用 python 编写一个使任意一元四次函数的程序，要求进退法单谷ξ＜0.01黄金分割法求极小值

要编写一个Python程序，使用进退法（也称为Brent's method）或黄金分割搜索法（Golden Section Search）寻找一元四次函数的最小值，首先需要定义一个函数来表示四次函数，并实现这两种数值优化算法。以下是一个基本的示例：

import math

# 定义四次函数 f(x) = ax^4 + bx^3 + cx^2 + dx + e
def quartic_function(a, b, c, d, e, x):
    return a * x**4 + b * x**3 + c * x**2 + d * x + e

# 进退法（Brent's method）
def bisection_search(f, lower_bound, upper_bound, tolerance=0.01):
    if lower_bound > upper_bound:
        raise ValueError("Lower bound must be less than upper bound")

    while (upper_bound - lower_bound) >= tolerance:
        mid = (lower_bound + upper_bound) / 2
        if f(mid) == 0 or (f(lower_bound) * f(mid) <= 0):  # 如果满足零点条件或区间改变方向
            upper_bound = mid
        else:
            lower_bound = mid
    return mid

# 黄金分割法
def golden_section_search(f, a, b, tolerance=0.01, golden_ratio=(math.sqrt(5) - 1) / 2):
    while abs(b - a) >= tolerance:
        ratio = 1 - golden_ratio
        pa = a + golden_ratio * (b - a)
        pb = a + ratio * (b - a)

        if f(pa) < f(pb):
            b = pb
        else:
            a = pa

    return (a + b) / 2

# 示例一元四次函数
a = 1
b = -3
c = 2
d = -1
e = 0
func = lambda x: quartic_function(a, b, c, d, e, x)

# 使用进退法求解
min_value_brents = bisection_search(func, -10, 10)
print(f"Using Brent's method: Min value is {min_value_brents}, at x={func(min_value_brents)}")

# 使用黄金分割法求解
min_value_golden = golden_section_search(func, -10, 10)
print(f"Using Golden Section search: Min value is {min_value_golden}, at x={func(min_value_golden)}")