下面是一个在Python中使用共轭梯度和回溯线搜索函数实现的不精确（近似）牛顿算法

import numpy as np
from scipy.linalg import solve

def backtracking_line_search(f, grad_f, x, d, alpha=1.0, rho=0.5, c=1e-4):
    """
    Backtracking line search to find a step size that satisfies the Armijo condition.
    f: the function to be minimized
    grad_f: the gradient of f
    x: the current iterate
    d: the search direction
    alpha: the initial step size
    rho: the backtracking parameter (0 < rho < 1)
    c: the Armijo condition parameter (0 < c < 1)
    """
    while f(x + alpha*d) > f(x) + c*alpha*np.dot(grad_f(x), d):
        alpha *= rho
    return alpha

def newton_cg(f, x0, grad_f, hess_f, tol=1e-6, max_iter=100):
    """
    Inexact Newton method with conjugate gradient for solving f(x) = 0.
    f: the function to be solved
    x0: the initial guess
    grad_f: the gradient of f
    hess_f: the Hessian matrix of f
    tol: the tolerance for convergence
    max_iter: the maximum number of iterations
    """
    x = x0
    for i in range(max_iter):
        g = grad_f(x)
        if np.linalg.norm(g) < tol:
            print(f"Converged in {i} iterations.")
            return x
        H = hess_f(x)
        # Compute the Newton direction using conjugate gradient
        d, _ = solve(H, -g, method='cg', tol=tol)
        # Perform a line search using backtracking
        alpha = backtracking_line_search(f, grad_f, x, d)
        # Update x
        x = x + alpha*d
    print("Failed to converge.")
    return x

backtracking_line_search函数将待最小化的函数f、f的梯度grad_f、当前迭代x、搜索方向d以及用于初始步长、回溯参数和Armijo条件参数的可选参数alpha、rho和c作为输入，它使用回溯来找到满足Armijo条件的步长。
newton_cg函数类似于前面的实现，但是使用backtracking_line_search函数而不是固定步长来执行线搜索，这应该提高算法的收敛性。
下面是如何使用函数求解方程x3 - x2 - 1 = 0的示例：

def f(x):
    return x**3 - x**2 - 1

def grad_f(x):
    return 3*x**2 - 2*x

def hess_f(x):
    return 6*x - 2*np.eye(len(x))

x0 = np.array([1.0, 1.0])
x = newton_cg(f, x0, grad_f, hess_f)
print(x)