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

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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?


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