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 usingn 2/4 processors, takingO(n log2 n) time for random matrices.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bischof, C.: Computing the singular value decomposition on a distributed system of vector processors. Technical report 87 869, Department of Computer Science, Cornell University 1987
Brent, R.P., Luk, F.T.: The solution of singular-value and symmetric eigenvalue problems on multiprocessor arrays. SIAM J. Sci. Stat. Comput.6, 69–84 (1985)
Causey, R.L.: Computing eigenvalues of non hermitian matrices by methods of Jacobi type. J. SIAM6, 172–181 (1958)
Eberlein, P.J.: A Jacobi method for the automatic computation of eigenvalues and eigenvectors of an arbitrary matrix. J. SIAM10, 74–88 (1962)
Eberlein, P.J.: On the Schur decomposition of a matrix for parallel computation. IEEE Trans. Comput.36, 167–174 (1987)
Eberlein, P.J., Moothroyd, J.: Solution to the eigenproblem by a normreducing Jacobi-type method. Numer. Math.4, 24–40 (1968)
Fan, K., Hoffman, A.J.: Lower bounds for the rank and location of eigenvalues of a matrix. National Bureau of Standards Applied Math. Series39, 117–130 (1954)
Forsythe, G.E., Henrici, P.: The cyclic Jacobi method for computing the principal values of a complex matrix. Trans. Am. Math. Soc.94, 1–23 (1960)
Goldstine, H.H., Horwitz, L.P.: A procedure for the diagonalization of normal matrices. J ACM6, 176–195 (1959)
Goldstine, H.H., Murray, F.J., Neumann, J. von: The Jacobi method for real symmetric matrices. J ACM6, 59–96 (1959)
Golub, G., Van Loan, C.: Matrix computations. Johns Hopkins University Press (1983)
Hansen, E.R.: On Jacobi methods and block Jacobi methods for computing matrix eigenvalues. PhD thesis, Stanford University 1960
Jacobi, C.G.J.: Über ein Verfahren die in Theorie der Säkularstörungen vorkommenden Gleichungen numerisch aufzulösen. J. Reine Angew. Math.30, 51–95 (1846)
Luk, F.T., Park, H.T.: On parallel Jacobi orderings. Technical report EE-CEG-86-5, School of Electrical Engineering, Cornell University 1986
Luk, F.T., Park, H.T.: A proof of convergence for two parallel Jacobi SVD algorithms. Technical Report EE-CEG-86-12, School of Electrical Engineering, Cornell University 1986
Osborne, E.E.: On pre-conditioning of matrices. J. ACM7, 338–345 (1960)
Pardekooper, M.H.C.: An eigenvalue algorithm based on norm-reducing transformations. PhD thesis, Technische Universiteit Eindhoven 1969
Pardekooper, M.H.C.: A quadratically convergent parallel Jacobi process for diagonally dominant matrices with distinct eigenvalues. J. Comput. Appl. Math.27, 3–16 (1989)
Ruhe, A.: On the quadratic convergence of a generalization of the Jacobi method to arbitrary matrices. BIT8, 210–231 (1968)
Rutishauser, H.: Une methode pour le calcul des values propres des matrices nonsymetriques. Comptes Rendus259, 2758 (1964)
Sameh, A.H.: On Jacobi and Jacobi-like algorithms for a parallel computer. Math. Comput.25, 579–590 (1971)
Schreiber, R.: Solving eigenvalue and singular value problems on an undersized systolic array. SIAM J. Sci. Stat. Comput.7, 441–451 (1986)
Shroff, G.: Parallel Jacobi algorithms for the algebraic eigenvalue problem. PhD thesis, Rensselaer Polytechnic Institute 1990
Shroff, G., Schreiber, R.: On the convergence of the cyclic Jacobi method for parallel block orderings. SIAM J. Matrix Anal. Appl.10 (1989)
Stewart, G.W.: A Jacobi-like algorithm for computing the schur decomposition of a nonhermitian matrix. SIAM J. Sci. Stat. Comput.6, 853–864 (1985)
Veselic, K.: On a class of Jacobi-like procedures for diagonalizing arbitrary real matrices. Numer. Math.33, 157–172 (1979)
Veselic, K.: A quadratically convergence Jacobi-like method for real matrices with complex conjugate eigenvalues. Numer. Math.33, 425–435 (1979)
Wilkinson, J.H.: Almost diagonal matrices with multiple or close eigenvalues. Linear Algebra Appl.1, 1–12 (1968)
Author information
Authors and Affiliations
Additional information
This research was supported by the Office of Naval Research under Contract N00014-86-k-0610 and by the U.S. Army Research Office under Contract DAAL 03-86-K-0112. A portion of this research was carried out while the author was visiting RIACS, Nasa Ames Research Center
Rights and permissions
About this article
Cite this article
Shroff, G.M. A parallel algorithm for the eigenvalues and eigenvectors of a general complex matrix. Numer. Math. 58, 779–805 (1990). https://doi.org/10.1007/BF01385654
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01385654