Recent content by garrus

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?

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...

1)They are subscripted by natural numbers in general ,i presume for simplicity and countability.

Homework Statement Show that ||A||_1 \le \sqrt{n} ||A||_2 , ||A||_2 \le \sqrt{n} ||A||_1 , where ||A||_1 = \max_{1\le j\le n}\sum_{i=1}^n |a_{ij}| \\ ||A||_2 = (p(A^TA))^\frac{1}{2} \\ p(B) = \max|\lambda_B| with A,B\in \mathbb{R}^{n,n}, i,j\in[1...n] , \lambda_Athe eigenvalues of matrix A...

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) ...

Homework Statement xn+1 = xn + cosxn , n>=1 where x0 E [π/4 , 3π/4] = D. Show it converges, find rate of convergence.Homework Equations contraction theoremThe Attempt at a Solution Setting a function f(x) = x+cosx we have f'(x) = 1 - sinx, f''(x)= -cosx. Now f' >= 0, so f is increasing. For...

Got it. Thanks for your help, should've noticed that =[