Dekker 的数值分析方法未正确收敛

Question

我正在尝试实现 Dekker 的求根方法，但我似乎无法找出算法似乎无法正确收敛到解决方案并且有时会突然停止的问题。

我的实现基于 Wiki 上 Dekker 方法的方程 Wiki

我对 python 和编程还很陌生，如果您能深入了解我可能出错的地方，我将不胜感激。

def dekker(func, x_bounds, tolerance, max_iterations):
# Initial bounds
a0, b0 = x_bounds
bk = b0
b_minus_1 = a0 # previous bk but start with a

k = 0

print(' k xk        f(xk)')

while k < max_iterations:
    fk = eval(fnon)(bk)
    f_minus_1 = eval(fnon)(b_minus_1)

    # check for division by zero when attemping secant method
    if fk - f_minus_1 == 0:
        # implement bisection method when division by zero 
        bk_plus_1 = (a0 + bk) / 2
    else:
        sk = bk - (bk - b_minus_1) / (fk - f_minus_1) * fk #secant
        mk = (a0 + bk) / 2 # bisection 
        
        if sk >= mk and sk <= bk:
            bk_plus_1 = sk
        else:
            bk_plus_1 = mk

    fk_plus_1 = eval(fnon)(bk_plus_1)

    if fk * fk_plus_1 < 0:
        a_k_plus_1 = bk
        b_k_plus_1 = bk_plus_1
    else:
        a_k_plus_1 = a0
        b_k_plus_1 = bk_plus_1

    if abs(eval(fnon)(a_k_plus_1)) < abs(eval(fnon)(b_k_plus_1)):
        best_solution = a_k_plus_1
    else:
        best_solution = b_k_plus_1

    k += 1
    print('{0:2.0f}  {1:2.8f}  {2:2.2e}'.format(k, bk_plus_1, abs(fk_plus_1)))

    if(abs(bk - b_minus_1) < tolerance):
        print('Converged')
        break

    bk = bk_plus_1
    b_minus_1 = bk

if k == maxk:
    print('Not converged')

测试方法：

def function(x):
    return -np.cos(x) + x*x*x + 2*x*x + 1
dekker('function', [-3,1], 1e-6, 100)

Answer 1

一些可能有帮助的更改：

1. 将整个代码中的

   fnon

   更改为

   func

   。
2. 通过将

   abs(bk - b_minus_1)

   更改为

   abs(bk_plus_1 - bk)

   来纠正收敛检查中的条件。