Preconditioning Strategy to Solve Fuzzy Linear Systems (FLS)

Transcription

1 From the SelectedWorks of SA Edalatpanah March Preconditioning Strategy to Solve Fuzzy Linear Systems (FLS) SA Edalatpanah University of Guilan Available at:

2 IRFM: Vol. 7 No. 2 July-December 2012 pp Serials Publications ISSN: Preconditioning Strategy to Solve Fuzzy Linear Systems (FLS) H. Saberi Najafi 1 S.A.Edalatpanah 1.2 * 1 Department of Mathematics Lahijan Branch Islamic Azad University Lahijan Iran 12 Young Researchers Club Lahijan Branch Islamic Azad University Lahijan Iran Abstract. In this article the preconditioning methods are used for fuzzy linear systems and especially some new preconditioners are introduced. Moreover the preconditioned iterative methods are studied from the point of view of rate of convergence and the convergence properties of the proposed methods have been analyzed and compared with the classical methods. Finally the methods are tested by numerical example that shows a good improvement on the convergence speed. Keywords: Fuzzy linear system (FLS)iterative methods preconditioning methods H-matrix Spectral radius 1. Introduction Science history shows the substantial improvements and huge jumps in science and technology require interaction between mathematicians and other different scientists. Meanwhile solving the linear equation systems plays the role of a catalyst for further connection of this interaction between mathematics and other sciences. There are many methods to solve usually linear systems [1-8]. Nevertheless when coefficients of a system are ambiguous and there is some inexplicit information about the exact amount of parameters how can the linear equation system be solved? To solve this problem first attempt researchers made was through accidental events and caring to probabilities. However yet scientists believe that the only way to improve efficiency is to increase accuracy. Thus fuzzy logic was proposed by Zadeh in1965 [9]. Professor Zadeh discussed fuzzy set and fuzzy number concept and arithmetical operation in [10-12] and following it many articles and books have since appeared that deal with various types of fuzzy models. Especially solving the fuzzy linear system was studied in several literatures. For example Friedman et al[13-14] proposed a generic model to solve a fuzzy linear system in which coefficient matrix is crisp and the right-hand side is the arbitrary fuzzy number vector. They used the embedding technique and replaced the n n fuzzy linear system by 2n 2n crisp linear system and studied uniqueness of the fuzzy solution. However this method had some limitations its designers were aware of it. In [15] we introduced some of the present challenges in this model and explained some appropriate conditions to for the avoiding of it. * Corresponding author. addresses: saedalatpanah@gmail.com saedalat@yahoo.com (S.A.Edalatpanah).

3 66 H. Saberi Najafi & S.A. Edalatpanah There are some iterative methods based on Friedman et al s model which were explained in several studies; see [15-21] and references therein. However in contrast to real attractiveness of iterative methods lack of strength in these methods over direct methods and slow rate of convergence where unexpected will make their real applications doubtful. The key to this problem is preconditioning techniques. Hence we use the preconditioning techniques in fuzzy logic and especially for solving the fuzzy linear systems. A preconditioner is an auxiliary approximate solver which will be combined with an iterative method. According to critical importance of spectral radius in preconditioning we want to find a more desired spectral radius. In other words for the iterative solution of a linear system Ax=b to accelerate the convergence of iterative solvers preconditioning transforms the system to PAx=Pb where P is a linear operator called the preconditioner ; see[22-26] and references therein. The structure of this article is organized as follows: In section2 we recall preliminaries for an n n fuzzy linear systems and general model for solving the system and some definitions of iterative methods. Some new preconditioners of (I+S)-type for solving iterative methods in FLS with various point of views are presented in section3. Finally in section. 4 numerical examples are given followed by concluding theorems in previous sections. 2.Background An arbitrary fuzzy number is represented in parametric form by an ordered pair of functions ( u ( u( ) 0 r 1 which satisfy the following requirements(see [27-28]): (i) u ( is a bounded monotonic increasing left continuous function over[01]; (ii) u( is a bounded monotonic decreasing left continuous function over[01]; (iii) u ( u( 0 r 1. A crisp number α can be simply expressed as u( = u( = α 0 r 1. The addition and scalar multiplication of fuzzy numbers previously defined by using Zadeh s extension principle can be described as follows: (i) x = y if and only if x ( = y( and x ( = y( (ii)x+y=( x ( + y( x( + y( ) ( K x K x) (iii) Kx = ( K x K x) K 0 K < 0 K R Definition 2.1: Consider the n n linear system of equations: a11x1 + a12x2 a1 nxn = b1 a21x1 + a22x2 a2nxn = b2 an1x1 + an2x2 annxn = bn (2.1) Where the coefficient matrix A=( a ij ) 1 ij n is a crisp matrix and linear system(fls). b i E 1 ; 1 i n is called a fuzzy

4 Preconditioning Strategy to Solve Fuzzy Linear Systems (FLS) 79 x 5 x x = x x = ( r = ( r = ( r = ( r Also in SOR with parameter (w=.9) after 8 iterations and in AOR with taking (w=1 r=.9) after 11 iterations we obtain this solution. 5. Conclusion In this paper we have proposed preconditioning methods based on the iterative methods for solving Fuzzy linear systems (FLS). We have studied how the iterative methods are affected if the Fuzzy linear system is preconditioned by our model. Finally from theoretical speaking and numerical example it may be concluded that the convergence rate of our proposed method are superior to the basic iterative methods References [1] A. Berman and R.J. Plemmons Nonnegative Matrices in the Mathematical Sciences SIAM Press Philadelphia [2] A. Frommer and D. B. Szyld H-splitting and two-stage iterative methods Numer. Math. 63(1992) [3] A. Gunawardena S. Jain and L. Snyder Modified iterative methods for consistent linear systems Linear Algebra Appl. 154/156 (1991) [4] A. Hadjidimos Accelerated Overrelaxation method Math. Comput. 32 (1978) [5] H.Saberi Najafi and S.A. Edalatpanah Fast Iterative Method-FIM. Application to the Convection- Diffusion Equation J.Info. Comp.Sci 6(2011) [6] H.Saberi Najafi and S.A. Edalatpanah On the Convergence Regions of Generalized AOR Methods for Linear Complementarity Problems. J. Optim. Theory. Appl doi: /s [7] H.Saberi Najafi S.Kordrostami and S.A. Edalatpanah The Convergence Analysis Of Preconditioned AOR Method For M- H-Matrices Journal of Applied Mathematics IAUL 5(2008) [8] H.Saberi Najafi and S.A. Edalatpanah Comparison Analysis for Improving Preconditioned SOR-type Iterative Method Numerical Analysis and Applications accepted. [9] L.A. Zadeh Fuzzy sets Information and Control 8 (1965) [10] L.A. Zadeh The concept of a linguistic variable and its application to approximation reasoning Information Sciences 3 (1975) [11] L.A. Zadeh Fuzzy sets as a basis for a possibility theory. Fuzzy Sets Syst. 1(1978) 3 28 [12] S.S.L. Chang and L.A. Zadeh On fuzzy mapping and conterol IEEE Trans. Syst. Man Cybern. 2 (1972) pp [13] M. Friedman and Ma. Ming and A. Kandel Fuzzy linear systems Fuzzy Sets and Systems 96 (1998)

5 80 H. Saberi Najafi & S.A. Edalatpanah [14] A. Kandel M. Friedamn and M. Ming Fuzzy linear systems and their solution IEEE (1996) [15] H. Saberi Najafi S.A. Edalatpanah and B. Parsamoghaddam H-matrices in fuzzy linear systems submitted. [16] B. Asady S. Abbasbandy and M. Alavi Fuzzy general linear systems Appl. Math. Comput. 169 (2005) pp [17] M. Dehghan and B. Hashemi Iterative solution of fuzzy linear systems Appl. Math. Comput. 175 (2006) [18] S. Abbasbandy A. Jafarian and R. Ezzati Conjugate gradient method for fuzzy symmetric positive definite system of linear equations Appl. Math. Comput.171 (2005) [19] S.Miao B. Zheng and K.Wang Block SOR methods for fuzzy linear systems J Appl Math Comput 26(2008) [20] T. Allahviranloo M. Ghanbari A.A. Hosseinzadeh E. Haghi and R. Nuraei A note on Fuzzy linear systems Fuzzy Sets and Systems 177 (2011) [21] H.Saberi Najafi and S.A.Edalatpanah An Improved Model for Iterative Algorithms in Fuzzy Linear Systems Computational Mathematics and Modeling accepted. [22] Evans D.J. Martins M.M. and Trigo M.E The AOR iterative method for new preconditioned linear systems Comput. Appl. Math.132(2001) [23] Wang Land Song Y. Preconditioned AOR iterative methods for M-matrices. Journal of Computational and Applied Mathematics 2009; 226: [24] H.Saberi Najafi and S.A.Edalatpanah Some Improvements In PMAOR Method For Solving Linear Systems Journal of Information and Computing Science 6(2011) [25] H.Saberi Najafi and S.A.Edalatpanah On the Two SAOR Iterative Formats for Solving Linear Complementarity Problems.IJITCS 5(2011) [26] H.Saberi Najafi and S.A.Edalatpanah New Model for Preconditioning Techniques with Application to the Boundary Value Problems International Journal of Computational Mathematics and Numerical Simulation accepted. [27] R. Goetschell and W. Voxman Elementary calculus Fuzzy Sets and Systems 18 (1986) [28] W. Cong-Xin and M. Ming Embeddingproblem of fuzzy number space: Part I Fuzzy Sets and Systems 44 (1991) [29] R.S. Varga Matrix Iterative Analysis second ed. Springer Berlin [30] A. Neumaier On the comparison of H-matrices with M-matrices. Linear Algebra Appl. 83(1986) [31] I.Marek D.B. Szyld Comparison theorems for weak splittings of bounded operators Numer. Math.58 (1990) This paper is copyrighted. Please visit to buy this article.