Tridiagonal Interval Matrix: Exploring New Perspectives and Application
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Keywords:
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Tridiagonal Interval Matrix, Generalized Interval Arithmetic, Interval LU Decomposition, Interval Determinants, Interval Matrix Inversion
Abstract
Tridiagonal interval matrices are relevant in diverse applications, especially in dealing with parameter estimation, optimization and circuit analysis uncertainties. This research paper aims to improve the computational efficiency of obtaining the inverse of a general tridiagonal interval matrix. This matrix is pivotal in electric circuit analysis. We achieve this by employing interval arithmetic operations in the LU decomposition process, enabling effective handling of circuit parameter uncertainties. This approach generates an inverse interval matrix that addresses uncertainties in circuit analyses.
References
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Caratelli, D. and P. E. Ricci (2021). Inversion of Tridiagonal Matrices Using the Dunford-Taylor’s Integral. Symmetry, 13(5); 870
Chu, W., Y. Zhao, and H. Yuan (2022). A Modified Inverse Iteration Method for Computing the Symmetric Tridiagonal Eigenvectors. Mathematics, 10(19); 3636
El-Mikkawy, M. and A. Karawia (2022). A Breakdown Free Numerical Algorithm for Inverting General Tridiagonal Matrices. arXiv preprint arXiv:2208.12843; 1–14
El-Mikkawy, M. E. (2004). On the Inverse of a General Tridiagonal Matrix. Applied Mathematics and Computation, 150(3); 669–679
Encinas, A. M. and M. J. Jiménez (2019). Explicit Inverse of Nonsingular Jacobi Matrices. Discrete Applied Mathematics, 263; 130–139
Fathi, F., M. A. Fariborzi Araghi, and S. A. Shahzadeh Fazeli (2020). Two Different Inverse Eigenvalue Problems for Nonsymmetric Tridiagonal Matrices. Journal of Algorithms and Computation, 52(2); 137–148
Furtado, S., C. R. Johnson, C. Marijuán, and M. Pisonero (2023). Square Matrices with the Inverse Diagonal Property. Kuwait Journal of Science; 1–4
Ganesan, K. and P. Veeramani (2005). On Arithmetic Operations of Interval Numbers. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 13(06); 619–631
Hartman, D., M. Hladík, and D. Říha (2021). Computing the Spectral Decomposition of Interval Matrices and a Study on Interval Matrix Powers. Applied Mathematics and Computation, 403; 126174
Heydari, M., S. A. S. Fazeli, and S. M. Karbassi (2019). On the Inverse Eigenvalue Problem for a Special Kind of Acyclic Matrices. Applications of Mathematics, 64; 351–366
Hopkins, T. and E. Kilic (2022). Analytically Explicit Inverse of a Kind of Periodic Tridiagonal Matrix Using a Backward Continued Fraction Approach. Journal of Applied Analysis and Computation, 12(6); 2299–2313
Kaucher, E. (1980). Interval Analysis in the Extended Interval Space IR. Fundamentals of Numerical Computation (Computer Oriented Numerical Analysis); 33–49
Mallik, R. K. (2001). The Inverse of a Tridiagonal Matrix. Linear Algebra and its Applications, 325(1 3); 109–139
Nirmala, T., D. Datta, H. Kushwaha, and K. Ganesan (2011). Inverse Interval Matrix: A New Approach. Applied Mathematical Sciences, 5(13); 607–624
Parker, J. T., P. A. Hill, D. Dickinson, and B. D. Dudson (2022). Parallel Tridiagonal Matrix Inversion with a Hybrid Multigrid-Thomas Algorithm Method. Journal of Computational and Applied Mathematics, 399; 113706
Qi, F., Q. Zou, and B.-N. Guo (2019). The Inverse of a Triangular Matrix and Several Identities of the Catalan Numbers. Applicable Analysis and Discrete Mathematics, 13(2); 518–541
Rohn, J. (1993). Inverse interval matrix. SIAM Journal on Numerical Analysis, 30(3); 864–870
Sabri, R. I. and B. Ahmed (2023). Best Proximity Point Results in Fuzzy Normed Spaces. Science and Technology Indonesia, 8(2); 298–304
Sengupta, A. and T. K. Pal (2000). On Comparing Interval Numbers. European Journal of Operational Research, 127(1); 28–43
Spellacy, L., D. Golden, and I. Rungger (2019). Performance Analysis of a Pairwise Method for Partial Inversion of Complex Block Tridiagonal Matrices. Concurrency and Computation: Practice and Experience, 31(19); e4918
Susanti, E., F. M. Puspita, S. S. Supadi, E. Yuliza, and A. F. Ramadhan (2023). Improve Fuzzy Inventory Model of Fractal Interpolation with Vertical Scaling Factor. Science and Technology Indonesia, 8(4); 654–659
Talibi, B., A. D. A. Hadj, and D. Sarsri (2022). On the Factorization and the Inverse of a Tridiagonal Matrix. Journal of Discrete Mathematical Sciences and Cryptography, 25(8); 2479– 2489
Tan, L. S. (2019). Explicit Inverse of Tridiagonal Matrix with Applications in Autoregressive Modelling. IMA Journal of Applied Mathematics, 84(4); 679–695
Thirupathi, S. and N. Thamaraiselvan (2023). Symbolic Algorithm for Inverting General k Tridiagonal Interval Matrices. International Journal of Analysis and Applications, 21; 1–12
Wang, X. T. and M. L. Jin (2019). Inverse Eigenvalue Problem for Quasi-Tridiagonal Matrices. Bulletin of the Iranian Mathematical Society, 45; 1697–1712
Wei, Y., X. Jiang, Z. Jiang, and S. Shon (2019). Determinants and Inverses of Perturbed Periodic Tridiagonal Toeplitz Matrices. Advances in Difference Equations, 2019(1); 1–11
Yuan, Q. and Z. Yang (2023). A Fast Algorithm for the Eigen value Bounds of a Class of Symmetric Tridiagonal Interval Matrices. AppliedMath, 3(1); 90–97
Bala, B., M. D. Manafov, and A. Kablan (2019). Inverse Spectral Problems for Spectral Data and Two Spectra of N by N Tridiagonal Almost-Symmetric Matrices. Applications and Applied Mathematics: An International Journal (AAM), 14(2); 33
Caratelli, D. and P. E. Ricci (2021). Inversion of Tridiagonal Matrices Using the Dunford-Taylor’s Integral. Symmetry, 13(5); 870
Chu, W., Y. Zhao, and H. Yuan (2022). A Modified Inverse Iteration Method for Computing the Symmetric Tridiagonal Eigenvectors. Mathematics, 10(19); 3636
El-Mikkawy, M. and A. Karawia (2022). A Breakdown Free Numerical Algorithm for Inverting General Tridiagonal Matrices. arXiv preprint arXiv:2208.12843; 1–14
El-Mikkawy, M. E. (2004). On the Inverse of a General Tridiagonal Matrix. Applied Mathematics and Computation, 150(3); 669–679
Encinas, A. M. and M. J. Jiménez (2019). Explicit Inverse of Nonsingular Jacobi Matrices. Discrete Applied Mathematics, 263; 130–139
Fathi, F., M. A. Fariborzi Araghi, and S. A. Shahzadeh Fazeli (2020). Two Different Inverse Eigenvalue Problems for Nonsymmetric Tridiagonal Matrices. Journal of Algorithms and Computation, 52(2); 137–148
Furtado, S., C. R. Johnson, C. Marijuán, and M. Pisonero (2023). Square Matrices with the Inverse Diagonal Property. Kuwait Journal of Science; 1–4
Ganesan, K. and P. Veeramani (2005). On Arithmetic Operations of Interval Numbers. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 13(06); 619–631
Hartman, D., M. Hladík, and D. Říha (2021). Computing the Spectral Decomposition of Interval Matrices and a Study on Interval Matrix Powers. Applied Mathematics and Computation, 403; 126174
Heydari, M., S. A. S. Fazeli, and S. M. Karbassi (2019). On the Inverse Eigenvalue Problem for a Special Kind of Acyclic Matrices. Applications of Mathematics, 64; 351–366
Hopkins, T. and E. Kilic (2022). Analytically Explicit Inverse of a Kind of Periodic Tridiagonal Matrix Using a Backward Continued Fraction Approach. Journal of Applied Analysis and Computation, 12(6); 2299–2313
Kaucher, E. (1980). Interval Analysis in the Extended Interval Space IR. Fundamentals of Numerical Computation (Computer Oriented Numerical Analysis); 33–49
Mallik, R. K. (2001). The Inverse of a Tridiagonal Matrix. Linear Algebra and its Applications, 325(1 3); 109–139
Nirmala, T., D. Datta, H. Kushwaha, and K. Ganesan (2011). Inverse Interval Matrix: A New Approach. Applied Mathematical Sciences, 5(13); 607–624
Parker, J. T., P. A. Hill, D. Dickinson, and B. D. Dudson (2022). Parallel Tridiagonal Matrix Inversion with a Hybrid Multigrid-Thomas Algorithm Method. Journal of Computational and Applied Mathematics, 399; 113706
Qi, F., Q. Zou, and B.-N. Guo (2019). The Inverse of a Triangular Matrix and Several Identities of the Catalan Numbers. Applicable Analysis and Discrete Mathematics, 13(2); 518–541
Rohn, J. (1993). Inverse interval matrix. SIAM Journal on Numerical Analysis, 30(3); 864–870
Sabri, R. I. and B. Ahmed (2023). Best Proximity Point Results in Fuzzy Normed Spaces. Science and Technology Indonesia, 8(2); 298–304
Sengupta, A. and T. K. Pal (2000). On Comparing Interval Numbers. European Journal of Operational Research, 127(1); 28–43
Spellacy, L., D. Golden, and I. Rungger (2019). Performance Analysis of a Pairwise Method for Partial Inversion of Complex Block Tridiagonal Matrices. Concurrency and Computation: Practice and Experience, 31(19); e4918
Susanti, E., F. M. Puspita, S. S. Supadi, E. Yuliza, and A. F. Ramadhan (2023). Improve Fuzzy Inventory Model of Fractal Interpolation with Vertical Scaling Factor. Science and Technology Indonesia, 8(4); 654–659
Talibi, B., A. D. A. Hadj, and D. Sarsri (2022). On the Factorization and the Inverse of a Tridiagonal Matrix. Journal of Discrete Mathematical Sciences and Cryptography, 25(8); 2479– 2489
Tan, L. S. (2019). Explicit Inverse of Tridiagonal Matrix with Applications in Autoregressive Modelling. IMA Journal of Applied Mathematics, 84(4); 679–695
Thirupathi, S. and N. Thamaraiselvan (2023). Symbolic Algorithm for Inverting General k Tridiagonal Interval Matrices. International Journal of Analysis and Applications, 21; 1–12
Wang, X. T. and M. L. Jin (2019). Inverse Eigenvalue Problem for Quasi-Tridiagonal Matrices. Bulletin of the Iranian Mathematical Society, 45; 1697–1712
Wei, Y., X. Jiang, Z. Jiang, and S. Shon (2019). Determinants and Inverses of Perturbed Periodic Tridiagonal Toeplitz Matrices. Advances in Difference Equations, 2019(1); 1–11
Yuan, Q. and Z. Yang (2023). A Fast Algorithm for the Eigen value Bounds of a Class of Symmetric Tridiagonal Interval Matrices. AppliedMath, 3(1); 90–97
Authors
Thirupathi, S., & Thamaraiselvan, N. . (2024). Tridiagonal Interval Matrix: Exploring New Perspectives and Application. Science and Technology Indonesia, 9(1), 77–85. https://doi.org/10.26554/sti.2024.9.1.77-85
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