AIMS Mathematics, 2020, 5(3): 1680-1692. doi: 10.3934/math.2020113

Research article

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

An efficient hyperpower iterative method for computing weighted MoorePenrose inverse

School of Mathematics, Thapar Institute of Engineering and Technology, Patiala 147004, India

In this paper, we propose a new hyperpower iterative method for approximating the weighted Moore-Penrose inverse of a given matrix. The main objective of the current work is to minimize the computational complexity of the hyperpower iterative method using some transformations. The proposed method attains the fifth-order of convergence using four matrix multiplications per iteration step. The theoretical convergence analysis of the method is discussed in detail. A wide range of numerical problems is considered from scientific literature, which demonstrates the applicability and superiority of the proposed method.
  Figure/Table
  Supplementary
  Article Metrics

References

1. S. Chandrasekaran, M. Gu, A. H. Sayed, A stable and efficient algorithm for the indefinite linear least-squares problem, SIAM J. Matrix Anal. Appl., 20 (1998), 354-362.    

2. S. F. Wang, B. Zheng, Z. P. Xiong, et al. The condition numbers for weighted Moore-Penrose inverse and weighted linear least squares problem, Appl. Math. Comput., 215 (2009), 197-205.

3. R. Penrose, A generalized inverse for matrices, In: Mathematical proceedings of the Cambridge philosophical society, 51 (1955), 406-413.    

4. R. Penrose, On best approximate solutions of linear matrix equations, In: Mathematical Proceedings of the Cambridge Philosophical Society, 52 (1956), 17-19.    

5. L. Van, F. Charles, Generalizing the singular value decomposition, SIAM J. Numer. Anal., 13 (1976), 76-83.    

6. G. Wang, Y. Wei, S. Qiao, et al. Generalized Inverses: Theory and Computations, Springer, 2018.

7. T. N. E. Greville, Some applications of the pseudoinverse of a matrix, SIAM Rev., 2 (1960), 15-22.    

8. G. R. Wang, A new proof of Grevile's method for computing the weighted MP inverse, J. Shangai Normal Uni. (Nat. Sci. Ed.), 1985.

9. M. D. Petković, P. S. Stanimirović, M. B. Tasić, Effective partitioning method for computing weighted Moore-Penrose inverse, Comput. Math. Appl., 55 (2008), 1720-1734.    

10. G. Wang, B. Zheng, The weighted generalized inverses of a partitioned matrix, Appl. Math. Comput., 155 (2004), 221-233.

11. X. Liu, Y. Yu, H. Wang, Determinantal representation of weighted generalized inverses, Appl. Math. Comput., 208 (2009), 556-563.

12. I. Kyrchei, Weighted singular value decomposition and determinantal representations of the quaternion weighted Moore-Penrose inverse, Appl. Math. Comput., 309 (2017), 1-16.    

13. I. Kyrchei, Determinantal representations of the quaternion weighted moore-penrose inverse and its applications, In: A. R. Baswell, Editor, Advances in Mathematics Research, Nova Science Publications, New York, 23 (2017), 35-96.

14. I. Kyrchei, Explicit determinantal representation formulas for the solution of the two-sided restricted quaternionic matrix equation, J. Appl. Math. Comput., 58 (2018), 335-365.    

15. M. Altman, An optimum cubically convergent iterative method of inverting a linear bounded operator in Hilbert space, Pacific J. Math., 10 (1960), 1107-1113.    

16. A. Ben-Israel, A note on an iterative method for generalized inversion of matrices, Math. Comput., 20 (1966), 439-440.    

17. H. Hotelling, Some new methods in matrix calculation, Ann. Math. Statist., 14 (1943), 1-34.    

18. G. Schulz, Iterative berechung der reziproken matrix, Z. Angew. Math. Mech., 13 (1933), 57-59.    

19. T. Söderström, G. W. Stewart, On the numerical properties of an iterative method for computing the Moore-Penrose generalized inverse, SIAM J. Numer. Anal., 11 (1974), 61-74.    

20. H. Esmaeili, A. Pirnia, An efficient quadratically convergent iterative method to find the Moore- Penrose inverse, Int. J. Comput. Math., 94 (2017), 1079-1088.    

21. H. B. Li, T. Z. Huang, Y. Zhang, et al. Chebyshev-type methods and preconditioning techniques, Appl. Math. Comput., 218 (2011), 260-270.

22. C. Chun, A geometric construction of iterative functions of order three to solve nonlinear equations, Comput. Math. Appl., 53 (2007), 972-976.    

23. H. Esmaeili, R. Erfanifar, M. Rashidi, A fourth-order iterative method for computing the MoorePenrose inverse, J. Hyperstruct., 6 (2017), 52-67.

24. F. Toutounian, F. Soleymani, An iterative method for computing the approximate inverse of a square matrix and the Moore-Penrose inverse of a non-square matrix, Appl. Math. Comput., 224 (2013), 671-680.

25. F. Soleymani, On finding robust approximate inverses for large sparse matrices, Linear Multilinear A., 62 (2014), 1314-1334.    

26. V. Pan, R. Schreiber, An improved Newton iteration for the generalized inverse of a matrix, with applications, SIAM J. Sci. Stat. Comput., 12 (1991), 1109-1130.    

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved