Re: Re: Mathematica, ARPACK and implicit matrices

*To*: mathgroup at smc.vnet.net*Subject*: [mg96620] Re: [mg96592] Re: Mathematica, ARPACK and implicit matrices*From*: Fernando Cucchietti <fernando.cucchietti at icfo.es>*Date*: Tue, 17 Feb 2009 06:27:37 -0500 (EST)*References*: <20090214221813.9l7ub5g1cs8c4g44@www.icfo.es> <4997A3D9.3060801@gmail.com> <gnbkea$3g2$1@smc.vnet.net> <200902162141.QAA16086@smc.vnet.net>

That's a good idea, but still I/d like to know if I can get a few eigenvalues and vectors in Mathematica without writing the matrix down. This is not the first time I face this problem, but it is the first one I cannot avoid it. I was trying to find a more general solution, and using Mathematica's internal algorithms is the only way I know of keeping efficiency good. Fernando On Feb 16, 2009, at 10:41 PM, dh wrote: > > > 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 > >> > > >

**References**:**Re: Mathematica, ARPACK and implicit matrices***From:*dh <dh@metrohm.com>