## Journal of Applied Mathematics

• J. Appl. Math.
• Volume 2014, Special Issue (2014), Article ID 741368, 7 pages.

### A High-Order Iterate Method for Computing ${A}_{T,S}^{(2)}$

#### Abstract

We investigate a new higher order iterative method for computing the generalized inverse ${A}_{T,S}^{(2)}$ for a given matrix $A$. We also discuss how the new method could be applied for finding approximate inverses of nonsingular square matrices. Analysis of convergence is included to show that the proposed scheme has at least fifteenth-order convergence. Some tests are also presented to show the superiority of the new method.

#### Article information

Source
J. Appl. Math., Volume 2014, Special Issue (2014), Article ID 741368, 7 pages.

Dates
First available in Project Euclid: 1 October 2014

https://projecteuclid.org/euclid.jam/1412178030

Digital Object Identifier
doi:10.1155/2014/741368

Mathematical Reviews number (MathSciNet)
MR3208636

#### Citation

Liu, Xiaoji; Zuo, Zemeng. A High-Order Iterate Method for Computing ${A}_{T,S}^{(2)}$. J. Appl. Math. 2014, Special Issue (2014), Article ID 741368, 7 pages. doi:10.1155/2014/741368. https://projecteuclid.org/euclid.jam/1412178030

#### References

• F. Hsuan, P. Langenberg, and A. Getson, “The $\{2\}$-inverse with applications in statistics,” Linear Algebra and Its Applications, vol. 70, pp. 241–248, 1985.
• M. Z. Nashed, Generalized Inverses and Applications, Academic Press, New York, NY, USA, 1976.
• Y.-L. Chen, “Iterative methods for computing the generalized inverses ${A}_{T,S}^{(2)}$ of a matrix $A$,” Applied Mathematics and Computation, vol. 75, no. 2-3, pp. 207–222, 1996.
• Y.-L. Chen and X. Chen, “Representation and approximation of the outer inverse ${A}_{T,S}^{(2)}$ of a matrix $A$,” Linear Algebra and Its Applications, vol. 308, no. 1–3, pp. 85–107, 2000.
• X. Sheng, G. Chen, and Y. Gong, “The representation and computation of generalized inverse ${A}_{T,S}^{(2)}$,” Journal of Computational and Applied Mathematics, vol. 213, no. 1, pp. 248–257, 2008.
• D. S. Djordjević, P. S. Stanimirović, and Y. Wei, “The representation and approximations of outer generalized inverses,” Acta Mathematica Hungarica, vol. 104, no. 1-2, pp. 1–26, 2004.
• D. S. Djordjević and P. S. Stanimirović, “Iterative methods for computing generalized inverses related with optimization methods,” Journal of the Australian Mathematical Society, vol. 78, no. 2, pp. 257–272, 2005.
• S. Miljković, M. Miladinović, P. S. Stanimirović, and Y. Wei, “Gradient methods for computing the Drazin-inverse solution,” Journal of Computational and Applied Mathematics, vol. 253, pp. 255–263, 2013.
• F. Soleymani, “A fast convergent iterative solver for approximate inverse of matrices,” Numerical Linear Algebra with Applications, vol. 21, no. 3, pp. 439–452, 2013.
• P. S. Stanimirović, D. Pappas, V. N. Katsikis, and I. P. Stanimirović, “Full-rank representations of outer inverses based on the QR decomposition,” Applied Mathematics and Computation, vol. 218, no. 20, pp. 10321–10333, 2012.
• P. S. Stanimirović and M. D. Petković, “Gauss-Jordan elimination method for computing outer inverses,” Applied Mathematics and Computation, vol. 219, no. 9, pp. 4667–4679, 2013.
• F. Soleymani and P. S. Stanimirović, “A higher order iterative method for computing the Drazin inverse,” The Scientific World Journal, vol. 2013, Article ID 708647, 11 pages, 2013.
• Y. Wei, “A characterization and representation of the generalized inverse ${A}_{T,S}^{(2)}$ and its applications,” Linear Algebra and Its Applications, vol. 280, no. 2-3, pp. 87–96, 1998.
• Y. Wei and H. Wu, “On the perturbation and subproper splittings for the generalized inverse ${A}_{T,S}^{(2)}$ of rectangular matrix $A$,” Journal of Computational and Applied Mathematics, vol. 137, no. 2, pp. 317–329, 2001.
• Y. Wei and D. S. Djordjević, “On integral representation of the generalized inverse ${A}_{T,S}^{(2)}$,” Applied Mathematics and Computation, vol. 142, no. 1, pp. 189–194, 2003.
• Y. Wei and N. Zhang, “Condition number related with generalized inverse ${A}_{T,S}^{(2)}$ and constrained linear systems,” Journal of Computational and Applied Mathematics, vol. 157, no. 1, pp. 57–72, 2003.
• G. Schulz, “Iterative Berechnung der reziproken matrix,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 13, no. 1, pp. 57–59, 1933.
• H.-B. Li, T.-Z. Huang, Y. Zhang, X.-P. Liu, and T.-X. Gu, “Chebyshev-type methods and preconditioning techniques,” Applied Mathematics and Computation, vol. 218, no. 2, pp. 260–270, 2011.
• E. V. Krishnamurthy and S. K. Sen, Numerical Algorithms: Computations in Science and Engineering, Affiliated East-West Press, New Delhi, India, 1986.
• Y. Wei and H. Wu, “The representation and approximation for Drazin inverse,” Journal of Computational and Applied Mathematics, vol. 126, no. 1-2, pp. 417–432, 2000.
• X. Li and Y. Wei, “Iterative methods for the Drazin inverse of a matrix with a complex spectrum,” Applied Mathematics and Computation, vol. 147, no. 3, pp. 855–862, 2004. \endinput