Solve for roots in given interval using scipy.optimize
<p>I have the function <code>f1 = lambda x: 1 - 1.12 * (x ** 0.5) * ((1-x) ** 0.02)</code>, and I wish to solve for its roots in the interval (0, 1). I have tried using the <code>scipy.optimize.brentq</code> and <code>scipy.optimize.fsolve</code> to do this, but both methods run into issues. Based on some experimentation, I got that the roots of this equation are approximately equal to 0.86322414 and 0.9961936895432034 (we know there are at most 2 roots because the function has one inflection point in this interval):</p>

<pre><code>f1 = lambda x: 1 - 1.12 * (x ** 0.5) * ((1-x) ** 0.02)
print(fsolve(f1, 0.5))
print(f1(0.99))
print(f1(0.999))
print(brentq(f1, 0.99, 0.999))
</code></pre>

<p>Output: </p>

<pre><code>[ 0.86322414]
-0.016332046983897452
0.025008640855473052
0.9961936895432034
</code></pre>

<p>The issue here is that in order for brentq to work, the values of the function must be of opposite signs at the specified endpoints. Furthermore, when I started fsolve at values of <code>x</code> close to 1, I got runtime warning messages:</p>

<pre><code>print(fsolve(f1, 0.97))
print(fsolve(f1, 0.98))
</code></pre>

<p>Output:</p>

<pre><code>[ 0.97]
[ 0.98]
C:/Users/Alexander/Google Drive/Programming/Projects/Root Finding/roots.py:6: RuntimeWarning: invalid value encountered in power

C:\Users\Alexander\Anaconda3\lib\site-packages\scipy\optimize\minpack.py:161: RuntimeWarning: The iteration is not making good progress, as measured by the 
  improvement from the last ten iterations.
</code></pre>

<p>Does anyone if there is a more systematic way to solve for roots of this equation, and why fsolve is not working on <code>x = 0.97, 0.98</code>?</p>

<p>If you take the derivative of your function and set it equal to 0, after a little algebra you'll find that the derivative is 0 at x0 = 0.5/0.52.  (In a calculus class, this point is called a <em>critical point</em>, not an inflection point.)  The function has a minimum at this point, and the value there is negative. The values at x=0 and x=1 are positive, so you can use [0, x0] and [x0, 1] as bracketing intervals in <code>brentq</code>:</p>

<pre><code>In [17]: from scipy.optimize import brentq

In [18]: f1 = lambda x: 1 - 1.12 * (x ** 0.5) * ((1-x) ** 0.02)

In [19]: x0 = 0.5/0.52

In [20]: brentq(f1, 0, x0)
Out[20]: 0.8632241390303161

In [21]: brentq(f1, x0, 1)
Out[21]: 0.9961936895432096
</code></pre>

python optimization scipy polynomial-math