Giuseppe Vettigli works at the Cybernetics Institute of the Italian National Reasearch Council. He is mainly focused on scientific software design and development. His main interests are in Artificial Intelligence, Data Mining and Multimedia applications. He is a Linux user and his favorite programming languages are Java and Python. You can check his blog about Python programming or follow him on Twitter. Giuseppe is a DZone MVB and is not an employee of DZone and has posted 35 posts at DZone. You can read more from them at their website. View Full User Profile

Fixed point iteration

  • submit to reddit
A fixed point for a function is a point at which the value of the function does not change when the function is applied. More formally, x is a fixed point for a given function f if

and the fixed point iteration

converges to the a fixed point if f is continuous.
The following function implements the fixed point iteration algorithm:
from pylab import plot,show
from numpy import array,linspace,sqrt,sin
from numpy.linalg import norm

def fixedp(f,x0,tol=10e-5,maxiter=100):
 """ Fixed point algorithm """
 e = 1
 itr = 0
 xp = []
 while(e > tol and itr < maxiter):
  x = f(x0)      # fixed point equation
  e = norm(x0-x) # error at the current step
  x0 = x
  xp.append(x0)  # save the solution of the current step
  itr = itr + 1
 return x,xp
Let's find the fixed point of the square root funtion starting from x = 0.5 and plot the result
f = lambda x : sqrt(x)

x_start = .5
xf,xp = fixedp(f,x_start)

x = linspace(0,2,100)
y = f(x)

The result of the program would appear as follows:




The red dot is the starting point, the blue ones are the sequence x_1,x_2,x_3,... and the green is the fixed point found.
In a similar way, we can compute the fixed point of function of multiple variables:

# 2 variables function
def g(x):
 x[0] = 1/4*(x[0]*x[0] + x[1]*x[1])
 x[1] = sin(x[0]+1)
 return array(x)

x,xf = fixedp(g,[0, 1])
print '   x =',x
print 'f(x) =',g(xf[len(xf)-1])
In this case g is a function of two variables and x is a vector, so the fixed point is a vector and the output is as follows:
   x = [ 0.          0.84147098]
f(x) = [ 0.          0.84147098]

Published at DZone with permission of Giuseppe Vettigli, author and DZone MVB.

(Note: Opinions expressed in this article and its replies are the opinions of their respective authors and not those of DZone, Inc.)