code: def test_event_dense_output_LSODA():
      def event_lsoda(t, y):
          return y[0] - 2.02e-5
  
      rtol = 1e-3
      atol = 1e-6
      y0 = [0.05]
      t_span = [-2, 2]
      first_step = 1e-3
      res = solve_ivp(
          fun_event_dense_output_LSODA,
          t_span,
          y0,
          method="LSODA",
          dense_output=True,
          events=event_lsoda,
          first_step=first_step,
          max_step=1,
          rtol=rtol,
          atol=atol,
          jac=jac_event_dense_output_LSODA,
      )
  
      assert_equal(res.t[0], t_span[0])
      assert_equal(res.t[-1], t_span[-1])
      assert_allclose(first_step, np.abs(res.t[1] - t_span[0]))
      assert res.success
      assert_equal(res.status, 0)
  
      y_true = sol_event_dense_output_LSODA(res.t)
      e = compute_error(res.y, y_true, rtol, atol)
      assert_array_less(e, 5)
  
      tc = np.linspace(*t_span)
      yc_true = sol_event_dense_output_LSODA(tc)
      yc = res.sol(tc)
      e = compute_error(yc, yc_true, rtol, atol)
      assert_array_less(e, 5)
  
      assert_allclose(res.sol(res.t), res.y, rtol=1e-15, atol=1e-15)
  
imports: from itertools import product
  from numpy.testing import assert_, assert_allclose, assert_array_less, assert_equal, assert_no_warnings, suppress_warnings
  import pytest
  from pytest import raises
  import numpy
  from scipy.optimize._numdiff import group_columns
  from scipy.integrate import solve_ivp, RK23, RK45, DOP853, Radau, BDF, LSODA
  from scipy.integrate import OdeSolution
  from scipy.integrate._ivp.common import num_jac
  from scipy.integrate._ivp.base import ConstantDenseOutput
  from scipy.sparse import coo_matrix, csc_matrix
label_desc: Use image node_id 31 for calling a global function with example usage: test_event_dense_output_LSODA() and returns: unknown