[eigen] Eigenvalues and eigenvectors of 2x2 self-adjoint matrix

[Alexey Korepanov <khumarahn@xxxxxxxxx>]

Hello.

I am comparing precision of computation of eigenvectors and eigenvalues 
of eigen and matlab. I started with a simple 2x2 self-adjoint case, 
working with long double datatype.  The best method seems to be.. "by 
hand": solving the quadratic equation for eigenvalues, and then 
computing eigenvectors. Both matlab and eigen are slower and give less 
precise results. As a measure of precision I take Frobenius norm of 
AV-VD, where A is original matrix, V is matrix of eigenvectors, D is 
diagonal matrix of eigenvalues.

Difference in precision is probably not a very big deal (like 4 last 
bits in long double), but it would be interesting to understand what 
eigen does to compute the decomposition for self-adjoint 2x2 matrix. It 
looks like eigen gives least precise results when discriminant of 
equation for eigenvalues is large. Can somebody comment on this?

Best