Skip to main content
SpringerLink
Log in

A tridiagonalization-based numerical algorithm for computing the inverses of (p, q)-pentadiagonal matrices

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

Matrix inverse computation is one of the fundamental mathematical problems of numerical linear algebra and has been widely used in various fields of computer science and engineering. In the current paper, a reliable and efficient algorithm is presented for numerically computing the inverses of n-square (\(p,\) \(q\))-pentadiagonal matrices. The algorithm is based on the combination of a tridiagonalization approach which preserves the banded structure and sparsity of the original matrix, and a recursive algorithm for inverting general tridiagonal matrices. The experimental results of some representative numerical examples are provided to demonstrate the performance and effectiveness of the proposed algorithm and its competitiveness with MATLAB built-in function.

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

Similar content being viewed by others

Data Availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

  • Abderramán Marrero J, Tomeo V (2015) Some results on determinants and inverses of nonsingular pentadiagonal matrices. J Comput Appl Math 275:447–455

    Article  MathSciNet  MATH  Google Scholar 

  • Anđelić M, da Fonseca CM (2019) Some comments on tridiagonal \((p, r)\)-Toeplitz matrices. Linear Algebra Appl 572:46–50

    Article  MathSciNet  MATH  Google Scholar 

  • Anđelić M, da Fonseca CM (2021) Some determinantal considerations for pentadiagonal matrices. Linear Multilinear Algebra 69(16):3121–3129

    Article  MathSciNet  MATH  Google Scholar 

  • Burden RL, Faires JD, Burden AM (2015) Numerical analysis. Cengage Learning, Boston

    MATH  Google Scholar 

  • da Fonseca CM, Yılmaz F (2015) Some comments on \(k\)-tridiagonal matrices: determinant, spectra, and inversion. Appl Math Comput 270:644–647

    MathSciNet  MATH  Google Scholar 

  • da Fonseca CM, Kowalenko V, Losonczi L (2020) Ninety years of \(k\)-tridiagonal matrices. Stud Sci Math Hung 57(3):298–311

    MathSciNet  MATH  Google Scholar 

  • Diele F, Lopez L (1998) The use of the factorization of five-diagonal matrices by tridiagonal Toeplitz matrices. Appl Math Lett 11(3):61–69

    Article  MathSciNet  MATH  Google Scholar 

  • El-Mikkawy MEA, Atlan F (2015) A new recursive algorithm for inverting general \(k\)-tridiagonal matrices. Appl Math Lett 44:34–39

    Article  MathSciNet  MATH  Google Scholar 

  • Encinas AM, Jiménez MJ (2019a) Boundary value problems for second order linear difference equations: application to the computation of the inverse of generalized Jacobi matrices. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas 113(4):3795–3828

    Article  MathSciNet  MATH  Google Scholar 

  • Encinas AM, Jiménez MJ (2019b) Explicit inverse of nonsingular Jacobi matrices. Discret Appl Math 263:130–139

    Article  MathSciNet  MATH  Google Scholar 

  • Fukuda A, Ishiwata E, Iwasaki M, Nakamura Y (2008) The discrete hungry Lotka–Volterra system and a new algorithm for computing matrix eigenvalues. Inverse Probl 25(1):015007

    Article  MathSciNet  MATH  Google Scholar 

  • Gutknecht MH (1993) Variants of BiCGSTAB for matrices with complex spectrum. SIAM J Sci Comput 14(5):1020–1033

    Article  MathSciNet  MATH  Google Scholar 

  • Jia J, Li S (2015) Symbolic algorithms for the inverses of general \(k\)-tridiagonal matrices. Comput Math Appl 70(12):3032–3042

    Article  MathSciNet  MATH  Google Scholar 

  • Jia J, Sogabe T (2020) Generalized Sherman–Morrison–Woodbury formula based algorithm for the inverses of opposite-bordered tridiagonal matrices. J Math Chem 58(7):1466–1480

    Article  MathSciNet  MATH  Google Scholar 

  • Jia J, Sogabe T, El-Mikkawy MEA (2013) Inversion of \(k\)-tridiagonal matrices with Toeplitz structure. Comput Math Appl 65(1):116–125

    Article  MathSciNet  MATH  Google Scholar 

  • Jia J, Yan Y, He Q (2021) A block diagonalization based algorithm for the determinants of block \(k\)-tridiagonal matrices. J Math Chem 59(3):745–756

    Article  MathSciNet  MATH  Google Scholar 

  • Kanal ME (2012) Parallel algorithm on inversion for adjacent pentadiagonal matrices with MPI. J Supercomput 59:1071–1078

    Article  Google Scholar 

  • Kanal ME, Baykara NA, Demiralp M (2012) Theory and algorithm of the inversion method for pentadiagonal matrices. J Math Chem 50:289–299

    Article  MathSciNet  MATH  Google Scholar 

  • Lv XG, Huang TZ, Le J (2008) A note on computing the inverse and the determinant of a pentadiagonal Toeplitz matrix. Appl Math Comput 206(1):327–331

    MathSciNet  MATH  Google Scholar 

  • Marr RB, Vineyard GH (1988) Five-diagonal Toeplitz determinants an their relation to Chebyshev polynomials. SIAM J Matrix Anal Appl 9(4):579–586

    Article  MathSciNet  MATH  Google Scholar 

  • McMillen T (2009) On the eigenvalues of double band matrices. Linear Algebra Appl 431(10):1890–1897

    Article  MathSciNet  MATH  Google Scholar 

  • Ohashi A, Sogabe T, Usuda T (2015) On decomposition of \(k\)-tridiagonal \({\ell }\)-Toeplitz matrices and its applications. Spec Matrices 3(1):200–206

    MathSciNet  MATH  Google Scholar 

  • Sogabe T (2008) A fast numerical algorithm for the determinant of a pentadiagonal matrix. Appl Math Comput 196:835–841

    MathSciNet  MATH  Google Scholar 

  • Sogabe T, El-Mikkawy MEA (2011) Fast block diagonalization of \(k\)-tridiagonal matrices. Appl Math Comput 218(6):2740–2743

    MathSciNet  MATH  Google Scholar 

  • Sogabe T, Yılmaz F (2014) A note on a fast breakdown-free algorithm for computing the determinants and the permanents of \(k\)-tridiagonal matrices. Appl Math Comput 249:98–102

    MathSciNet  MATH  Google Scholar 

  • Takahira S, Sogabe T, Usuda TS (2019) Bidiagonalization of \((k, k+ 1)\)-tridiagonal matrices. Spec Matrices 7(1):20–26

    Article  MathSciNet  MATH  Google Scholar 

  • Wittenburg J (1998) Inverses of tridiagonal Toeplitz and periodic matrices with applications to mechanics. J Appl Math Mech 62(4):575–587

    Article  MathSciNet  Google Scholar 

  • Yamamoto T (2001) Inversion formulas for tridiagonal matrices with applications to boundary value problems. Numer Funct Anal Optim 22(3–4):357–385

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao XL, Huang TZ (2008) On the inverse of a general pentadiagonal matrix. Appl Math Comput 202(2):639–646

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions that substantially enhanced the quality of the manuscript. This work was supported by the Fundamental Research Funds for the Central Universities (Grant No. JB210720) and the Young Talent Fund of Association for Science and Technology in Shaanxi, China (Grant No. 20220506).

Author information

Author notes

  1. Jie Wang and Rong Xie have contributed equally to this work.

Authors and Affiliations

  1. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, Shaanxi, China

    Jie Wang, Ji-Teng Jia & Rong Xie

Authors
  1. Jie Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ji-Teng Jia
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rong Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ji-Teng Jia.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by Jinyun Yuan.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, J., Jia, JT. & Xie, R. A tridiagonalization-based numerical algorithm for computing the inverses of (p, q)-pentadiagonal matrices. Comp. Appl. Math. 42, 166 (2023). https://doi.org/10.1007/s40314-023-02305-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-023-02305-x

Keywords

Mathematics Subject Classification

Search

Navigation