Consider finding roots of $$ f(x) = x^2 - a, \qquad a > 0 $$ using following fixed point maps \begin{eqnarray*} x &=& x + c(x^2 - a) \\ x &=& \frac{a}{x} \\ x &=& \frac{1}{2}\left( x + \frac{a}{x} \right) \end{eqnarray*}
a = 3.0
c = 0.1
x1, x2, x3 = 2.0, 2.0, 2.0
print("%6d %18.10e %18.10e %18.10e" % (0,x1,x2,x3))
for i in range(10):
x1 = x1 + c*(x1**2 - a)
x2 = a/x2
x3 = 0.5*(x3 + a/x3)
print("%6d %18.10e %18.10e %18.10e" % (i+1,x1,x2,x3))
0 2.0000000000e+00 2.0000000000e+00 2.0000000000e+00 1 2.1000000000e+00 1.5000000000e+00 1.7500000000e+00 2 2.2410000000e+00 2.0000000000e+00 1.7321428571e+00 3 2.4432081000e+00 1.5000000000e+00 1.7320508100e+00 4 2.7401346820e+00 2.0000000000e+00 1.7320508076e+00 5 3.1909684895e+00 1.5000000000e+00 1.7320508076e+00 6 3.9091964797e+00 2.0000000000e+00 1.7320508076e+00 7 5.1373781913e+00 1.5000000000e+00 1.7320508076e+00 8 7.4766436594e+00 2.0000000000e+00 1.7320508076e+00 9 1.2766663700e+01 1.5000000000e+00 1.7320508076e+00 10 2.8765433904e+01 2.0000000000e+00 1.7320508076e+00
The first iteration diverges, the second oscillates while the last (Newton method) converges.