It is well-known that one can extract Ritz approximations for eigenpairs from Krylov subspace information. In the 1990s much attention has been given to the so-called harmonic Ritz approximations. The harmonic Ritz pairs can be interpreted as to correspond with the inverse of A, restricted to A times the Krylov subspace. These new approximations have been suggested in the context of the methods of Lanczos, Arnoldi, and Jacobi-Davidson. From the two subspaces generated with the two-sided Lanczos algorithm we can also extract approximations for the eigenpairs associated with the inverse of A: the harmonic Petrov approximations. In this case, we obtain approximations for left and right eigenvectors of A.
We will give an overview of the various harmonic approximations for eigenpairs and we we will discuss how they can be computed. It turns out that the harmonic approximations may have an advantage for the computation of eigenpairs corresponding to interior eigenvalues.