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Spam deleted by QuickTopic 08-18-2010 02:02 AM
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06-14-2010
08:11 PM ET (US)
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06-11-2010
05:09 AM ET (US)
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Messages 15-14 deleted by topic administrator between 10-07-2008 02:27 AM and 02-22-2008 04:21 PM |
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Valintino
05-07-2007
11:44 AM ET (US)
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Hello, Your site is great. Regards, Valintino Guxxi
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Deleted by topic administrator 07-22-2006 10:23 AM
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Rasit Topaloglu
10-09-2004
05:22 PM ET (US)
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It is interesting that this technique was used until 1960's, then were forgotten until recently. I think the remedy that brought it back is the introduction of periodic restarts. Otherwise, error build-up would cause the search vectors to lose their A-orthogonality for complex problems.
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Gary Tedeschi
10-07-2004
03:53 PM ET (US)
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Unfortunately, upon first exposure to a topic I usually need a couple of readings to digest the material. So my comments/questions are probably not very insightful. 1. How would one show that performing conjugation on the axial unit vectors makes Conjugate Directions equivalent to Gaussian Elimination? Is there any geometric intuition here? (p. 26) 2. I think the key to understanding the CG method as presented in this paper is understanding section 7.3 Optimality of the Error Term. Luck has it, though, that I do not yet completely follow the author's discussion. Edited 10-07-2004 03:54 PM
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Stephen Krotosky
10-07-2004
03:22 PM ET (US)
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Interesting. How does it compare to other optimization techniques, such as EM?
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Sameer
10-07-2004
04:33 AM ET (US)
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To answer Robin's question about the size of the problems that conjugate gradient is used to handle. Think BIG, by big here we are talking about simulations containing millions of variables and going upto potentially a billion variables.
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Sameer
10-07-2004
04:25 AM ET (US)
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Since Vincent brought up Bi-Conjugate gradient, the folks who brought us the various pieces of software like LINPACK and LAPACK, also bring us
Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM Press
http://netlib2.cs.utk.edu/linalg/html_templates/Templates.html
This is a comprehensive reference and comparison of the various techniques that go into solving large linear systems.
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Sameer
10-07-2004
04:21 AM ET (US)
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In case the history of Conjugate Gradient is of interest, there is a rather remarkable article in SIAM reviews (part of JStor)
Gene H. Golub; Dianne P. O'Leary
SIAM Review, Vol. 31, No. 1. (Mar., 1989), pp. 50-102.
Stable URL: http://links.jstor.org/sici?sici=0036-1445...OTCG%3E2.0.CO%3B2-S
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Sameer
10-07-2004
04:18 AM ET (US)
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Well since you asked,
you do not need to do any hacks for to get symmetric positive semi-definiteness. Instead of solving
Ax = b
consider the system
A^\top A x = A^\top b
Section 13 of the paper discusses this. In case you are wondering if A^\top A is still sparse or not, turns out there is no need to ever actually form A^\top A explicitly. Simple Matrix vector products suffice.
So what is the downside?, well condition number for A^\top A is the square of the condition number of A, and hence convergence is slower.
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Vincent Rabaud
10-07-2004
04:09 AM ET (US)
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cool paper. A bit long, but half of it is just basics of linear algebra so that is fine.
So, this seems cool to solve Ax=b when A is symmetric positive definite, but what is done when it is not the case ? Is there a way of modifying a bit the method like it was done in this fantastic talk on tuesday on LM ? (like you "add" enough to A to have it symmetric positive definite; you find a solution; then you do something with the original A). Or maybe something can only be done if the matrix is purely symmetric, purely positive...
Briefly: how generalizable is it ? (I've heard of biconjugate gradient method but I am not sure of its application)
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Louka Dlagnekov
10-06-2004
11:59 PM ET (US)
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Robin, it is indeed a very well written explanation with a lot of pre-requisites well covered.
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