Task Based Robust Computation of Eigenvectors

Abstract. Dense eigenvalue problems are solved using orthogonal similarity transformations. In the case of the standard eigenvalue problem, one first reduces the matrix to upper Hessenberg form. Then the QR algorithm is applied until the real Schur form emerges. Eigenvectors of the Schur matrix for the user’s selection of eigenvalues can now be computed and transformed to the original basis. In this talk, we focus on the problem of computing eigenvectors from matrices in real Schur form. The problem is interesting for the following reasons. If the eigenvalues are clustered, then the calculation is likely to overflow. In LAPACK, overflow is prevented by xLATRS and its descendants. We have recently shown how to protect against overflow in a parallel setting. Moreover, while the calculation of a single eigenvector is memory-bound, the computation of several eigenvectors can be interleaved to increase the overall arithmetic intensity.