In [1]:

from __future__ import division

import numpy as np
import matplotlib.pyplot as pt

In [2]:

x = np.linspace(0, 4.5, 200)

def f(x):
    return x**2 - x - 2

pt.plot(x, f(x))
pt.grid()

Actual roots: \(2\) and \(-1\)

In [3]:

def fp1(x): return x**2-2
def fp2(x): return np.sqrt(x+2)
def fp3(x): return 1+2/x
def fp4(x): return (x**2+2)/(2*x-1)

fixed_point_functions = [fp1, fp2, fp3, fp4]

In [4]:

for fp in fixed_point_functions:
    plot(x, fp(x), label=fp.__name__)
pt.ylim([0, 3])
pt.legend(loc="best")

-c:3: RuntimeWarning: divide by zero encountered in true_divide

Out[4]:

<matplotlib.legend.Legend at 0x471db10>

Common feature?

In [5]:

for fp in fixed_point_functions:
    ...

2
2.0
2.0
2.0

(Edit this cell for solution.)

In [6]:

z = 2.1; fp = fp1
#z = 1; fp = fp2
#z = 1; fp = fp3
#z = 1; fp = fp4

n_iterations = 4

pt.figure(figsize=(8,8))
pt.plot(x, fp(x))
pt.plot(x, x, "--")
pt.gca().set_aspect("equal")
pt.ylim([-0.5, 4])

for i in range(n_iterations):
    z_new = fp(z)
    
    pt.arrow(z, z, 0, z_new-z)
    pt.arrow(z, z_new, z_new-z, 0)
    
    z = z_new
    print z

2.41
3.8081
12.50162561
154.290642893

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