On Solving Centrosymmetric Linear Systems

ABSTRACT

The current paper is mainly devoted for solving centrosymmetric linear systems of equations. Formulae for the determinants of tridiagonal centrosymmetric matrices are obtained explicitly. Two efficient computational algorithms are established for solving general centrosymmetric linear systems. Based on these algorithms, a MAPLE procedure is written. Some illustrative examples are given.

Cite this paper

M. El-Mikkawy and F. Atlan, "On Solving Centrosymmetric Linear Systems,"*Applied Mathematics*, Vol. 4 No. 12, 2013, pp. 21-32. doi: 10.4236/am.2013.412A003.

M. El-Mikkawy and F. Atlan, "On Solving Centrosymmetric Linear Systems,"

References

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http://dx.doi.org/10.1137/S0036144597328341

[2] A. Cantoni and P. Butler, “Eigenvalues and Eigenvectors of Symmetric Centrosymmetric Matrices,” Linear Algebra and Its Applications, Vol. 13, No. 3, 1976, pp. 275288.

[3] A. Cantoni and P. Butler, “Properties of the Eigenvectors of Persymmetric Matrices with Applications to Communication Theory,” IEEE Transactions on Communications, Vol. 24, No. 8, 1976, pp. 804-809.

http://dx.doi.org/10.1109/TCOM.1976.1093391

[4] W. Chen, Y. Yu and X. Wang, “Reducing the Computational Requirement of Differential Quadrature Method,” Numerical Methods for Partial Differential Equations, Vol. 12, 1996, pp. 565-577.

[5] L. Datta and S. Morgera, “Some Results on Matrix Symmetries and a Pattern Recognition Application,” IEEE Transactions on Signal Processing, Vol. 34, No. 4, 1986, pp. 992-994.

[6] L. Datta and S. Morgera, “On the Reducibility of Centrosymmetric Matrices—Applications in Engineering Problems,” Circuits, Systems and Signal Processing, Vol. 8, No. 1, 1989, pp. 71-96.

[7] J. Delmas, “On Adaptive EVD Asymptotic Distribution of Centro-Symmetric Covariance Matrices,” IEEE Transactions on Signal Processing, Vol. 47, No. 5, 1999, pp. 1402-1406.

[8] I. J. Good, “The Inverse of a Centrosymmetric Matrix,” Technometrics, Vol. 12, No. 4, 1970, pp. 925-928.

[9] Z.-Y. Liu, “Some Properties of Centrosymmetric Matrices,” Applied Mathematics and Computation, Vol. 141, No. 2-3, 2003, pp. 297-306.

[10] R. B. Mattingly, “Even Order Regular Magic Squares Are Singular,” The American Mathematical Monthly, Vol. 107, No. 9, 2000, pp. 777-782.

[11] F. Stenger, “Matrices of Sinc Methods,” Journal of Computational and Applied Mathematics, Vol. 86, No. 1, 1997, pp. 297-310.

[12] J. Weaver, “Centrosymmetric (Cross-Symmetric) Matrices, Their Basic Properties, Eigenvalues, and Eigenvectors,” The American Mathematical Monthly, Vol. 92, No. 10, 1985, pp. 711-717.

[13] A. C. Aitken, “Determinants and Matrices,” Oliver and Boyd, Edinburgh, 1956.

[14] H.-T. Gao, C.-H. You and Y. Yang, “An Iterative Method for Generalized Centro-Symmetric Solution of Matrix Equation ” 2008.

http://www.paper.edu.cn/en_releasepaper/content/21924.

[15] A. Graovac, O. Ori, M. Faghani and A. R. Ashrafi, “Distance Property of Fullerenes,” Iranian Journal of Mathematical Chemistry, Vol. 2, No. 1, 2011, pp. 99-107.

[16] I. T. Abu-Jeib, “Algorithms for Centrosymmetric and Skew-Centrosymmetric Matrices,” Missouri Journal of Mathematical Sciences, Vol. 18, No. 1, 2006, pp. 46-53.

[17] O. Krafft and M. Schaefer, “Centrogonal Matrices,” Linear Algebra and its Applications, Vol. 306, No., 2000, pp. 145-154.

[18] Z.-Y. Liu, “Some Properties of Centrosymmetric Matrices and Its Applications,” Numerical Mathematics: A Journal of Chinese Universities, Vol. 14, No. 2, 2005, pp. 136-148.

[19] Z. Tian and C. Gu, “The Iterative Methods for Centrosymmetric Matrices,” Applied Mathematics and Computation, Vol. 187, No. 2, 2007, pp. 902-911.

[20] R. Vein and P. Dale, “Determinants and Their Applications in Mathematical Physics,” Springer, New York, 1999.

[21] A. M. Yasuda, “Some Properties of Commuting and AntiCommuting m-Involutions,” Acta Mathematica Scientia, Series B (English Edition), Vol. 32, No. 2, 2012, pp. 631-644.

[22] M. E. A. El-Mikkawy, “A Generalized Symbolic Thomas algoriThm,” Applied Mathematics, Vol. 3, No. 4, 2012, pp. 342-345. http://dx.doi.org/10.4236/am.2012.34052

[23] T. Sugimoto, “On an Inverse Formula of a Tridiagonal Matrix,” Operators and Matrices, Vol. 6, No. 3, 2012, pp. 465-480. http://dx.doi.org/10.7153/oam-06-30

[24] M. E. A. El-Mikkawy, “A Note on a Three-Term Recurrence for a Tridiagonal Matrix,” Journal of Applied Mathematics and Computing, Vol. 139, No. 2-3, 2003, pp. 503-511.

[25] M. E. A. El-Mikkawy, “An Algorithm for Solving Tridiagonal Systems,” Journal of Institute of Mathematics and Computer Science, Vol. 4, No. 2, 1991, pp. 205-210.

[26] M. E. A. El-Mikkawy, “On the Inverse of a General Tridiagonal Matrix,” Applied Mathematics and Computation, Vol. 150, No. 3, 2004, pp. 669-679.

[27] M. E. A. El-Mikkawy, “A New Computational Algorithm for Solving Periodic Tri-Diagonal Linear Systems,” Applied Mathematics and Computation, Vol. 161, No. 2, 2005, pp. 691-696.

[28] M. E. A. El-Mikkawy and A. Karawia, “Inversion of General Tridiagonal Matrices,” Applied Mathematics Letters, Vol. 19, No. 8, 2006, pp. 712-720.

[29] M. E. A. El-Mikkawy and E.-D. Rahmo, “A New Recursive Algorithm for Inverting General Tridiagonal and Anti-Tridiagonal Matrices,” Applied Mathematics and Computation, Vol. 204, No. 1, 2008, pp. 368-372.

[30] R. L. Burden and J. D. Faires, “Numerical Analysis,” 7th ed., Books & Cole Publishing, Pacific Grove, 2001.

[31] M. E. A. El-Mikkawy, “A Fast Algorithm for Evaluating nth Order Tri-Diagonal Determinants,” Journal of Computational and Applied Mathematics, Vol. 166, No. 2, 2004, pp. 581-584.

[1] A. L. Andrew, “Centrosymmetric Matrices,” SIAM Review, Vol. 40, No. 3, 1998, pp. 697-699.

http://dx.doi.org/10.1137/S0036144597328341

[2] A. Cantoni and P. Butler, “Eigenvalues and Eigenvectors of Symmetric Centrosymmetric Matrices,” Linear Algebra and Its Applications, Vol. 13, No. 3, 1976, pp. 275288.

[3] A. Cantoni and P. Butler, “Properties of the Eigenvectors of Persymmetric Matrices with Applications to Communication Theory,” IEEE Transactions on Communications, Vol. 24, No. 8, 1976, pp. 804-809.

http://dx.doi.org/10.1109/TCOM.1976.1093391

[4] W. Chen, Y. Yu and X. Wang, “Reducing the Computational Requirement of Differential Quadrature Method,” Numerical Methods for Partial Differential Equations, Vol. 12, 1996, pp. 565-577.

[5] L. Datta and S. Morgera, “Some Results on Matrix Symmetries and a Pattern Recognition Application,” IEEE Transactions on Signal Processing, Vol. 34, No. 4, 1986, pp. 992-994.

[6] L. Datta and S. Morgera, “On the Reducibility of Centrosymmetric Matrices—Applications in Engineering Problems,” Circuits, Systems and Signal Processing, Vol. 8, No. 1, 1989, pp. 71-96.

[7] J. Delmas, “On Adaptive EVD Asymptotic Distribution of Centro-Symmetric Covariance Matrices,” IEEE Transactions on Signal Processing, Vol. 47, No. 5, 1999, pp. 1402-1406.

[8] I. J. Good, “The Inverse of a Centrosymmetric Matrix,” Technometrics, Vol. 12, No. 4, 1970, pp. 925-928.

[9] Z.-Y. Liu, “Some Properties of Centrosymmetric Matrices,” Applied Mathematics and Computation, Vol. 141, No. 2-3, 2003, pp. 297-306.

[10] R. B. Mattingly, “Even Order Regular Magic Squares Are Singular,” The American Mathematical Monthly, Vol. 107, No. 9, 2000, pp. 777-782.

[11] F. Stenger, “Matrices of Sinc Methods,” Journal of Computational and Applied Mathematics, Vol. 86, No. 1, 1997, pp. 297-310.

[12] J. Weaver, “Centrosymmetric (Cross-Symmetric) Matrices, Their Basic Properties, Eigenvalues, and Eigenvectors,” The American Mathematical Monthly, Vol. 92, No. 10, 1985, pp. 711-717.

[13] A. C. Aitken, “Determinants and Matrices,” Oliver and Boyd, Edinburgh, 1956.

[14] H.-T. Gao, C.-H. You and Y. Yang, “An Iterative Method for Generalized Centro-Symmetric Solution of Matrix Equation ” 2008.

http://www.paper.edu.cn/en_releasepaper/content/21924.

[15] A. Graovac, O. Ori, M. Faghani and A. R. Ashrafi, “Distance Property of Fullerenes,” Iranian Journal of Mathematical Chemistry, Vol. 2, No. 1, 2011, pp. 99-107.

[16] I. T. Abu-Jeib, “Algorithms for Centrosymmetric and Skew-Centrosymmetric Matrices,” Missouri Journal of Mathematical Sciences, Vol. 18, No. 1, 2006, pp. 46-53.

[17] O. Krafft and M. Schaefer, “Centrogonal Matrices,” Linear Algebra and its Applications, Vol. 306, No., 2000, pp. 145-154.

[18] Z.-Y. Liu, “Some Properties of Centrosymmetric Matrices and Its Applications,” Numerical Mathematics: A Journal of Chinese Universities, Vol. 14, No. 2, 2005, pp. 136-148.

[19] Z. Tian and C. Gu, “The Iterative Methods for Centrosymmetric Matrices,” Applied Mathematics and Computation, Vol. 187, No. 2, 2007, pp. 902-911.

[20] R. Vein and P. Dale, “Determinants and Their Applications in Mathematical Physics,” Springer, New York, 1999.

[21] A. M. Yasuda, “Some Properties of Commuting and AntiCommuting m-Involutions,” Acta Mathematica Scientia, Series B (English Edition), Vol. 32, No. 2, 2012, pp. 631-644.

[22] M. E. A. El-Mikkawy, “A Generalized Symbolic Thomas algoriThm,” Applied Mathematics, Vol. 3, No. 4, 2012, pp. 342-345. http://dx.doi.org/10.4236/am.2012.34052

[23] T. Sugimoto, “On an Inverse Formula of a Tridiagonal Matrix,” Operators and Matrices, Vol. 6, No. 3, 2012, pp. 465-480. http://dx.doi.org/10.7153/oam-06-30

[24] M. E. A. El-Mikkawy, “A Note on a Three-Term Recurrence for a Tridiagonal Matrix,” Journal of Applied Mathematics and Computing, Vol. 139, No. 2-3, 2003, pp. 503-511.

[25] M. E. A. El-Mikkawy, “An Algorithm for Solving Tridiagonal Systems,” Journal of Institute of Mathematics and Computer Science, Vol. 4, No. 2, 1991, pp. 205-210.

[26] M. E. A. El-Mikkawy, “On the Inverse of a General Tridiagonal Matrix,” Applied Mathematics and Computation, Vol. 150, No. 3, 2004, pp. 669-679.

[27] M. E. A. El-Mikkawy, “A New Computational Algorithm for Solving Periodic Tri-Diagonal Linear Systems,” Applied Mathematics and Computation, Vol. 161, No. 2, 2005, pp. 691-696.

[28] M. E. A. El-Mikkawy and A. Karawia, “Inversion of General Tridiagonal Matrices,” Applied Mathematics Letters, Vol. 19, No. 8, 2006, pp. 712-720.

[29] M. E. A. El-Mikkawy and E.-D. Rahmo, “A New Recursive Algorithm for Inverting General Tridiagonal and Anti-Tridiagonal Matrices,” Applied Mathematics and Computation, Vol. 204, No. 1, 2008, pp. 368-372.

[30] R. L. Burden and J. D. Faires, “Numerical Analysis,” 7th ed., Books & Cole Publishing, Pacific Grove, 2001.

[31] M. E. A. El-Mikkawy, “A Fast Algorithm for Evaluating nth Order Tri-Diagonal Determinants,” Journal of Computational and Applied Mathematics, Vol. 166, No. 2, 2004, pp. 581-584.