L|x_{k-1} - h|<L*L|x_{k-2}-h}|<L*L*L|x_{k-3}-h}|< ...
So, I am reading in a introductory numerical analysis book, but I believe my question fits into the calculus forums.
First, the definition of a simple iteration as given in my book:
Suppose that is a real-valued function, defined and continuous on a bounded closed interval of the real line, and assume that for all . Given that , the recursion defined by
is called a simple iteration.
Contraction Mapping Theorem:
Let g be as in the definition above. Suppose further, that g is a contraction on . Then, has a unique fixed point in the interval . Moreover the sequence defined above converges to as for any starting value in .
First of all, I don't really understand the simple iteration. I see that because for all , then there exists a number such that . It would be great if someone had the time to explain this in more detail.
As for the contraction mapping theorem, I understand how has a unique fixed point in the interval . However, I do not understand how it implies that the recursive sequence converges to as .
The proof of this is given in the book and is as follows;
we then deduce by induction that
I do not understand how they come to that last part by induction.
Any help is appreciated. Thanks!
Since g is continuous is continuous as well. Moreover and . By the itermediate value theorem for some . Hence is a fixpoint of g.First of all, I don't really understand the simple iteration. I see that because for all , then there exists a number such that . It would be great if someone had the time to explain this in more detail.
Since g is a contraction there exists a real number such that for all we have .I do not understand how they come to that last part by induction.
The fact that is important and the reason why this works.
If we let our fixpoint then:
And since and it's clear that , hence why
Hi,
to me that looks like a proof for Brouwer's Fixed Point Theorem. Perhaps that explains all there is to explain about simple iteration, but I unfortunately don't get it.
tells me that if then . I don't see why ..
The rest of what you wrote is crystal clear, thank you very much.
That's exactly what it was. I thought you wanted more clarification on the existence of a fixpoint.to me that looks like a proof for Brouwer's Fixed Point Theorem.
About the simple iteration I don't quite understand your problem. It's just a definition.
Given that for all . Starting with some point we can define a sequence by .
We get a sequence:
And the contraction mapping theorem then gives that this sequence converges to , the fixpoint of g