Higher Level BLAS in Symmetric Eigensolvers

In this report a way to apply high level Blas to the tridiagonalization process of a symmetric matrix A is investigated. Tridiagonalization is a very important and work-intensive preprocessing step in eigenvalue computations. It also arises as a very central part of the material sciences code Wien 97 (Blaha et al. [12]). After illustrating the drawbacks and limitations of the tridiagonalization implemented in Lapack (Anderson et al. [1]), the dependency structure in the tridiagonalization process is analyzed. It is shown that data references cannot be localized for tridiagonalization in one sweep and that Level 2 operations cannot be avoided in a one-sweep tridiagonalization algorithm. Thus a different method is considered, by which the inherent potential for using higher level Blas can be exploited. By performing tridiagonalization in two successive sweeps, it is possible to utilize Level 3 Blas in the first sweep, which transforms A into a band matrix B. In a second sweep, B is transformed to a tridiagonal matrix T. The second sweep represents only a very small portion of the overall work, and for this reason the new method yields very good performance. Investigation of the backtransformation process for the corresponding eigenvectors reveals that it is preferable to calculate the eigenvectors from the band matrix B and not from the tridiagonal matrix T