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

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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.
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Keywords weighted Moore-Penrose inverse; rank-deficient matrix; hyperpower iterative method; matrix multiplication; generalized inverse

Citation: Manpreet Kaur, Munish Kansal, Sanjeev Kumar. An efficient hyperpower iterative method for computing weighted MoorePenrose inverse. AIMS Mathematics, 2020, 5(3): 1680-1692. doi: 10.3934/math.2020113


  • 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.    


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