[SOLVED] fixed point iteration algorithm

fixed point iteration algorithm

I am asked to write a program to solve this equation ( x^3 + x -1 = 0 ) using fixed point iteration.

What is the algorithm for fixed point iteration? Is there any fixed point iteration code sample in Python? (not a function from any modules, but the code with algorithms)

Thank You

Solution

First, read this: Fixed point iteration:Applications

I chose Newton's Method.

Now if you'd like to learn about generator functions, you could define a generator function, and instance a generator object as follows

def newtons_method(n):
    n = float(n)  #Force float arithmetic
    nPlusOne = n - (pow(n,3) + n - 1)/(3*pow(n,2) +1)
    while 1:
        yield nPlusOne
        n = nPlusOne
        nPlusOne = n - (pow(n,3) + n - 1)/(3*pow(n,2) +1)

approxAnswer = newtons_method(1.0)   #1.0 can be any initial guess...

Then you can gain successively better approximations by calling:

approxAnswer.next()

see: PEP 255 or Classes (Generators) - Python v2.7 for more info on Generators

For example

approx1 = approxAnswer.next()
approx2 = approxAnswer.next()

Or better yet use a loop!

As for deciding when your approximation is good enough... ;)