code: def test_compute_fun_jac():
      x = np.linspace(0, 1, 5)
      y = np.empty((2, x.shape[0]))
      y[0] = 0.01
      y[1] = 0.02
      p = np.array([])
      df_dy, df_dp = estimate_fun_jac(
          lambda x, y, p: exp_fun(x, y), x, y, p
      )
      df_dy_an = exp_fun_jac(x, y)
      assert_allclose(df_dy, df_dy_an)
      assert_(df_dp is None)
  
      x = np.linspace(0, np.pi, 5)
      y = np.empty((2, x.shape[0]))
      y[0] = np.sin(x)
      y[1] = np.cos(x)
      p = np.array([1.0])
      df_dy, df_dp = estimate_fun_jac(sl_fun, x, y, p)
      df_dy_an, df_dp_an = sl_fun_jac(x, y, p)
      assert_allclose(df_dy, df_dy_an)
      assert_allclose(df_dp, df_dp_an)
  
      x = np.linspace(0, 1, 10)
      y = np.empty((2, x.shape[0]))
      y[0] = (3 / 4) ** 0.5
      y[1] = 1e-4
      p = np.array([])
      df_dy, df_dp = estimate_fun_jac(
          lambda x, y, p: emden_fun(x, y), x, y, p
      )
      df_dy_an = emden_fun_jac(x, y)
      assert_allclose(df_dy, df_dy_an)
      assert_(df_dp is None)
  
imports: import sys
  import numpy
  from numpy.testing import assert_, assert_array_equal, assert_allclose, assert_equal
  from pytest import raises
  from scipy.sparse import coo_matrix
  from scipy.special import erf
  from scipy.integrate._bvp import modify_mesh, estimate_fun_jac, estimate_bc_jac, compute_jac_indices, construct_global_jac, solve_bvp
label_desc: Use image node_id 34 for calling a global function with example usage: test_compute_fun_jac() without return types