Abstract
ABSTRACT A new parallel Jacobi-like algorithm for computing the eigenvalues of a general complex matrix is presented.The asymptotic convergence rate of this algorithm is provably quadratic and this is also demonstrated in numericalexperiments. The algorithm promises to be suitable for real-time signal processing applications. In particular, thealgorithm can be implemented using n2/4 processors, taking O(n log2 n) time for random matrices.1. INTRODUCTIONMany signal processing applications involve computing the eigenvalues and eigenvectors of matrices. Tradi-tionally, the Jacobi method has been popular for real-time applications involving the symmetric (or Hermitian)eigenvalue problem because its inherent parallelism (as opposed to the QR algorithm) which allows for its efficientimplementation on architectures such as systolic arrays. Sometimes however, the eigenvalues of non-symmetricmatrices are required, such as in the ESPRIT algorithm for high resolution beamforming [1].Many attempts have been made to extend the Jacobi method to the general case [2, 3, 4, 5, 6, 7] and some ofthese are suited for parallel implementation {2, 4, 5, 7]. However, most of these parallel algorithms do not possessthe quadratic convergence property typical of the Jacobi method for hermitian matrices.In this paper we present a parallel Jacobi-like algorithm for general complex matrices based on a method firstintroduced by Eberlein [3] which can be proven to have a quadratic rate of convergence. It is also shown that thealgorithm may be implemented in parallel using n2/4 processors and can be expected to take O(n log2 n) time forrandom matrices.The rest of the paper is organized as follows. In we give a brief history of Jacobi-like methods for general
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