Alexander Whatley - 5 months ago 25

Python Question

I have the function

`f1 = lambda x: 1 - 1.12 * (x ** 0.5) * ((1-x) ** 0.02)`

`scipy.optimize.brentq`

`scipy.optimize.fsolve`

`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))

Output:

`[ 0.86322414]`

-0.016332046983897452

0.025008640855473052

0.9961936895432034

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

`x`

`print(fsolve(f1, 0.97))`

print(fsolve(f1, 0.98))

Output:

`[ 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.

Does anyone if there is a more systematic way to solve for roots of this equation, and why fsolve is not working on

`x = 0.97, 0.98`

Answer

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 *critical point*, 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 `brentq`

:

```
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
```