Fixed Point Iteration

Fixed Point Iteration method for finding roots of f(x)

Category:

Science & Technology

Tags:

• fixed
• point
• iteration
• newton
• newton's
• method
• numerical analysis
• numerical
• analysis
• educational
• intstructions

[91418300]

how should we come up with our first guess? Im working on a function g(x) = 3 / x. What should be my first guess? I changed it into x = (3 / x) + x. So what now?

[oscarsveliz]

@91418300 your pick for x1 should be a reasonable guess close to the root of your original function f(x) for it converge the fastest. If you find that your g(x) is not converging (your numbers keep growing to infinity [or negative infinity]) then you'll need to figure out a different g(x). What was your original f(x)?

[91418300]

@oscarsveliz my original f(x) is : 3 / x. How can I find the root to it? And how can I make it look like f(x) = x?

[oscarsveliz]

@91418300 The function f(x) = 3/x is undefined when x is zero (3/0) and is equal to zero only as x -> ∞ (3 / ∞ = 0) or as x -> -∞ (3 / -∞ = 0). So in this case there's nothing you can do for fixed point iteration to work because your f(x) has no root. Does this make sense?

[91418300]

@oscarsveliz what you are saying makes sense. Thanks for replying to my questions so far. In my textbook, the answer in the back of the book says the fixed points are -sqrt(3) and sqrt(3). Well I can skip that for now.. I have yet to solve any of the problems on my hw. I have another function g(x) = x^2 - 2x + 2. I find that the root is -1. But I cant find a way to make it g(x) = x where g'(x) < 1. I hope you dont mind me asking you another question

[oscarsveliz]

@91418300 The function where the roots are -sqrt(3) and sqrt(3) could be f(x) = x^2 - 3 or f(x) = -(x^2) + 3.

When abs(g'(root)) < 1 it guarantees that fixed point iteration will converge however if it isn't < 1 that does not mean it won't converge, it just means there's no guarantee.

[Darklink224]

Thank you so much! I wish my professor would lay it out like this after doing all the proofs and derivations.

[91418300]

@oscarsveliz ok i will use your advice and do my best to solve the rest of the problems. Thanks a lot for the help. greatly appreciated

[oscarsveliz]

@Anandtrceg It is sort of a catch-22 isnt it. Saying less than 1 guarantees that the function converges. However if greater/equal 1 it does not say that the function will diverge; it might still converge but that it is not guaranteed to converge. This just happens to be one of the properties of fixed point iteration. The other way to figure out if it is converging or diverging is to run it and check if x is converging to a number or bouncing around everywhere in a loop or sipping to infinity.

[Anandtrceg]

The aim is to find the roots of the equation.Using those roots to check if the method converges is not right.If we already know the roots, why do we even do the fixed point method?