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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 105301, 9 pages
http://dx.doi.org/10.1155/2014/105301
Research Article

Approximating the Matrix Sign Function Using a Novel Iterative Method

1Department of Mathematics, Islamic Azad University, Shahrekord Branch, Shahrekord, Iran
2Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, 18000 Niš, Serbia
3Department of Mathematics and Applied Mathematics, University of Venda, Private Bag X5050, Thohoyandou 0950, South Africa

Received 19 April 2014; Accepted 6 July 2014; Published 17 July 2014

Academic Editor: Juan R. Torregrosa

Copyright © 2014 F. Soleymani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. N. J. Higham, Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  2. J. L. Howland, “The sign matrix and the separation of matrix eigenvalues,” Linear Algebra and Its Applications, vol. 49, pp. 221–232, 1983. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. J. Leyva-Ramos, “A note on mode decoupling of linear time-invariant systems using the generalized sign matrix,” Applied Mathematics and Computation, vol. 219, no. 22, pp. 10817–10821, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. J. D. Roberts, “Linear model reduction and solution of the algebraic Riccati equation by use of the sign function,” International Journal of Control, vol. 32, no. 4, pp. 677–687, 1980. View at Publisher · View at Google Scholar
  5. Z. Bai and J. Demmel, “Using the matrix sign function to compute invariant subspaces,” SIAM Journal on Matrix Analysis and Applications, vol. 19, no. 1, pp. 205–225, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. R. Byers, C. He, and V. Mehrmann, “The matrix sign function method and the computation of invariant subspaces,” SIAM Journal on Matrix Analysis and Applications, vol. 18, no. 3, pp. 615–632, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. C. S. Kenney, A. J. Laub, and P. M. Papadopoulos, “Matrix-sign algorithms for Riccati equations,” IMA Journal of Mathematical Control and Information, vol. 9, no. 4, pp. 331–344, 1992. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. P. Benner and E. S. Quintana-Ortí, “Solving stable generalized Lyapunov equations with the matrix sign function,” Numerical Algorithms, vol. 20, no. 1, pp. 75–100, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. S. Barrachina, P. Benner, and E. S. Quintana-Ort, “Efficient algorithms for generalized algebraic Bernoulli equations based on the matrix sign function,” Numerical Algorithms, vol. 46, no. 4, pp. 351–368, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. A. N. Norris, A. L. Shuvalov, and A. A. Kutsenko, “The matrix sign function for solving surface wave problems in homogeneous and laterally periodic elastic half-spaces,” Wave Motion, vol. 50, no. 8, pp. 1239–1250, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. B. Laszkiewicz and K. Zietak, “Algorithms for the matrix sector function,” Electronic Transactions on Numerical Analysis, vol. 31, pp. 358–383, 2008. View at MathSciNet · View at Scopus
  12. C. Kenney and A. J. Laub, “Rational iterative methods for the matrix sign function,” SIAM Journal on Matrix Analysis and Applications, vol. 12, no. 2, pp. 273–291, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. B. Iannazzo, Numerical solution of certain nonlinear matrix equations [Ph.D. thesis], Universita degli Studi di Pisa, Facolta di Scienze Matematiche, Fisiche e Naturali, Pisa, Italy, 2007.
  14. E. Schröder, “Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen,” Mathematische Annalen, vol. 2, no. 2, pp. 317–365, 1870. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. E. Schröder, “On infinitely many algorithms for solving equations,” Tech. Rep. TR-92-121, Department of Computer Science, University of Maryland, College Park, Md, USA, 1992.
  16. F. Soleymani, E. Tohidi, S. Shateyi, and F. Khaksar Haghani, “Some matrix iterations for computing matrix sign function,” Journal of Applied Mathematics, vol. 2014, Article ID 425654, 9 pages, 2014. View at Publisher · View at Google Scholar
  17. P. Jarratt, “Some fourth order multi-point iterative methods for solving equations,” Mathematics of Computation, vol. 20, no. 95, pp. 434–437, 1966.
  18. A. Iliev and N. Kyurkchiev, Nontrivial Methods in Numerical Analysis: Selected Topics in Numerical Analysis, LAP Lambert, 2010.
  19. C. S. Kenney and A. J. Laub, “The matrix sign function,” IEEE Transactions on Automatic Control, vol. 40, no. 8, pp. 1330–1348, 1995. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. B. Iannazzo, “A family of rational iterations and its application to the computation of the matrix pth root,” SIAM Journal on Matrix Analysis and Applications, vol. 30, no. 4, pp. 1445–1462, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. M. Trott, The Mathematica Guide-Book for Numerics, Springer, New York, NY, USA, 2006.