Understanding Big O Notation: Limit Definition and Reconciling Statements

[ericm1234]

I am trying to reconcile the following statement:

" ||u||<=eps*||f|| means ||u||=O(eps) ("||u|| is order eps")... "

with the limit definition of "big O"; considering it's not clear that ||u|| here even depends on eps:

" lim as eps goes to 0 of ||u||/eps, by definition of "big O", should equal some constant "; is it necessarily ||f||?

I understand that, for example, sin(x)=O(x) because lim as x goes to 0 of sin(x)/x is bounded.

So to casually say something is "order epsilon", if it doesn't depend on epsilon, I am not sure how to reconcile that with the definition of "big O" given by the limit one should be able to take, as above.

 

[Simon Bridge]

The relation $|u|\leq \epsilon |f| \implies |u|=O(\epsilon)$ ... looks like a typo.
Where did you get this from?

We'd normally say: $|u|\leq \epsilon |f| \implies u=O(f)$

http://web.mit.edu/16.070/www/lecture/big_o.pdf
http://en.wikipedia.org/wiki/Big_O_notation

 

[ericm1234]

I paraphrased for the sake of drawing attention to my (pedantic) question;
The original statement, from a book on finite element, was:
||u-u_s||<=eps*||u-u_s||_E <=eps^2*||f|| (the middle norm is energy norm)

the following text then goes:
"The point of course is that ||u-u_s||_E (energy norm) is of order eps, whereas ||u-u_s|| is of order eps^2."

So maybe you can explain based on this context then?

And sorry for lack of "tex" skills.

 

[Simon Bridge]

It looks like the author is not using regular big-O notation and/or conventions here.
Please provide a citation.

Possibly the author just refers to the order in "epsilon".
What is epsilon supposed to stand for?

 

[ericm1234]

Well, that was sort of what I figured..but anyway, Brenner, Mathematical Theory of Finite Element, p.6 bottom.

Thanks

 

[Simon Bridge]

I don't know that one - the best way to be sure is to look for the same lesson in another text and compare.

If I said that |y| = x|k| while |z|=x^2|k| then you could say that y = O(x) while z=O(x^2).
I'm thinking that the dependent variable in the inequalities above is epsilon.

 

[ericm1234]

Thanks; do you believe (as in, is it plausible) that the author means: ||u-u_s||/||f||<=eps means that thing less than eps is "of order epsilon"?

 

[Simon Bridge]

Really need the context.
But, especially if ||f|| is just a number, it would be fair to say that ||u-u_s||=O(eps)
i.e. it seems one could expand the expressions as $\sum_n a_n\epsilon^n$ ... whatever epsilon is supposed to represent. You are the one with the book in front of you.