Re: Mathematica, ARPACK and implicit matrices

• To: mathgroup at smc.vnet.net
• Subject: [mg96592] Re: Mathematica, ARPACK and implicit matrices
• From: dh <dh at metrohm.com>
• Date: Mon, 16 Feb 2009 16:41:53 -0500 (EST)
• References: <20090214221813.9l7ub5g1cs8c4g44@www.icfo.es> <4997A3D9.3060801@gmail.com> <gnbkea\$3g2\$1@smc.vnet.net>

```
Hi Fernando,

if the searched for eigenvalue is the largest in magnitude, your problem

is easily solved by the "Power method".

Take an arbitrary vector, keep on multiplying it by your matrix until it

cnoverges. This gives the eigenvector (provided you did not pick a start

vector perpendicular to the eigenvector).

hope this helps, Daniel

Fernando Cucchietti wrote:

> I am trying to find the largest eigenvalue and associated eigenvector

> of a very large matrix, mildly sparse but without a simple structure

> -- that is, zero elements are arranged in a seemingly random way.

>

> Mathematica uses ARPACK routines for this task. However, it appears

> that it wants to construct the matrix explicitly before starting. This

> is strange when compared to how ARPACK works, and in fact it is the

> one thing I was trying to avoid doing: Writing down the matrix takes

> too much memory and time, even if it is sparse.

>

> ARPACK is designed to require not the matrix, but just a function that

> gives the result of multiplying the matrix with an arbitrary vector. I

> call this an implicit definition of the matrix, hence the subject of

> the email. This would work very well with me since I have a compact

> expression of the matrix in the form of a function that returns the

> product with a vector, and I would not need to define the array

> explicitly.

>

> I have been looking but I cannot find an option or a way to make

> Mathematica give me the eigenvalues without writing the matrix

> explicitly, any suggestions?

>

> Fernando Cucchietti

>

```

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