Talk:Lanczos algorithm: Difference between revisions

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== Old comments that did not have a section of their own ==

[[http://en.wikipedia.org/wiki/Lanczos_algorithm]] - an unfocused variety of Lanczos algorithm

<span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:134.219.166.104|134.219.166.104]] ([[User talk:134.219.166.104|talk]] • [[Special:Contributions/134.219.166.104|contribs]]) 21:23, 1 September 2005</span><!-- Template:Unsigned -->

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:PS: also, it should be said what is 'm'...

*'''Seconded''' — the article spends a lot of ink on the Lanczos iteration (but could do a better job at explaining it) for producing a tridiagonal matrix, says various other algorithms can be used for calculating eigenvalues and eigenvectors of that tridiagonal matrix, but is almost silent on how the two are related. As far as I can tell, early steps of the iteration tends to put most of the weight on the extreme eigenvalues (largest and smallest both, ''regardless of their absolute values''), meaning those are fairly accurately reproduced in the tridiagonal matrix, and the algorithm proceeds towards less extreme eigenvalues the longer it is run; it's 'tends' because the initial weight distribution depends on the initial vector, which is chosen at random. What is not clear from mere thought experiments is how concentrated the distribution is … [[Special:Contributions/130.243.68.202|130.243.68.202]] ([[User talk:130.243.68.202|talk]]) 14:22, 2 May 2017 (UTC)

:Thinking some more about this, I find it desirable to modify the way the algorithm is stated — partly to address the case <math>\beta_{j+1}=0</math>, and partly to do something about the "changes to" remark at the end, which is confusing in that no variable assigned in the algorithm is ever changed. It turns out the remark is boilerplate text in the {{tl|algorithm-end}} template, and there is no option to omit it. Since [[Wikipedia:WikiProject_Computer_science/Manual_of_style#Algorithms_and_data_structures]] does not recommend using that template, and instead recommends using (numbered) lists with explicit '''input''' and '''output''' headings, a rewrite from scratch seems in order.

:Said rewrite is now live in the article. It also addresses the gap of where to go when you've got the tridiagonal matrix. A remaining gap is the result that eigenvalues of <math>T</math> approximates eigenvalues of <math>A</math>. [[Special:Contributions/130.243.68.122|130.243.68.122]] ([[User talk:130.243.68.122|talk]]) 15:48, 26 May 2017 (UTC)

Following up on the previous comment, the section "Application to eigenproblem" misleadingly implies that the eigenvalues of the mxm matrix T are exactly the eigenvalues of A, which is obviously false. (Indeed the equation in parentheses at the end of the first paragraph is only valid if m =n). The point is that these eigenvalues approximate the eigenvalues of A; more work needs to be done to understand why and how good the approximation is.

== Define variables ==

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: It's not stated explicitly at that point, but presumably <math>A</math> is already taken to be Hermitian (as it needs to be for the Lanczos algorithm to work), which means it ''has'' an eigendecomposition of the form stated. Instead using the SVD decomposition in this argument won't work, because the entire point is that <math>U' U = I</math> so that the product telescopes! Possibly it would be clearer to just use <math>A = U \operatorname{diag}(\sigma_i) U^{-1}</math>, i.e., hold off on requiring orthogonality — the reason being that the paragraph in question is about the plain power method, which applies in a greater generality. [[Special:Contributions/130.243.68.202|130.243.68.202]] ([[User talk:130.243.68.202|talk]]) 13:01, 2 May 2017 (UTC)

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