牛顿迭代法解非线性方程组python代码

牛顿迭代法是一种常用的数值优化方法，用于求解非线性方程组的根。在Python中，我们可以使用该算法来逐步逼近方程组的解。这里是一个基本的牛顿迭代法的实现示例：

import numpy as np

def newton_raphson(funcs, x0, Jacobian, tol=1e-6, max_iter=100):
    """
    Newton-Raphson method for solving a system of nonlinear equations.
    
    Args:
    - funcs (list of functions): List of functions that represent the equations f_i(x) = 0.
    - x0 (numpy array): Initial guess for the solution.
    - Jacobian (function): Function that returns the Jacobian matrix of the functions with respect to x.
    - tol (float): Tolerance for convergence (default 1e-6).
    - max_iter (int): Maximum number of iterations (default 100).

    Returns:
    - numpy array: Approximate solution after convergence or max_iter reached.
    """
    x = x0.copy()
    for _ in range(max_iter):
        J = Jacobian(x)
        delta_x = np.linalg.solve(J, -funcs(x))  # Update using the inverse Jacobian
        if np.all(np.abs(delta_x) < tol):
            break  # Converged
        x += delta_x
    else:
        print("Warning: Max iterations reached without convergence.")
    return x

# Example usage:
# Define your functions f_1 and f_2
def func1(x):
    return x[0]**2 + x[1] - 1

def func2(x):
    return x[0] + x[1]**2 - 2

# Jacobian matrix calculation
def jacobian(x):
    dfdx = np.array([[2*x[0], 1], [1, 2*x[1]]])
    return dfdx

# Initial guess
x0 = np.array([1, 1])

# Solve the system
solution = newton_raphson([func1, func2], x0, jacobian)
solution,