python - 如何在 Python 中执行二分法

Question

我想编写一个 Python 程序来运行二分法来确定根:

f(x) = -26 + 85x - 91x2 +44x3 -8x4 + x5

二分法是一种用于估计多项式 f(x) 的根的数值方法。

是否有任何可用的伪代码、算法或库可以用来告诉我答案？

Answer 1

基本技术

下面是一些显示基本技术的代码:

>>> def samesign(a, b):
        return a * b > 0

>>> def bisect(func, low, high):
    'Find root of continuous function where f(low) and f(high) have opposite signs'

    assert not samesign(func(low), func(high))

    for i in range(54):
        midpoint = (low + high) / 2.0
        if samesign(func(low), func(midpoint)):
            low = midpoint
        else:
            high = midpoint

    return midpoint

>>> def f(x):
        return -26 + 85*x - 91*x**2 +44*x**3 -8*x**4 + x**5

>>> x = bisect(f, 0, 1)
>>> print(x, f(x))
0.557025516287 3.74700270811e-16

公差

要在达到给定容差时提前退出，请在循环末尾添加一个测试:

def bisect(func, low, high, tolerance=None):
    assert not samesign(func(low), func(high))   
    for i in range(54):
        midpoint = (low + high) / 2.0
        if samesign(func(low), func(midpoint)):
            low = midpoint
        else:
            high = midpoint
        if tolerance is not None and abs(high - low) < tolerance:
            break   
    return midpoint