#!/usr/bin/env python
"""Root finding using SciPy's Newton's method routines.
"""
from math import sin
import scipy, scipy.integrate, scipy.optimize
quad = scipy.integrate.quad
newton = scipy.optimize.newton
# test input function
def f(t):
return t*(sin(t))**2
# exact \int_0^t f(s) ds - u
def g(t):
u = 0.25
return .25*(t**2-t*sin(2*t)+(sin(t))**2)-u
# now let's construct g(t) via numerical integration
def gn(t):
u = 0.25
return quad(f,0.0,t)[0] - u
# main
tguess = 10.0
print '"Exact" solution (knowing the analytical form of the integral)'
t0 = newton(g,tguess,f)
print "t0, g(t0) =",t0,g(t0)
print
print "Solution using the numerical integration technique"
t1 = newton(gn,tguess,f)
print "t1, g(t1) =",t1,g(t1)
print
print "To six digits, the answer in this case is t==1.06601."
