Preconditioned Jacobi Method for Fuzzy Linear Systems
A preconditioned Jacobi (PJ) method is provided for solving fuzzy linear systems whose coefficient matrices are crisp Mmatrices and the right-hand side columns are arbitrary fuzzy number vectors. The iterative algorithm is given for the preconditioned Jacobi method. The convergence is analyzed with convergence theorems. Numerical examples are given to illustrate the procedure and show the effectiveness and efficiency of the method.
<p>[1] S. Abbasbandy, R. Ezzati, A. Jafarian, LU decomposition method for
solving fuzzy system of linear equations, Appl. Math. Comput. 172
(2006) 633-643.
[2] S. Abbasbandy, A. Jafarian, Steepest descent method for system of fuzzy
linear equations, Appl. Math. Comput. 175 (2006) 823-833.
[3] S. Abbasbandy, A. Jafarian, R. Ezzati, Conjugate gradient method for
fuzzy symmetric positive definite system of linear equations, Appl.
Math. Comput. 171 (2005) 1184-1191.
[4] T. Allahviranloo, Numerical methods for fuzzy system of linear equations,
Appl. Math. Comput. 155 (2004) 493-502.
[5] T. Allahviranloo, Successive over relaxation iterative method for fuzzy
system of linear equations, Appl. Math. Comput. 162 (2005) 189-196.
[6] T. Allahviranloo, The Adomian decomposition method for fuzzy system
of linear equations, Appl. Math. Comput. 163 (2005) 553-563.
[7] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical
Sciences, SIAM, Philadelphia, 1994.
[8] M. Dehghan, B. Hashemi, Iterative solution of fuzzy linear systems,
Appl. Math. Comput. 175 (2006) 645-674.
[9] R. Ezzati, Solving fuzzy linear systems, Soft Comput. 15 (2011) 193-
197.
[10] M.A. Fariborzi Araghi, A. Fallahzadeh, Inherited LU factorization for
solving fuzzy system of linear equations, Soft Comput. 17 (2013) 159-
163.
[11] M. Friedman, M. Ma, A. Kandel, Fuzzy linear systems, Fuzzy Sets and
Systems 96 (1998) 201-209.
[12] H.-K. Liu, On the solution of fully fuzzy linear systems, World Academy
of Science, Engineering and Technology 43 (2010) 310-314.
[13] S.-X. Miao, Block homotopy perturbation method for solving fuzzy
linear systems, World Academy of Science, Engineering and Technology
51 (2011) 1062-1065.
[14] S.-X. Miao, B. Zheng, K. Wang, Block SOR methods for fuzzy linear
systems, J. Appl. Math. Comput. 26 (2008) 201-218.
[15] S.H. Nasseri, M. Matinfar, M. Sohrabi, QR-decomposition method for
solving fuzzy system of linear equations, Int. J. Math. Comput. 4 (2009)
129-136.
[16] G. Simons, Y. Yao, Approximating the inverse of a symmetric positive
definite matrix, Linear Algebra Appl. 281 (1998) 97-103.
[17] K. Wang, Y. Wu, Uzawa-SOR method for fuzzy linear system, International
Journal of Information and Computer Science 1 (2012) 36-39.
[18] K. Wang, B. Zheng, Symmetric successive overrelaxation methods for
fuzzy linear systems, Appl. Math. Comput. 175 (2006) 891-901.
[19] K. Wang, B. Zheng, Block iterative methods for fuzzy linear systems,
J. Appl. Math. Comput. 25 (2007) 119-136.
[20] Y. Zhu, J. Joutsensalo, T. H¨am¨al¨ainen, Solutions to fuzzy linear systems,
Information 13 (2010) 23-30.</p>
<p>[1] S. Abbasbandy, R. Ezzati, A. Jafarian, LU decomposition method for
solving fuzzy system of linear equations, Appl. Math. Comput. 172
(2006) 633-643.
[2] S. Abbasbandy, A. Jafarian, Steepest descent method for system of fuzzy
linear equations, Appl. Math. Comput. 175 (2006) 823-833.
[3] S. Abbasbandy, A. Jafarian, R. Ezzati, Conjugate gradient method for
fuzzy symmetric positive definite system of linear equations, Appl.
Math. Comput. 171 (2005) 1184-1191.
[4] T. Allahviranloo, Numerical methods for fuzzy system of linear equations,
Appl. Math. Comput. 155 (2004) 493-502.
[5] T. Allahviranloo, Successive over relaxation iterative method for fuzzy
system of linear equations, Appl. Math. Comput. 162 (2005) 189-196.
[6] T. Allahviranloo, The Adomian decomposition method for fuzzy system
of linear equations, Appl. Math. Comput. 163 (2005) 553-563.
[7] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical
Sciences, SIAM, Philadelphia, 1994.
[8] M. Dehghan, B. Hashemi, Iterative solution of fuzzy linear systems,
Appl. Math. Comput. 175 (2006) 645-674.
[9] R. Ezzati, Solving fuzzy linear systems, Soft Comput. 15 (2011) 193-
197.
[10] M.A. Fariborzi Araghi, A. Fallahzadeh, Inherited LU factorization for
solving fuzzy system of linear equations, Soft Comput. 17 (2013) 159-
163.
[11] M. Friedman, M. Ma, A. Kandel, Fuzzy linear systems, Fuzzy Sets and
Systems 96 (1998) 201-209.
[12] H.-K. Liu, On the solution of fully fuzzy linear systems, World Academy
of Science, Engineering and Technology 43 (2010) 310-314.
[13] S.-X. Miao, Block homotopy perturbation method for solving fuzzy
linear systems, World Academy of Science, Engineering and Technology
51 (2011) 1062-1065.
[14] S.-X. Miao, B. Zheng, K. Wang, Block SOR methods for fuzzy linear
systems, J. Appl. Math. Comput. 26 (2008) 201-218.
[15] S.H. Nasseri, M. Matinfar, M. Sohrabi, QR-decomposition method for
solving fuzzy system of linear equations, Int. J. Math. Comput. 4 (2009)
129-136.
[16] G. Simons, Y. Yao, Approximating the inverse of a symmetric positive
definite matrix, Linear Algebra Appl. 281 (1998) 97-103.
[17] K. Wang, Y. Wu, Uzawa-SOR method for fuzzy linear system, International
Journal of Information and Computer Science 1 (2012) 36-39.
[18] K. Wang, B. Zheng, Symmetric successive overrelaxation methods for
fuzzy linear systems, Appl. Math. Comput. 175 (2006) 891-901.
[19] K. Wang, B. Zheng, Block iterative methods for fuzzy linear systems,
J. Appl. Math. Comput. 25 (2007) 119-136.
[20] Y. Zhu, J. Joutsensalo, T. H¨am¨al¨ainen, Solutions to fuzzy linear systems,
Information 13 (2010) 23-30.</p>
@article{"International Journal of Engineering, Mathematical and Physical Sciences:52319", author = "Lina Yan and Shiheng Wang and Ke Wang", title = "Preconditioned Jacobi Method for Fuzzy Linear Systems", abstract = "A preconditioned Jacobi (PJ) method is provided for solving fuzzy linear systems whose coefficient matrices are crisp Mmatrices and the right-hand side columns are arbitrary fuzzy number vectors. The iterative algorithm is given for the preconditioned Jacobi method. The convergence is analyzed with convergence theorems. Numerical examples are given to illustrate the procedure and show the effectiveness and efficiency of the method.
", keywords = "preconditioning, M-matrix, Jacobi method, fuzzy linear
system (FLS).", volume = "7", number = "2", pages = "216-4", }
