Generalized AOR Method for Solving System of Linear Equations. Davod Khojasteh Salkuyeh. Department of Mathematics, University of Mohaghegh Ardabili,

Transcription

1 Australian Journal of Basic and Applied Sciences, 5(3): , 20 ISSN Generalized AOR Method for Solving Syste of Linear Equations Davod Khojasteh Salkuyeh Departent of Matheatics, University of Mohaghegh Ardabili, P.O.Box. 79, Ardabil, Iran Abstract: he accelerated overrelaxation (AOR) iterative ethod is a stationary iterative ethod for solving linear syste of equations. In this paper, a generalization of the AOR iterative ethod is presented and its convergence properties are studied. Soe nuerical experients are given to show the efficiency of the proposed ethod. AMS Matheatics Subject Classification: 65F0. Key words: AOR, generalized AOR, M-atrix, regular splitting, convergence. Consider the linear syste of equations INRODUCION Ax = b, () where the atrix nn and n. Let A be a nonsingular atrix with nonzero diagonal A xb, entries and A = D-E-F, where D is the diagonal of A, -E its strict lower part, and -F its strict upper part. hen the Jacobi and the Gauss-Seidel ethods for solving Eq. () are defined as ( k) ( k) x D ( EF) x D b, ( k) ( k) x ( DE) Fx ( DE) b, respectively. In the accelerated overrelaxation (AOR) iterative ethod (Hadjidios, A., 978), syste () is Ax b 0 A written as, where is a paraeter. hen the coefficient atrix is decoposed in the for A ( D E) [( ) D( ) EF], where Next, the syste Ax b is written as x ( D E) [( ) D( ) EF] x( DE) b, and then the AOR iterative ethod is defined as ( k ) ( k) x D E D EF x DE b ( ) [( ) ( ) ] ( ). It is well-known that for specific values of ω and γ the AOR iterative ethod reduces to Jacobi, Gauss- Seidel and successive overrelaxation iterative (SOR) ethods: 0, : Jacobi ethod, Corresponding Author: Davod Khojasteh Salkuyeh, Departent of Matheatics, University of Mohaghegh Ardabili, P.O.Box. 79, Ardabil, Iran E-ail: khojaste@ua.ac.ir 35

2 Aust. J. Basic & Appl. Sci., 5(3): , 20 : : Gauss-Seidel ethod, SOR ethod. Although there are several iterative ethods such as GMRES (Saad, Y., M.H. Schultz, 986) and Bi- CGSAB (van der Vorst, H. A., 992) algoriths for solving Eq. () which are ore effective than these four stationary iterative ethods, they have been used as preconditioners for coon iterative solvers (see for exaple (DeLong, M., J.M. Ortega, 995; DeLong, M., J.M. Ortega, 996; DeLong, M., J.M. Ortega, 998)). In (Salkuyeh, D.K., 2007), Salkuyeh proposed the generalized Jacobi (GJ) and Gauss-Seidel (GGS) ethods and studied their convergence properties. In this paper, we propose a generalization of the AOR (hereafter denoted by GAOR) ethod and verify its convergence properties. his paper is organized as follows. In section 2, we review the GJ and GGS iterative ethods and propose the GAOR iterative ethod. Section 2 also provides the background for the convergence of the proposed ethod. Convergence properties of the GAOR ethod are presented in section 3. Soe nuerical experients are given in section 4. Concluding rearks are given in section he GJ, GGS and GAOR ethods: A ( a ij ) n n ( t ) 2 Let be an atrix and ij be a banded atrix of bandwidth defined as t ij aij, i j, 0, otherwise. We consider the decoposition A, E F where E and F are the strict lower and upper part of the atrix A, respectively. In other words, atrices, E, and F, are defined as following (2). a a, a a ann, a n, n nn, E a a 2, a n, n, n F a a, 2, n a n, n Now, siilar to the classical AOR ethod its generalized version is defined as following 352

3 Aust. J. Basic & Appl. Sci., 5(3): , 20 x ( E ) [( ) ( ) E F ] x ( E ) b. ( k ) ( k ) (3) Note that, if = 0, then the GAOR iterative ethod results in the classical AOR ethod. As the AOR ethod, for soe specific values of γ and ω the GAOR ethod reduces to GJ, GGS and the GSOR (generalized SOR) ethods. In this section, we focus our attention on the GAOR ethod and refer the readers to (Salkuyeh, D.K., 2007) for ore details about GJ and GGS ethods. Evidently, results for the GAOR ethod also are valid for the GSOR ethod. Let G D E D E F G (, ) ( E ) [( ) ( ) E F ] AOR (, ) ( ) [( ) ( ) ], ( ) AOR be the iteration atrix of the AOR and GAOR ethods, respectively. For convenience, soe notations, definitions and results that will be used in the next section are given below. A atrix is called nonnegative, sei-positive and positive if each entry of A is nonnegative, nonnegative but A 0 A 0 A 0 at least a positive entry and positive, respectively. We denote the by, and. Siilarly, for n- diensional vectors, by identifying the with x 0 x 0 A 0 that n atrices, we can also define, and. A atrix A is said to be reducible if there exists a perutation atrix P such X Y P AP 0 Z, where X and Z are both square atrices. Otherwise, A is said to be irreducible. Additionally, we denote the ( A) spectral radius of A by. Definition : A ( a ij ) a 0 i,, n a 0 i j A atrix is said to be an M-atrix if ii for, ij, for, A is nonsingular and A 0. heore : (Saad, Y., 995) Let ( ) and B ( b ij ) be two atrices such that A B and bij 0 for i j all. A a ij hen, if A is an M-atrix, so is the atrix B. Definition 2: Let nn. he splitting is called: A A M N (a) weak regular if and ; M 0 M M 0 N 0 (b) regular if and. N 0 353

4 Aust. J. Basic & Appl. Sci., 5(3): , 20 heore 2: (Wang, L., Y. Song, 2009) Let be an irreducible atrix. If for soe, then ( A). A 0 Ax x x 0 heore 3: (Wang, L., Y. Song, 2009) If is irreducible, then is a siple eigenvalue and A has an x 0 ( A) eigenvector corresponding to. A 0 ( A) heore 4: (Wang, L., Y. Song, 2009) Let A be an M-atrix and ( M N) of A. hen,. A M N In the next section, the convergence properties of the GAOR ethod are studied. be a regular or weak regular splitting 3. Main Results: Lea : Let A be an M-atrix and be the splitting defined as (2). hen is an atrix and ( E). A E F Proof: Let S E. Obviously, we have A S. herefore, fro heore, S is an M-atrix. 0 E 0 Siilarly, it is easy see that is also an M-atrix. Hence. On the other hand,. his S E shows that is a regular splitting. Now, by heore 4,.. ( E) heore 5: (Wu, M., L. Wang, Y. Song, 2007) If A is an M-atrix and, with then the AOR iterative ethod is convergent, i.e.,. G AOR (, ) 0 0, heore 6: If A is an M-atrix and, with then the GAOR ethod is convergent, i.e., ( ) G GAOR (, ). 0 0, Proof: In the GAOR iterative ethod we have A M N where M E, and A M N ( ) ( ) E F. Obviously, we have. herefore, by heore, is an M-atrix, and as a result M 0. On the other hand, fro Lea we have ( E). 0 Since we have ( E ), and therefore M N ( E ) [( ) ( ) E F] ( I E ) [( ) I ( ) E F] ( E) [( ) I ( ) E F] 0. j0 Hence, we conclude that A M N is a weak regular splitting of A. M 354

5 Aust. J. Basic & Appl. Sci., 5(3): , 20 Now, fro heore 4, we observe that ( M N) and this copletes the proof. Exaple : Consider the M-atrix A ( ) Fro heore 6, we have G GAOR () () G GAOR (0.5,0.9) (, ). For exaple, we have G GAOR (0.4, 0.7) ,, (2) (2) G GAOR (0.5,0.9) hese results show that heore 6 holds here. Exaple 2: Consider the atrix A 7, his atrix is not an M-atrix. Here we have G GAOR (0.4,0.7) 0.627,, () G GAOR (0.6,0.8) (2) G GAOR (0.6, 0.8) G GAOR (0.6,0.8) 0.772,, p q G GAOR ( p) ( q) his exaple shows that, if, then in general (, ) is not less than (,. ) he next theore shows that this is true in the special case. G GAOR heore 7: Let A be an irreducible M-atrix, 0, with 0, and p. If is an irreducible atrix, then ( ) ( p) GGAOR GGAOR (, ) (, ). G ( p) GAOR (, ) Proof: ( p) ( ) For the sake of siplicity let Sp GGAOR(, ) and S GGAOR(, ). Siilar to the proof of heore 6, we have S p 0. Since S p is an irreducible atrix, fro heore 3, ( S p ) is an S x 0, eigenvalue of p corresponding to the eigenvector i.e., 355

6 Spx x. Fro heore 6 we have Aust. J. Basic & Appl. Sci., 5(3): , 20 0 ( ). S p [( ) ( ) E F ] x( E ) x, or p p p p p ( ) E xf x( ) x p p p Eq. (4) is equivalent to (4) (5) Now, we have Sxx( E) [( ) x( ) E xf x x Ex] ( E) [( ) x( ) E xf x] p 0 p p L U L Evidently,. We split the atrix as, where and are strictly lower and strictly upper triangular atrices, respectively. Note that hand, we have E E L, F F U. p p herefore, for (5) and (6) we obtain L, U 0 S x x E L U L U x ( ) [ ( )( ) ( ) ] ( E) [ ( ) L ( ) U] x ( E) [ (( ) ( )) L ( ) U] x ( )( E) [( ) L U] x. (6) U. On the other Fro the proof of heore 6 we have ( ) 0. E On the other hand, we have 0, 0, L, U 0 S x x 0. ( ) ( ), and. herefore, Now, fro heore 2 we have S Sp and this copletes the proof. Reark : Let A be an irreducible M-atrix,, with and. hen ( ) GGAOR GAOR (, ) (, ). 0 0, Proof: Fro heore 7 it is enough to show that the atrix in heore 2.5 in (Wang, L., Y.Song, 2009). G (, ) AOR Exaple 3: Consider the atrix of the Exaple. his atrix is an M-atrix and we have is irreducible. his has been proved (2) () GGAOR GGAOR GAOR (0.5,0.9) (0.5,0.9) (0.5,0.9) , 356

7 Aust. J. Basic & Appl. Sci., 5(3): , 20 (2) () GGAOR GGAOR GAOR (0.4, 0.7) (0.4, 0.7) (0.4, 0.7) 0.872, hese results show that heore 7 and Reark hold here. In the next section, we give soe nuerical experients to show the effectiveness of the proposed ethod. 4. Nuerical Experients: All the nuerical experients presented in this section were coputed in double precision with soe MALAB codes on a personal coputer Pentiu MHz. In all the experients, vector b A(,,,) was taken to be the right-hand side of the linear syste and a null vector as an initial guess. he stopping 0 criterion used was always baxk / b 0, where k is the coputed solution at step k of each k 2 ethod. We present two exaples to copare the nuerical results of the GAOR ethod with that of the AOR ethod. x Exaple 4: In the this exaple, we consider the n n banded atrix A his atrix is strictly diagonally doinant with positive diagonal and nonpositive off-diagonal entries. herefore, the atrix A is an M-atrix (Axelsson, O., 996). We let γ = 0.4 and ω = 0.8 Nuerical results of the AOR and the GAOR iterative ethods for =,2, for different values of n (n = 25000, 50000, 75000, 00000) are given in able. In each iteration of the GAOR ethod vector of the for ( ) should be coputed. Hence, to copute y we ay solve ( ) y E z for y. In the ipleentation of the GAOR ethod we used the LU factorization of syste. Here, we ention that the LU factorization of E E y z E to solve this is coputed before starting the iterations of the GAOR ethod. In able the nuber of iterations of the ethod and the CPU tie (in parenthesis) for convergence are given (tiings are in seconds). he tie for the GAOR ethod is the su of the tie for coputing the LU factorization of E ethod is ore effective than the AOR ethod. and the tie for the convergence. As we observe the GAOR Exaple 5: (Wang, L., Y. Song, 2009) We consider the two diensional convection-diffusion equation x y ( u u ) 2 e ( xu yu ) f( x, y), in (0,) (0,), xx yy x y 357

8 Aust. J. Basic & Appl. Sci., 5(3): , 20 with the hoogeneous Dirichlet boundary conditions. Discretization of this equation on a ( p) ( p) grid, by using the second order centered differences for the second and first order differentials gives a linear syste of equations of order n = p 2 with n unknowns. As the previous exaple we consider the right hand b A(,,,). side the syste as All of the assuptions are as Exaple. Assuing p = 70, 80, 90, 00, we obtain four systes of linear equations of diension n = 4900, 6400, 800, In able 2 the nuerical results obtained by the AOR and GAOR with γ = 0.5 and ω = 0.9 are reported. In the GAOR ethod we assued =. As we observe, for this exaple, the GAOR ethod reduces the nuber of the iterations of the AOR ethod for the convergence by a factor of two. able 2 also shows the the CPU ties of the GAOR ethod is slightly less that that of the AOR ethod. able : Nuerical results of the AOR and GAOR for Exaple 4. Method n = n = n = n =00000 AOR 570 (8.48) 570 (7.06) 570 (27.53 ) 570 (36.78 ) GAOR = 294 (6.64) 294 (3.53) 294 (22.05) 294 (29.28) = 2 09 (2.56) 09 (5.3) 09 (8.34) 09 (.08) able 2: Nuerical results of the AOR and GAOR for Exaple 5. Method n = 4900 n = 6400 n = 800 n = 0000 AOR 647 (2.02) 838 (3.48) 052 (5.45) 29 (8.52) GAOR 330 (.8) 425 (3.20) 532 (5.7) 65 (8.05) Conclusion: In this paper, we proposed a generalization of the AOR ethod say GAOR ethod and studied its convergence properties for M-atrices. We presented soe nuerical experients to show the effectiveness of the proposed ethod. Nuerical results show that the GAOR ethod is ore effective than the AOR ethod. REFERENCES Axelsson, O., 996. Iterative solution ethods, Cabridge University Press, Cabridge. Datta, B. N., 995. Nuerical linear algebra and applications, Brooks/Cole Publishing Copany. DeLong, M., J.M. Ortega, 995. SOR as a preconditioner, Appl. Nuer. Math., 8: DeLong, M., J.M. Ortega, 996. SOR as a parallel preconditioner, in: L. Adas and J. Nazareth, eds., Linear and Nonlinear Conjugate Gradient-Related Methods (SIAM, Philadelphia, PA) DeLong, M., J.M. Ortega, 998. SOR as a preconditioner II, Appl. Nuer. Math., 26: Hadjidios, A., 978. Accelerated overrelaxation ethod, Math. Coput. 32: Salkuyeh, D.K., Generalized Jacobi and the Gauss-Seidel ethods for solving linear syste of equations, Nuer. Math. J. Chinese Univ., 6: Meyer, C.D., Matrix analysis and applied linear algebra, SIAM. Meurant, G., 999. Coputer solution of large linear systes, Elsevier Ltd., North Holland. Saad, Y., 995. Iterative Methods for Sparse Linear Systes, PWS press, New York. Saad, Y., M. H. Schultz, 986. GMRES: a generalized inial residual algorith for solving nonsyetric linear systes, SIAM J. Sci. Stat. Coput., 7: Stoer, J., Bulirsch, 993. Introduction to nuerical analysis, Springer-Verlag, Second edition. an der Vorst, H.A., 992. Bi-CGSAB: a fast and soothly converging variant of Bi-CG for the solution of nonsyetric linear systes, SIAM J. Sci. Stat. Coput., 2: Wu, M., L. Wang, Y. Song, Preconditioned AOR iterative ethod for linear systes, Applied Nuerical Matheatics, 57: Wang, L., Y. Song, Preconditioned AOR iterative ethods for M-atrices, Journal of Coputational and Applied Matheatics, 226: Young, D. M., 97. Iterative solution of large linear systes, Acadeic press, NY. 358