高斯牛顿法 Python 实现

 

import numpy as np


def generate_noise_data(num=1000, noise_sigma=1.0):
    x = np.linspace(0, 1, num)
    a, b, c = 1, 2, 3
    func = lambda x: np.exp(a * x**2 + b * x + c)
    y = func(x) + np.random.normal(0, noise_sigma, num)
    return x, y

def gauss_newton():
    x, y = generate_noise_data()
    a, b, c = 1, 1, 1
    detla_a = lambda a, b, c: - x**2 * np.exp(a * x**2 + b * x + c)
    detla_b = lambda a, b, c: - x * np.exp(a * x**2 + b * x + c)
    detla_c = lambda a, b, c: - np.exp(a * x**2 + b * x + c)
    func = lambda a, b, c: np.exp(a * x**2 + b * x + c)

    for iter in range(101):
        error = y - func(a, b, c)
        cost = np.sum(error ** 2)

        J = np.vstack([detla_a(a, b, c), detla_b(a, b, c), detla_c(a, b, c)])
        H = J @ J.T
        bb = np.sum(-J * error, axis=1).reshape(-1, 1)
        delta = np.linalg.inv(H) @ bb

        a += delta[0, 0]
        b += delta[1, 0]
        c += delta[2, 0]

        if iter % 10 == 0:
            print(f"iter: {iter}, cost: {cost}, a: {a}, b: {b}, c: {c}")


if __name__ == "__main__":
    gauss_newton()

'''
iter: 0, cost: 19451883.170121048, a: 7.778319737506763, b: 6.646686710088488, c: 7.615218392754173
iter: 10, cost: 4952005071479.72, a: 8.788212680472725, b: 4.652260498746273, c: -1.3984130105756094
iter: 20, cost: 113097.87399971182, a: 1.2484731861459692, b: 1.6092037161265653, c: 3.1479602802393742
iter: 30, cost: 930.2174621037896, a: 1.0005171031754931, b: 1.9993132909257523, c: 3.0000523922709754
iter: 40, cost: 930.2174621037897, a: 1.0005171031754931, b: 1.9993132909257525, c: 3.0000523922709754
iter: 50, cost: 930.2174621037914, a: 1.0005171031754936, b: 1.9993132909257518, c: 3.0000523922709754
iter: 60, cost: 930.2174621037896, a: 1.0005171031754931, b: 1.9993132909257523, c: 3.0000523922709754
iter: 70, cost: 930.2174621037897, a: 1.0005171031754931, b: 1.9993132909257525, c: 3.0000523922709754
iter: 80, cost: 930.2174621037914, a: 1.0005171031754936, b: 1.9993132909257518, c: 3.0000523922709754
iter: 90, cost: 930.2174621037896, a: 1.0005171031754931, b: 1.9993132909257523, c: 3.0000523922709754
iter: 100, cost: 930.2174621037897, a: 1.0005171031754931, b: 1.9993132909257525, c: 3.0000523922709754
'''