Thanks for your responses, but i think I'm way out of my league :/
I want to apply a diffeomorphism in image analysis and I'm looking for a way to build a function to map pixel positions.
edit: disregard that.
I'm a complete rookie here, and i'd like some help.
For starters , can a diffeomorphic mapping be represented via a matrix , like say a transformation?
If so, how would it be parameterised?
Homework Statement
a,b\in R, a<b, n\in N\\ h=\frac{b-a}{n} , x_i = a+ih , i=0..n \\
f\in C^1[a,b]
we approximate the integral of f in a,b with Q_n(f) = h\left[f(x_1) + f(x_1) + ... + f(x_n)\right]
Find the error R_n(f) = \int_a^bf(x)dx - Q_n(f), as function of the first derivative of f...
Noone?
One another norm problem, I'm given.If you could verify / correct:
A\in\mathbb{R}^{n,n} , x\in \mathbb{R}^n. \|Ax\|\ge \frac{\|x\|}{10}
and ask to show that \|A^{-1}\| \le 10
where ||\cdot|| a norm and the corresponding matrix norm derived by it.
\|Ax\|\ge...
Ah, i solved an identical one yesterday,so i can't have the "i figured this out on my own" satisfaction.
Via taylor expansion:
f(x_n) = x_{n+1} = f(x^*) + (x_n-x^*)f'(x^*)+...+(x_n-x^*)^{(m-1)}\frac{f^{(m-1)}(x^*) }{(m-1)!} +(x_n-x^*)^m \frac{f^{(m)}(k) }{m!} =\\
=x^* + (x_n-x^*)^m...
Well, you just unveiled a pretty tragic gap in my knowledge.Sloppiness, sloppiness everywhere.
okay, the last fraction you apply de l hospital's again and you're there.
So to summarize :
To show that a sequence's convergence is of order p, you have to show that...
I showed superilnear convergence with the 3rd formula above.
The limit:
\lim_{n\rightarrow\infty}\frac{\left|x_{n+1} - x^*\right|}{\left|x_{n} - x^*\right|}=
\left|\lim_{n\rightarrow\infty}\frac{x_{n+1} - x^*}{x_{n}-x^*}\right|=\left|\lim_{n\rightarrow\infty}\phi^'(k_n)\right| = \phi(x^*)...
Actually i got that from a worked example.
My textbook instructions on RoC are:
----------------------------------------
At least linear convergence , when exist C<1 , N such that:
\left| x_{n+1} - x^*\right|\le C \left|x_n - x^* \right| , \forall n\in \mathbb N
Convergence of...
Thank you so much.
I guess i was confused with the output of h(x) being also the input (points in the x axis) for f.
Your illustrations have cleared things out a bit :)
Edit again =(
I have forgotten to ask, in the problem in the 1st post, about the rate of convergence r.
f(x) = x + cosx ...
I hope it's ok to add more questions on sequences.. (i reckon it's better forum-wise than making new threads.If not , I'm sorry + close this thread).
Maybe a moderator could append " and general sequence questions".
When examining the (global) convergence of...
Got it.
In general, since the max|f'(x)| < 1 , there will be a point b in the inverse where f'(b) = 1/f'(a) >1, since f'(a) <= max|f'(x)| < 1
Makes sense. Thanks again.
Ah okay thanks.
Strangely enough, i have done the same exact thing in other identical exercises, and that question didn't arise there.
Thank you.
So my proof is consistent?
Edit: may i add another question on sequences.
if you have an "backwards" sequence, of the form yn-1 = f(yn) ...