为什么我的自制泰勒多项式函数不能像SciPy那样起作用？

Question

作为练习，我编写了一个脚本，该脚本为函数function oddOneOut(arr) { let word = '' if (arr[0] === arr[1]) { word = arr[0] } if (arr[1] === arr[2]) { word = arr[1] } if (arr[2] === arr[3]) { word = arr[2] } function wrong(element) { return element !== word } return arr.findIndex(wrong) } console.log(oddOneOut(["sword", "sword", "sword", "sword", "drows", "sword"])) //returns 4 （此处为n（此处为1）生成度x0（此处为8）的泰勒多项式的系数，f(x)（此处为1）。 sqrt（x）^ sqrt（x））。

我使用了for循环将这种近似转换为独立函数T(x)。然后，我使用Ts(x)定义了scipy.interpolate.approximate_taylor_polynomial。通过测试0到1之间的值，我确定了SciPy函数的scale参数的最佳值约为0.3（您不能超过1，因为该函数未定义为负x）。

正如我预期的那样，我的函数T(x)的性能明显比Ts(x)差，如下面的图所示，从x = 0到3：

我不是做什么的SciPy？

我想知道答案是否可能与SciPy dx函数中的derivative参数有关。我选择了e1-2，因为较高的值会产生nans，而较低的值会产生明显的废话。我也对order参数进行了估算：错误消息告诉我它必须是大于n的奇数值，所以我在for循环中使用了2*i+1。

这是我的完整代码。

# Imports
import numpy as np
from scipy.interpolate import approximate_taylor_polynomial
from scipy.misc import derivative
import matplotlib.pyplot as plt
import pandas as pd
%matplotlib inline


# Function I'm trying to approximate
def f(x):
    return np.sqrt(x) ** np.sqrt(x)


degree=8      # I am making an 8th degree polynomial
x0=1          # Centered about x=1

dvs=[f(x0)]   # This will hold the values of the nth degree derivative 
coef=[f(x0)]  # And this will hold the polynomial coefs (dvs divided by the corresponding factorials)

for i in range(1,degree+1):
    # I'm not entirely sure what the dx parameter does here, but 1e-2 produced the most accurate results.
    a = derivative(f,x0=x0,dx=1e-2,n=i,order=2*i+1)  # Compute the derivative
    dvs.append(a)                                    # Append to dvs
    coef.append(a/np.math.factorial(i))              # Compute the coefficient and append to coef

# Pretty pandas dataframe
out=pd.DataFrame({"degree":range(0,degree+1),"value":dvs,"coef":coef})


# Define Taylor polynomial using my coefs
def T(x):
    r = 0
    for i in range(0,degree+1):
        r = r + out['coef'][i] * ((x - x0) ** i)
    return r

# Use the scipy function for comparison. It is a poly1d object which can behave like a function.
# Not shown: I tried scale values on np.linspace(0,1,101) and 0.3 was about the best.
Ts = approximate_taylor_polynomial(f,x=x0,degree=8,scale=0.3)


# Test x values to display our functions
x = np.linspace(0,3,301)


fig = plt.figure(figsize=(10,10))
ax = fig.add_subplot(111)
# Actual function
ax.plot(x,f(x),c="black",label="f(x)")
# My approximation
ax.plot(x,T(x),c="deepskyblue",label="T(x)")
# Scipy approximation
# Note that scipy doesn't account for the horizontal translation
ax.plot(x,Ts(x-x0),c="forestgreen",label="Ts(x) (scale 0.3)") 
ax.legend()

这将输出上面显示的图形。

如何提高我的近似度？