A new Approximation to the solution of the linear matrix equation AXB = C
الموضوعات :
1 - Department of Mathematics, Robat Karim Branch, Islamic Azad University, Tehran, Iran
الکلمات المفتاحية: Homotopy Perturbation method, matrix equation, Diagonally dominant matrix, Convergence, Iterative method,
ملخص المقالة :
It is well-known that the matrix equations play a significant role in several applications in science and engineering. There are various approaches either direct methods or iterative methods to evaluate the solution of these equations. In this research article, the homotopy perturbation method (HPM) will employ to deduce the approximated solution of the linear matrix equation in the form AXB=C. Furthermore, the conditions will be explored to check the convergence of the homotopy series. Numerical examples are also adapted to illustrate the properties of the modified method.
[1] P. Benner, Factorized solution Of Sylvester equations with applications in Control. In: Proceedings of international symposium of mathematics. Theory networks and systems, MTNS 2004.
[2] P. Benner, Large-scale matrix equations of special type, Numer Linear Algebra Appl, 15 (2008), 747-754.
[3] J. Cai, G. Chen, An iterative algorithm for the least squares bisymmetric solutions of the matrix equations A1XB1 = C1 and A2XB2 = C2, Mathematical and Computer Modelling 50 (8), (2009), 1237-1244.
[4] B. N. Datta, K. Datta, Theoretical and computational aspects of some linear algebra problems in Control theory. In: Byrnes CI, Lindquist A (eds) Computational and combinatorial methods in systems theory. Elsevier, Amsterdam, pp. 201-212, 1986.
[5] J. Ding, Y.J. Liu, F. Ding, Iterative solutions to matrix equations of form AiXBi = Fi. Comput. Math. Appl. 59 (11), (2010), 3500-3507.
[6] A. J. Laub, M. T. Heath, C. Paige, and R. C. Ward, Computation of system balancing transformations and other applications of simultaneous diagonalisation algorithms. IEEE Trans Automat Control 32 (1987), 115-122.
[7] S. J. Liao. The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D. Thesis, Shanghai Jiao Tong University, 1992.
[8] A. Liao, Y. Lei, Least-squares solution with the minimum-norm for the matrix equation (AXB, GXH) = (C, D), Computers and Mathematics with Applications 50 (3), (2005), 539-549.
[9] J. H. He. Homotopy perturbation technique. Comput Methods Appl Mech Eng, (1999), 57-62.
[10] J. H. He. A coupling method of a homotopy technique and a perturbation technique for non-linear problems. Int J Non-linear Mech, 35 (1), (2000), 37-43.
[11] J. H. He. Homotopy perturbation method: a new non-linear analytical technique. Appl Math Comput, 135 (1), (2003), 73-79.
[12] B. Keramati, An approach to the solution of linear system of equations by he’s HPM, Chaos Solitons Fract. doi:10.1016/j.chaos.2007.11.020.
[13] H. K. Liu, Application of homotopy perturbation methods for solving systems of linear equations, Appl. Math. Comput, 217(12), (2011), 5259-5264.
[14] Y.H. Liu, Ranks of least squares solutions of the matrix equation AXB = C, Computers and Mathematics with Applications, 55 (6), (2008), 1270-1278.
[15] S.K. Mitra, A pair of simultaneous linear matrix equations A1XB1 = C1 and A2XB2 = C2 and a programming problems, Linear Algebra Appl. 131 (1990), 107-123.
[16] A. Navarra, P.L. Odell, D.M. Young, A representation of the general common solution to the matrix equations A1XB1 = C1 and A2XB2 = C2 with applications, Computers and Mathematics with Applications 41 (8), (2001), 929-935.
[17] M. A. Noor, K. I. Noor, S. Khan, and M. Waseem, Modified homotopy perturbation method for solving system of linear equations, Journal of the Association of Arab Universities for Basic and Applied Sciences, 13 (2013), 35-37.
[18] S. A. Edalatpanah, and M.M. Rashidi, On the application of homotopy perturbation method for solving systems of linear equations, International Scholarly Research Notices, (2014), doi.10.1155/2014/143512.
[19] A. J. Laub, Matrix analysis for scientists and engineers, SIAM, Philadelphia, PA, 2005.
[20] Y. Saad, Iterative methods for sparse linear systems, second ed., SIAM, 2003.
[21] A. Sadeghi, M. I. Ahmad, A. Ahmad, and M. E. Abbasnejad, A note on solving the fuzzy Sylvester matrix equation, journal of computational analysis and applications, 15 (1), (2013), 10-22.
[22] A. Sadeghi, S. Abbasbandy, and M. E. Abbasnejad, The common solution of the pair of fuzzy matrix equations, World Applied Sciences, 15 (2), (2011), 232-238.
[23] A. Sadeghi, M. E. Abbasnejad, and M. I. Ahmad, On solving systems of fuzzy matrix equation, Far East Journal of Applied Mathematics, 59 (1), (2011), 31-44.
[24] J. Saeidian, E. Babolian, and A. Aziz, On a homotopy based method for solving systems of linear equations, TWMS J. Pure Appl. Math., 6(1), (2015), 15-26.
[25] Y. Tian, Ranks of solutions of the matrix equation AXB = C, Linear Multilinear Algebra. 51 (2003), 111-125.
[26] S. Yuan, A. Liao, Y. Lei, Least squares Hermitian solution of the matrix equation (AXB, CXD) = (E, F) with the least norm over the skew field of quaternions, Mathematical and Computer Modelling, 48 (2): (2008), 91-100.
[27] E. Yusufoglu, An improvement to homotopy perturbation method for solving system of linear equations. Comput Math Appl, 58 (2009), 2231-2235.
