A PARALLEL ALGORITHM FOR THE REDUCTION TO TRIDIAGONAL FORM FOR EIGENDECOMPOSITIONy

A new algorithm for the orthogonal reduction of a symmetric matrix to tridiagonal form is developed and analysed. It uses a Cholesky factorization of the original matrix and the rotations are applied to the factors. The idea is similar to the one used for the one-sided Jacobi algorithms B. The algorithm uses little communication , accesses data with stride one and is to a large extent independent of data distribution. It has been implemented on the Fujitsu VPP 500. The algorithm is designed to be the rst step of an eigensolver so the procedure for accumulating transforms for eventual calculation of eigenvectors is given. 1. Introduction Symmetric eigenvalue problems appear in many applications ranging from computational chemistry to structural engineering. Algorithms for symmetric eigenvalue problems have been extensively discussed in the literature 11, 9] and implemented in various software packages (e.g. LAPACK 1]). With the broader introduction of parallel computers in scientiic computing new parallel algorithms have been suggested 7, 2]. In the following another new parallel algorithm is suggested which is particularly well adapted to vector parallel computers and has low operation counts. Eigenvalue problems can only be solved by iterative algorithms in general as they are in an algebraic sense equivalent to nding the n zeros of a polynomial. There are, however, two main classes of methods to solve the symmetric eigenvalue problem. The rst class only requires matrix vector products and does not inspect nor alter the matrix elements of the matrix. This class includes the Lanczos method 9] and