如何在我的寻根算法中添加Aitken序列

Question

from numpy import log as ln                 
def g(x):
    return ln(4+x-x**2)

def FixedPoint(p0,tolerance):                     
    p = g(p0)                                
    abs_error = abs(p-p0)                    
    p0 = p                          
    while abs_error>=tolerance:                 
        p = g(p0)                                
        abs_error = abs(p-p0)                    
        p0 = p                                   
    return p

starting_point = 2
tolerance = 10**-10
fixed_point = FixedPoint(starting_point,tolerance)

print('Fixed-point of g(x) = {0} is x = {1:.7f}'.format(formula,fixed_point))

所以，我有一个定点求根算法来求ln(4+x-x^2)的根，我如何添加aitken序列程序来提高求ln(4+x-x^2)根的收敛速度。我在保持每次迭代的值时遇到了麻烦。

Answer 1

我将通过以下方式减少定点迭代中的行数

def FixedPoint(p0,tolerance):                     
    p1 = g(p0)                                
    while abs(p1-p0)>=tolerance:                 
        p0,p1 = p1,g(p1) 
        print(p0)                               
    return p1

对于给定的测试用例，这需要29次g评估

Aitken增量平方过程可以实现为

def FixedPointAitken(p0,tolerance):                     
    while True:                 
        p1=g(p0); p2=g(p1);   
        print(p0,p1,p2)
        if abs(p1-p0)<tolerance: break
        p0 = p0 - (p1-p0)**2/(p0+p2-2*p1) 
    return p0

这需要5个步骤，对g进行15次评估才能达到目标精度。