On the Orthogonality of Eigenvectors Computed by Divide-and-Conquer Techniques

A detailed analysis on the accuracy issues in calculating the eigensystems of rank 1 perturbed diagonal systems is presented. Such calculations are the core of the divide-and-conquer technique proposed by Bunch, Nielsen, and Sorensen and refined by Dongarra and Sorensen. In particular, the computed eigenvectors are proved to be guaranteed orthogonality provided the secular equation is evaluated in a precision that doubles the working one. An efficient algorithm that simulates such "doubled precision" in working precision is also provided. Numerical results that confirm the analysis and implementation are presented.