  We describe algorithms for computing eigenpairs (eigenvalue-eigenvector
pairs) of a complex $n\times n$ matrix $A$. These algorithms are numerically
stable, strongly accurate, and theoretically efficient (i.e., polynomial-time).
We do not believe they outperform in practice the algorithms currently used for
this computational problem. The merit of our paper is to give a positive answer
to a long-standing open problem in numerical linear algebra.
