I have n degree polynomial system ,just I want to learn number of root between previously determined interval .But I do not want to find root.I need number of root.I need to write python code. For example : x^8+2x^6+5x^4+x^2+45x+1=0 How many root have we between 3-5? emphasize=I do not want to find root,just I want to learn how many root I have.
You can do this with numpy
import numpy as np
coeff = [1,0,2,0,5,0,1,45,1] #for the polynomial x^8+2x^6+5x^4+x^2+45x+1=0
np.roots(coeff)
# array([ 1.37234708+0.91495949j, 1.37234708-0.91495949j,
# 0.43013459+1.75225934j, 0.43013459-1.75225934j,
# -1.06175643+1.53395567j, -1.06175643-1.53395567j,
# -1.45921726+0.j , -0.02223323+0.j ])
Thus, as you can see, this one has zero real roots.
Edit: If you want to find the number of roots between an interval without finding the roots explicitly, you can use sturm's theorem. Using sympy,
from sympy import Sturm,Symbol
from sympy.abc import x
sturm_seq = sturm(x**6-3*x**5+2*x**4)
limits = [2,4]; x = Symbol('x')
values_at_start = [polynomial.subs(x,limits[0]).evalf() for polynomial in sturm_seq]
values_at_end = [polynomial.subs(x,limits[1]).evalf() for polynomial in sturm_seq]
import itertools
# Count number of sign changes in values_at_start
count_start = len(list(itertools.groupby(values_at_start, lambda values_at_start: values_at_start > 0)))
count_end = len(list(itertools.groupby(values_at_end, lambda values_at_end: values_at_end > 0)))
ans = count_start - count_end
print ans # ans = 1