Re: Re: Eigenvalues of sparse arrays

*To*: mathgroup at smc.vnet.net*Subject*: [mg102277] Re: [mg102238] Re: Eigenvalues of sparse arrays*From*: Mark McClure <mcmcclur at unca.edu>*Date*: Mon, 3 Aug 2009 05:49:26 -0400 (EDT)*References*: <200907310952.FAA19221@smc.vnet.net> <h50sjj$6pl$1@smc.vnet.net>

On Sun, Aug 2, 2009 at 5:59 AM, gopher <gophergoon at gmail.com> wrote: > Thanks for your reply. I read those docs but IMO they are somewhat > misleading. They state that the Arnoldi method may not converge but > not that it may converge and still give spurious eigenvalues. I agree; a warning should be issued. There are simple tests based on condition numbers for this sort of thing. The matrix I was using is here (it is only 156x156): > https://netfiles.uiuc.edu:443/aroy2/www/sparse%20array/bugmatrix.dat > Arnoldi iteration is a technique to find the *largest* eigenvalue of a matrix. We can find other eigenvalues of the matrix A by applying Arnoldi iteration to the matrix (A-sI)^(-1), where s is a number close to the eigenvalue we want. This works since s+1/m is an eigenvalue of A close to s, whenever m is a large eigenvalue of (A-sI)^(-1). Not surprisingly, there is a problem if s is too close to an exact eigenvalue, for then A-sI is singular. A simple test for this possibility is the condition number of A-sI, which should not be too large. In your case, you are looking for eigenvalues close to zero of a matrix that is singular to start with. The condition number of this matrix is huge, since it's singular. dat = N[Import[ "https://netfiles.uiuc.edu/aroy2/www/sparse%20array/bugmatrix.dat";]]; sp = SparseArray[dat]; LinearAlgebra`MatrixConditionNumber[sp] 7.22615*10^17 We can fix the situation by using the Shift sub-option of the Arnoldi method. This options specifies the number s and should be set close to zero, but certainly not equal to zero. We can do this as follows. Eigenvalues[sp, 6, Method -> {"Arnoldi", "Shift" -> 0.01}] {0.382326, -0.382326, 0.350062, -0.350062, 4.74967*10^-15, 9.95731*10^-16} Eigenvalues[dat, -6] {0.382326, -0.382326, -0.350062, 0.350062, 2.29823*10^-15, 7.57673*10^-16} Hope that helps, Mark