Skip to main content
SpringerLink
Log in

On Matrix Inversions Through Difference Operators

  • Research Paper
  • Published:
Iranian Journal of Science and Technology, Transactions A: Science Aims and scope Submit manuscript

Abstract

In the present article, using difference operators, we develop a novel recursion-free algorithm for inverting any arbitrary non-singular matrix of finite order having LU factorization directly. The entire algorithm is based on LU factorization and individual inversions of triangular matrices L and U using difference operators. The pseudo codes for the complete process including decomposition, inversions of L and U has been constructed and also been verified by Mat lab implementation. The theoretical treatment along with appropriateness and computational effectiveness of the new algorithm is provided.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  • Ahmad ZU, Mursaleen M (1987) Köthe–Toeplitz duals of some new sequence spaces and their martix maps. Publ Inst Math (Beograd) 42(56):57–61

    MathSciNet  MATH  Google Scholar 

  • Altay B, Başar F (2004) On the fine spectrum of the difference operator \(\Delta\) on \(c_0\) and \(c\). Inform Sci 168:217–224

    Article  MathSciNet  Google Scholar 

  • Altay B, Başar F (2007) The fine spectrum and the matrix domain of the difference operator \(\Delta\) on the sequence space \(\ell _p, (0 < p < 1)\). Commun Math Anal 2(2):1–11

    MathSciNet  MATH  Google Scholar 

  • Baliarsingh P, Dutta S (2015) On an explicit formula for inverse of triangular matrices. J Egypt Math Soc 23:297–302

    Article  MathSciNet  Google Scholar 

  • Baliarsingh P, Dutta S (2015) On the classes of fractional order difference sequence spaces and their matrix transformations. Appl Math Comput 250:665–674

    MathSciNet  MATH  Google Scholar 

  • Baliarsingh P (2013) Some new difference sequence spaces of fractional order and their dual spaces. Appl Math Comput 219(18):9737–9742

    MathSciNet  MATH  Google Scholar 

  • Başar F, Altay B (2003) On the space of sequences of p-bounded variation and related matrix mappings. Ukr Math J 55(1):136–147

    Article  MathSciNet  Google Scholar 

  • Başar F (2012) Summability theory and its applications, Bentham Science Publishers, e-books. Monographs, İstanbul

  • Bini DA, Gemignani L, Meini B (2002) Computations with infinite Toeplitz matrices and polynomials. Linear Algebra Appl 343(344):21–61

    Article  MathSciNet  Google Scholar 

  • Ben-Artzi A, Shalom T (1986) On inversion of Toeplitz and close to Toeplitz matrices. Linear Algebra Appl 75:173–192

    Article  MathSciNet  Google Scholar 

  • Demmel JW (1997) Applied numerical linear algebra. SIAM, Philadelphia, PA

  • Dutta S, Baliarsingh P (2014) On some Toeplitz matrices and their inversion. J Egypt Math Soc 22(3):420–424

    Article  MathSciNet  Google Scholar 

  • Dutta S, Baliarsingh P (2012) On the fine spectra of the generalized rth difference operator \(\Delta _{\nu }^{r}\) on the sequence space \(\ell _{1}\). Appl Math Comput 219(18):1776–1784

    MathSciNet  MATH  Google Scholar 

  • Et M, Çolak R (1995) On some generalized difference sequence spaces. Soochow J Math 21(4):377–386

    MathSciNet  MATH  Google Scholar 

  • Farden DC, Scharf LL (1974) Statistical design of non recursive digital filter. IEEE Trans Acoust Spheech Signal Process 22:188–196

    Article  Google Scholar 

  • Gantmacher FR (2005) Applicatons of the theory of matrices, 1st edn. Dover Publications, New York

    Google Scholar 

  • Jain AK (1977) An operator factorization method for restoration of blurred images. IEEE Trans Comput C–26:1061–1071

    Article  MathSciNet  Google Scholar 

  • Kızmaz H (1981) On certain sequence spaces. Can Math Bull 24(2):169–176

    Article  MathSciNet  Google Scholar 

  • Lv XG, Huang TZ (2007) A note on inversion of Toeplitz matrices. Appl Math Lett 20(12):1189–1193

    Article  MathSciNet  Google Scholar 

  • Mursaleen M, Noman AK (2010) On some new difference sequence spaces of non-absolute type. Math Comput Model 52:603–617

    Article  MathSciNet  Google Scholar 

  • Mursaleen M, Noman AK (2011) On generalized means and some related sequence spaces. Comput Math Appl 61:988–999

    Article  MathSciNet  Google Scholar 

  • Ng MK, Rost K, Wen YW (2002) On inversion of Toeplitz matrices. Linear Algebra Appl 348:145–151

    Article  MathSciNet  Google Scholar 

  • Sharma G, Agarwala A, Bhattacharya B (2013) A fast parallel Gauss Jordan algorithm for matrix inversion using CUDA. Comput Struct 128:31–37

    Article  Google Scholar 

  • Strassen V (1969) Gaussian elimination is not optimal. Numer Math 13:354–356

    Article  MathSciNet  Google Scholar 

  • Wilansky A (1984) Summability through functional analysis. North-Holland Mathematics Studies, Amsterdam

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, School of Applied Sciences, KIIT University, Bhubaneswar, 751024, India

    P. Baliarsingh & L. Nayak

  2. School of Electronics, KIIT University, Bhubaneswar, 751024, India

    V. Kumar

Authors
  1. P. Baliarsingh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. Nayak
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Nayak.

Appendix

Appendix

Finally, adapting new algorithm proposed in the present work, we develop the entire Mat lab codes for inversion of a non-singular matrix as follows:

figure b

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baliarsingh, P., Nayak, L. & Kumar, V. On Matrix Inversions Through Difference Operators. Iran J Sci Technol Trans Sci 42, 2069–2077 (2018). https://doi.org/10.1007/s40995-017-0161-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40995-017-0161-9

Keywords

Mathematics Subject Classification

Search

Navigation