Abstract
In the current paper, we present a novel symbolic algorithm for solving periodic tridiagonal linear systems without imposing any restrictive conditions. The computational cost of the algorithm is less than or almost equal to those of three well-known algorithms given by Chawla and Khazal (Int. J. Comput. Math. 79(4):473–484, 2002) and by El-Mikkawy (Appl. Math. Comput. 161:691–696, 2005), respectively. In addition, the solution of periodic anti-tridiagonal linear systems is also discussed. Two numerical experiments are provided in order to illustrate the performance and effectiveness of our algorithm. All of the experiments were performed on a computer with aid of programs written in MATLAB.
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References
M. Znojil, J. Math. Chem. 28, 140 (2000)
A. Bjorck, G.H. Golub, SIAM Rev. 19, 5 (1977)
F. Iachello, A. Del Sol Mesa, J. Math. Chem. 25, 345 (1999)
Y.J. Ye, J. Math. Chem. 14, 121 (1993)
J. Verkaik, H.X. Lin, Parallel Comput. 31, 563 (2005)
M.M. Chawla, K. Passi, P.N. Shivakumar, Int. J. Comput. Math. 45, 113 (1992)
M. Alfaro, J.M. Montaner, Numer. Algor. 10, 137 (1995)
C.F. Fischer, R.A. Usmani, SIAM J. Numer. Anal. 6, 127 (1969)
J.T. Jia, T. Sogabe, J. Math. Chem. 51, 881 (2013)
J.T. Jia, T. Sogabe, M. El-Mikkawy, Comput. Math. Appl. 65, 116 (2013)
A.A. Karawia, Appl. Math. Comput. 194, 534 (2007)
D.J. Evans, Int. J. Comput. Math. 72, 375 (1999)
C. Temperton, J. Comput. Phys. 19, 317 (1975)
M.A.L. Marques, A. Castro, G.F. Bertsch, A. Rubio, Comput. Phys. Commun. 151, 60 (2003)
T.A. Davis, Direct Methods for Sparse Linear Systems (SIAM, Philadelphia, 2006)
M.M. Chawla, R.R. Khazal, Int. J. Comput. Math. 79, 473 (2002)
M. El-Mikkawy, Appl. Math. Comput. 161, 691 (2005)
G.H. Golub, C.F. Van Loan, Matrix Computations, 3rd edn. (Johns Hopkins University Press, Baltimore and London, 1996)
M. El-Mikkawy, Appl. Math. 3, 342 (2012)
M.B. Allen III, E.L. Isaacson, Numerical Analysis for Applied Science (Wiley-Interscience, Wiley, New York, 1997)
Y.L. Jiang, J.T. Jia, J. Math. Chem. 51, 2503 (2013)
M. El-Mikkawy, J. Comput. Appl. Math. 166, 581 (2004)
Acknowledgments
This work was supported by the Natural Science Foundation of China (NSFC) under Grant 11371287 and the International Science and Technology Cooperation Program of China under Grant 2010DFA14700.
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Jia, JT., Kong, QX. A symbolic algorithm for periodic tridiagonal systems of equations. J Math Chem 52, 2222–2233 (2014). https://doi.org/10.1007/s10910-014-0378-1
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DOI: https://doi.org/10.1007/s10910-014-0378-1
Keywords
- Periodic tridiagonal matrices
- Periodic anti-tridiagonal matrices
- Matrix decomposition
- Linear systems
- Computational cost
- Computer Algebra Systems (CASs)