Jacobi-Based Methods in Solving Fuzzy Linear Systems
Linear systems are widely used in many fields of science and engineering. In many applications, at least some of the parameters of the system are represented by fuzzy rather than crisp numbers. Therefore it is important to perform numerical algorithms or procedures that would treat general fuzzy linear systems and solve them using iterative methods. This paper aims are to solve fuzzy linear systems using four types of Jacobi based iterative methods. Four iterative methods based on Jacobi are used for solving a general n × n fuzzy system of linear equations of the form Ax = b , where A is a crisp matrix and b an arbitrary fuzzy vector. The Jacobi, Jacobi Over-Relaxation, Refinement of Jacobi and Refinement of Jacobi Over-Relaxation methods was tested to a five by five fuzzy linear system. It is found that all the tested methods were iterated differently. Due to the effect of extrapolation parameters and the refinement, the Refinement of Jacobi Over-Relaxation method was outperformed the other three methods.
<p>[1] S. Abbasbandy, R. Ezzati & A. Jafarian, “LU decomposition method for
solving fuzzy system of linear equations,” Applied Mathematics and
Computation, vol. 172, pp. 633-643, 2006.
[2] T. Allahviranloo, “A comment on fuzzy linear systems,” Fuzzy Sets and
Systems, vol. 140, pp. 559-559, 2003.
[3] T. Allahviranloo, “Numerical methods for fuzzy system of linear
equations,” Applied Mathematics and Computation, vol. 155, pp. 493-
502, 2004.
[4] T. Allahviranloo, “Successive over relaxation iterative method for fuzzy
system of linear equations,” Applied Mathematics and Computation, vol.
162, pp. 189-196, 2005.
[5] T. Allahviranloo, “The Adomian decomposition method for fuzzy
system of linear equations,” Applied Mathematics and Computation, vol.
163, pp. 553-563, 2005.
[6] T. Allahviranloo & M.A. Kermani, “ Solution of a fuzzy system of
linear equation,” Applied Mathematics and Computation, vol. 175, pp.
519-531, 2006.
[7] B. Asady, S. Abbasbandy & M. Alavi, “Fuzzy general linear systems,”
Applied Mathematics and Computation, vol. 169, pp. 34-40, 2005.
[8] E. Babolian & M. Paripour, “Numerical solving of general fuzzy linear
systems,” Tarbiat Moallem University, 20th Seminar on Algebra, pp. 40-
43, 2009.
[9] F.N. Dafchahi, “A new refinement of Jacobi method for solution of
linear system equations AX = b ,” International Journal of
Contemporary Mathematical Sciences, vol. 3, no. 17, pp. 819-827,
2008..
[10] M. Dehgan & B. Hashemi, “Iterative solution of fuzzy linear systems,”
Applied Mathematics and Computation, vol. 175, pp. 645-674, 2006.
[11] M. Friedman, M. Ming and A. Kandel, “Fuzzy linear systems,” Fuzzy
Sets and Systems 96, pp. 201-209, 1998.
[12] S.J.H. Ghoncheh & M. Paripour, “Numerical solving of general fuzzy
linear systems by Huang’s method,” International Journal of
Computational and Mathematical Sciences, vol. 3, pp. 25-27, 2009.
[13] R. Goetschell & W. Voxman, “Elementary calculus,” Fuzzy Sets and
Systems, vol. 18, pp. 31-43, 1986.
[14] L.A. Hageman & D.M. Young, “Applied iterative methods” New York:
Academic Press, 1981.
[15] O. Kaleva, “Fuzzy differential equations,” Fuzzy Sets and Systems, vol.
24, pp. 301-317, 1987.
[16] A. Kandel, M. Friedman & M. Ming, “Fuzzy linear systems and their
solution,” IEEE, pp. 336-338, 1996.
[17] G.J. Klir, U.S Clair & B. Yuan, “Fuzzy sets theory: foundations and
applications,” Prentice Hall Incorporated, 1997.
[18] M. Ma, M. Friedman & A. Kandel, “A new fuzzy arithmetic,” Fuzzy
Sets and Systems, vol. 108, pp. 83-90, 1999.
[19] M. Ma, M. Friedman & A. Kandel, “Duality in fuzzy linear systems,”
Fuzzy Sets and Systems, vol. 109, pp. 55-58, 2000.
[20] S.H. Nasseri and M. Khorramizadeh, “A new method for solving fuzzy
linear systems,” International Journal of Applied Mathematics, vol. 20,
pp. 507-516, 2007.
[21] N. Ujevic, “A new iterative method for solving linear systems,” Applied
Mathematics and Computation, vol. 179, pp. 725-730, 2006.
[22] D.M. Young, “Iterative solution of large linear systems,” New York:
Academic Press, 1971.</p>
<p>[1] S. Abbasbandy, R. Ezzati & A. Jafarian, “LU decomposition method for
solving fuzzy system of linear equations,” Applied Mathematics and
Computation, vol. 172, pp. 633-643, 2006.
[2] T. Allahviranloo, “A comment on fuzzy linear systems,” Fuzzy Sets and
Systems, vol. 140, pp. 559-559, 2003.
[3] T. Allahviranloo, “Numerical methods for fuzzy system of linear
equations,” Applied Mathematics and Computation, vol. 155, pp. 493-
502, 2004.
[4] T. Allahviranloo, “Successive over relaxation iterative method for fuzzy
system of linear equations,” Applied Mathematics and Computation, vol.
162, pp. 189-196, 2005.
[5] T. Allahviranloo, “The Adomian decomposition method for fuzzy
system of linear equations,” Applied Mathematics and Computation, vol.
163, pp. 553-563, 2005.
[6] T. Allahviranloo & M.A. Kermani, “ Solution of a fuzzy system of
linear equation,” Applied Mathematics and Computation, vol. 175, pp.
519-531, 2006.
[7] B. Asady, S. Abbasbandy & M. Alavi, “Fuzzy general linear systems,”
Applied Mathematics and Computation, vol. 169, pp. 34-40, 2005.
[8] E. Babolian & M. Paripour, “Numerical solving of general fuzzy linear
systems,” Tarbiat Moallem University, 20th Seminar on Algebra, pp. 40-
43, 2009.
[9] F.N. Dafchahi, “A new refinement of Jacobi method for solution of
linear system equations AX = b ,” International Journal of
Contemporary Mathematical Sciences, vol. 3, no. 17, pp. 819-827,
2008..
[10] M. Dehgan & B. Hashemi, “Iterative solution of fuzzy linear systems,”
Applied Mathematics and Computation, vol. 175, pp. 645-674, 2006.
[11] M. Friedman, M. Ming and A. Kandel, “Fuzzy linear systems,” Fuzzy
Sets and Systems 96, pp. 201-209, 1998.
[12] S.J.H. Ghoncheh & M. Paripour, “Numerical solving of general fuzzy
linear systems by Huang’s method,” International Journal of
Computational and Mathematical Sciences, vol. 3, pp. 25-27, 2009.
[13] R. Goetschell & W. Voxman, “Elementary calculus,” Fuzzy Sets and
Systems, vol. 18, pp. 31-43, 1986.
[14] L.A. Hageman & D.M. Young, “Applied iterative methods” New York:
Academic Press, 1981.
[15] O. Kaleva, “Fuzzy differential equations,” Fuzzy Sets and Systems, vol.
24, pp. 301-317, 1987.
[16] A. Kandel, M. Friedman & M. Ming, “Fuzzy linear systems and their
solution,” IEEE, pp. 336-338, 1996.
[17] G.J. Klir, U.S Clair & B. Yuan, “Fuzzy sets theory: foundations and
applications,” Prentice Hall Incorporated, 1997.
[18] M. Ma, M. Friedman & A. Kandel, “A new fuzzy arithmetic,” Fuzzy
Sets and Systems, vol. 108, pp. 83-90, 1999.
[19] M. Ma, M. Friedman & A. Kandel, “Duality in fuzzy linear systems,”
Fuzzy Sets and Systems, vol. 109, pp. 55-58, 2000.
[20] S.H. Nasseri and M. Khorramizadeh, “A new method for solving fuzzy
linear systems,” International Journal of Applied Mathematics, vol. 20,
pp. 507-516, 2007.
[21] N. Ujevic, “A new iterative method for solving linear systems,” Applied
Mathematics and Computation, vol. 179, pp. 725-730, 2006.
[22] D.M. Young, “Iterative solution of large linear systems,” New York:
Academic Press, 1971.</p>
@article{"International Journal of Engineering, Mathematical and Physical Sciences:65303", author = "Lazim Abdullah and Nurhakimah Ab. Rahman", title = "Jacobi-Based Methods in Solving Fuzzy Linear Systems", abstract = "Linear systems are widely used in many fields of science and engineering. In many applications, at least some of the parameters of the system are represented by fuzzy rather than crisp numbers. Therefore it is important to perform numerical algorithms or procedures that would treat general fuzzy linear systems and solve them using iterative methods. This paper aims are to solve fuzzy linear systems using four types of Jacobi based iterative methods. Four iterative methods based on Jacobi are used for solving a general n × n fuzzy system of linear equations of the form Ax = b , where A is a crisp matrix and b an arbitrary fuzzy vector. The Jacobi, Jacobi Over-Relaxation, Refinement of Jacobi and Refinement of Jacobi Over-Relaxation methods was tested to a five by five fuzzy linear system. It is found that all the tested methods were iterated differently. Due to the effect of extrapolation parameters and the refinement, the Refinement of Jacobi Over-Relaxation method was outperformed the other three methods.
", keywords = "Fuzzy linear systems, Jacobi, Jacobi Over-
Relaxation, Refinement of Jacobi, Refinement of Jacobi Over-
Relaxation.", volume = "7", number = "3", pages = "476-7", }
