The problem states : Let . The equation has a unique root approximately equals to in .
Show that the following function has a fixed point in : .
I'm trying to understand the fixed point signification since about 3 days but I really don't get the concept, and much less how to do exercises related to it. So if you could do this problem explaining, I will be very very grateful.
I didn't know that. So all what I had to do is to replace by in ?Essentially you are asking us to show there exists an x such that g(x)=x
Because I don't understand it well.This needs only the definition of fixed point... why are you getting stuck?
I didn't! It was a given information.But you have already found out a value for this! Yes, x=1.365230013 is the fixed point of g(x).
Thanks a lot, if I could I'd push twice on the thanks button!
Correct me if I'm wrong : a fixed point of a function is the point where the function cross the y=x line?
The b part of the exercise ask me to do 4 iterations using the fixed point method starting with . I think I know how to do so, but I don't understand why it will converges to the root. Is this method fast, compared to the Newton's one?
And a big question : if wasn't given, how could I have guessed a function that would work for the fixed point method? (I guess the method to find such a function is more or less as complicated as to approximate the root of g!)
What is r?
YesCorrect me if I'm wrong : a fixed point of a function is the point where the function cross the y=x line?
I dont know actually, since I have not studied these numerical methods. But Wiki says that Newtons method is equivalent to fixed point. I am not sure though but I am sure CaptainBlack can clarify. He is very good at such Numerical Algorithms.The b part of the exercise ask me to do 4 iterations using the fixed point method starting with . I think I know how to do so, but I don't understand why it will converges to the root. Is this method fast, compared to the Newton's one?
Just a thought: Cant you just take g(x) = f(x) - x ?And a big question : if wasn't given, how could I have guessed a function that would work for the fixed point method?
To, I answerWhat is r?from my first post.The equation f(x)=0 has a unique root r
Hmm, I take this as a yes, without understanding well! But I know that many functions can do the job, it's the case of g in the exercise, which is not .Just a thought: Cant you just take g(x) = f(x) - x ?
Thanks very much, I feel I've understood at least a little bit about this method that was totally unknown some hours ago.