Skip to main content
Log in

An interval extension of SMS method for computing weighted Moore–Penrose inverse

  • Published:
Calcolo Aims and scope Submit manuscript

Abstract

An interval extension of successive matrix squaring (SMS) method for computing the weighted Moore–Penrose inverse \(A^{\dagger }_{MN}\) along with its rigorous error bounds is proposed for given full rank \(m \times n\) complex matrices A, where M and N be two Hermitian positive definite matrices of orders m and n, respectively. Starting with a suitably chosen complex interval matrix containing \(A^{\dagger }_{MN}\), this method generates a sequence of complex interval matrices each enclosing \(A^{\dagger }_{MN}\) and converging to it. A new method is developed for constructing initial complex interval matrix containing \(A^{\dagger }_{MN}\). Convergence theorems are established. The R-order convergence is shown to be equal to at least l, where \(l \ge 2\). A number of numerical examples are worked out to demonstrate its efficiency and effectiveness. Graphs are plotted to show variations of the number of iterations and computational times compared to matrix dimensions. It is observed that ISMS is more stable compared to SMS.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Alefeld, G., Herzberger, J.: Introduction to Interval Computation. Academic press, London (2012)

    MATH  Google Scholar 

  2. Ben-Israel, A., Greville, T.N.: Generalized Inverses: Theory and Applications, vol. 15. Springer Science & Business Media, Berlin (2003)

    MATH  Google Scholar 

  3. Ilić, Dragana S.Cvetković, Wei, Yimin: Algebraic Properties of Generalized Inverses. Springer, Berlin (2003)

    MATH  Google Scholar 

  4. Bojanczyk, A., Higham, N.J., Patel, H.: Solving the indefinite least squares problem by hyperbolic QR factorization. SIAM J. Matrix Anal. Appl. 24(4), 914–931 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Li, Z., Sun, J.: Mixed and componentwise condition numbers for weighted Moore–Penrose inverse and weighted least squares problems. Filomat 23(1), 43–59 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lin, L., Lu, T.T., Wei, Y.: On level-2 condition number for the weighted Moore–Penrose inverse. Comput. Math. Appl. 55(4), 788–800 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Frommer, A., Hashemi, B., Sablik, T.: Computing enclosures for the inverse square root and the sign function of a matrix. Linear Algebra Appl. 456, 199–213 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ma, J., Qi, L., Li, Y.: The representations and computations of generalized inverses \({A}^{(1)}_{TS}\), \({A}^{(1,2)}_{TS}\) and the group inverse. Calcolo 54(4), 1147–1168 (2017)

    Article  MathSciNet  Google Scholar 

  11. Sheng, Xingping: Computation of weighted Moore–Penrose inverse through Gauss–Jordan elimination on bordered matrices. Appl. Math. Comput. 323, 64–74 (2017)

    MathSciNet  Google Scholar 

  12. Martins, M.: Some improvements on the convergence of the interval Maor method. Int. J. Comput. Math. 74(4), 493–507 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Miladinović, M., Miljković, S., Stanimirović, P.: Modified SMS method for computing outer inverses of Toeplitz matrices. Appl. Math. Comput. 218(7), 3131–3143 (2011)

    MathSciNet  MATH  Google Scholar 

  14. Moore, R.E.: Methods and Applications of Interval Analysis. SIAM, Philadelphia (1979)

    Book  MATH  Google Scholar 

  15. Neumaier, A.: Interval Methods for Systems of Equations, vol. 37. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  16. Oseledets, I.V., Tyrtyshnikov, E.E.: Approximate inversion of matrices in the process of solving a hypersingular integral equation. Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki 45(2), 315–326 (2005)

    MathSciNet  MATH  Google Scholar 

  17. Petkovic, M.D., Petkovic, M.S.: Iterative methods for the inclusion of the inverse matrix. arXiv preprint arXiv:1406.5343 (2014)

  18. Petković, M.D., Stanimirović, P.S., Tasić, M.B.: Effective partitioning method for computing weighted Moore–Penrose inverse. Comput. Math. Appl. 55(8), 1720–1734 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Potra, F.: On Q-order and R-order of convergence. J. Optim. Theory Appl. 63(3), 415–431 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rump, S.M.: INTLAB–INTerval LABoratory. In: Cesndes, T. (ed.) Developments in Reliable Computing, pp. 77–104. Kluwer Academic Publishers, Dordrecht (1999)

    Chapter  Google Scholar 

  21. Rump, S.M.: Verification methods: rigorous results using floating-point arithmetic. Acta Numer. 19, 287–449 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Rump, S.M., Graillat, S.: Verified error bounds for multiple roots of systems of nonlinear equations. Numer. Algorithms 54(3), 359–377 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ullah, M.Z., Soleymani, F., Al-Fhaid, A.S.: An efficient matrix iteration for computing weighted Moore–Penrose inverse. Appl. Math. Comput. 226, 441–454 (2014)

    MathSciNet  MATH  Google Scholar 

  24. Soheili, A.R., Soleymani, F., Petkovic, M.D.: On the computation of weighted Moore–Penrose inverse using a high-order matrix method. Comput. Math. Appl. 66(11), 2344–2351 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Saraev, P.V.: Interval pseudo-inverse matrices and interval Greville algorithm. Reliab. Comput. 18, 147–156 (2013)

    MathSciNet  Google Scholar 

  26. Sheng, X., Chen, G.: A note of computation for MP inverse \(A^{\dagger }\). Int. J. Comput. Math. 87(10), 2235–2241 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. 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 

  28. Van Loan, C.F.: Generalizing the singular value decomposition. SIAM J. Numer. Anal. 13(1), 76–83 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang, H., Cao, D., Liu, H., Qiu, L.: Numerical validation for systems of absolute value equations. Calcolo 54(3), 669–683 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  32. Zhang, N., Wei, Y.: A note on the perturbation of an outer inverse. Calcolo 45(4), 263–273 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhang, X., Cai, J., Wei, Y.: Interval iterative methods for computing Moore–Penrose inverse. Appl. Math. Comput. 183(1), 522–532 (2006)

    MathSciNet  MATH  Google Scholar 

  34. Chen, Y., Shi, X., Wei, Y.: Convergence of Rump’s method for computing the Moore–Penrose inverse. Czechoslov. Math. J. 66(3), 859–879 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Falguni Roy.

Additional information

Third author gratefully acknowledges support from the Research Project 174013 of the Serbian Ministry of Science.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Roy, F., Gupta, D.K. & Stanimirović, P.S. An interval extension of SMS method for computing weighted Moore–Penrose inverse. Calcolo 55, 15 (2018). https://doi.org/10.1007/s10092-018-0257-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10092-018-0257-4

Keywords

Mathematics Subject Classification

Navigation