code: def test_newton_combined(self):
      f1 = lambda x: x**2 - 2 * x - 1
      f1_1 = lambda x: 2 * x - 2
      f1_2 = lambda x: 2.0 + 0 * x
  
      def f1_and_p_and_pp(x):
          return x**2 - 2 * x - 1, 2 * x - 2, 2.0
  
      sol0 = root_scalar(f1, method="newton", x0=3, fprime=f1_1)
      sol = root_scalar(
          f1_and_p_and_pp, method="newton", x0=3, fprime=True
      )
      assert_allclose(sol0.root, sol.root, atol=1e-8)
      assert_equal(2 * sol.function_calls, sol0.function_calls)
  
      sol0 = root_scalar(
          f1, method="halley", x0=3, fprime=f1_1, fprime2=f1_2
      )
      sol = root_scalar(
          f1_and_p_and_pp, method="halley", x0=3, fprime2=True
      )
      assert_allclose(sol0.root, sol.root, atol=1e-8)
      assert_equal(3 * sol.function_calls, sol0.function_calls)
  
imports: import pytest
  from math import sqrt, exp, sin, cos
  from functools import lru_cache
  from numpy.testing import assert_warns, assert_, assert_allclose, assert_equal, assert_array_equal, suppress_warnings
  import numpy
  from numpy import finfo, power, nan, isclose
  from scipy.optimize import zeros, newton, root_scalar
  from scipy._lib._util import getfullargspec_no_self
  from scipy.optimize._tstutils import get_tests, functions, fstrings
label_desc: Use image node_id 30 for calling the TestBasic obj's underlying member method code with example usage: obj.test_newton_combined() and returns: unknown,  unknown,  int