Finding Eigenvectors with Lanczos Algorithm

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In summary, the speaker is seeking help in finding an algorithm to generate the eigenvector corresponding to the minimal eigenvalue of a huge matrix using the Lanczos algorithm. They have looked through multiple sources but have not found a clear explanation or example. The speaker also mentions their limitations in storing only 2-3 vectors and their frustration with the lack of accessible information on this topic.
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maverick_starstrider
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Hi, I'm applying the Lanczos algorithm to find the minimal eigenvalue of some huge matrix. Now that I've got it I'm trying to find the eigenvector corresponding to this eigenvalue. Now I have looked through book after book after book and I have yet to find an explanation of how to do this that is even the slightest bit approachable and I don't have the time or patience to read through a whole 300 page textbook to understand the authors notations and terminology well enough to glean a probably 10 line algorithm to generate these things. So I'm wondering if someone could simply tell me a bare-bones algorithm or point me to one that accomplishes this (an algorithm not a linear algebra proof from willoughby and callum or some such).

I have a lanczos algorithm that uses two vectors and the standard lanczos algorithm (http://en.wikipedia.org/wiki/Lanczos_algorithm#The_algorithm). I cannot store more than 2 (maybe 3) vectors (so I can't have something like V=[v1,v2,...,vm]). I generate my tridiagonal lanczos matrix Tm, I can solve for the eigenvalue I want now what do I do? Any help is greatly appreciated, this is driving me mad. It's amazing how many books/internet sources "discuss" this but other then a 5 page linear algebra proof fail to provide any clear explanation, much less an algorithm or an example.
 
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Did you solve this issue eventually?
 

FAQ: Finding Eigenvectors with Lanczos Algorithm

What is the Lanczos algorithm?

The Lanczos algorithm is a method used to find a small number of eigenvalues and corresponding eigenvectors of a large sparse symmetric square matrix. It is an iterative algorithm that uses orthogonal projection to reduce the matrix to a smaller size, making it more efficient to compute the eigenvalues and eigenvectors.

How does the Lanczos algorithm work?

The Lanczos algorithm starts by selecting a random vector and using it to generate a Krylov subspace. It then iteratively applies orthogonal projection to this subspace to generate a smaller subspace, while keeping the important information about the eigenvalues and eigenvectors. This process is repeated until the desired number of eigenvalues and eigenvectors are found.

What is the importance of finding eigenvectors with the Lanczos algorithm?

Eigenvectors are important in many scientific fields, such as physics, engineering, and data analysis. They represent the direction in which a matrix transforms an input vector, and can provide insights into the behavior of a system. The Lanczos algorithm allows for the efficient computation of eigenvectors, making it a valuable tool for many applications.

What are the advantages of using the Lanczos algorithm?

Compared to other methods for finding eigenvectors, the Lanczos algorithm is particularly efficient for large sparse matrices. This is because it only requires matrix-vector multiplications, which can be easily parallelized. Additionally, the Lanczos algorithm can find the largest or smallest eigenvalues, making it versatile for a range of applications.

Are there any limitations to the Lanczos algorithm?

While the Lanczos algorithm is efficient for large sparse matrices, it may not be the best choice for smaller dense matrices. Additionally, it may not always converge to the exact eigenvalues and eigenvectors due to rounding errors. However, it is still a powerful and widely used method for finding eigenvectors in various scientific and engineering fields.

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