Sufficient Conditions for Existence of the LU Factorization of Toeplitz Symmetric Tridiagonal Matrices
DOI:
https://doi.org/10.5540/tcam.2022.024.01.00177Keywords:
Toeplitz tridiagonal matrix, Crout's method, tridiagonal and diagonally dominant matrixAbstract
The characterization of inverses of symmetric tridiagonal and block tridiagonal matrices and the development of algorithms for finding the inverse of any general non-singular tridiagonal matrix are subjects that have been studied by many authors. The results of these research usually depend on the existence of the LU factorization of a non-sigular matrix A, such that A = LU. Besides, the conditions that ensure the nonsingularity of A and its LU factorization are not promptly obtained. Then, we are going to present in this work two extremely simple sufficient conditions for existence of the LU factorization of a Toeplitz symmetric tridiagonal matrix A. We take into consideration the roots of the modified Chebyshev polynomial, and we also present an analysis based on the parameters of the Crout’s method.
References
R. L. Burden and J. D. Faires}, Numerical Analysis, 9th ed., Springer, New York, USA, 2014.
P. Z. Revesz, Cubic spline interpolation by solving a recurrence equation instead of a tridiagonal matrix, Proceedings of the First International Conference on Mathematical Methods and Computational Techniques in Science and Engineering, WSEAS Press, 2014, Athens, Greece, 21--25.
P. Z. Revesz, Introduction to Databases: From Biological to Spatio-Temporal, 2010, Springer, New York, USA.
M. E. A. El-Mikkawy, Notes on linear systems with positive definite tridiagonal coefficient matrices, Indian Journal of Pure and Applied Mathematics, vol. 33, 8, 1285 - 1293, 2002.
M. E. A. El-Mikkawy, On the inverse of a general tridiagonal matrix, Applied Mathematics and Computation, vol. 150, 3, 669 - 679, 2004, issn: 0096-3003, doi: https://doi.org/10.1016/S0096-3003(03)00298-4.
M. El-Mikkawy and A. Karawia, Inversion of general tridiagonal matrices, Applied Mathematics Letters, vol. 19, 712 - 720, 2006, doi: 10.1016/j.aml.2005.11.012.
C. F. Fischer and R. A. Usmani, Properties of some Tridiagonal Matrices and their application to Boundary Value
Problems, SIAM Journal on Numerical Analysis, vol. 6, 1, 127 - 142, 1969, url: https://www.jstor.org/stable/2156523.
G. Meurant, A review on the inverse of symmetric tridiagonal and block tridiagonal matrices, SIAM Journal Matrix Anal. Appl., vol. 13, 3, 707 - 728, 1992.
R. E. Bank and D. J. Rose, Marching algorithms for elliptic boundary value problems. I: The constant coefficient case, SIAM Journal on Numerical Analysis, vol. 14, 5, 792 - 829, 1977.
C. M. da Fonseca and J. Petronilho, Explicit inverses of some tridiagonal matrices, Linear Algebra and its Applications, vol. 325, 7 - 21, 2001.
Xi-Le Zhao and Ting-Zhu Huang, On the inverse of a general pentadiagonal matrix, Applied Mathematics and Computation, vol. 202, 2, 639 - 646, 2008, issn: 0096-3003, doi: https://doi.org/10.1016/j.amc.2008.03.004.
J. M. McNally, A fast algorithm for solving diagonally dominant symmetric pentadiagonal Toeplitz systems, Journal of Computational and Applied Mathematics, vol. 234, 4, 995 - 1005, 2010, issn: 0377-0427, doi: https://doi.org/10.1016/j.cam.2009.03.001.
S. S. Askar and A. A. Karawia, On Solving Pentadiagonal Linear Systems via Transformations, Mathematical Problems in Engineering, George S. Dulikravich, Hindawi Publishing Corporation, vol. 2015, 2015,
doi: https://doi.org/10.1155/2015/232456.
J. M. McNally and L. E. Garey and R. E. Shaw, A split-correct parallel algorithm for solving tridiagonal symmetric Toeplitz systems, Int. J. Comput. Math., vol. 75, 303--313, 2000.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish in this journal agree to the following terms:
Authors retain copyright and grant the journal the right of first publication, with the work simultaneously licensed under the Creative Commons Attribution License that allows the sharing of the work with acknowledgment of authorship and initial publication in this journal.
Authors are authorized to assume additional contracts separately, for non-exclusive distribution of the version of the work published in this journal (eg, publish in an institutional repository or as a book chapter), with acknowledgment of authorship and initial publication in this journal.
Authors are allowed and encouraged to publish and distribute their work online (eg, in institutional repositories or on their personal page) at any point before or during the editorial process, as this can generate productive changes as well as increase impact and the citation of the published work (See The effect of open access).
This is an open access journal which means that all content is freely available without charge to the user or his/her institution. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the
author. This is in accordance with the BOAI definition of open access
Intellectual Property
All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License under attribution BY.