A parallel algorithm for computing the eigenvalues of a symmetric tridiagonal matrix

Abstract:

A parallel algorithm, called polysection, is presented for computing the eigenvalues of a symmetric tridiagonal matrix. The method is based on a quadratic recurrence in which the characteristic polynomial is constructed on a binary tree from polynomials whose degree doubles at each level. Intervals that contain exactly one zero are determined by the zeros of polynomials at the previous level which ensures that different processors compute different zeros. The signs of the polynomials at the interval endpoints are determined a priori and used to guarantee that all zeros are found. The use of finite-precision arithmetic may result in multiple zeros; however, in this case, the intervals coalesce and their number determines exactly the multiplicity of the zero. For an $N \times N$ matrix the eigenvalues can be determined in $O({\log ^2}N)$ time with ${N^2}$ processors and $O(N)$ time with N processors. The method is compared with a parallel variant of bisection that requires $O({N^2})$ time on a single processor, $O(N)$ time with N processors, and $O(\log N)$ time with ${N^2}$ processors.