An Iterative Method for Solving Unsymmetric System of Fuzzy Linear Equations

Transcription

1 The SIJ Trasactios o Computer Sciece Egieerig & its Applicatios (CSEA) Vol. No. 5 November-December 03 A Iterative Method for Solvig Usymmetric System of Fuzzy Liear Equatios Majid Hasazadeh* & Hossei Zareamoghaddam** *Torbat-e-Jam Brach Islamic Azad Uiversity Torbat-e-Jam IRAN. hassazadeh.majid@gmail.com **Youg Researchers ad Elite Club Kashmar Brach Islamic Azad Uiversity Kashmar IRAN. zareamoghaddam@yahoo.com Abstract Recetly the solutio of Fuzzy Liear System of Equatios (FLSE) has bee studied by may authors i literature. I may cases some suitable methods of Crisp Liear System of Equatios (CLSE) are exteded to compute the solutio of FLSE. Here we suggest a ew iterative method for solvig FLSE based o the Hermitia ad Skew-Hermitia Splittig (HSS) method which HSS is a recetly proposed iterative method for CLSE where the coefficiet matrix is a crisp usymmetric matrix. I HSS method the coefficiet matrix is ito two Hermitia ad Skew-Hermitia matrices ad the iterative solutios are obtaied by solvig two well-coditioed systems of equatios istead of the origial problem sequetially. Our suggested method is coverged to the solutio for all kids of FLSE problems with triagular or rectagular fuzzy umbers. To examie this method umerical results of some problems are illustrated at the ed of this paper. Keywords FLSE; Fuzzy Liear System; HSS; Iterative Methods; Triagular Fuzzy Numbers. Abbreviatios Cojugate Gradiet (CG); Crisp Liear System of Equatios (CLSE); Fuzzy Liear System of Equatios (FLSE); Geeralized Miimal Residual (GMRES); Hermitia ad Skew-Hermitia Splittig (HSS). I. INTRODUCTION THERE are may applicatios for the solutio of liear system of equatios. Due to the importace of these solutios for various problems scietists have paid especial attetio to solve such problems by iterative methods as ease ad fast as possible. So it is importat to study more about variety of efficiet techiques for solvig liear system of equatios. There are may iterative methods such as GMRES differet CG variats etc. which have bee proposed for the solutio of such systems [Datta 995; Saad 996; Barrett et al. 000]. May authors have studied differet properties of curret methods i literature. For geeral iformatio about such methods refer to [Friedma et al. 998; Barrett et al. 000]. I may cases at least some of the system's parameters are represeted by fuzzy rather tha crisp umbers to fit more the problem with its ature that this problem is extracted from discritizatio of some PDEs or other applicable equatios. The solutio of such fuzzy liear systems is applicable i differet areas such as ecoomic egieerig physics etc. Therefore it is essetial to develop solvig methods which appropriately treat fuzzy liear systems for fidig suitable solutios. For the first time the cocept of fuzzy umbers ad associated arithmetic operatios was proposed by Zadeh. Next a geeral model for solvig a FLSE where the coefficiet matrix is crisp ad the right had side is a fuzzy vector was proposed by Friedma et al. (998). They exteded their work by cosiderig duality for fuzzy liear systems i Friedma et al. (000). Wag et al. (00) proposed some iterative methods for the solutio of such systems. After that may scietists have focused o FLSE ot oly because of the variety of its applicatios eve i other majors but also because it may be possible to exted some suggested methods used for computig the solutio of ordiary liear systems for fuzzy problems uder some especial circumstaces. So i literature there are may techiques such as Jacobi Gauss-Seidel SOR ad steepest decet as iterative methods [Wag et al. 00; Allahviraloo ; Abbasbady et al. 005; Dehgha & Hashemi 006; Abbasbady & Jafaria 006] which exteded from ordiary problems to fuzzy systems. Abbasbadi et al. (005) proposed a Cojugate Gradiet method for symmetric fuzzy liear systems Allahviraloo (005A) suggested a Adomia decompositio method to compute the solutio of such system. Some direct methods like LU decompositio i Abbasbady et al. (006) as well as block iterative methods [Wag & Zheg 007] ad lots of other approaches [Nasseri & Khorramizadeh 007; Allahviraloo & Ghabari 0; Sethilkumar & Rajedra 03; Allahviraloo et al. 03; Najafi & Edalatpaad 03; Najafi et al. 03] for solvig FLSE have bee proposed. The works of Saberi ad his colleagues [Najafi & Edalatpaad 03; Najafi et al. 03] ad Allahviraloo ad his colleagues [Allahviraloo & Ghabari 0; Allahviraloo et al. 03] are some recetly proposed methods for the solutio of such problems. ISSN: Published by The Stadard Iteratioal Jourals (The SIJ) 8

2 The SIJ Trasactios o Computer Sciece Egieerig & its Applicatios (CSEA) Vol. No. 5 November-December 03 Bai ad his colleagues (003; 005) developed some ew iterative methods for the solutio of CLSE. I these works they split the coefficiet matrix ito two Hermitia ad Skew-Hermitia matrices. The two differet liear systems are geerated. So the solutio of origial system is computed by solvig these two systems. I this work we exted this idea for the solutio of FLSE. Thus we deal with the problem Ax = y () where A R ad y is a fuzzy vector. More explaatio about this method is i sectio 3. This paper is orgaized as follows. I sectio we discuss some basic defiitios ad results o fuzzy umbers ad FLSE. The ew proposed method for solvig such fuzzy systems is discussed i sectio 3. Numerical tests ad coclusio are draw i the ext sectios. II. BASIC CONCEPTS AND DEFINITIONS Here some primary defiitios ad otes which are required i this study have bee idicated from [Friedma et al. 998; 000; Allahviraloo 004; 005; 005A; Abbasbady et al. 005; 006]. Defiitio : The r-level set of a fuzzy set u is defied as a ordiary set u r of which the degree of membership fuctio exceeds the level r i.e. u u x R u x r r 0 () r r Defiitio : A fuzzy set u defied o the uiversal set of real umber R is said to be a fuzzy umber if its membership fuctio has the followig characteristics: u is covex i.e. u x x mi u x u x x x R 0 (3) u is ormal i.e. x 0 R such that μ u x 0 =. μ u is piecewise cotiuous. Defiitio 3: A fuzzy umber u i parametric form is a pair u u of fuctios u r u r 0 r that satisfies the followig requiremet: u r is a bouded mootoically icreasig left cotiuous fuctio; u r is a bouded mootoically decreasig left cotiuous fuctio; u r u r 0 r. Defiitio 4: The additio ad scalar multiplicatio of fuzzy umbers are defied by the extesio priciple ad ca be equivaletly represeted as follows see Friedma et al. (998). For arbitrary u = u u v = v v ad k R the additio ad the scalar multiplicatio are defied as follows: u = v iff u r = v r ad u r = v r u ± v r = u r ± v r u r ku ku r k 0 ku = ku ku r k < 0. Remark : A crisp umber α is simply represeted by u r = u r = α 0 r. Defiitio 5: The triagular fuzzy umber u = u u u 3 is a fuzzy set where the membership fuctio is as x u u x u u u u3 x u x u x u3 (4) u3 u 0 Otherwise; ad its parametric form is u u r u r u r u u r u (5) u r u u u r. 3 3 Defiitio 6: A triagular fuzzy umber u is said to be o-egative fuzzy umber if ad oly if u x = 0 x < 0. For solvig a fuzzy liear system Ax = b (6) with a crisp square matrix A ad a triagular fuzzy vector b differet iterative methods have bee proposed. The ith row of fuzzy liear system with the solutio x = x x T x i = x i r x i r i = is as a x a x b i j j i j j i j j a x a x b i... i j j i j j i j j From the above it is extracted that a x a x b () r i i i a x a x b () r i i i Based o the idea of Friedma et al. the system (6) is coverted ito a crisp fuctio liear system EX = Y (7) where matrix S = S ij is obtaied as follows: aij 0 eij aij ei j aij (8) aij 0 ei j aij ei j aij ad each e ij which is ot determied by (8) is zero Y = b b b b T ad X = x x x x T. The matrix E is determied as a symmetric block matrix E = F G G F where f ij = e ij ad g ij = e i+j. A approximate solutio of (7) that is ofte used is the least square solutio of (7) defied as a vector X which miimizes the Euclidea orm of Y EX. Defiitio 7: Let x = x j r x j r j deote the solutio of EX = Y. The triagular fuzzy vector U = u j r u j r j defied by u j r = mi x j r x j r x j ad u j r = max x j r x j r x j is called the fuzzy solutio of EX = Y. If j: u j = x j ad u j = x j U is called a strog fuzzy solutio. ISSN: Published by The Stadard Iteratioal Jourals (The SIJ) 8

3 The SIJ Trasactios o Computer Sciece Egieerig & its Applicatios (CSEA) Vol. No. 5 November-December 03 III. SOLVING FUZZY LINEAR SYSTEMS There are various iterative methods for solvig differet crisp liear system of equatios [Datta 995; Saad 996; Barrett et al. 000]. I may papers to solve the crisp system Bu f (9) where B R ad u f R the coefficiet matrix B has bee split as B = M N ad the origial problem is replaced by iterative method k k u M Nu M f for k 0. (0) It proved that (0) is coverges if M N < where is the iduced Euclidea orm for the square matrices (see [Datta 995; Saad 996]). Recetly Bai et al. (003; 005) suggested some ew iterative methods based o similar splittig patter. They supposed to split the coefficiet matrix B ito two differet matrices H ad S so that B = H + S where H = B + B T ad S = B B T with B T as the traspose of B. I fact B ad S are Hermitia ad Skew-Hermitia matrices respectively. I this way the origial problem Bu = f is trasformed ito two systems of equatios i. I H u I S u b ii. I S u I H u b where α is a give positive costat. Suppose u (0) is the iitial guess. To compute the solutio of origial system at first a approximatio of the first equatio is obtaied by a appropriate iterative method. The the computed solutio of first equatio is cosidered as the iitial vector of the secod equatio. By applyig the iterative method o the secod system the ext approximatio of origial problem is resulted. This procedure is cotiued util the computed approximatio satisfied. So the iterative sequece u (k) for iteger k 0 is computed by ( k ) ( k ) ii. I S u I H u b util u (k) coverges. If we set i. I H u I S u b M I H I S N I H I S () the the matrix splittig B = M N as the key poit of the above iterative algorithm yields so that we have Mu = Nu + b. Now we use this techique to fid the solutio of FLSE (). For this aim it is more coveiet to apply HSS algorithm o the parametric crisp liear system (7). So if X (0) is the iitial vector for the solutio of EX = Y the iterative sequece X (k) is obtaied by ( k ) i. I H X I S X Y ( k ) ii. I S X I H X Y where H ad S are Hermitia ad Skew-Hermitia matrices of E respectively. This procedure is cotiued util X (k) coverges. To kow more about HSS algorithm the followig two theorems are helpful. Theorem : [Bai et al. 003] Let E R ad E = M i N i (i = ) be two matrix splittig of matrix E ad X (0) be a give iitial vector. Suppose X (k) k 0 is a twostep iteratio sequece defied by ( k ) i. MX NX b ( k ) ii. M X NX b the ( k X M N M N X ) M I N M b k 0. Moreover if the spectral radius ρ M N M N is less tha the the iterative sequece X (k) coverges to the uique solutio of (7). Theorem : [Bai et al. 003] Let E R be a positive defiite matrix ad H = E E T ad S = E E T ad α be a give positive costat. The the iteratio matrix M(α) of the HSS iteratio is give by M( ) I S I H I H I S ad its spectral radius ρ(m(α)) is bouded by max ( H ) i i where λ H is the spectral set of the matrix H. Therefore it holds that M( ) ( ) 0 ; ad hece HSS iteratio coverges to the uique solutio of (6). IV. NUMERICAL EXAMPLES To examie the covergece of our suggested method for solvig FLSE the followig fuzzy systems are solved by this method. Here we suppose X 0 = 0 is the iitial guess ad α = 0 is the costat factor for both problems. Example : Cosider the fuzzy liear system x x r r x 3x 4 r7 r. From this fuzzy system we have 0 0 r E 4 r Y 0 0 r r 7 i ISSN: Published by The Stadard Iteratioal Jourals (The SIJ) 83

4 The SIJ Trasactios o Computer Sciece Egieerig & its Applicatios (CSEA) Vol. No. 5 November-December 03 By applyig the above method o this system the exact solutio X r 0.875r r.375 T by eps = e 6 is estimated after 5 iteratios. For this example we have T H E E ad T S E E I this example we fid the solutio of a FLSE by solvig two parametric problems of order 4. The ext example is to show the performace of this algorithm o a fuzzy liear system with 6 fuzzy variables. Example : Now the followig 6 6 fuzzy liear system is cosidered x x r3 r x x r4 r r68 r x6 x Here the coefficiet matrix E ad right had side vector Y are geerated based o (7). We apply HSS algorithm to solve this problem. This method fids the solutio of this system after 34 iteratios for eps = e 6 while the exact solutio is 0.434r r x x r r x x 0.493r r x r r 0.983r r x6 x6 0.96r r V. CONCLUSION To fid the solutio of a FLSE we used HSS algorithm o the exteded parametric system of equatios extracted from origial system. By this approach the origial parametric system is replaced by two well-coditioed systems ad the iterative approximatios are obtaied i two steps. From the first equatio a vector is computed ad this vector is cosidered as the iitial vector of the secod system. I fact the approximatios of this method are the vectors computed by the secod equatio. As the umerical tests show our suggested method is practical for solvig fuzzy liear system of equatios with a crisp square coefficiets matrix ad a fuzzy right had side vector. So we recommed this method for solvig such problems. ACKNOWLEDGMENT Authors thak Islamic Azad Uiversity (Torbat-e-Jam Brach) for fiacial support. REFERENCES [] B.N. Datta (995) Numerical Liear Algebra ad Applicatios ITP Press New York. [] Y. Saad (996) Iterative Methods for Sparse Liear Systems PWS Press New York. [3] M. Friedma M. Mig & A. Kadel (998) Fuzzy Liear Systems Fuzzy Sets ad Systems Vol. 96 Pp [4] M. Friedma M. Mig & A. Kadel (000) Duality i Fuzzy Liear Systems Fuzzy Sets ad Systems Vol. 09 Pp [5] R. Barrett Michael Berry Toy F. Cha James Demmel Jue M. Doato Jack Dogarra Victor Eijkhout Rolda Pozo Charles Romie & Hek Va der Vorst (000) Templates for the Solutio of Liear Systems: Buildig Blocks for Iterative Methods Society for Idustrial ad Applied Mathematics. [6] X.Z. Wag Z.M. Zhog & M.H. Ha (00) Iteratio Algorithms for Solvig a System of Fuzzy Liear Equatios Fuzzy Sets ad Systems Vol. 9 Pp. 8. [7] Z.Z. Bai G.H. Golub & M.K. Ng (003) Hermitia ad Skew-Hermitia Splittig Methods for No-Hermitia Positive Defiite Liear Systems SIAM Joural o Matrix Aalysis ad Applicatios Vol. 4 No. 3 Pp [8] T. Allahviraloo (004) Numerical Methods for Fuzzy System of Liear Equatios Applied Mathematics ad Computatio Vol. 55 Pp [9] Z.Z. Bai G.H. Golub L.Z. Lu & J.F. Yi (005) Block Triagular ad Skew-Hermitia Splittig Methods for Positive- Defiite Liear Systems SIAM Joural o Scietific Computig Vol. 6 No. 3 Pp [0] S. Abbasbady A. Jafaria & R. Ezzati (005) Cojugate Gradiet Method for Fuzzy Symmetric Positive-Defiite System of Liear Equatios Applied Mathematics ad Computatio Vol. 7 Pp [] T. Allahviraloo (005) Successive over Relaxatio Iterative Method for Fuzzy System of Liear Equatios Applied Mathematics ad Computatio Vol. 6 Pp [] T. Allahviraloo (005A) The Adomia Decompositio Method for Fuzzy System of Liear Equatios Applied Mathematics ad Computatio Vol. 63 Pp [3] S. Abbasbady R. Ezzati & A. Jafaria (006) LU Decompositio Method for Solvig Fuzzy System of Liear Equatios Applied Mathematics ad Computatio Vol. 7 Pp [4] M. Dehgha & B. Hashemi (006) Iterative Solutio of Fuzzy Liear Systems Applied Mathematics ad Computatio Vol. 75 Pp [5] S. Abbasbady & A. Jafaria (006) Steepest Descet Method for System of Fuzzy Liear Equatios Applied Mathematics ad Computatio Vol. 75 Pp [6] S.H. Nasseri & M. Khorramizadeh (007) A New Method for Solvig Fuzzy Liear Systems Iteratioal Joural of Applied Mathematics Vol. 0 No. 4 Pp [7] K. Wag & B. Zheg (007) Block Iterative Methods for Fuzzy Liear Systems Joural of Applied Mathematics ad Computig Vol. 5 Pp [8] T. Allahviraloo & M. Ghabari (0) A New Approach to Obtai Algebraic Solutio of Iterval Liear Systems Soft Computig Vol. 6 Pp. 33. ISSN: Published by The Stadard Iteratioal Jourals (The SIJ) 84

5 The SIJ Trasactios o Computer Sciece Egieerig & its Applicatios (CSEA) Vol. No. 5 November-December 03 [9] P. Sethilkumar & G. Rajedra (03) A Algorithmic Approach to Solve Fuzzy Liear Systems Joural of Iformatio & Computatioal Sciece Vol. 8 No. 3 Pp [0] T. Allahviraloo F. Hosseizadeh Lotfi M. Khorasai & M. Khezerloo (03) O the Solutio of Fuzzy Liear System Applied Mathematical Modellig Vol. 37 Pp [] H.S. Najafi & S.A. Edalatpaad (03) O Applicatio of Liao s Method for System of Liear Equatios Ai Shams Egieerig Joural Vol. 4 Pp [] H.S. Najafi S.A. Edalatpaah & A.H. Refahi Sheikhai (03) Applicatio of Homotopy Perturbatio Method for Fuzzy Liear Systems ad Compariso with Adomia s Decompositio Method Chiese Joural of Mathematics Vol. 03 Article ID Pp. 7. ISSN: Published by The Stadard Iteratioal Jourals (The SIJ) 85