python: fsolve with unknown inside the upper limit of an integral

Question

Is it possible to apply fsolve method for an integral with the unknown in the upper limit and in the integrand??? I am ussing quad method for integration in python.

Thanks in advance!!!

Answer 1

Sure.

Suppose you want to find x such that the integral over t from t=0 to t=x of t*(1-x*t) is 0. You can do this by defining two functions. integrand(t, x) will evaluate t*(1-x*t), and func(x) will integrate integrand using quad, with x as both the upper limit of the integration, and as the extra argument of the integrand. Here's a demo:

import numpy as np
from scipy.integrate import quad
from scipy.optimize import fsolve


def integrand(t, x):
    return t*(1 - x*t)


def func(x):
    y, err = quad(integrand, 0, x, args=(x,))
    return y


# Use 1.0 as the initial guess.  Note that a bad initial guess
# might generate a warning and return the degenerate solution at x=0.
sol = fsolve(func, 1.0)

print "Solution:      ", sol[0]

# The exact solution that we want is sqrt(3/2)
print "Exact solution:", np.sqrt(1.5)

Output:

Solution:       1.22474487139
Exact solution: 1.22474487139