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An efficient class of iterative methods for computing generalized outer inverse \({M_{T,S}^{(2)}}\)

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Abstract

In this paper, we propose a new matrix iteration scheme for computing the generalized outer inverse for a given complex matrix. The convergence analysis of the proposed scheme is established under certain necessary conditions, which indicates that the methods possess at least fourth-order convergence. The theoretical discussions show that the convergence order improves from 4 to 5 for a particular parameter choice. We prove that the sequence of approximations generated by the family satisfies the commutative property of matrices, provided the initial matrix commutes with the matrix under consideration. Some real-world and academic problems are chosen to validate our methods for solving the linear systems arising from statically determinate truss problems, steady-state analysis of a system of reactors, and elliptic partial differential equations. Moreover, we include a wide variety of large sparse test matrices obtained from the matrix market library. The performance measures used are the number of iterations, computational order of convergence, residual norm, efficiency index, and the computational time. The numerical results obtained are compared with some of the existing robust methods. It is demonstrated that our method gives improved results in terms of computational speed and efficiency.

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References

  1. Matrix Market. https://math.nist.gov/MatrixMarket/

  2. Abidi, O., Jbilou, K.: Balanced truncation-rational Krylov methods for model reduction in large scale dynamical systems. Comput. Appl. Math. 37(1), 525–540 (2018)

    MathSciNet  MATH  Google Scholar 

  3. Ben-Israel, A., Greville, T.N.: Generalized Inverses: Theory and Applications, vol. 15. Springer, Berlin (2003)

    MATH  Google Scholar 

  4. Chen, L., Krishnamurthy, E., Macleod, I.: Generalised matrix inversion and rank computation by successive matrix powering. Parallel Comput. 20(3), 297–311 (1994)

    MathSciNet  MATH  Google Scholar 

  5. Chun, C.: A geometric construction of iterative functions of order three to solve nonlinear equations. Comput. Math. Appl. 53(6), 972–976 (2007)

    MathSciNet  MATH  Google Scholar 

  6. Climent, J.J., Thome, N., Wei, Y.: A geometrical approach on generalized inverses by Neumann-type series. Linear Algebra Appl. 332, 533–540 (2001)

    MathSciNet  MATH  Google Scholar 

  7. Codevico, G., Pan, V.Y., Van Barel, M.: Newton-like iteration based on a cubic polynomial for structured matrices. Numer. Algorithms 36(4), 365–380 (2004)

    MathSciNet  MATH  Google Scholar 

  8. Cordero, A., Franques, A., Torregrosa, J.R.: Chaos and convergence of a family generalizing Homeier’s method with damping parameters. Nonlinear Dyn. 85(3), 1939–1954 (2016)

    MathSciNet  MATH  Google Scholar 

  9. Esmaeili, H., Pirnia, A.: An efficient quadratically convergent iterative method to find the Moore–Penrose inverse. Int. J. Comput. Math. 94(6), 1079–1088 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Forsythe, G.E., Malcolm, M.A., Moler, C.B.: Computer Methods for Mathematical Computations, vol. 259. Prentice-Hall, Englewood Cliffs (1977)

    MATH  Google Scholar 

  11. Grosz, L.: Preconditioning by incomplete block elimination. Numer. Linear Algebra Appl. 7(7–8), 527–541 (2000)

    MathSciNet  MATH  Google Scholar 

  12. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (2012)

    Google Scholar 

  13. Hotelling, H.: Some new methods in matrix calculation. Ann. Math. Stat. 14(1), 1–34 (1943)

    MathSciNet  MATH  Google Scholar 

  14. Katsaggelos, A., Biemond, J., Mersereau, R., Schafer, R.: A general formulation of constrained iterative restoration algorithms. In: ICASSP’85. IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 10, pp. 700–703. IEEE (1985)

  15. Kumar, A., Stanimirović, P.S., Soleymani, F., Krstić, M., Rajković, K.: Factorizations of hyperpower family of iterative methods via least squares approach. Comput. Appl. Math. 37(3), 3226–3240 (2018)

    MathSciNet  MATH  Google Scholar 

  16. Li, H.B., Huang, T.Z., Zhang, Y., Liu, X.P., Gu, T.X.: Chebyshev-type methods and preconditioning techniques. Appl. Math. Comput. 218(2), 260–270 (2011)

    MathSciNet  MATH  Google Scholar 

  17. Ma, J., Gao, F., Li, Y.: An efficient method for computing the outer inverse \({A}_{{T},{S}}^{(2)}\) through Gauss–Jordan elimination. Numer. Algorithms. (2019). https://doi.org/10.1007/s11075-019-00803-w

  18. Moriya, K., Nodera, T.: A new scheme of computing the approximate inverse preconditioner for the reduced linear systems. J. Comput. Appl. Math. 199(2), 345–352 (2007)

    MathSciNet  MATH  Google Scholar 

  19. Pan, V., Schreiber, R.: An improved Newton iteration for the generalized inverse of a matrix, with applications. SIAM J. Sci. Stat. Comput. 12(5), 1109–1130 (1991)

    MathSciNet  MATH  Google Scholar 

  20. Pan, V., Soleymani, F., Zhao, L.: An efficient computation of generalized inverse of a matrix. Appl. Math. Comput. 316, 89–101 (2018)

    MathSciNet  MATH  Google Scholar 

  21. Penrose, R.: A generalized inverse for matrices. Mathematical Proceedings of the Cambridge Philosophical Society. 51(3), 406–413 (1955)

    MATH  Google Scholar 

  22. Petković, M.D.: Generalized Schultz iterative methods for the computation of outer inverses. Comput. Math. Appl. 67(10), 1837–1847 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Qiao, S., Wang, X.Z., Wei, Y.: Two finite-time convergent Zhang neural network models for time-varying complex matrix Drazin inverse. Linear Algebra Appl. 542, 101–117 (2018)

    MathSciNet  MATH  Google Scholar 

  24. Schulz, G.: Iterative berechung der reziproken matrix. ZAMM Z. Angew. Math. Mech. 13(1), 57–59 (1933)

    MATH  Google Scholar 

  25. Söderström, T., Stewart, G.: On the numerical properties of an iterative method for computing the Moore–Penrose generalized inverse. SIAM J. Numer. Anal. 11(1), 61–74 (1974)

    MathSciNet  MATH  Google Scholar 

  26. Soleymani, F., Stanimirović, P.S., Haghani, F.K.: On hyperpower family of iterations for computing outer inverses possessing high efficiencies. Linear Algebra Appl. 484, 477–495 (2015)

    MathSciNet  MATH  Google Scholar 

  27. Srivastava, S., Gupta, D.: A higher order iterative method for \({A}^{(2)}_{{T},{S}}\). J. Appl. Math. Comput. 46(1–2), 147–168 (2014)

    MathSciNet  MATH  Google Scholar 

  28. Srivastava, S., Gupta, D.K.: The iterative methods for \({A}^{(2)}_{{T},{S}}\) of the bounded linear operator between Banach spaces. J. Appl. Math. Comput. 49(1–2), 383–396 (2015)

    MathSciNet  MATH  Google Scholar 

  29. Stanimirović, P., Bogdanović, S., Ćirić, M.: Adjoint mappings and inverses of matrices. Algebra Colloq. 13(3), 421–432 (2006)

    MathSciNet  MATH  Google Scholar 

  30. Stanimirović, P., Stojanović, I., Chountasis, S., Pappas, D.: Image deblurring process based on separable restoration methods. Comput. Appl. Math. 33(2), 301–323 (2014)

    MathSciNet  MATH  Google Scholar 

  31. Stanimirović, P.S., Cvetković-Ilić, D.S.: Successive matrix squaring algorithm for computing outer inverses. Appl. Math. Comput. 203(1), 19–29 (2008)

    MathSciNet  MATH  Google Scholar 

  32. Stanimirović, P.S., Živković, I.S., Wei, Y.: Neural network approach to computing outer inverses based on the full rank representation. Linear Algebra Appl. 501, 344–362 (2016)

    MathSciNet  MATH  Google Scholar 

  33. Toutounian, F., Soleymani, F.: 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, 671–680 (2013)

    MathSciNet  MATH  Google Scholar 

  34. Traub, J.: Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs, NJ (1964)

    MATH  Google Scholar 

  35. Trott, M.: The Mathematica Guidebook for Programming. Springer, New York (2013)

    MATH  Google Scholar 

  36. Wang, X.Z., Stanimirović, P.S., Wei, Y.: Complex ZFs for computing time-varying complex outer inverses. Neurocomputing 275, 983–1001 (2018)

    Google Scholar 

  37. Wei, Y.: A characterization and representation of the generalized inverse \({A}_{{T},{S}}^{(2)}\) and its applications. Linear Algebra Appl. 280(2–3), 87–96 (1998)

    MathSciNet  MATH  Google Scholar 

  38. Wei, Y.: Successive matrix squaring algorithm for computing the Drazin inverse. Appl. Math. Comput. 108(2–3), 67–75 (2000)

    MathSciNet  MATH  Google Scholar 

  39. Wei, Y., Djordjević, D.S.: On integral representation of the generalized inverse \({A}_{{T},{S}}^{(2)}\). Appl. Math. Comput. 142(1), 189–194 (2003)

    MathSciNet  MATH  Google Scholar 

  40. Wei, Y., Stanimirović, P., Petković, M.: Numerical and Symbolic Computations of Generalized Inverses. World Scientific, Singapore (2018)

    MATH  Google Scholar 

  41. Wei, Y., Wu, H.: On the perturbation and subproper splittings for the generalized inverse \({A}_{{T},{S}}^{(2)}\) of rectangular matrix \({A}\). J. Comput. Appl. Math. 137(2), 317–329 (2001)

    MathSciNet  MATH  Google Scholar 

  42. Wei, Y., Wu, H.: (T, S) splitting methods for computing the generalized inverse and rectangular systems. Int. J. Comput. Math. 77(3), 401–424 (2001)

    MathSciNet  MATH  Google Scholar 

  43. Wei, Y., Wu, H., Wei, J.: Successive matrix squaring algorithm for parallel computing the weighted generalized inverse \({A}^{\dagger }_{{M}{N}}\). Appl. Math. Comput. 116(3), 289–296 (2000)

    MathSciNet  MATH  Google Scholar 

  44. Xia, Y., Zhang, S., Stanimirović, P.S.: Neural network for computing pseudoinverses and outer inverses of complex-valued matrices. Appl. Math. Comput. 273, 1107–1121 (2016)

    MathSciNet  MATH  Google Scholar 

  45. Živković, I.S., Stanimirović, P.S., Wei, Y.: Recurrent neural network for computing outer inverse. Neural Comput. 28(5), 970–998 (2016)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to sincerely thank the referees for their very detailed comments and valuable suggestions, which significantly improved the quality of the presented manuscript.

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

    Manpreet Kaur & Munish Kansal

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  1. Manpreet Kaur
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  2. Munish Kansal
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Correspondence to Munish Kansal.

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Kaur, M., Kansal, M. An efficient class of iterative methods for computing generalized outer inverse \({M_{T,S}^{(2)}}\). J. Appl. Math. Comput. 64, 709–736 (2020). https://doi.org/10.1007/s12190-020-01375-y

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