A parallel algorithm for determining all eigenvalues of large real symmetric tridiagonal matrices

Abstract

A method for determining all eigenvalues of large real symmetric tridiagonal matrices on multiprocessor system with vector facilities is presented. For finding the eigenvalues of a tridiagonal matrix, the method of the Sturm sequence is a standard method. The method uses bisection first to isolate all eigenvalues, bisection is and then to extract the eigenvalues to a predefined accuracy. For extracting the eigenvalues, bisection is accelerated by a superlinearly convergent zero finder, the Pegasus method. The evaluation of the Sturm sequence is the central component for both isolation and extraction. Some new ideas are presented, such as a method for weighting the values of the characteristics polynomial to avoid under- or overflow, a method for combining the Pegasus method with preceding bisection steps and a vectorization and parallelization strategy over intervals. The method was implemented and the results were measured on a SUPRENUM multiprocessor system with 16 processors and on a CRAY Y-MP8/832 with 8 processors. On the latter machine, both the sequential and parallel execution time of our algorithm ALLEV (ALL Eigen Values) presented in this paper are considerably shorter than the execution times of the vectorized EISPACK-routine TQL1 which uses the QL method.