Over a million developers have joined DZone.
{{announcement.body}}
{{announcement.title}}

Fixed point iteration

DZone's Guide to

Fixed point iteration

· Web Dev Zone ·
Free Resource

Deploying code to production can be filled with uncertainty. Reduce the risks, and deploy earlier and more often. Download this free guide to learn more. Brought to you in partnership with Rollbar.

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)
plot(x,y,xp,f(xp),'bo',
     x_start,f(x_start),'ro',xf,f(xf),'go',x,x,'k')
show()

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]

Source:  http://glowingpython.blogspot.com/2012/01/fixed-point-iteration.html

Deploying code to production can be filled with uncertainty. Reduce the risks, and deploy earlier and more often. Download this free guide to learn more. Brought to you in partnership with Rollbar.

Topics:

Opinions expressed by DZone contributors are their own.

{{ parent.title || parent.header.title}}

{{ parent.tldr }}

{{ parent.urlSource.name }}