Re: [Bug-gsl] Bug in eigenvalue decomposition of a nonsymmetric matrix

[Patrick Alken]

Sorry for the late reply. I had a chance to look into this, and you are 
correct that your matrix causes GSL nonsymmv to fail.

In fact, if you check the return value of gsl_eigen_nonsymmv() it will 
be 11, or GSL_EMAXITER, meaning the solver failed to find all 
eigenvalues within the allotted number of iterations. In this case, 
according to the documentation, the eigenvectors are not even computed, 
so the eigenvectors you printed out are just garbage.

The eigenvalue solver failed to converge, most likely because your 
matrix is ill-conditioned and the entries vary wildly in magnitude, and 
the double-shift algorithm was unable to find appropriate shifts to 
break off a new eigenvalue block. The LAPACK library uses a more 
sophisticated algorithm than the GSL which is able to handle your 
matrix, so I'd recommend using that for these types of matrices. In the 
mean time, I'll try tinkering with the shift calculation to see if GSL 
can be improved to work with this type of matrix.

Thanks for the report and as a future note, always check the return 
value of gsl_eigen_nonsymmv() to tell immediately if there was a 
convergence problem.

Patrick

On 11/13/2012 01:07 PM, Youna Hu wrote:
> Hi, there:
>
> I was using *gsl_eigen_nonsymmv *to get the eigenvalue decomposition of the
> matrix in the attached file but found that the result is not right. Theory
> says that there is an eigenvalue with value 1.0 and the corresponding
> eigenvectors should be real. I've also tested this in the package R.
>
> Could you please check? Thanks,
>
> Youna
>
> *Eigenvalue decompositon result from R:*
>
> eigen(a)$values
>
>   [1]   1.000000e+00   1.110223e-16  -2.691078e-17  -1.540744e-33
> -1.022873e-48
>
>   [6]  -3.798227e-64   9.041170e-80 -4.313994e-112  1.043900e-127
> 2.451083e-129
>
> [11]   0.000000e+00   0.000000e+00   0.000000e+00   0.000000e+00
> 0.000000e+00
>
> [16]   0.000000e+00   0.000000e+00   0.000000e+00   0.000000e+00
> 0.000000e+00
>
> [21]   0.000000e+00   0.000000e+00   0.000000e+00   0.000000e+00
> 0.000000e+00
>
> [26]   0.000000e+00   0.000000e+00   0.000000e+00   0.000000e+00
> 0.000000e+00
>
> [31]   0.000000e+00
>
> eigen(a)$vectors[,1]
>
>   [1] -0.1796053 -0.1796053 -0.1796053 -0.1796053 -0.1796053 -0.1796053
> -0.1796053
>
>   [8] -0.1796053 -0.1796053 -0.1796053 -0.1796053 -0.1796053 -0.1796053
> -0.1796053
>
> [15] -0.1796053 -0.1796053 -0.1796053 -0.1796053 -0.1796053 -0.1796053
> -0.1796053
>
> [22] -0.1796053 -0.1796053 -0.1796053 -0.1796053 -0.1796053 -0.1796053
> -0.1796053
>
> [29] -0.1796053 -0.1796053 -0.1796053
>
>
> *EigenValue decomposition from GSL:*
>
> eigenvalue[0] = 1.00000000000000066613 + 0.00000000000000000000 i
>
> eigenvalue[1] = -0.00000000000000003534 + 0.00000000000000000000 i
>
> eigenvalue[2] = 0.00000000000000000413 + 0.00000000000000000000 i
>
> eigenvalue[3] = -0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[4] = -0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[5] = -0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[6] = -0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[7] = -0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[8] = -0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[9] = -0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[10] = 0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[11] = -0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[12] = 0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[13] = 0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[14] = -0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[15] = 0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[16] = -0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[17] = 0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[18] = -0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[19] = 0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[20] = 0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[21] = 0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[22] = 0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[23] = 0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[24] = 0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[25] = 0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[26] = 0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[27] = 0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[28] = 0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[29] = 0.00000000000000000000 + 0.00000000000000000000 i
>
> eigenvalue[30] = 0.00000000000000000000 + 0.00000000000000000000 i
>
> *The eigenvectors:*
>
> The eigenvector [0] = 0,0
>
> The eigenvector [1] = 0.158848,-0.000727072
>
> The eigenvector [2] = 2.0093e-07,-9.19685e-10
>
> The eigenvector [3] = 1.87534e-06,-8.58373e-09
>
> The eigenvector [4] = 1.26586e-05,-5.79402e-08
>
> The eigenvector [5] = 6.58246e-05,-3.01289e-07
>
> The eigenvector [6] = 0.000274269,-1.25537e-06
>
> The eigenvector [7] = 0.000940351,-4.30413e-06
>
> The eigenvector [8] = 0.00270351,-1.23744e-05
>
> The eigenvector [9] = 0.00660858,-3.02484e-05
>
> The eigenvector [10] = 0.013878,-6.35217e-05
>
> The eigenvector [11] = 0.0252328,-0.000115494
>
> The eigenvector [12] = 0.0399519,-0.000182866
>
> The eigenvector [13] = 0.055318,-0.000253199
>
> The eigenvector [14] = 0.0671718,-0.000307455
>
> The eigenvector [15] = 0.0716499,-0.000327952
>
> The eigenvector [16] = 0.0671718,-0.000307455
>
> The eigenvector [17] = 0.055318,-0.000253199
>
> The eigenvector [18] = 0.0399519,-0.000182866
>
> The eigenvector [19] = -0.974767,-0.000115494
>
> The eigenvector [20] = 0.013878,0.025667
>
> The eigenvector [21] = 0.00660858,0.0349227
>
> The eigenvector [22] = 0.00270351,-1.15514e-05
>
> The eigenvector [23] = 0.000940351,-0.625902
>
> The eigenvector [24] = 0.000274269,0.262268
>
> The eigenvector [25] = 6.58244e-05,0.65909
>
> The eigenvector [26] = 1.26585e-05,1.9919e-05
>
> The eigenvector [27] = 1.87525e-06,0.351666
>
> The eigenvector [28] = 2.0093e-07,-9.19685e-10
>
> The eigenvector [29] = 1.38572e-08,-6.34266e-11
>
> The eigenvector [30] = 4.61907e-10,-2.11422e-12