A parallel algorithm for the eigenvalues and eigenvectors of a general complex matrix

Summary

A new parallel Jacobi-like algorithm is developed for computing the eigenvalues of a general complex matrix. Most parallel methods for this problem typically display only linear convergence, Sequential ‘norm-reducing’ algorithms also exist and they display quadratic convergence in most cases. The new algorithm is a parallel form of the ‘norm-reducing’ algorithm due to Eberlein. It is proven that the asymptotic convergence rate of this algorithm is quadratic. Numerical experiments are presented which demonstrate the quadratic convergence of the algorithm and certain situations where the convergence is slow are also identified. The algorithm promises to be very competitive on a variety of parallel architectures. In particular, the algorithm can be implemented usingn²/4 processors, takingO(n log²n) time for random matrices.