Solve this equation with fixed point iteration

Question

How can I solve this equation

x³ + x - 1 = 0

using fixed point iteration?

Is there any fixed-point iteration code (especially in Python) I can find online?

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 is installed. You can find that 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 online 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, though.