# Eigenvector for Large Matrix

1. Jul 1, 2008

### senorbum

Hi, this is my first post here, so bare with me.

So I need to compute the eigenvectors of a large matrix (1000x1000) to (10000x10000x) or so. However, I already have the eigenvalues and diagonal/superdiagonal form of the matrix. The equation (A-lambda*I)*v = 0, where A is the matrix, lambda is the given eigenvalue, and v is the eigenvector to solve for. However, with such large matrices there would be an absurd amount of equations to solve for, correct? A) Are there any BLAS algorithms to solve for this(seems unlikely) and B) Would it be easier to ignore the eigenvalues that I can get from a different program and just start at the beginning with something like a jacobi transformation? I know I can use programs like MATLAB to calculate these eigenvectors, but my goal is to implement a parallel version to increase the calculation time of the eigenvectors.

If any clarification is needed, please let me know. I will check this thread/forum at least a couple times during the work day (8am-5pm EST)

Thanks,

Joe

2. Jul 1, 2008

3. Jul 2, 2008

4. Jul 2, 2008

### maze

Iterative methods like Gauss-Seidel are great if $(A-\lambda I)$ is symmetric and positive definite, and possibly under other conditions. On the other hand, for a general matrix it is not guaranteed to converge.

5. Jul 7, 2008

### senorbum

Do you happen to have any resources (such as a good website) that describe this algorithm? The matrix will be square/symmetric so it should converge fine.

Edit: I found some resources on it, but would still find others helpful if you know of any.
Edit2: This method is probably not going to work for me, as it is very serial. I am in need of something that can at least be somewhat parallel. If I can't find anything like that, I might as well just use a matlab implementation. What about a Jacobi implementation? The problem I see with it is that there are numerous matrix calculations.

So, the Jacobi method makes sense to me except for one thing. How do you choose x to start with? I'm probably missing something easy here...

Last edited: Jul 7, 2008
6. Jul 7, 2008

### HallsofIvy

Staff Emeritus
Here's another guy asking us to "bare" with him! This forum is getting downright kinky!

7. Jul 7, 2008

### senorbum

I couldn't remember which spelling to use. That and its really hot in an office with 3 computers and 5 monitors running all day every day ;)

8. Jul 7, 2008

### maze

Ahh, too bad. I just would like to point out (maybe you already realized this), that it isn't completely serial - although the computation for a single eigenvector is serial, you can compute all the different eigenvectors in parallel.

The more I think about it though, the more I am skeptical if it will work for your matrix. The method would be applied to the matrix $A-\lambda I$ (not A), which is certainly not going to be positive definite. The method could probably be salvaged with some trickery, but I'm not seeing it right now.

The Jacobi method is basically a weaker form of Gauss-Seidel, so I would be surprised if you could get something out of the Jacobi method you couldn't with Gauss-Seidel.

9. Jul 8, 2008

### senorbum

Yeah, I figured this out as well.

So, the new plan after discussing some things with my adviser is to go a slightly different route, which will hopefully still involve a speedup. Instead of directly speeding up the eigen decomposition, the plan is to hopefully speedup the preconditioning enough to impact the problem.

10. Jul 17, 2008

### JPRitchie

The area of parallel diagonalization of large matrices is of great interest. To get started, take a look at: http://www.cse.scitech.ac.uk/arc/diags.shtml . If you have additional questions about the software, contact me privately.

Jim Ritchie