Find the Roots of a Polynomial.

If you want roots to arbitrary precision, a very interesting library is rpncalc.ratfun (you will need another module for arbitrary precision called clnum). Here is an example with the same polynomial on a linux machine:

from rpncalc.ratfun import Polynomial

P = Polynomial(-7.1, 1, 2, -5.22, 3.4)
print("Here is a polynomial\n")
print P
print("\nIts complex roots as a list of multiprecision floating points numbers:\n")
print P.roots(eps=1e-30)

"""  my output ---->
Here is a polynomial

3.4*x^4-5.22*x^3+2*x^2+x-7.1

Its complex roots as a list of multiprecision floating points numbers:

[cmpf('-0.913412053509803965200954224858149742316-5.23858817606991616489822814712984590827e-312j',prec=36), cmpf('0.436014015782063442817853261862879518184-1.12245145548416568676706088778633149012j',prec=36), cmpf('0.436014015782063442817853261862879518184+1.12245145548416568676706088778633149012j',prec=36), mpf('1.57667813959273587005688087837686649466',36)]
"""

If you have the sympy module, this is another way to find the roots. If the polynomial has rational coefficients and small degree, sympy can also compute exact roots. However I had some memory leak problems with sympy.

import sympy

print("Floating point roots:n")
print(sympy.roots([3.4, -5.22, 2, 1, -7.1]))

from sympy import Integer as Int

print("Exact roots as sympy expressions:")
print(sympy.roots([Int(34)/10, -Int(522)/100, 2, 1, -Int(71)/10]))

"""my output --->
Floating point roots:

{-0.913412053509803: 1, 0.436014015782064 + 1.12245145548416*I: 1, 0.436014015782064 - 1.12245145548416*I: 1, 1.57667813959273: 1}
Exact roots as sympy expressions:
{261/680 - (68363/346800 - 13501/(10404*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3)) + 2*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3))**(1/2)/2 - (68363/173400 + 11528419/(19652000*(68363/346800 - 13501/(10404*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3)) + 2*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3))**(1/2)) + 13501/(10404*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3)) - 2*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3))**(1/2)/2: 1, 261/680 + (68363/346800 - 13501/(10404*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3)) + 2*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3))**(1/2)/2 + (68363/173400 - 11528419/(19652000*(68363/346800 - 13501/(10404*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3)) + 2*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3))**(1/2)) + 13501/(10404*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3)) - 2*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3))**(1/2)/2: 1, 261/680 + (68363/173400 + 11528419/(19652000*(68363/346800 - 13501/(10404*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3)) + 2*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3))**(1/2)) + 13501/(10404*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3)) - 2*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3))**(1/2)/2 - (68363/346800 - 13501/(10404*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3)) + 2*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3))**(1/2)/2: 1, 261/680 + (68363/346800 - 13501/(10404*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3)) + 2*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3))**(1/2)/2 - (68363/173400 - 11528419/(19652000*(68363/346800 - 13501/(10404*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3)) + 2*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3))**(1/2)) + 13501/(10404*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3)) - 2*(-38639957/424483200 + 12534291435179913031527229440000**(1/2)/6673555077120000)**(1/3))**(1/2)/2: 1}
"""

You can also use the Gnu Scientific Library (http://www.gnu.org/software/gsl/). There is a python interface to gsl called pygsl http://pygsl.sourceforge.net/ . To install it you need an ansi C compiler and the development files of gsl and python-numeric. On a linux system, this is not a problem. Here is how to compute the roots using pygsl:

from pygsl.poly import poly_complex

L = [-7.1, 1, 2, -5.22, 3.4]
roots = poly_complex(len(L)).solve(L)

print("Using the Gnu Scientific Library\n")
print("Complex roots as a numpy array:")
print(roots)
print("\nComplex roots as a python list:")

""" my output ---->
Using the Gnu Scientific Library

Complex roots as a numpy array:
[-0.91341205+0.j          0.43601402+1.12245146j  0.43601402-1.12245146j
  1.57667814+0.j        ]

Complex roots as a python list:
[(-0.91341205350980292+0j), (0.43601401578206328+1.122451455484166j), (0.43601401578206328-1.122451455484166j), (1.5766781395927358+0j)]
"""

The PARI computer algebra system also has a function to compute the roots. To use it with python you need the python interface to pari http://code.google.com/p/pari-python/ . To install it on my Mandriva system, I had to install the packages pari and pari-devel first. It should work in Windows too with cygwin. Here are the roots of the same polynomial computed with pari

from pari import polroots
print polroots("-7.1 + x + 2*x^2 -5.22*x^3 + 3.4*x^4", 0)

""" My output --->
[-0.9134120535098039691852671888 + 0.E-38*I, 1.576678139592735906083229898 + 0.E-38*I, 0.4360140157820634433157245276 + 1.122451455484165684444607827*I, 0.4360140157820634433157245276 - 1.122451455484165684444607827*I]~
"""

Pari seems able to compute multiprecision roots too.

Yet another module to find the roots: mpmath.polyroots() finds the roots with arbitrary precision and can also give an error estimate, using the Durand-Kerner method:

>>> from mpmath import *
>>> mp.dps = 60
>>> for r in polyroots([1, 0, -10, 0, 1]): # x^4 -10 x^2 + 1
...     print r
...
-3.14626436994197234232913506571557044551247712918732870123249
-0.317837245195782244725757617296174288373133378433432554879127
0.317837245195782244725757617296174288373133378433432554879127
3.14626436994197234232913506571557044551247712918732870123249
>>>
>>> sqrt(3) + sqrt(2)
3.14626436994197234232913506571557044551247712918732870123249
>>> sqrt(3) - sqrt(2)
0.317837245195782244725757617296174288373133378433432554879127

It seems that polyroot()is also available in sympy as sympy.mpmath.polyroot().

It's been 2 years since I last updated this collection, but today, I found a new way to compute the roots of a polynomial from python: one can use the python interface to scilab, called sciscipy. Here is the code

>>> from scilab import scilab as sci
>>> sci.roots([3.4, -5.22, 2, 1, -7.1])
array([-0.91341205+0.j        ,  1.57667814+0.j        ,
        0.43601402+1.12245146j,  0.43601402-1.12245146j])

In ubuntu, install sciscipy with

sudo aptitude install python-sciscipy

I did not test sciscipy in windows.