Dear All,

At

http://eigen.tuxfamily.org/dox/group__T ... tions.html

one is informed that the algorithm reliability and accuracy of

EigenSolver

depends on condition number.

Can one improve the accuracy of a eigenvalue returned by Eigen?

Thanks in advance!

hm, this table seems to be obsolete. the implementation of EigenSolver has proven to be pretty stable. Nonetheless, don't miss the SelfAdjointEigenSolver class for symmetric/hermitian matrices.

Thanks for your reply.

Unfortunately, my matrix is not symmetric and, moreover, is defective.

Eigen returns the following eigenvalues:

(2,0),
(-1.83128e-06,3.17189e-06),
(-1.83128e-06,-3.17189e-06),
(3.66256e-06,0),
(-1.16573e-15,9.85619e-09),
(-1.16573e-15,-9.85619e-09).

And theoretically the eigenvalues are:

2 (with algebraic multiplicity 1)

and

0 (with algebraic multiplicity 5).

So, my question is: How can one increase accuracy in order to obtain the last 5 eigenvalues much closer to zero and conclude with safety that all 5 eigenvalues are equal to zero?

Again, many thanks!

could you provide your matrix so that we can reproduce.

Thanks again, Gael.

The matrix is the following:

1,0,0,0,0,1
0,1,0,0,1,0
0,1,0,0,1,0
1,0,1,0,0,0
1,0,1,0,0,0
0,1,0,1,0,0

And, please, notice that the proper subspace associated to the zero eigenvalue of the above matrix has dimension 2 and the eigenvectors that Eigen returns for this eigenvalue do not form a basis of the mentioned proper subspace.

Alright, I confirm. I also see that Matlab is returning the same as Eigen.

Thanks, Gael. But is there some way of circumventing the reported problem?

Looking back at this entry, the "problem" is that this matrix is defective; it is not diagonalizable. The eigenvalue 0 has only 2 corresponding eigenvectors, whereas its algebraic multiplicity is 5.