Symbolic Algorithm for Inverting General k-Tridiagonal Interval Matrices

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Sivakumar Thirupathi, Nirmala Thamaraiselvan


The k-tridiagonal matrices have received much attention in recent years. Many different algorithms have been proposed to improve the efficiency of k-tridiagonal matrix estimation. A novel method based on interval analysis has been identified to improve the efficiency of the calculation. This paper presents efficient and reliable computational algorithms for determining the determinant and inverse of general k-tridiagonal interval matrices built on generalized interval arithmetic. This study is based on the Doolittle LU factorization of the interval matrix. Finally, examples are presented to illustrate the algorithms.

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  1. J. Alberto, J. Brox, Inverses of k-Toeplitz Matrices With Applications to Resonator Arrays With Multiple Receivers, Appl. Math. Comput. 377 (2020), 125185.
  2. J. Brox, H. Albuquerque, The Determinant, Spectral Properties, and Inverse of a Tridiagonal k-Toeplitz Matrix Over a Commutative Ring, arXiv, (2021).
  3. C.M. Da Fonseca, V. Kowalenko, Eigenpairs of a Family of Tridiagonal Matrices: Three Decades Later, Acta Math. Hungar. 160 (2019), 376–389.
  4. C.M. da Fonseca, V. Kowalenko, L. Losonczi, Ninety Years of k-Tridiagonal Matrices, Stud. Sci. Math. Hung. 57 (2020), 298–311.
  5. Y. Fu, X. Jiang, Z. Jiang, S. Jhang, Inverses and Eigenpairs of Tridiagonal Toeplitz Matrix With Opposite-Bordered Rows, J. Appl. Anal. Comput. 10 (2020), 1599–1613.
  6. K. Ganesan, P. Veeramani, on Arithmetic Operations of Interval Numbers, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 13 (2005), 619–631.
  7. A. Iampan, V. Vijaya Bharathi, M. Vanishree, N. Rajesh, Interval-Valued Intuitionistic Fuzzy Subalgebras/Ideals of Hilbert Algebras, Int. J. Anal. Appl. 20 (2022), 25.
  8. J.T. Jia, J. Wang, T.F. Yuan, K.K. Zhang, B.M. Zhong, An Incomplete Block-Diagonalization Approach for Evaluating the Determinants of Bordered k-Tridiagonal Matrices, J. Math. Chem. 60 (2022), 1658–1673.
  9. J.T. Jia, Y.C. Yan, Q. He, A Block Diagonalization Based Algorithm for the Determinants of Block k-Tridiagonal Matrices, J. Math. Chem. 59 (2021), 745–756.
  10. A. Kucuk Zahid, M. Ozen, H. Ince, Recursive and Combinational Formulas for Permanents of General k-Tridiagonal Toeplitz Matrices, Filomat. 33 (2019), 307–317.
  11. 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.
  12. V.A. Khan, E. Evren Kara, U. Tuba, K.M.A.S. Alshlool, A. Ahmad, Sequences of Fuzzy Star-Shaped Numbers, J. Math. Computer Sci. 23 (2020), 321–327.
  13. M. El-Mikkawy, F. Atlan, A New Recursive Algorithm for Inverting General k-Tridiagonal Matrices, Appl. Math. Lett. 44 (2015), 34–39.
  14. M. El-Mikkawy, F. Atlan, A Novel Algorithm for Inverting a General k-Tridiagonal Matrix, Appl. Math. Lett. 32 (2014), 41–47.
  15. M. El-Mikkawy, A. Karawia, A Breakdown Free Numerical Algorithm for Inverting General Tridiagonal Matrices, arXiv, (2022).
  16. T. Nirmala, D. Datta, H.S. Kushwaha, K. Ganesan, Inverse Interval Matrix: A New Approach, Appl. Math. Sci. 5 (2011), 607-624.
  17. K. Palanivel, P. Muralikrishna, P. Hemavathi, R. Chinram, P. Singavananda, Interval Valued Intuitionistic Fuzzy β-Filters on β-Algebras, Int. J. Anal. Appl. 20 (2022), 50.
  18. J. Rohn, Inverse Interval Matrix, SIAM J. Numer. Anal. 30 (1993), 864–870.
  19. M.S. Solary, M. Rasouli, Inverting a K-Heptadiagonal Matrix Based on Doolitle LU Factorization, Appl. Math. J. Chin. Univ. 37 (2022), 340–349.
  20. S. Takahira, T. Sogabe, T.S. Usuda, Bidiagonalization of (k, k+1)-Tridiagonal Matrices, Spec. Matrices. 7 (2019), 20–26.
  21. A. Tanasescu, M. Carabaş, F. Pop, P.G. Popescu, Scalability of k-Tridiagonal Matrix Singular Value Decomposition, Mathematics. 9 (2021), 3123.
  22. A. Tanasescu, P.G. Popescu, A Fast Singular Value Decomposition Algorithm of General k-Tridiagonal Matrices, J. Comput. Sci. 31 (2019), 1–5.
  23. A. Yalciner, the Lu Factorizations and Determinants of the k-Tridiagonal Matrices, Asian-Eur. J. Math. 04 (2011), 187–197.
  24. Y. Wei, Y. Zheng, Z. Jiang, S. Shon, The Inverses and Eigenpairs of Tridiagonal Toeplitz Matrices With Perturbed Rows, J. Appl. Math. Comput. 68 (2021), 623–636.