DOI QR코드

DOI QR Code

A SIMPLE AUGMENTED JACOBI METHOD FOR HERMITIAN AND SKEW-HERMITIAN MATRICES

  • Min, Cho-Hong (Department of Mathematics, Ewha Womans University) ;
  • Lee, Soo-Joon (Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University) ;
  • Kim, Se-Goo (Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University)
  • Received : 2010.03.02
  • Accepted : 2011.07.15
  • Published : 2011.08.31

Abstract

In this paper, we present a new extended Jacobi method for computing eigenvalues and eigenvectors of Hermitian matrices which does not use any complex arithmetics. This method can be readily applied to skew-Hermitian and real skew-symmetric matrices as well. An example illustrating its computational efficiency is given.

Keywords

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

References

  1. P. J. Eberlein: A Jacobi-like method for the automatic computation of eigenvalues and eigenvectors of an artibrary matrix. J. Soc. Indust. Appl. Math. (1962) 10, no. 1, 74-88. https://doi.org/10.1137/0110007
  2. H. H. Goldstine & L.P. Horwitz: A procedure for the diagonalization of normal matrices. J. Assoc. Comput. Mach. (1959) 6, 176-195. https://doi.org/10.1145/320964.320975
  3. G. Golub & C. V. Loan: Matrix Computations. Johns Hopkins University Press, 1983.
  4. D. Hacon: Jacobi's method for skew-symmetric matrices. SIAM J. Matrix Anal. Appl. (1993) 14, no. 3, 619-628. https://doi.org/10.1137/0614043
  5. P. Henrici: On the speed of convergence of cyclic and quasicyclic Jacobi methods for computing eigenvalues of Hermitian matrices. J. Soc. Indust. Appl. Math. (1958) 6, no. 2, 144-162. https://doi.org/10.1137/0106008
  6. J. Jacobi: Ber ein leichtes Verfahren, die in der Theorie der SdtularstSrungen vorkom-menden Gleichungen numerisch aufzulSsen. J. Reine Angew. Math. (1846) 30, 51-95.
  7. D. James & V. Kresimir: Jacobi's method is more accurate than QR. SIAM J. Matrix Anal. Appl. (1992) 13, no. 4, 1204-1245. https://doi.org/10.1137/0613074
  8. A. Ruhe: On the quadratic convergence of a generalization of the Jacobi method to arbitrary matrices. BIT Numerical Mathematics (1968) 8, no. 3, 210-231. https://doi.org/10.1007/BF01933422
  9. G. L. G. Sleijpen & H. A. Van der Vorst: A Jacobi-Davidson iteration method for linear eigenvalue problems. SIAM Review (2000) 42, no. 2, 267-293. https://doi.org/10.1137/S0036144599363084