A Symbolic Algorithm for Solving Doubly Bordered k-Tridiagonal Interval Linear Systems

Main Article Content

Sivakumar Thirupathi, Nirmala Thamaraiselvan


Doubly bordered k-tridiagonal interval linear systems play a crucial role in various mathematical and engineering applications where uncertainty is inherent in the system’s parameters. In this paper, we propose a novel symbolic algorithm for solving such systems efficiently. Our approach combines symbolic computation techniques with interval arithmetic to provide rigorous solutions in the form of tight interval enclosures. By exploiting the tridiagonal structure and employing a divide-and-conquer strategy, our algorithm achieves significantly reduced computational complexity compared to existing numerical methods. We also present theoretical analysis and provide numerical experiments to demonstrate the effectiveness and accuracy of our algorithm. The proposed symbolic algorithm offers a valuable tool for handling doubly bordered k-tridiagonal interval linear systems and opens up possibilities for addressing uncertainty in real-world problems with improved efficiency and reliability.

Article Details


  1. M. Andelic, C.M. da Fonseca, T. Koledin, Z. Stanic, An Extended Eigenvalue-Free Interval for the Eccentricity Matrix of Threshold Graphs, J. Appl. Math. Comput. 69 (2022), 491-503. https://doi.org/10.1007/s12190-022-01758-3.
  2. C.M. da Fonseca, V. Kowalenko, L. Losonczi, Ninety Years of k-Tridiagonal Matrices, Stud. Sci. Math. Hung. 57 (2020), 298-311. https://doi.org/10.1556/012.2020.57.3.1466.
  3. Y. Fan, X. Huang, Z. Wang, Y. Li, Global Dissipativity and Quasi-Synchronization of Asynchronous Updating Fractional-Order Memristor-Based Neural Networks via Interval Matrix Method, J. Franklin Inst. 355 (2018), 5998- 6025. https://doi.org/10.1016/j.jfranklin.2018.05.058.
  4. Y. Fan, X. Huang, Y. Li, J. Xia, G. Chen, Aperiodically Intermittent Control for Quasi-Synchronization of Delayed Memristive Neural Networks: An Interval Matrix and Matrix Measure Combined Method, IEEE Trans. Syst. Man Cybern, Syst. 49 (2019), 2254-2265. https://doi.org/10.1109/tsmc.2018.2850157.
  5. K. Ganesan, P. Veeramani, On Arithmetic Operations of Interval Numbers, Int. J. Uncertain. Fuzziness Knowl.- Based Syst. 13 (2005), 619-631. https://doi.org/10.1142/s0218488505003710.
  6. Hoffman JD, Numerical Methods for Engineers and Scientists, first Editions, Mcgraw-Hill Education, New York, 1992.
  7. D. Hartman, M. Hladík, D. Říha, Computing the Spectral Decomposition of Interval Matrices and a Study on Interval Matrix Powers, Appl. Math. Comput. 403 (2021), 126174. https://doi.org/10.1016/j.amc.2021.126174.
  8. X. Huang, J. Jia, Y. Fan, Z. Wang, J. Xia, Interval Matrix Method Based Synchronization Criteria for FractionalOrder Memristive Neural Networks With Multiple Time-Varying Delays, J. Franklin Inst. 357 (2020), 1707-1733. https://doi.org/10.1016/j.jfranklin.2019.12.014.
  9. J.T. Jia, A Breakdown-Free Algorithm for Computing the Determinants of Periodic Tridiagonal Matrices, Numer. Algorithms. 83 (2019), 149-163. https://doi.org/10.1007/s11075-019-00675-0.
  10. E. Kaucher, Interval Analysis in the Extended Interval Space IR, in: G. Alefeld, R.D. Grigorieff (Eds.), Fundamentals of Numerical Computation (Computer-Oriented Numerical Analysis), Springer Vienna, Vienna, 1980: pp. 33-49. https://doi.org/10.1007/978-3-7091-8577-3_3.
  11. L. Losonczi, Eigenpairs of Some Imperfect Pentadiagonal Toeplitz Matrices, Linear Algebra Appl. 608 (2021), 282-298. https://doi.org/10.1016/j.laa.2020.09.014.
  12. M. El-Mikkawy, T. Sogabe, A New Family of k-Fibonacci Numbers, Appl. Math. Comput. 215 (2010), 4456-4461. https://doi.org/10.1016/j.amc.2009.12.069.
  13. J. Abderramán Marrero, A Reliable Givens-Lu Approach for Solving Opposite-Bordered Tridiagonal Linear Systems, Computers Math. Appl. 76 (2018), 2409-2420. https://doi.org/10.1016/j.camwa.2018.08.038.
  14. T. Nirmala, D. Datta, H.S. Kushwaha, K. Ganesan, Inverse Interval Matrix: A New Approach, Appl. Math. Sci. 5 (2011), 607-624.
  15. J.T. Parker, P.A. Hill, D. Dickinson, B.D. Dudson, Parallel Tridiagonal Matrix Inversion With a Hybrid MultigridThomas Algorithm Method, J. Comput. Appl. Math. 399 (2022), 113706. https://doi.org/10.1016/j.cam.2021.113706.
  16. A. Sengupta, T.K. Pal, On Comparing Interval Numbers, Eur. J. Oper. Res. 127 (2000), 28-43. https://doi.org/10.1016/s0377-2217(99)00319-7.
  17. M.S. Solary, Eigenvalues for Tridiagonal 3-Toeplitz Matrices, J. Mahani Math. Res. 10 (2021), 63-72.
  18. N. Shehab, M. El-Mikkawy, M. El-Shehawy, A Generalized Symbolic Thomas Algorithm for Solving Doubly Bordered k-Tridiagonal Linear Systems, J. Appl. Math. Phys. 03 (2015), 1199-1206. https://doi.org/10.4236/jamp.2015.39147.
  19. S. Thirupathi, N. Thamaraiselvan, Symbolic Algorithm for Inverting General k-Tridiagonal Interval Matrices, Int. J. Anal. Appl. 21 (2023), 20. https://doi.org/10.28924/2291-8639-21-2023-20.
  20. A. Tănăsescu, P.G. Popescu, A Fast Singular Value Decomposition Algorithm of General K-Tridiagonal Matrices, J. Comput. Sci. 31 (2019), 1-5. https://doi.org/10.1016/j.jocs.2018.12.009.
  21. Tanasescu A, Carabas M, Pop F, Popescu PG, Scalability of k-Tridiagonal Matrix Singular Value Decomposition, Math. 9 (2021), 3123. https://doi.org/10.3390/math9233123.
  22. F. Wei, G. Chen, W. Wang, Finite-Time Stabilization of Memristor-Based Inertial Neural Networks With TimeVarying Delays Combined With Interval Matrix Method, Knowledge-Based Syst. 230 (2021), 107395. https://doi.org/10.1016/j.knosys.2021.107395.
  23. S. Xiao, Z. Wang, C. Wang, Passivity Analysis of Fractional-Order Neural Networks With Interval Parameter Uncertainties via an Interval Matrix Polytope Approach, Neurocomputing. 477 (2022), 96-103. https://doi.org/10.1016/j.neucom.2021.12.106.