text: from __future__ import division, print_function, absolute_import
  from itertools import product
  from numpy.testing import (assert_, assert_allclose,
                             assert_equal, assert_no_warnings)
  from pytest import raises as assert_raises
  from scipy._lib._numpy_compat import suppress_warnings
  import numpy as np
  from scipy.optimize._numdiff import group_columns
  from scipy.integrate import solve_ivp, RK23, RK45, 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
  
  
  def fun_linear(t, y):
      return np.array([-y[0] - 5 * y[1], y[0] + y[1]])
  
  
  def jac_linear():
      return np.array([[-1, -5], [1, 1]])
  
  
  def sol_linear(t):
      return np.vstack((-5 * np.sin(2 * t),
                        2 * np.cos(2 * t) + np.sin(2 * t)))
  
  
  def fun_rational(t, y):
      return np.array([y[1] / t,
                       y[1] * (y[0] + 2 * y[1] - 1) / (t * (y[0] - 1))])
  
  
  def fun_rational_vectorized(t, y):
      return np.vstack((y[1] / t,
                        y[1] * (y[0] + 2 * y[1] - 1) / (t * (y[0] - 1))))
  
  
  def jac_rational(t, y):
      return np.array([
          [0, 1 / t],
          [-2 * y[1] ** 2 / (t * (y[0] - 1) ** 2),
           (y[0] + 4 * y[1] - 1) / (t * (y[0] - 1))]
      ])
  
  
  def jac_rational_sparse(t, y):
      return csc_matrix([
          [0, 1 / t],
          [-2 * y[1] ** 2 / (t * (y[0] - 1) ** 2),
           (y[0] + 4 * y[1] - 1) / (t * (y[0] - 1))]
      ])
  
  
  def sol_rational(t):
      return np.asarray((t / (t + 10), 10 * t / (t + 10) ** 2))
  
  
  def fun_medazko(t, y):
      n = y.shape[0] // 2
      k = 100
      c = 4
  
      phi = 2 if t <= 5 else 0
      y = np.hstack((phi, 0, y, y[-2]))
  
      d = 1 / n
      j = np.arange(n) + 1
      alpha = 2 * (j * d - 1) ** 3 / c ** 2
      beta = (j * d - 1) ** 4 / c ** 2
  
      j_2_p1 = 2 * j + 2
      j_2_m3 = 2 * j - 2
      j_2_m1 = 2 * j
      j_2 = 2 * j + 1
  
      f = np.empty(2 * n)
      f[::2] = (alpha * (y[j_2_p1] - y[j_2_m3]) / (2 * d) +
                beta * (y[j_2_m3] - 2 * y[j_2_m1] + y[j_2_p1]) / d ** 2 -
                k * y[j_2_m1] * y[j_2])
      f[1::2] = -k * y[j_2] * y[j_2_m1]
  
      return f
  
  
  def medazko_sparsity(n):
      cols = []
      rows = []
  
      i = np.arange(n) * 2
  
      cols.append(i[1:])
      rows.append(i[1:] - 2)
  
      cols.append(i)
      rows.append(i)
  
      cols.append(i)
      rows.append(i + 1)
  
      cols.append(i[:-1])
      rows.append(i[:-1] + 2)
  
      i = np.arange(n) * 2 + 1
  
      cols.append(i)
      rows.append(i)
  
      cols.append(i)
      rows.append(i - 1)
  
      cols = np.hstack(cols)
      rows = np.hstack(rows)
  
      return coo_matrix((np.ones_like(cols), (cols, rows)))
  
  
  def fun_complex(t, y):
      return -y
  
  
  def jac_complex(t, y):
      return -np.eye(y.shape[0])
  
  
  def jac_complex_sparse(t, y):
      return csc_matrix(jac_complex(t, y))
  
  
  def sol_complex(t):
      y = (0.5 + 1j) * np.exp(-t)
      return y.reshape((1, -1))
  
  
  def compute_error(y, y_true, rtol, atol):
      e = (y - y_true) / (atol + rtol * np.abs(y_true))
      return np.sqrt(np.sum(np.real(e * e.conj()), axis=0) / e.shape[0])
  
  
  def test_integration():
      rtol = 1e-3
      atol = 1e-6
      y0 = [1/3, 2/9]
  
      for vectorized, method, t_span, jac in product(
              [False, True],
              ['RK23', 'RK45', 'Radau', 'BDF', 'LSODA'],
              [[5, 9], [5, 1]],
              [None, jac_rational, jac_rational_sparse]):
  
          if vectorized:
              fun = fun_rational_vectorized
          else:
              fun = fun_rational
  
          with suppress_warnings() as sup:
              sup.filter(UserWarning,
                         "The following arguments have no effect for a chosen solver: `jac`")
              res = solve_ivp(fun, t_span, y0, rtol=rtol,
                              atol=atol, method=method, dense_output=True,
                              jac=jac, vectorized=vectorized)
          assert_equal(res.t[0], t_span[0])
          assert_(res.t_events is None)
          assert_(res.success)
          assert_equal(res.status, 0)
  
          assert_(res.nfev < 40)
  
          if method in ['RK23', 'RK45', 'LSODA']:
              assert_equal(res.njev, 0)
              assert_equal(res.nlu, 0)
          else:
              assert_(0 < res.njev < 3)
              assert_(0 < res.nlu < 10)
  
          y_true = sol_rational(res.t)
          e = compute_error(res.y, y_true, rtol, atol)
          assert_(np.all(e < 5))
  
          tc = np.linspace(*t_span)
          yc_true = sol_rational(tc)
          yc = res.sol(tc)
  
          e = compute_error(yc, yc_true, rtol, atol)
          assert_(np.all(e < 5))
  
          tc = (t_span[0] + t_span[-1]) / 2
          yc_true = sol_rational(tc)
          yc = res.sol(tc)
  
          e = compute_error(yc, yc_true, rtol, atol)
          assert_(np.all(e < 5))
  
          # LSODA for some reasons doesn't pass the polynomial through the
          # previous points exactly after the order change. It might be some
          # bug in LSOSA implementation or maybe we missing something.
          if method != 'LSODA':
              assert_allclose(res.sol(res.t), res.y, rtol=1e-15, atol=1e-15)
  
  
  def test_integration_complex():
      rtol = 1e-3
      atol = 1e-6
      y0 = [0.5 + 1j]
      t_span = [0, 1]
      tc = np.linspace(t_span[0], t_span[1])
      for method, jac in product(['RK23', 'RK45', 'BDF'],
                                 [None, jac_complex, jac_complex_sparse]):
          with suppress_warnings() as sup:
              sup.filter(UserWarning,
                         "The following arguments have no effect for a chosen solver: `jac`")
              res = solve_ivp(fun_complex, t_span, y0, method=method,
                              dense_output=True, rtol=rtol, atol=atol, jac=jac)
  
          assert_equal(res.t[0], t_span[0])
          assert_(res.t_events is None)
          assert_(res.success)
          assert_equal(res.status, 0)
  
          assert_(res.nfev < 25)
          if method == 'BDF':
              assert_equal(res.njev, 1)
              assert_(res.nlu < 6)
          else:
              assert_equal(res.njev, 0)
              assert_equal(res.nlu, 0)
  
          y_true = sol_complex(res.t)
          e = compute_error(res.y, y_true, rtol, atol)
          assert_(np.all(e < 5))
  
          yc_true = sol_complex(tc)
          yc = res.sol(tc)
          e = compute_error(yc, yc_true, rtol, atol)
  
          assert_(np.all(e < 5))
  
  
  def test_integration_sparse_difference():
      n = 200
      t_span = [0, 20]
      y0 = np.zeros(2 * n)
      y0[1::2] = 1
      sparsity = medazko_sparsity(n)
  
      for method in ['BDF', 'Radau']:
          res = solve_ivp(fun_medazko, t_span, y0, method=method,
                          jac_sparsity=sparsity)
  
          assert_equal(res.t[0], t_span[0])
          assert_(res.t_events is None)
          assert_(res.success)
          assert_equal(res.status, 0)
  
          assert_allclose(res.y[78, -1], 0.233994e-3, rtol=1e-2)
          assert_allclose(res.y[79, -1], 0, atol=1e-3)
          assert_allclose(res.y[148, -1], 0.359561e-3, rtol=1e-2)
          assert_allclose(res.y[149, -1], 0, atol=1e-3)
          assert_allclose(res.y[198, -1], 0.117374129e-3, rtol=1e-2)
          assert_allclose(res.y[199, -1], 0.6190807e-5, atol=1e-3)
          assert_allclose(res.y[238, -1], 0, atol=1e-3)
          assert_allclose(res.y[239, -1], 0.9999997, rtol=1e-2)
  
  
  def test_integration_const_jac():
      rtol = 1e-3
      atol = 1e-6
      y0 = [0, 2]
      t_span = [0, 2]
      J = jac_linear()
      J_sparse = csc_matrix(J)
  
      for method, jac in product(['Radau', 'BDF'], [J, J_sparse]):
          res = solve_ivp(fun_linear, t_span, y0, rtol=rtol, atol=atol,
                          method=method, dense_output=True, jac=jac)
          assert_equal(res.t[0], t_span[0])
          assert_(res.t_events is None)
          assert_(res.success)
          assert_equal(res.status, 0)
  
          assert_(res.nfev < 100)
          assert_equal(res.njev, 0)
          assert_(0 < res.nlu < 15)
  
          y_true = sol_linear(res.t)
          e = compute_error(res.y, y_true, rtol, atol)
          assert_(np.all(e < 10))
  
          tc = np.linspace(*t_span)
          yc_true = sol_linear(tc)
          yc = res.sol(tc)
  
          e = compute_error(yc, yc_true, rtol, atol)
          assert_(np.all(e < 15))
  
          assert_allclose(res.sol(res.t), res.y, rtol=1e-14, atol=1e-14)
  
  
  def test_events():
      def event_rational_1(t, y):
          return y[0] - y[1] ** 0.7
  
      def event_rational_2(t, y):
          return y[1] ** 0.6 - y[0]
  
      def event_rational_3(t, y):
          return t - 7.4
  
      event_rational_3.terminal = True
  
      for method in ['RK23', 'RK45', 'Radau', 'BDF', 'LSODA']:
          res = solve_ivp(fun_rational, [5, 8], [1/3, 2/9], method=method,
                          events=(event_rational_1, event_rational_2))
          assert_equal(res.status, 0)
          assert_equal(res.t_events[0].size, 1)
          assert_equal(res.t_events[1].size, 1)
          assert_(5.3 < res.t_events[0][0] < 5.7)
          assert_(7.3 < res.t_events[1][0] < 7.7)
  
          event_rational_1.direction = 1
          event_rational_2.direction = 1
          res = solve_ivp(fun_rational, [5, 8], [1 / 3, 2 / 9], method=method,
                          events=(event_rational_1, event_rational_2))
          assert_equal(res.status, 0)
          assert_equal(res.t_events[0].size, 1)
          assert_equal(res.t_events[1].size, 0)
          assert_(5.3 < res.t_events[0][0] < 5.7)
  
          event_rational_1.direction = -1
          event_rational_2.direction = -1
          res = solve_ivp(fun_rational, [5, 8], [1 / 3, 2 / 9], method=method,
                          events=(event_rational_1, event_rational_2))
          assert_equal(res.status, 0)
          assert_equal(res.t_events[0].size, 0)
          assert_equal(res.t_events[1].size, 1)
          assert_(7.3 < res.t_events[1][0] < 7.7)
  
          event_rational_1.direction = 0
          event_rational_2.direction = 0
  
          res = solve_ivp(fun_rational, [5, 8], [1 / 3, 2 / 9], method=method,
                          events=(event_rational_1, event_rational_2,
                                  event_rational_3), dense_output=True)
          assert_equal(res.status, 1)
          assert_equal(res.t_events[0].size, 1)
          assert_equal(res.t_events[1].size, 0)
          assert_equal(res.t_events[2].size, 1)
          assert_(5.3 < res.t_events[0][0] < 5.7)
          assert_(7.3 < res.t_events[2][0] < 7.5)
  
          res = solve_ivp(fun_rational, [5, 8], [1 / 3, 2 / 9], method=method,
                          events=event_rational_1, dense_output=True)
          assert_equal(res.status, 0)
          assert_equal(res.t_events[0].size, 1)
          assert_(5.3 < res.t_events[0][0] < 5.7)
  
          # Also test that termination by event doesn't break interpolants.
          tc = np.linspace(res.t[0], res.t[-1])
          yc_true = sol_rational(tc)
          yc = res.sol(tc)
          e = compute_error(yc, yc_true, 1e-3, 1e-6)
          assert_(np.all(e < 5))
  
      # Test in backward direction.
      event_rational_1.direction = 0
      event_rational_2.direction = 0
      for method in ['RK23', 'RK45', 'Radau', 'BDF', 'LSODA']:
          res = solve_ivp(fun_rational, [8, 5], [4/9, 20/81], method=method,
                          events=(event_rational_1, event_rational_2))
          assert_equal(res.status, 0)
          assert_equal(res.t_events[0].size, 1)
          assert_equal(res.t_events[1].size, 1)
          assert_(5.3 < res.t_events[0][0] < 5.7)
          assert_(7.3 < res.t_events[1][0] < 7.7)
  
          event_rational_1.direction = -1
          event_rational_2.direction = -1
          res = solve_ivp(fun_rational, [8, 5], [4/9, 20/81], method=method,
                          events=(event_rational_1, event_rational_2))
          assert_equal(res.status, 0)
          assert_equal(res.t_events[0].size, 1)
          assert_equal(res.t_events[1].size, 0)
          assert_(5.3 < res.t_events[0][0] < 5.7)
  
          event_rational_1.direction = 1
          event_rational_2.direction = 1
          res = solve_ivp(fun_rational, [8, 5], [4/9, 20/81], method=method,
                          events=(event_rational_1, event_rational_2))
          assert_equal(res.status, 0)
          assert_equal(res.t_events[0].size, 0)
          assert_equal(res.t_events[1].size, 1)
          assert_(7.3 < res.t_events[1][0] < 7.7)
  
          event_rational_1.direction = 0
          event_rational_2.direction = 0
  
          res = solve_ivp(fun_rational, [8, 5], [4/9, 20/81], method=method,
                          events=(event_rational_1, event_rational_2,
                                  event_rational_3), dense_output=True)
          assert_equal(res.status, 1)
          assert_equal(res.t_events[0].size, 0)
          assert_equal(res.t_events[1].size, 1)
          assert_equal(res.t_events[2].size, 1)
          assert_(7.3 < res.t_events[1][0] < 7.7)
          assert_(7.3 < res.t_events[2][0] < 7.5)
  
          # Also test that termination by event doesn't break interpolants.
          tc = np.linspace(res.t[-1], res.t[0])
          yc_true = sol_rational(tc)
          yc = res.sol(tc)
          e = compute_error(yc, yc_true, 1e-3, 1e-6)
          assert_(np.all(e < 5))
  
  
  def test_max_step():
      rtol = 1e-3
      atol = 1e-6
      y0 = [1/3, 2/9]
      for method in [RK23, RK45, Radau, BDF, LSODA]:
          for t_span in ([5, 9], [5, 1]):
              res = solve_ivp(fun_rational, t_span, y0, rtol=rtol,
                              max_step=0.5, atol=atol, method=method,
                              dense_output=True)
              assert_equal(res.t[0], t_span[0])
              assert_equal(res.t[-1], t_span[-1])
              assert_(np.all(np.abs(np.diff(res.t)) <= 0.5))
              assert_(res.t_events is None)
              assert_(res.success)
              assert_equal(res.status, 0)
  
              y_true = sol_rational(res.t)
              e = compute_error(res.y, y_true, rtol, atol)
              assert_(np.all(e < 5))
  
              tc = np.linspace(*t_span)
              yc_true = sol_rational(tc)
              yc = res.sol(tc)
  
              e = compute_error(yc, yc_true, rtol, atol)
              assert_(np.all(e < 5))
  
              # See comment in test_integration.
              if method is not LSODA:
                  assert_allclose(res.sol(res.t), res.y, rtol=1e-15, atol=1e-15)
  
              assert_raises(ValueError, method, fun_rational, t_span[0], y0,
                            t_span[1], max_step=-1)
  
              if method is not LSODA:
                  solver = method(fun_rational, t_span[0], y0, t_span[1],
                                  rtol=rtol, atol=atol, max_step=1e-20)
                  message = solver.step()
  
                  assert_equal(solver.status, 'failed')
                  assert_("step size is less" in message)
                  assert_raises(RuntimeError, solver.step)
  
  
  def test_first_step():
      rtol = 1e-3
      atol = 1e-6
      y0 = [1/3, 2/9]
      first_step = 0.1
      for method in [RK23, RK45, Radau, BDF, LSODA]:
          for t_span in ([5, 9], [5, 1]):
              res = solve_ivp(fun_rational, t_span, y0, rtol=rtol,
                              max_step=0.5, atol=atol, method=method,
                              dense_output=True, first_step=first_step)
  
              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] - 5))
              assert_(res.t_events is None)
              assert_(res.success)
              assert_equal(res.status, 0)
  
              y_true = sol_rational(res.t)
              e = compute_error(res.y, y_true, rtol, atol)
              assert_(np.all(e < 5))
  
              tc = np.linspace(*t_span)
              yc_true = sol_rational(tc)
              yc = res.sol(tc)
  
              e = compute_error(yc, yc_true, rtol, atol)
              assert_(np.all(e < 5))
  
              # See comment in test_integration.
              if method is not LSODA:
                  assert_allclose(res.sol(res.t), res.y, rtol=1e-15, atol=1e-15)
  
              assert_raises(ValueError, method, fun_rational, t_span[0], y0,
                            t_span[1], first_step=-1)
              assert_raises(ValueError, method, fun_rational, t_span[0], y0,
                            t_span[1], first_step=5)
      
  
  def test_t_eval():
      rtol = 1e-3
      atol = 1e-6
      y0 = [1/3, 2/9]
      for t_span in ([5, 9], [5, 1]):
          t_eval = np.linspace(t_span[0], t_span[1], 10)
          res = solve_ivp(fun_rational, t_span, y0, rtol=rtol, atol=atol,
                          t_eval=t_eval)
          assert_equal(res.t, t_eval)
          assert_(res.t_events is None)
          assert_(res.success)
          assert_equal(res.status, 0)
  
          y_true = sol_rational(res.t)
          e = compute_error(res.y, y_true, rtol, atol)
          assert_(np.all(e < 5))
  
      t_eval = [5, 5.01, 7, 8, 8.01, 9]
      res = solve_ivp(fun_rational, [5, 9], y0, rtol=rtol, atol=atol,
                      t_eval=t_eval)
      assert_equal(res.t, t_eval)
      assert_(res.t_events is None)
      assert_(res.success)
      assert_equal(res.status, 0)
  
      y_true = sol_rational(res.t)
      e = compute_error(res.y, y_true, rtol, atol)
      assert_(np.all(e < 5))
  
      t_eval = [5, 4.99, 3, 1.5, 1.1, 1.01, 1]
      res = solve_ivp(fun_rational, [5, 1], y0, rtol=rtol, atol=atol,
                      t_eval=t_eval)
      assert_equal(res.t, t_eval)
      assert_(res.t_events is None)
      assert_(res.success)
      assert_equal(res.status, 0)
  
      t_eval = [5.01, 7, 8, 8.01]
      res = solve_ivp(fun_rational, [5, 9], y0, rtol=rtol, atol=atol,
                      t_eval=t_eval)
      assert_equal(res.t, t_eval)
      assert_(res.t_events is None)
      assert_(res.success)
      assert_equal(res.status, 0)
  
      y_true = sol_rational(res.t)
      e = compute_error(res.y, y_true, rtol, atol)
      assert_(np.all(e < 5))
  
      t_eval = [4.99, 3, 1.5, 1.1, 1.01]
      res = solve_ivp(fun_rational, [5, 1], y0, rtol=rtol, atol=atol,
                      t_eval=t_eval)
      assert_equal(res.t, t_eval)
      assert_(res.t_events is None)
      assert_(res.success)
      assert_equal(res.status, 0)
  
      t_eval = [4, 6]
      assert_raises(ValueError, solve_ivp, fun_rational, [5, 9], y0,
                    rtol=rtol, atol=atol, t_eval=t_eval)
  
  
  def test_t_eval_dense_output():
      rtol = 1e-3
      atol = 1e-6
      y0 = [1/3, 2/9]
      t_span = [5, 9]
      t_eval = np.linspace(t_span[0], t_span[1], 10)
      res = solve_ivp(fun_rational, t_span, y0, rtol=rtol, atol=atol,
                      t_eval=t_eval)
      res_d = solve_ivp(fun_rational, t_span, y0, rtol=rtol, atol=atol,
                        t_eval=t_eval, dense_output=True)
      assert_equal(res.t, t_eval)
      assert_(res.t_events is None)
      assert_(res.success)
      assert_equal(res.status, 0)
  
      assert_equal(res.t, res_d.t)
      assert_equal(res.y, res_d.y)
      assert_(res_d.t_events is None)
      assert_(res_d.success)
      assert_equal(res_d.status, 0)
  
      # if t and y are equal only test values for one case
      y_true = sol_rational(res.t)
      e = compute_error(res.y, y_true, rtol, atol)
      assert_(np.all(e < 5))
  
  
  def test_no_integration():
      for method in ['RK23', 'RK45', 'Radau', 'BDF', 'LSODA']:
          sol = solve_ivp(lambda t, y: -y, [4, 4], [2, 3],
                          method=method, dense_output=True)
          assert_equal(sol.sol(4), [2, 3])
          assert_equal(sol.sol([4, 5, 6]), [[2, 2, 2], [3, 3, 3]])
  
  
  def test_no_integration_class():
      for method in [RK23, RK45, Radau, BDF, LSODA]:
          solver = method(lambda t, y: -y, 0.0, [10.0, 0.0], 0.0)
          solver.step()
          assert_equal(solver.status, 'finished')
          sol = solver.dense_output()
          assert_equal(sol(0.0), [10.0, 0.0])
          assert_equal(sol([0, 1, 2]), [[10, 10, 10], [0, 0, 0]])
  
          solver = method(lambda t, y: -y, 0.0, [], np.inf)
          solver.step()
          assert_equal(solver.status, 'finished')
          sol = solver.dense_output()
          assert_equal(sol(100.0), [])
          assert_equal(sol([0, 1, 2]), np.empty((0, 3)))
  
  
  def test_empty():
      def fun(t, y):
          return np.zeros((0,))
  
      y0 = np.zeros((0,))
  
      for method in ['RK23', 'RK45', 'Radau', 'BDF', 'LSODA']:
          sol = assert_no_warnings(solve_ivp, fun, [0, 10], y0,
                                   method=method, dense_output=True)
          assert_equal(sol.sol(10), np.zeros((0,)))
          assert_equal(sol.sol([1, 2, 3]), np.zeros((0, 3)))
  
      for method in ['RK23', 'RK45', 'Radau', 'BDF', 'LSODA']:
          sol = assert_no_warnings(solve_ivp, fun, [0, np.inf], y0,
                                   method=method, dense_output=True)
          assert_equal(sol.sol(10), np.zeros((0,)))
          assert_equal(sol.sol([1, 2, 3]), np.zeros((0, 3)))
  
  
  def test_ConstantDenseOutput():
      sol = ConstantDenseOutput(0, 1, np.array([1, 2]))
      assert_allclose(sol(1.5), [1, 2])
      assert_allclose(sol([1, 1.5, 2]), [[1, 1, 1], [2, 2, 2]])
  
      sol = ConstantDenseOutput(0, 1, np.array([]))
      assert_allclose(sol(1.5), np.empty(0))
      assert_allclose(sol([1, 1.5, 2]), np.empty((0, 3)))
  
  
  def test_classes():
      y0 = [1 / 3, 2 / 9]
      for cls in [RK23, RK45, Radau, BDF, LSODA]:
          solver = cls(fun_rational, 5, y0, np.inf)
          assert_equal(solver.n, 2)
          assert_equal(solver.status, 'running')
          assert_equal(solver.t_bound, np.inf)
          assert_equal(solver.direction, 1)
          assert_equal(solver.t, 5)
          assert_equal(solver.y, y0)
          assert_(solver.step_size is None)
          if cls is not LSODA:
              assert_(solver.nfev > 0)
              assert_(solver.njev >= 0)
              assert_equal(solver.nlu, 0)
          else:
              assert_equal(solver.nfev, 0)
              assert_equal(solver.njev, 0)
              assert_equal(solver.nlu, 0)
  
          assert_raises(RuntimeError, solver.dense_output)
  
          message = solver.step()
          assert_equal(solver.status, 'running')
          assert_equal(message, None)
          assert_equal(solver.n, 2)
          assert_equal(solver.t_bound, np.inf)
          assert_equal(solver.direction, 1)
          assert_(solver.t > 5)
          assert_(not np.all(np.equal(solver.y, y0)))
          assert_(solver.step_size > 0)
          assert_(solver.nfev > 0)
          assert_(solver.njev >= 0)
          assert_(solver.nlu >= 0)
          sol = solver.dense_output()
          assert_allclose(sol(5), y0, rtol=1e-15, atol=0)
  
  
  def test_OdeSolution():
      ts = np.array([0, 2, 5], dtype=float)
      s1 = ConstantDenseOutput(ts[0], ts[1], np.array([-1]))
      s2 = ConstantDenseOutput(ts[1], ts[2], np.array([1]))
  
      sol = OdeSolution(ts, [s1, s2])
  
      assert_equal(sol(-1), [-1])
      assert_equal(sol(1), [-1])
      assert_equal(sol(2), [-1])
      assert_equal(sol(3), [1])
      assert_equal(sol(5), [1])
      assert_equal(sol(6), [1])
  
      assert_equal(sol([0, 6, -2, 1.5, 4.5, 2.5, 5, 5.5, 2]),
                   np.array([[-1, 1, -1, -1, 1, 1, 1, 1, -1]]))
  
      ts = np.array([10, 4, -3])
      s1 = ConstantDenseOutput(ts[0], ts[1], np.array([-1]))
      s2 = ConstantDenseOutput(ts[1], ts[2], np.array([1]))
  
      sol = OdeSolution(ts, [s1, s2])
      assert_equal(sol(11), [-1])
      assert_equal(sol(10), [-1])
      assert_equal(sol(5), [-1])
      assert_equal(sol(4), [-1])
      assert_equal(sol(0), [1])
      assert_equal(sol(-3), [1])
      assert_equal(sol(-4), [1])
  
      assert_equal(sol([12, -5, 10, -3, 6, 1, 4]),
                   np.array([[-1, 1, -1, 1, -1, 1, -1]]))
  
      ts = np.array([1, 1])
      s = ConstantDenseOutput(1, 1, np.array([10]))
      sol = OdeSolution(ts, [s])
      assert_equal(sol(0), [10])
      assert_equal(sol(1), [10])
      assert_equal(sol(2), [10])
  
      assert_equal(sol([2, 1, 0]), np.array([[10, 10, 10]]))
  
  
  def test_num_jac():
      def fun(t, y):
          return np.vstack([
              -0.04 * y[0] + 1e4 * y[1] * y[2],
              0.04 * y[0] - 1e4 * y[1] * y[2] - 3e7 * y[1] ** 2,
              3e7 * y[1] ** 2
          ])
  
      def jac(t, y):
          return np.array([
              [-0.04, 1e4 * y[2], 1e4 * y[1]],
              [0.04, -1e4 * y[2] - 6e7 * y[1], -1e4 * y[1]],
              [0, 6e7 * y[1], 0]
          ])
  
      t = 1
      y = np.array([1, 0, 0])
      J_true = jac(t, y)
      threshold = 1e-5
      f = fun(t, y).ravel()
  
      J_num, factor = num_jac(fun, t, y, f, threshold, None)
      assert_allclose(J_num, J_true, rtol=1e-5, atol=1e-5)
  
      J_num, factor = num_jac(fun, t, y, f, threshold, factor)
      assert_allclose(J_num, J_true, rtol=1e-5, atol=1e-5)
  
  
  def test_num_jac_sparse():
      def fun(t, y):
          e = y[1:]**3 - y[:-1]**2
          z = np.zeros(y.shape[1])
          return np.vstack((z, 3 * e)) + np.vstack((2 * e, z))
  
      def structure(n):
          A = np.zeros((n, n), dtype=int)
          A[0, 0] = 1
          A[0, 1] = 1
          for i in range(1, n - 1):
              A[i, i - 1: i + 2] = 1
          A[-1, -1] = 1
          A[-1, -2] = 1
  
          return A
  
      np.random.seed(0)
      n = 20
      y = np.random.randn(n)
      A = structure(n)
      groups = group_columns(A)
  
      f = fun(0, y[:, None]).ravel()
  
      # Compare dense and sparse results, assuming that dense implementation
      # is correct (as it is straightforward).
      J_num_sparse, factor_sparse = num_jac(fun, 0, y.ravel(), f, 1e-8, None,
                                            sparsity=(A, groups))
      J_num_dense, factor_dense = num_jac(fun, 0, y.ravel(), f, 1e-8, None)
      assert_allclose(J_num_dense, J_num_sparse.toarray(),
                      rtol=1e-12, atol=1e-14)
      assert_allclose(factor_dense, factor_sparse, rtol=1e-12, atol=1e-14)
  
      # Take small factors to trigger their recomputing inside.
      factor = np.random.uniform(0, 1e-12, size=n)
      J_num_sparse, factor_sparse = num_jac(fun, 0, y.ravel(), f, 1e-8, factor,
                                            sparsity=(A, groups))
      J_num_dense, factor_dense = num_jac(fun, 0, y.ravel(), f, 1e-8, factor)
  
      assert_allclose(J_num_dense, J_num_sparse.toarray(),
                      rtol=1e-12, atol=1e-14)
      assert_allclose(factor_dense, factor_sparse, rtol=1e-12, atol=1e-14)
  
  