Solve this equation with fixed point iteration

Question

How can I solve this equation

x ³ + x - 1 = 0

using fixed point iteration?

Is there any fixed point iteration code (especially in Python) that I can find on the web?

Answer 1

Using scipy.optimize.fixed_point :

import scipy.optimize as optimize def func(x): return -x**3+1 # This finds the value of x such that func(x) = x, that is, where # -x**3 + 1 = x print(optimize.fixed_point(func,0)) # 0.682327803828 

The Python code defining fixed_point is in scipy / optimize / minpack.py. The exact location depends on where scipy installed. You can find out by typing

 In [63]: import scipy.optimize In [64]: scipy.optimize Out[64]: <module 'scipy.optimize' from '/usr/lib/python2.6/dist-packages/scipy/optimize/__init__.pyc'> 

The current fixed_point source code can be found on the Internet by going to the documentation page and clicking the [source] link.

Answer 2

Try the SymPy library. Here's a relevant example :

 >>> solve(x**3 + 2*x**2 + 4*x + 8, x) [-2*I, 2*I, -2] 

I'm not sure which algorithm SymPy uses to solve the equation.