Mathematica, ARPACK and implicit matrices

*To*: mathgroup at smc.vnet.net*Subject*: [mg96562] Mathematica, ARPACK and implicit matrices*From*: Fernando Cucchietti <Fernando.Cucchietti at icfo.es>*Date*: Mon, 16 Feb 2009 06:59:21 -0500 (EST)*References*: <20090214221813.9l7ub5g1cs8c4g44@www.icfo.es> <4997A3D9.3060801@gmail.com>

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