code: def test_cauchypoint_equalsto_newtonpoint(self):
      A = np.array([[1, 8]])
      b = np.array([-16])
      _, _, Y = projections(A)
      newton_point = np.array([0.24615385, 1.96923077])
  
      # Newton point inside boundaries
      x = modified_dogleg(
          A, Y, b, 2, [-np.inf, -np.inf], [np.inf, np.inf]
      )
      assert_array_almost_equal(x, newton_point)
  
      # Spherical constraint active
      x = modified_dogleg(
          A, Y, b, 1, [-np.inf, -np.inf], [np.inf, np.inf]
      )
      assert_array_almost_equal(
          x, newton_point / np.linalg.norm(newton_point)
      )
  
      # Box Constraints active
      x = modified_dogleg(A, Y, b, 2, [-np.inf, -np.inf], [0.1, np.inf])
      assert_array_almost_equal(
          x, (newton_point / newton_point[0]) * 0.1
      )
  
imports: import numpy
  from scipy.sparse import csc_matrix
  from scipy.optimize._trustregion_constr.qp_subproblem import eqp_kktfact, projected_cg, box_intersections, sphere_intersections, box_sphere_intersections, modified_dogleg
  from scipy.optimize._trustregion_constr.projections import projections
  from numpy.testing import TestCase, assert_array_almost_equal, assert_array_equal, assert_array_less, assert_equal, assert_, run_module_suite, assert_allclose, assert_warns, dec
  import pytest
label_desc: Use image node_id 1 for calling the TestModifiedDogleg obj's underlying member method code with example usage: obj.test_cauchypoint_equalsto_newtonpoint() without return types