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eigenvector computations

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  • Subject: [mg117328] eigenvector computations
  • From: Bill Thurston <wpthurston at>
  • Date: Tue, 15 Mar 2011 06:05:13 -0500 (EST)

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

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