.error_estimate.shape == ()
      assert jnp.all(stepped_kronecker.error_estimate >= 0)
  
  
  def test_attempt_step_error_estimate_reference(stepped_reference, d):
  
      assert isinstance(stepped_reference.error_estimate, jnp.ndarray)
      assert stepped_reference.error_estimate.shape == (d,)
      assert jnp.all(stepped_reference.error_estimate >= 0)
  
  
  def test_attempt_step_reference_state_kronecker(stepped_kronecker, d):
  
      assert isinstance(stepped_kronecker.reference_state, jnp.ndarray)
      assert stepped_kronecker.reference_state.shape == (d,)
      assert jnp.all(stepped_kronecker.reference_state >= 0)
  
  
  def test_attempt_step_reference_state_reference(stepped_reference, d):
  
      assert isinstance(stepped_reference.reference_state, jnp.ndarray)
      assert stepped_reference.reference_state.shape == (d,)
      assert jnp.all(stepped_reference.reference_state >= 0)
  
  
  # Test for values of output
  
  
  def test_attempt_step_values_y_mean(stepped_kronecker, stepped_reference):
      m1, m2 = stepped_reference.y.mean, stepped_kronecker.y.mean
      assert jnp.allclose(m1, m2)
  
  
  def test_attempt_step_values_y_cov(stepped_kronecker, stepped_reference, d):
      c1, c2 = stepped_reference.y.cov, stepped_kronecker.y.dense_cov()
      assert jnp.allclose(c1, c2)
  
  
  def test_attempt_step_values_y_error_estimate(stepped_kronecker, stepped_reference, d):
      e1, e2 = stepped_reference.error_estimate, stepped_kronecker.error_estimate
      assert jnp.allclose(e1, e2)
  
  
  def test_attempt_step_values_y_reference_state(stepped_kronecker, stepped_reference, d):
      r1, r2 = stepped_reference.reference_state, stepped_kronecker.reference_state
      assert jnp.allclose(r1, r2)
  