A new multi-step ABS model to solve full row rank linear systems
Subject Areas : StatisticsMahmoud Paripour 1 , Esmaeil Babolian 2 , Leila Asadbeigi 3
1 - Department of Computer Engineering and Information Technology, Hamadan University of Technology, Hamadan, Iran
2 - Faculty of Mathematical and Computer Sciences, Kharazmi University, Karaj, Iran
3 - Faculty of Basic Sciences, Department of Mathematics, Hamadan Branch, Islamic Azad University, Hamadan, Iran
Keywords: روشهای ABS, روشهای ABS سهگامی, دستگاههای معادلات خطی تمام رتبه سطری, فشردگی فضای محاسبات, روشهای ABS دوگامی,
Abstract :
ABS methods are direct iterative methods for solving linear systems of equations, where the i-th iteration satisfies the first i equations. Thus, a system of m equations is solved in at most m ABS iterates. In 2004 and 2007, two-step ABS methods were introduced in at most [((m+1))/2] steps to solve full row rank linear systems of equations. These methods consuming less space, are more compress than corresponding Huang’s method. Also, these ABS-type models need less number of multiplications for a square system. In this paper, in order to economize and compress required space, we present a new three-step ABS procedure that is terminated in at most [((m+2))/3] steps. Computational complexity is considerable up to those corresponding Huang’s method and initial two-step ABS approaches. we present a new three-step ABS procedure that is terminated in at most [((m+2))/3] steps. Computational complexity is considerable up to those corresponding Huang’s method and initial two-step ABS approaches.
[1] Abaffy, J., Broyden, C.G., Spedicato, E., “A class of direct methods for linear systems”, Numerische Mathematik, 45 (1984) 361-376
[2] Abaffy, J., Spedicato, E., “ABS projection algorithms: mathematical techniques for linear and nonlinear equations”, Prentice-Hall, Inc., 1989
[3] Adrash, M., Sharma, S., “ABS methods to solve optimization problems: A review”, Research Journal of Mathematical and Statistical Sciences, 1(2) (2013) 19-21.
[4] J. Amini, K., Mahdavi-Amiri, N., Peyghami, M.R., “ABS-type methods for solving full row rank linear systems using a new rank two update”, Bulletin of the Australian Mathematical Society,
70(1) (2004), 17-34.
[5] Amini, K., Mahdavi-Amiri, N., Peyghami, M,R., “Extended reduced rank two Abaffian update schemes in the ABS-type methods”, Applied Mathematics and Compution, 185(1) (2007) 255-265.
[6] Asadbeigi, L., Paripour, M., “ A note on extended reduced rank-two Abaffian update schemes in the ABS-type methods”, Applied Mathematics and Computation, 326 (2018) 105-107.
[7] Asadbeigi, L., Paripour, M., Babolian, E., “General solution of full row rank linear systems of equations via a new extended ABS model”, U.P.B. Scientfic Bulletin., Series A, 79(4) (2017) 61-68.
[8] Asadbeigi, L., Paripour, M., Babolian, E., Javadi, Sh. “General solution of full row rank linear systems of equations using a new compression ABS model”, Mathematical Sciences, 11(4) (2017) 333-343.
[9] Emilio, S., Spedicato, E., Bodon, E., Xi, Z., Mahdavi-Amiri, N., “ABS methods for coninous and integer linear equations and optimization”, Central European Journal of Operation Research, 18(1) (2010) 73-95..
[10] Ghanbari, R., Mahdavi-Amiri, N., “New solutions of LR linear systems using ranking functions and ABS algorithms”, Applied Mathematical Modelling, 34 (2010) 3363-3375..
