python编写不用库最小二乘法非线性模型y=a*x**2+b*x+c迭代法求值代码

Python中没有内置的最小二乘法用于直接处理非线性模型，如`y = a * x^2 + b * x + c`。但是，我们可以自定义一个简单的迭代算法（如梯度下降）来求解这个问题。以下是一个使用基本迭代方法（牛顿法或梯度下降）求解三次多项式的简单示例：

def f(x, a, b, c):
    return a * x ** 2 + b * x + c

def gradient_descent(x_data, y_data, initial_guess, learning_rate=0.01, max_iterations=1000):
    a, b, c = initial_guess
    for _ in range(max_iterations):
        # 计算预测值和误差
        predictions = [f(xi, a, b, c) for xi in x_data]
        error = [yi - pi for yi, pi in zip(y_data, predictions)]
        
        # 计算梯度
        da = sum(2 * a * xi ** 2 + b for xi in x_data)
        db = sum(2 * a * xi + b for xi in x_data)
        dc = sum(-1 for ei in error)

        # 更新参数
        a -= learning_rate * da
        b -= learning_rate * db
        c -= learning_rate * dc
        
        # 判断是否收敛，这里通常会添加绝对或相对精度判断
        if abs(da) < 1e-6 and abs(db) < 1e-6 and abs(dc) < 1e-6:
            break
    
    return a, b, c

# 示例数据
x_data = [i for i in range(len(y_data))]
y_data = [a * i ** 2 + b * i + c for a, b, c in [(1, 2, 3), (4, 5, 6)] for i in x_data]  # 假设我们有两个真实的数据集

# 初始化猜测值
initial_guess = [1, 1, 1]

# 迭代求解
solution = gradient_descent(x_data, y_data, initial_guess)
print(f"参数为: a={solution}, b={solution}, c={solution}")