在python中找到切比雪夫多项式的根

Question

I want to find the roots of the Chebysev polynomial of any order using Python. I have seen similar threads for Legendre polynomials. However, I have constructed my polynomials using the method defined here as

import numpy as np 
import sympy as sp 

f0 = lambda x: chebyt(0,x)
f1 = lambda x: chebyt(1,x)
f2 = lambda x: chebyt(2,x)
f3 = lambda x: chebyt(3,x)
f4 = lambda x: chebyt(4,x)
plot([f0,f1,f2,f3,f4],[-1,1])

I have tried to use np.roots(f4) , but I receive the following error: TypeError: float() argument must be a string or a number, not 'function' . Additionally, it seems that even if I could, it wouldn't work for high order polynomials.

Answer 1

You can do this by finding the coefficients of the Chebyshev polynomials using the method under the heading "Basic Evaluation" here , and then using np.roots on the reversed list to generate the roots of the polynomial.

Using np.roots(f4) wasn't working because the roots function only accepts a list of polynomial coefficients, rather than a lambda function.

Code:

from mpmath import chebyt, chop, taylor
import numpy as np

for n in range(5):
    print(np.roots(chop(taylor(lambda x: chebyt(n, x), 0, n))[::-1]))

Output:

[]
[0.]
[ 0.70710678 -0.70710678]
[ 0.8660254 -0.8660254  0.       ]
[-0.92387953  0.92387953 -0.38268343  0.38268343]

Hope that helps.