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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>