Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2013, Special Issue (2013), Article ID 827826, 4 pages.

Comparison Theorems for Single and Double Splittings of Matrices

Cui-Xia Li, Qun-Fa Cui, and Shi-Liang Wu

Full-text: Open access

Abstract

Some comparison theorems for the spectral radius of double splittings of different matrices under suitable conditions are presented, which are superior to the corresponding results in the recent paper by Miao and Zheng (2009). Some comparison theorems between the spectral radius of single and double splittings of matrices are established and are applied to the Jacobi and Gauss-Seidel double SOR method.

Article information

Source
J. Appl. Math., Volume 2013, Special Issue (2013), Article ID 827826, 4 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394807695

Digital Object Identifier
doi:10.1155/2013/827826

Mathematical Reviews number (MathSciNet)
MR3039756

Zentralblatt MATH identifier
1266.65053

Citation

Li, Cui-Xia; Cui, Qun-Fa; Wu, Shi-Liang. Comparison Theorems for Single and Double Splittings of Matrices. J. Appl. Math. 2013, Special Issue (2013), Article ID 827826, 4 pages. doi:10.1155/2013/827826. https://projecteuclid.org/euclid.jam/1394807695


Export citation

References

  • Z. I. Woźnicki, “Estimation of the optimum relaxation factors in partial factorization iterative methods,” SIAM Journal on Matrix Analysis and Applications, vol. 14, no. 1, pp. 59–73, 1993.
  • R. S. Varga, Matrix Iterative Analysis, vol. 27 of Springer Series in Computational Mathematics, Springer, Berlin, Germany, 2000.
  • A. Berman and R. J. Plemons, Nonnegative Matrices in the Mathematics Sciences, SIAM, Philadelphia, Pa, USA, 1994.
  • D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, NY, USA, 1971.
  • Z. I. Woźnicki, “Nonnegative splitting theory,” Japan Journal of Industrial and Applied Mathematics, vol. 11, no. 2, pp. 289–342, 1994.
  • Z. I. Woźnicki, “Basic comparison theorems for weak and weaker matrix splittings,” Electronic Journal of Linear Algebra, vol. 8, pp. 53–59, 2001.
  • M. Benzi and D. B. Szyld, “Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods,” Numerische Mathematik, vol. 76, no. 3, pp. 309–321, 1997.
  • J. Song and Y. Song, “Convergence for nonnegative double splittings of matrices,” Calcolo, vol. 48, no. 3, pp. 245–260, 2011.
  • G. H. Golub and R. S. Varga, “Chebyshev semi-iterative methods, successive over-relaxation iterative methods, and second order Richardson iterative methods,” Numerische Mathematik, vol. 3, pp. 147–156, 1961.
  • S. Q. Shen and T. Z. Huang, “Convergence and comparison theorems for double splittings of matrices,” Computers & Mathematics with Applications, vol. 51, no. 12, pp. 1751–1760, 2006.
  • S. Q. Shen, T. Z. Huang, and J. L. Shao, “Convergence and comparison results for double splittings of Hermitian positive definite matrices,” Calcolo, vol. 44, no. 3, pp. 127–135, 2007.
  • S. X. Miao and B. Zheng, “A note on double splittings of different monotone matrices,” Calcolo, vol. 46, no. 4, pp. 261–266, 2009.
  • C. Y. Zhang, “On convergence of double splitting methods for non-Hermitian positive semidefinite linear systems,” Calcolo, vol. 47, no. 2, pp. 103–112, 2010.
  • L. Elsner, A. Frommer, R. Nabben, H. Schneider, and D. B. Szyld, “Conditions for strict inequality in comparisons of spectral radii of splittings of different matrices,” Linear Algebra and its Applications, vol. 363, pp. 65–80, 2003.
  • L. J. Cvetković, “Two-sweep iterative methods,” Nonlinear Analysis: Theory, Methods & Applications, vol. 30, no. 1, pp. 25–30, 1997.