Chebyshev Semi-iterative Method to Solve Fully Fuzzy linear Systems

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1 ISSN , England, UK Journal of Information and Computing Science Vol 9, No, 204, pp Chebyshev Semi-iterative Method to Solve Fully Fuy linear Systems E bdolmaleki and S Edalatpanah Department of pplied Mathematics, Tonekabon Branch, Islamic ad University, Tonekabon, Iran (Received may 27, 202, accepted October 29, 203) bstract In this paper, semi-iterative method is applied to find solution of the fully fuy linear systems The convergence of this method is discussed in details Furthermore, we show that in some situations that the existing methods such as Jacobi, Gauss-Seidel, JOR, SOR and are divergent, our proposed method is applicable Finally, numerical computations are presented based on a particular linear system, which clearly show the reliability and efficiency of our algorithms Keywords: iterative methods, semi-iterative methods, Chebyshev, fuy numbers, fuy arithmetic, fuy linear equations Introduction Let us consider the following linear systems x=b, () n n n where R, b, x R These method often occur in a wide variety of area including numerical differential equation, eigenvalue problems, economics models, design and computer analysis of circuits, power system networks, chemical engineering processes, physical and biological sciences ; see [-2] and the references therein However, when the estimation of the system coefficients is imprecise and only some vague knowledge about the actual values of the parameters is available, it may be convenient to represent some or all of them with fuy numbers [3] Fuy data is being used as a natural way to describe uncertain data Fuy concept was introduced by Zadeh [3-4] We refer the reader to [5] for more information on fuy numbers and fuy arithmetic Fuy systems are used to study a variety of problems including fuy metric spaces [6], fuy differential equations [7], particle physics [8-9], Game theory [20], optimiation [2] and fuy linear systems[22-25] Fuy number arithmetic is widely applied and useful in computation of linear system whose parameters are all or partially represented by fuy numbers Dubois and Prade [26-27] investigated two definitions of a system of fuy linear equations, consisting of system of tolerance constraints and system of approximate equalities The simplest method for finding a solution for this system is creating scenarios for the fuy system, which is a realiation of fuy systems Based on these actual scenarios, Buckley and Qu [28] extended several methods for this category and proved their approaches are not practicable, because infinite number of scenarios can be driven for a fully fuy linear system (FFLS) Friedman et al [22] introduced a general model for solving a fuy n n linear system whose coefficient matrix is crisp and the right-hand side column is an arbitrary fuy number vector They used the parametric form of fuy numbers and replaced the original fuy n n linear system by a crisp 2n 2n linear system and studied duality in fuy linear systems X = BX+Y where, B are real n n matrix, the unknown vector X is vector consisting of n fuy numbers and the constant Y is vector consisting of n fuy numbers, in [29] There are many other numerical methods for solving fuy linear systems such as Jacobi, Gauss-Seidel, domiam decomposition method and SOR iterative method [30-35] In addition, another important kind of fuy linear systems are the fully fuy linear systems (FFLS) in which all the parameters are fuy numbers Dehghan and Hashemi [36-37] proposed the domian decomposition method, and other iterative methods to find the positive fuy vector solution of n n fully fuy linear system Dehghan et al [38] proposed some computational methods Published by World cademic Press, World cademic Union

2 68 E bdolmaleki etal : Chebyshev Semi-iterative Method to Solve Fully Fuy linear Systems such as Cramer s rule, Gauss elimination method, LU decomposition method and linear programming approach for finding the approximated solution of FFLS Nasseri et al [39] used a certain decomposition methods of the coefficient matrix for solving fully fuy linear system of equations Kumar et al in [40] obtained exact solution of fully fuy linear system by solving a linear programming In this paper, we propose Semi-iterative method for solving fully fuy linear systems This paper is organied as follows: In Section 2 some basic definitions and arithmetic are reviewed In Section 3 a new method is proposed for solving FFLS and we respectively give the semi-iterative method and some convenient iterative methods In section 4 numerical results are considered to show the efficiency of the proposed method Section 5 ends this paper with a conclusion 2 Some Basic Definition and rithmetic Operations In this section, an appropriate brief introduction to preliminary topics such as fuy numbers and fuy calculus will be introduced and the definition for FFLS will be provided For details, we refer to [26, 37] Definition 2 Let X denote a universal set Then a fuy subset ~ of X is defined by its membership function μ ~ : X [0,] ; which assigns a real number μ ~ ( x) in the interval [0,], to each element x X, where the value of μ ~ ( x) at x shows the grade of membership of x in ~ fuy subset ~ can be characteried as a set of ordered pairs of element x and grade μ ~ ( x) and is ~ often written = {( x, μ ~ ( x)); x X} The class of fuy sets on X is denoted with Γ (X ) Definition 22 fuy set with the following membership function is named a triangular fuy number and in this paper we will use these fuy numbers m x, m α x m, α > 0, α x m μ ( x) =, m x m+, > 0, % β β β 0, else Definition 23 fuy number % is said to be positive (negative) by % > 0( % < 0) if its membership function μ ( x ) % satisfies μ ( x ) = 0, x 0( x 0) % Using its mean value and left and right spreads, and shape functions, such a fuy number % is symbolically written % = ( m, αβ, ) Obviously, % is positive, if and only if m α 0 Definition 24 Two fuy numbers % = ( m, αβ, ) and B % = ( n, γδ, ) are said to be equal, if and only if m = n, α = γ and β = δ Definition 25 Let % = ( m, αβ, ), B % = ( n, γδ, ) be two triangular fuy numbers then; (i) % B% = ( m, α, β) ( n, γ, δ) = ( m+ n, α + γβ, + δ), (ii) ) % = ( m, αβ, ) = ( m, βα, ), (iii)if %, B % be a positive fuy number then: ( m, αβ, ) ( n, γδ, ) ( mn, nα + mγ, nβ + mδ ), (iv) For scalar multiplication we have; JIC for contribution: editor@jicorguk

3 Journal of Information and Computing Science, Vol 9 (204) No pp ( λm, λα, λβ), λ 0, λ ( m, αβ, ) = ( λm, λβ, λα), λ < 0 Definition 26 matrix % = ( a% ij ) is called a fuy matrix, if each element of % is a fuy number fuy matrix % will be positive and denoted by % > 0, if each element of % be positive We may represent n n fuy matrix % = ( a % ij ) n n, such that a% (,, ) ij = aij αij βij, with the new notation % = ( M,, N), where =(a ij ), M =(α ij ) and N =(β ij ) are three n n crisp matrices 3 Chebyshev Semi-iterative Method for FFLS Consider Fully fuy linear system (FFLS) % x% = b % In this paper we are going to obtain a positive solution of FFLS, where, % = (, M, N) > 0, % b% = ( b, g, h) > 0 % and x% = ( x, y, ) > 0 % So we have; (, M, N) ( x, y, ) = ( b, g, h) (2) Then by Definition 25 we have; ( x, y+ Mx, + Nx) = ( bgh,, ) (3) nd by Definition 24, concludes that; x = b, y + Mx = g, + Nx = h Then, 0 0 x b M 0 y g = N 0 h { { So, by assuming that be a nonsingular matrix we have; Λ X Ξ (4) x = b, y = ( g Mx), (5) = ( h Nx) Dehghan et al [36] applied some iterative techniques such as Richardson, Jacobi, Jacobi overrelaxation (JOR), Gauss Seidel, successive overrelaxation (SOR), accelerated overrelaxation (OR), symmetric and unsymmetric SOR (SSOR and USSOR) and extrapolated modify ed itken (EM) for solving FFLS First, we review their work Consider Eq (4) and let =Q- P be a proper splitting of crisp matrix and Q, called the splitting matrix, be a nonsingular crisp matrix Thus, the iterative method for FFLS is as follows; JIC for subscription: publishing@wuorguk

4 70 E bdolmaleki etal : Chebyshev Semi-iterative Method to Solve Fully Fuy linear Systems ( k+ ) ( k) x x = +,( 0) (6) ( k+ ) ( k) y Ty ξ k ( k+ ) ( k) T is called the iteration matrix and ξ is a vector and; Q P 0 0 Q b = 0, = Q N 0 Q P Q h T Q M Q P ξ Q g (7) Therefore by choose special parameters in Q we can obtain the popular iterative method For example, if =D-L-U, where D is diagonal, L is lower triangular and U is upper triangular part of, then we have; ) Jacobi method for Q=D 2) JOR(Jacobi Overrelaxation) method for Q = D,( w R) w 3) Gauss-Seidel method for Q=D-L 4) SOR method for Q = ( D L),( w R) w For details, we refer to [36] Next, we apply another acceleration method called Chebyshev semi-iterative method for FFLS Based on above demonstration, semi-iterative method is as follows; ( k+ ) ( k) ( k ) ( k ) x Q P 0 0 x x x ( k+ ) ( k) ( k ) ( k ) y = ωk + ( Q M Q P 0 y + ξ y ) + y, (8) ( k+ ) ( k) ( k ) ( k ) Q N 0 Q P where, 2 Ck ( ) ρ( T ) ωk + =, ρ( TC ) K + ( ) ρ ( T ) ( k ) () (0) x x x ( k) 3 n () (0) y R, y = Ty + ξ, ( k ) () (0) and, Ck( x) = 2 xck ( x) Ck 2( x), C0( x) =, C( x) = x, are the Chebyshev polynomials of the first kind and also ρ( T ) is called spectral radius of T; see [- 2, 4] For example, by Eq (8), Chebyshev-SOR semi-iterative method is as follows; ( k+ ) ( k) ( k ) x = ωk+ ( κx + w( D wl) b) + ( ωk+ ) x, ( k+ ) ( k) ( k) ( k ) y = ωk + ( κy wd ( wl) Mx + wd ( wl) g) + ( ωk + ) y, ( k+ ) ( k) ( k) ( k ) = ωk + ( κy w( D wl) Nx + w( D wl) h) + ( ωk + ) y (9) JIC for contribution: editor@jicorguk

5 Journal of Information and Computing Science, Vol 9 (204) No pp Where, κ = ( D wl) [( w) D+ wu] However, since the spectral radius of T is not known in advance, ρ ( T ) is usually replaced by the lower and upper bounds (see [4]), that is; ( k+ ) ( k) ( k) ( k ) ( k ) x m x x x y = ω ( γ n + y y ) + y, (0) ( k+ ) ( k) ( k) ( k ) ( k ) k+ ( k+ ) ( k) ( k) ( k ) ( k ) o where, ω i+ Ci ( υ) = 2 υ, C ( υ) i+ ( k) ( k) m Q b x ( k) ( k) 2 2 ( β + α) n = Q g Λ y, γ =, υ =, 2 ( + ) ( ) ( k) ( k) o Q h β α β α α λ β, β > α, λ eigenvalues( T ) Furthermore, after some calculation, from Eq (0), we have; see [2, 42]: ( k+ ) ( k) ( k ) x x Q b x ( k+ ) ρ k ( k) ( k ) + y = T α + β I y + Q g + ρki+ y 2 ( α + β ) ( k+ ) ( k) ( k ) {[2 ( ) ] 2 } ( ), () Q h where, ρ =, ρ2 = 2 υ / (2υ ), for n 2; ρk+ = ( ρk / 4 υ ) Theorem 3 Chebyshev semi-iterative method (8) for solving fully fuy linear system % x% = b %, converges if and only if its classical version converges for solving the crisp linear system x = b derived from the corresponding FFLS Proof By above demonstrations and based on Eq (7) it is easy to see that spectrum of T is equal to spectrum of Q P Therefore, the proof is complete Theorem 32 Let P ( T ) = [2 T ( α + β ) I ]/[2 ( α + β )] Then Chebyshev semi-iterative method converges, if ρ ( P ( T)) < Proof Using Theorem 3 of this paper and Theorem 4[42], the result is trivial 4 Numerical Experiments In this section, we give some numerical experiments to illustrate the results obtained in previous sections ll the numerical experiments presented in this section were computed in double precision using a MTLB 7 on a PC with a 86GH 32-bit processor and GB memory JIC for subscription: publishing@wuorguk

6 72 E bdolmaleki etal : Chebyshev Semi-iterative Method to Solve Fully Fuy linear Systems Example 4 Consider the Consider the following FFLS: = tridiag(, 4,), % = (, M, N ); M = tridiag(0,,0), N = tridiag(02,,0) nd, i i b% = ( b, g, h); b = i i, g = i, hi n = n + 6 The following table shows the numerical results of above example with the tolerance ε = 0 and the initial approximation ero vector In the Table, we reported the number of iterations (Iter) and Elapsed time (ELP) for the SOR iterative method and Chebyshev-SOR semi-iterative method with different n and w= Table Shows the results of example 4 for SOR and Chebyshev-SOR methods Method SOR method Chebyshev-SOR method n Iter ELP Iter ELP Example 42 Consider the following FFLS: (,02,02) (,04,03) (2,03,04) (4,02,0) ( x, y, ) (69,5,4) (4,03,0) (3,04,02) (2,02,03) (,0,03) ( x2, y2, 2) (5,3,4) = (,03,02) (,05,02) (3,03,0) (,02,03) ( x3, y3, 3) (49, 4,32) (2,04,05) (4,05,02) (2,06,2) (3,03,03) ( x4, y4, 4) (7,5,6) If we use iterative methods [36] for this problem we can see that all of the Jacobi, Gauss-Sidel and JOR, SOR methods are divergent (since ρ ( Q P) > ) However, by using the initial values x = y = = s = (0,0,0,0) t and stopping criterion tol 0 Chebyshev-Gauss-Seidel semi-iterative method(q = D L) converges in 29 iterations to the following 6, the solution; ( x, y, ) (096,00072,0448) ( x3, y3, 3) (69,08533,05744) ( x4, y4, 4) (0390,06348,04386) ( x2, y2, 2) (0343,00393,0368) = 5 Conclusion JIC for contribution: editor@jicorguk

7 Journal of Information and Computing Science, Vol 9 (204) No pp In this paper, the fully fuy linear systems, ie, fuy linear systems with fuy coefficients involving fuy variables are investigated and semi-iterative method is applied for solving these systems The proposed method is easy to understand and apply in real life situations Furthermore, we show that our algorithm compare with some other algorithms works better Finally, from theoretical speaking and numerical examples, it may be concluded that this method is efficient and convenient 6 cknowledgements The authors would like to thank Tonekabon Branch of Islamic ad University of Iran for the financial support of this research, which is based on a research project contract 7 References [] RS Varga, Matrix iterative analysis, second ed, Springer, Berlin, 2000 [2] D M Young, Iterative solution of large linear systems, cademic Press, New York,97 [3] M M Martins, M E Trigo and D J Evans, n iterative method for positive real systems, Int J Comput Math, 84 (2007) [4] L-T Zhang, T-Z Huang, S-H Cheng,Y-PWang, Convergence of a generalied MSSOR method for augmented systems, J Comput ppl Math, 236 () (202) [5] H Saberi Najafi and S Edalatpanah, Fast iterative method-fim application to the convection- diffusion equation, J Inform Comp Sci, 6 (20), [6] H Saberi Najafi and S Edalatpanah, On the convergence regions of generalied OR methods for linear complementarity problems, J Optim Theory ppl, 56(203), [7] H Saberi Najafi and S Edalatpanah, new modified SSOR ieration method for solving augmented linear systems, Int J Comput Math, (203), doi: 0080/ [8] H Saberi Najafi, S Edalatpanah, Two stage mixed-type splitting iterative methods with applications, J Taibah Univ Sci, 7(203), [9] H Saberi Najafi, S Edalatpanah, Preconditioning Strategy to Solve Fuy Linear Systems (FLS), International Review of Fuy Mathematics, 7(202), [0] H Saberi Najafi, S Edalatpanah, Comparison analysis for improving preconditioned SOR-type iterative Method, Numerical nalysis and pplications, 6(203) [] H Saberi Najafi, S Edalatpanah, Iterative methods with analytical preconditioning technique to linear complementarity problems: application to obstacle problems RIRO - Operations Research, 47(203), 59-7 [2] H Saberi Najafi, S Edalatpanad, On application of Liao s method for system of linear equations, in Shams Eng J, 4 (203) [3] L Zadeh, Fuy sets Information and Control, 8(965) [4] L Zadeh, fuy-set-theoretic interpretation of linguistic hedges, Journal of Cybernetics, 2 (972) 4-34 [5] Kaufmann and MM Gupta, Introduction Fuy rithmetic, Van Nostrand Reinhold, New York, 985 [6] JH Park, Intuitionistic fuy metric spaces, Chaos Solitons & Fractals, 22 (2004) [7] M Freidman, Ma Ming and a Kandel, Numerical solutions of fuy differential and integral equations, Fuy Sets ans Systems 06 (999) [8] MS Elnaschie, review of E-infinity theory and the mass spectrum of high energy particle physics, Chaos, Solitons & Fractals, 9 (2004) [9] MS Elnaschie, The concepts of E infinity: n elementary introduction to thecantorian fractal theory of quantum physics, Chaos, Solitons & Fractals, 22 (2004)495-5 [20] H Saberi Najafi, S Edalatpanad,On the Nash equilibrium solution of fuy bimatrix games, International Journal of Fuy Systems and Rough Systems,5 (2) (202) [2] H Saberi Najafi, S Edalatpanad, Note on new method for solving fully fuy linear programming problems, pplied Mathematical Modelling, 37 (203), [22] M Friedman, Ma Ming, Kandel, Fuy linear systems, Fuy Sets and Systems, 96 (998) [23] T llahviranloo, Successive over relaxation iterative method for fuy system of linear equations, ppl Math Comput 62 (2005) [24] T llahviranloo, The domian decomposition method for fuy system of linear equations, ppl Math Comput JIC for subscription: publishing@wuorguk

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