Find the root of the transcendental equation with python

Question

I have to solve the following transcendental equation

cos (x) / x = s

for a given constant c.

For example, I made a short code in Mathematica, where I created a list of random values ​​for the constant c

const = Table[RandomReal[{0, 5}], {i, 1, 10}]

(*{1.67826, 0.616656, 0.290878, 1.10592, 0.0645222, 0.333932, 3.59584, \
2.70337, 3.91535, 2.78268}*)

How did I define the function

f[x_, i_] := Cos[x]/x - const[[i]]

and started looking for roots:

Table[FindRoot[f[x, i] == 0, {x, 0.1}][[1, 2]], {i, 1, Length[const]}]
(*{0.517757, 0.947103, 1.21086, 0.694679, 1.47545, 1.16956, 0.26816, \
0.347764, 0.247615, 0.338922}*)

---

Now I would like to program something like this in python (perhaps using numpy?), But I cannot find a good answer to a similar problem. Can anyone help?

Answer 1

One of the ways that I have achieved in the past is to use the scipy.optimize.minimizequadratic function to find the minima.

from scipy.optimize import minimize
from numpy import cos

def opt_fun(x, c):
    return (cos(x)/x - c)**2

const = 1.2
res = minimize(lambda x: opt_fun(x, const), x0=0.001)

# Check if the optimization was successful
print(res.success)
# >> True

# Extract the root from the minimization result
print(res.x[0])
# >> 0.65889256782472172

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, , , - . , c , . , x, 1, ()).

Answer 2

root:

import numpy as np
from scipy.optimize import root

def func_cos(x, c):
    return np.cos(x) / x - c

crange = range(1, 11)

res = [root(func_cos, 0.5, args=(ci, )).x[0] for ci in crange]

res :

[0.73908513321516056,
 0.45018361129487355,
 0.31675082877122118,
 0.24267468064089021,
 0.19616428118784215,
 0.16441893826043114,
 0.14143076140757282,
 0.12403961812459068,
 0.11043425911223313,
 0.099505342687387879]

root, .

Answer 3

, Python Chebfun. , Chebpy pychebfun, .

, Chebpy, cos(x)/x - 0.05 [0.5, 12]:

from chebpy import chebfun

x = chebfun('x', [0.5, 12])
c = 0.05
f = np.cos(x)/x - c

rts = f.roots()
print(rts)

[ 1.4959 4.9632 7.4711 11.6152]

Answer 4

sympy:

>>> from sympy import cos, Symbol, nsolve
>>> x = Symbol('x')
>>> consts = [random.random() for _ in range(10)]
>>> [nsolve(cos(x)/x - c, x, 1) for c in consts]
[mpf('0.89659506789294669'),
 mpf('0.96201114853313738'),
 mpf('0.74186728791161379'),
 mpf('1.1720944924353926'),
 mpf('0.92953351945607071'),
 mpf('0.96626530553984035'),
 mpf('1.4270719610604761'),
 mpf('0.85968954499458035'),
 mpf('0.86682911058530746'),
 mpf('0.91591678333479274')]