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Re: eigenvector computations

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  • Subject: [mg117362] Re: eigenvector computations
  • From: Daniel Lichtblau <danl at>
  • Date: Wed, 16 Mar 2011 06:27:16 -0500 (EST)

Bill Thurston wrote:
> I'm doing some computations where I need to find the size of the leading eigenvalue of various matrices that depend on a rational number between 0 and 1.  The matrices are mostly 0, with nonzero entries 1 or 2.  I'm doing this using
> sparse matrices and, for a matrix A,  asking for 
> First@Abs[ Eigenvalues[ N[ A],1]]
> which usually works fine, but sometimes I get error messages like this:
> Eigenvalues::maxit2 :  "Warning: maximum number of iterations, 1000, has been \
> reached by the Arnoldi algorithm without convergence to the specified \
> tolerance, but the current best computed value has been returned. You can use \
> method options with Method -> {Arnoldi, opts} to increase the size of basis \
> vectors, the maximum number of iterations, reduce the tolerance, or use an \
> estimate as a shift, any of which may help."
> This would be fine except that I can't find any Mathematica documentation for the Arnoldi method or its options.
> What gives?   
> 	Bill Thurston

This lack of documentation on the advanced features of numerical linear 
algebra was filed as a bug report several months back. I am ever hopeful 
that it will be addressed.

For your specific problem, I wonder if it might be worked around by 
adding a modestly large multiple m of an identity matrix, then 
subtracting m from the eigenvalue result? That is akin (or at least 
related) to a shift, but does not require intricate knowledge of Method 
-> {Arnoldi, opts} (which, without careful checking, I also do not have).

Daniel Lichtblau
Wolfram Research

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