What is a robust and reliable way/library for eigenvalues of 3x3 matrices? I use Eigen to compute the eigenvalues of symmetric matrices. 
The problem is, that sometimes the matrices not nice at all numerically. Because of this, I get NaN among the eigenvalues. I have tested the matrix with maxima and WolframAlpha as well. Maxima would give me two complex and one real eigenvalues, while apparently WolframAlpha is able to compute all three real eigenvalues.
Is there a method or library, that could guarantee the eigenvalues? If not, what can be done in these cases?
For the curious, this is the matrix I've tested:
$$\begin{bmatrix} 1.4580680&-0.0028459986&0.10822548 \\\\ -0.0028459986&5.5612186 \times 10^{-6}&-0.00012824166 \\\\ 0.10822548&-0.00012824166&1.3172486 \end{bmatrix}$$

I get all the real eigenvalues using SymPy, and it gives me three real eigenvalues for the matrix you gave. Unfortunately though, it cannot determine the eigenvector for $\lambda = 1.51677607024351$, but it does determine all three real eigenvalues.
Specifically, I obtain that $\lambda \in \{ 8.60027278426881*10^{-10}, 1.25854609011506, 1.51677607024351 \}$.
SymPy uses the Berkowitz algorithm to compute eigenvalues, so I imagine it can find eigenvalues as accurately as the algorithm allows.
