# A Modified Precondition in the Gauss-Seidel Method

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(2) A. M. NAZARI, S. ZIA BORUJENI 32 lower and strictly upper triangular parts of S Sm L respectively. Assume that the following inequalities are satisfied: 1 i < n 1 0 < aii 1ai 1i aili all i < 1 0 < aii 1ai 1i < 1 i = n 1. Then I Ds m L Es m is nonsingular. The preconditioned Gauss-Seidel iterative matrix Ts m for As m is then defined by Ts m = I Ds m L Es m 1 U S Sm SU SmU Fs m . Morimoto et al. [3] proved that Ts m Ts . To extend the preconditioning effect to the last row, Morimoto et al. [7] proposed the preconditioner Put AG = PG A = aijG = M G NG , and TG = M G1 N G . Replacing PG by PG = I S Sm G, and setting = 1, the Gauss-Seidel splitting of AG can be written as AG I Ds m L Es m G L U G U S Sm SU SmU Fs m , where G L U G is constructed by the elements anjG = g nj a jn . Thus, if the preconditioner PG is used, then all of the rows of A are subject to preconditioning. Niki et al. [5] proved that under the condition u > , TR TG , where u is the upper bound of those values of for which TG < 1 . By setting anjG = 0 , they obtained = anj g nj PR = I R, where R is defined by Niki et al. [5] proved that the preconditioner PG satisfies the Equation (5) unconditionally. Moreover, they reported that the convergence rate of the GaussSeidel method using preconditioner PG is better than that of the SOR method using the optimum found by numerical computation. They also reported that there is an optimum opt in the range u > opt > m , which produces an extremely small T opt , where m is the anj 1 j n 1 R = rnj = otherwise 0 The elements anjR of AR are given by AR = I R A = aijR , 1 i < n, 1 j n aij n 1 a = anj ank akj 1 j n. k =1 R ij And Morimoto et al. proved that TR T holds, where TR is the iterative matrix for AR . They also presented combined preconditioners, which are given by combinations of R with any upper preconditioner, and showed that the convergence rate of the combined methods are better than those of the Gauss-Seidel method applied with other upper preconditioners [7]. In [14], Niki et al. considered the preconditioner PSR = I S R . De- note ASR = M SR N SR . In [5], Niki et al. proved that if the following inequality is satisfied, anj anjR = anj n 1 j = 1, 2, , n 1, ank akj k =1, k j (5) then TSR TS holds, where TSR is the iterative matrix for ASR . For matrices that do not satisfy Equation (5), by putting R = rnj = n 1 ank akj anj ,1 j < n, k =1 , k j Equation (5) is satisfied. Therefore, Niki et al. [5] proposed a new preconditioner PG = I G 1 , where n 1 ank akj anj , 1 j n 1, G = g nj = k =1 , k j 0 otherwise. Copyright © 2012 SciRes. g nk akj . k =1, k j n 1 upper bound of the values of for which anjG 0 , for all j . In this paper we use different preconditions for solving (1) by Gauss-Siedel method, that assuming none of the components of the matrix A to be zero. If the largest component of the column j is not ai ,i 1 , then the value of T will be improved. 2. Main Result In this section we replace Sl by S Sm of Morimoto such that Sl = Sn Sm and define S n by aki j i = 1, 2,..., n 2 , j = i 1, Sn = sij = aij i = n 1, otherwise, 0 where aki j max akj s.t k = i, i 2, i 3, , n , and Sm has the same form as the Sm proposed by Morimoto et al. [3]. The precondition Matrix ASl I Sl A can then be written as ASl = I L U Sl Sl L SlU = I DSl L ESl U Sl SlU FSl , where DSl , ESl , and FSl are the diagonal, strictly lower and strictly upper triangular parts of Sl L , reALAMT

(3) A. M. NAZARI, S. ZIA BORUJENI spectively. Assume that the following inequalities are satisfied: aii 1 akii 1 aili alii 1 0 i = 1, 2, , n 2 i = 1, 2,, n 2 0 < akii 1ai 1i aili alii < 1 i = n 1, 0 < aii 1ai 1i < 1 (6) Therefore M Sl1 exists and the preconditioned GaussSeidel iterative matrix TSl for ASl is defined by TSl = M Sl1 N Sl U S S U F . For A = a and B = b R , we write A B whenever a b holds for all i, j = 1, 2, , n. A is nonnegative if A 0, a 0; i, j = 1, , n , and A B if = I DSl L ESl 1 l l Sl n n ij ij ij ij ij and only if A B 0 . Definition 2.1 (Young, [15]). A real n n matrix A = aij with aij 0 for all i j is called a Zmatrix. Definition 2.2 (Varga, [16]). A matrix A is irreducible if the directed graph associated to A is strongly connected. Lemma 2.3. If A is an irreducible diagonally dominant Z-matrix with unit diagonal, and if the assumption (6) holds, then the preconditioned matrix AS is a diagonally dominant Z-matrix. Proof. The elements aijSl of ASl are given by i = n 1, (7) i = n. 0 aki ,i 1ai 1, j 1 for j i 1, 0 ai ,li ali , j 1 for j li , (8) 1 aki ,i 1ai 1,i 1 0. Therefore, the following inequalities hold: pi = aki ,i 1ai 1,i 0, qi = ai ,li ali ,i 0, i 1 ri = aki ,i 1 ai 1, j 0, j =1 i 1 si = ai ,li ali , j 0, j =1 ai 1, j 0, j = i 1 n al , j 0. j = i 1 i We denote that pn = qn = rn = sn = tn = un = 0. Then Copyright © 2012 SciRes. n n j =1 j =1 = aki ,i 1 ai 1, j ai ,li ali , j 0 1 i < n. Furthermore, if ai ,i 1 0 , and i < n , then we have pi qi ri si ti ui < 0 n ai 1, j < 0 , for some j =1 for some i < n. (9) Let d Sl i , l Sl i and u Sl i be the sums of the elements in row i of DSl , LSl , and U Sl , respectively. The following equations hold: d Sl = aijSl = 1 pi qi , 1 i n, i 1 l Sl = aijSl = li ri si 1 i n, (10) j =1 n u Sl = aijSl = ui ti ui 1 i n, j =i 1 where li and ui are the sums of the elements in row i of L and U for A = I L U , respectively. Since A is a diagonally dominant Z-matrix, by (8) and by the condition (6) the following relations hold: 1 aki ,i 1ai 1, j ai ,li ali , j > 0, aij aki ,i 1 n for j = i n ai 1, j ai ,l al , j 0, j = i 1 i i j = i 1 aij aki ,i 1 ai ,li ali ,i 1 aki ,i 1 for i > j n n ai 1, j ai,l ali , j 0 i j =i 2 j =i 2 Therefore, l Sl 0 , u Sl 0 , and ASl is a Zmatrix. Moreover, by (9) and by the assumption, we can easily obtain d S l l S l u Sl = di li ui pi qi ri si ti ui > 0 (11) for all i. Therefore, ASl satisfies the condition of diagonal dominance. Lemma 2.4 [10, Lemma 2]. An upper bound on the spectral radius T for the Gauss-Seidel iteration matrix T is given by T max i n ui = ai ,li pi qi ri si ti ui for i < j. Since A is a diagonally dominant Z-matrix, so we have ti = aki ,i 1 the following inequality holds: 1 i < n 1, aij aki ,i 1ai 1, j ai ,li ali , j a aij ai ,i 1ai 1, j aij Sl ij 33 ui 1 li , where li and ui are the sums of the moduli of the elements in row i of the triangular matrices L and U, respectively. Theorem 2.5. Let A be a nonsingular diagonally dominant Z-matrix with unit diagonal elements and let the condition (6) holds, then TSl < 1. ALAMT

(4) A. M. NAZARI, S. ZIA BORUJENI 34 Proof. From (11) and u Sl 0 we have d Sl l Sl > u Sl 0 for all i. This implies that u Sl d Sl l Sl (12) < 1. Hence, by Lemma (2.4) we have TSl < 1. Definition 2.6. Let A be an n n real matrix. Then, A = M N is referred to as: 1) a regular splitting, if M is nonsingular, M 1 0 and N 0. 2) a weak regular splitting, if M is nonsingular, M 1 0 and M 1 N 0. 3) a convergent splitting, if M 1 N < 1. Lemma 2.7 (Varga, [10]). Let A R n n be a nonnegative and irreducible n n matrix. Then 1) A has a positive real eigenvalue equal to its spectral radius A ; 2) for A , there corresponds an eigenvector x > 0; 3) A is a simple eigenvalue of A; 4) A increases whenever any entry of A increases. Corollary 2.8 [16, Corollary 3.20]. If A = aij is a real, irreducibly diagonally dominant n n matrix with aij < 0 for all i j , and aii > 0 for all 1 i n , then A1 > O . Theorem 2.9 [16, Theorem 3.29]. Let A = MN be a regular splitting of the matrix A. Then, A is nonsingular with A1 > O if and only if M 1 N < 1 , where M N = 1 A1 N 1 M 1 N T1 < T2 . Now in the following lemma we prove that ASl = I Sn Sm A = M Sl N Sl is Gauss-Seidel convergent regular splitting. Theorem 2.12. Let A be an irreducibly diagonally dominant Z-matrix with unit diagonal, and let the condition (6) holds, then ASl = M Sl N Sl is Gauss-Seidel convergent regular splitting. Moreover TSl Ts m < 1. Proof. If A is an irreducibly diagonally dominant Zmatrix, then by Lemma (2.3), ASl is a diagonally dominant Z -matrix. So we have ASl1 0 . By hypothesis I D I . Thus the strictly lower trianmatrix L E has nonnegative elements. By 1 we have Sl gular Sl considering Neumanns series, the following inequality holds: I D I DSl L E 1 Sl 1 L E 2 Sl 1 Sl L E I D n 1 Sl 1 Sl 0. Direct calculation shows that N Sl 0 holds. Thus, by definition (2.6) ASl = M Sl N Sl is the Gauss-Seidel convergent regular splitting. Also in [3] we have As m = M s m N s m and M s1m = [ I I DS M I D I DS M 1 1 L ES M L ES M L ES M 1 S M 2 , n 1 1 I DS M 0. Direct comparison of the two matrix elements n As is remarked in [1,17], when B = R and K = R , all n n real matrices have the property “d”. Therefore Copyright © 2012 SciRes. Moreover, if M 11 > M 21 and N1 N 2 , then M Sl1 I I DSl for all x K . n T1 T2 . <1 Theorem 2.10 (Gunawardena et al. [2, Theorem 2.2]). Let A be a nonnegative matrix. Then 1) If x Ax for some nonnegative vector x, x 0, then A . 2) If Ax x for some positive vector x, then A . Moreover, if A is irreducible and if 0 x Ax x for some nonnegative vector x, then A and x is a positive vector. Let B be a real Banach space, B its dual and L B the space of all bounded linear operator mapping B into itself. We assume that B is generated by a normal cone K [17]. As is defined in [17], the operator A L B has the property “d” if its dual A possesses a Fro- benius eigenvector in the dual cone K which is de- fined by K = x B: x, x = x( x) 0 the case are discussing fulfills the property “d”. For the space of all n n matrices, the theorem of Marek and Szyld can be stated as follows: Theorem 2.11 (Marek and Szyld [17, Theorem 3.15]). Let A1 = M 1 N1 and A2 = M 2 N 2 be weak regular splitting with T1 = M 11 N1 , T2 = M 21 N 2 . Let x 0, y 0 be such that T1 x = T1 x and T2 y = T2 y . If M 11 M 21 , and if either A1 A2 x 0, A1 x 0, or A1 A2 y 0, A1 y 0, with y > 0 , then I DS M L E Sl 1 and I D Sl 1 also L ES M and we obtain ALAMT

(5) A. M. NAZARI, S. ZIA BORUJENI I DS M 1 I DSl L ES M L ES 1 0 0 TS 0 0 , . l Thus 0 M s1m M Sl1 . Furthermore, since Sn S , we have ASl x As m x Sn S Ax 0 . From Lemma (2.7), x is an eigenvector of TSl , and x is also a Perron vector of TSl . Therefore, from Theorem (2.11), TSl Ts m 35 and TS = 0.3205 . Using the preconditioner Ps m , we obtain As m 0.90 0.12 0.06 0.4 0.25 0.91 0.2 0.09 = 0.16 0.29 0.94 0 1 0.2 0.3 0.2 holds. Denote Asmr I S Sm R A, ASl r I Sl R A, AG I S Sm G A, AGSl I Sl G A, and also let Tsmr , TSl r , TG and TG Sl be the iterative matrix associated to Asmr , ASl r , AG and AGSl respectively. Then we can prove TSl r Tsmr and TG Sl TG , similarly. In summary, we have the following inequalities: T Ts Ts m TSl Tsmr TSl r TG TGSl . Remark 2.13. W. Li, in [18] used the M-matrix instead of irreducible diagonally dominant Z-matrix, therefore we can say that the Lemma 2.3 and the Theorems 2.5 and 2.12 are hold for M-matrices. Ts m 0.2 0.6 0.2 1 1 0.2 0.3 0.1 A 0.1 0.2 0.3 1 1 0.2 0.3 0.2 Applying the Gauss-Seidel method, we have T = 0.5099 . By using preconditioner PS = I S A, we find that AS and TS have the following forms: 0 0.66 0.22 0.96 0 0.23 0.94 0.19 AS 0.16 0.29 0.94 0 1 0.2 0.3 0.2 Copyright © 2012 SciRes. 0 0 = 0 0 0.133 0.067 0.444 0.037 0.040 0.221 0.034 0.023 0.144 0.044 0.030 0.184 and Ts m = 0.2581 . For PSl = I Sl , we have 0.88 0.02 0.09 0.41 0.25 0.91 0.2 0.09 ASl = 0.16 0.29 0.94 0 1 0.2 0.3 0.2 0 0 TSl = 0 0 0.465 0.227 0.006 0.033 0.149 0.007 0.042 0.1911 0.023 0.102 0.006 0.050 and TSl = 0.2328 . For Psmr = I S Sm R , we have 3. Numerical Results In this section, we test a simple example to compare and contrast the characteristics of the different preconditioners. Consider the matrix 0 0.6875 0.2292 0 0.1682 0.2582 0 0.1689 0.1187 0 0.2217 0.1470 Asmr 0.90 0.12 0.06 0.4 0.25 0.91 0.2 0.09 = 0.16 0.29 0.94 0 0.08 0.08 0.21 0.87 Tsmr 0 0 = 0 0 0.133 0.067 0.444 0.037 0.040 0.221 0.034 0.023 0.144 0.024 0.015 0.096 and Tsmr = 0.1694 . For PSl r = I Sl R, we have ASl r 0.88 0.02 0.09 0.41 0.25 0.91 0.02 0.09 = 0.16 0.29 0.94 0 0.08 0.08 0.21 0.87 ALAMT

(6) A. M. NAZARI, S. ZIA BORUJENI 36 0 0 = 0 0 TSl r 0.023 0.102 0.465 0.006 0.050 0.227 0.006 0.033 0.149 0.004 0.022 0.1 and TSl r = 0.1415. From the above results, we have Gnj 0.28, 0.38, 0.41, 0 . Then AG = 1 and TG have the forms: 0.90 0.12 0.06 0.4 0.25 0.91 0.02 0.09 AG = 0.16 0.29 0.94 0 0.04 0.06 0.07 0.78 0 0 TG = 0 0 0.133 0.067 0.444 0.037 0.040 0.221 0.034 0.024 0.144 0.012 0.008 0.050 and TG = 0.1218 . For PGSl = 1 = I Sl G , we have AGS l 0.88 0.02 0.09 0.41 0.25 0.91 0.02 0.09 = 0.16 0.29 0.94 0 0.04 0.06 0.07 0.78 TGS 0 0 = 0 0 l 0.023 0.102 0.465 0.006 0.050 0.227 0.006 0.033 0.149 0.002 0.011 0.052 and TGSl = 0.0940. Since the preconditioned matrices differ only in the values of their last rows, the related matrices also differ only in these values, as is shown in the above results. Thus the elements of new AG and AGSl are similar to elements of AG and AGSl , respectively than the elements of last rows. Therefore, we hereafter show only the last row. By putting 1 = 1.22699 , the matrices AG , TG , AGSl 1 1 1 and TGSl 1 have the following forms: AG = 0, 0.003, 0.042, 0.733 , AG = 0.0021, 0, 0.0413, 0.7310 , 2 TG = 0, 0.0015, 0.0011, 0.0069 , 2 = 0.0751 and and TG AGSl 2 = 0.0021, 0, 0.0413, 0.7310 , TGSl 2 = 0, 0.00026, 0.0016, 0.0071 , For 3 opt = 1.5625 we have: AG and TG TGSl Copyright © 2012 SciRes. and = 0.0547, 0.0781, 0, 0.6609 , opt = 0, 0.0026, 0.0144, 0.0654 , 3 opt = 0.0216 . REFERENCES [1] T. Z. Huang, G. H. Cheng and X. Y. Cheng, “Modified SOR-Type Ierative Method for Z-Matrices,” Applied Mathematics and Computation, Vol. 175, No. 1, 2006, pp. 258-268. doi:10.1016/j.amc.2005.07.050 [2] A. D. Gunawardena, S. K. Jain and L. Snyder, “Modified Iterative Method for Consistent Linear Systems,” Linear Algebra and Its Applications, Vol. 154-156, 1991, pp. 123143. doi:10.1016/0024-3795(91)90376-8 [3] M. Morimoto, K. Harada, M. Sakakihara and H. Sawami, “The Gauss-Seidel Iterative Method with the Preconditioning Matrix (I + S + Sm),” Japan Journal of Industrial and Applied Mathematics, Vol. 21, No. 1, 2004, pp. 2534. doi:10.1007/BF03167430 [4] D. Noutsos and M. Tzoumas, “On Optimal Improvements of Classical Iterative Schemes for Z-Matrices,” Journal of Computational and Applied Mathematics, Vol. 188, No. 1, 2006, pp. 89-106. doi:10.1016/j.cam.2005.03.057 [5] H. Niki, T. Kohno and M. Morimoto, “The Preconditioned Gauss-Seidel Method Faster than the SOR Method,” Journal of Computational and Applied Mathematics, Vol. 218, No. 1, 2008, pp. 59-71. doi:10.1016/j.cam.2007.07.002 [6] H. Niki, T. Kohno and K. Abe, “An Extended GS Method for Dense Linear System,” Journal of Computational and Applied Mathematics, Vol. 231, No. 1, 2009, pp. 177-186. doi:10.1016/j.cam.2009.02.005 TGSl 1 = 0, 0.00036, 0.0021, 0.0096 , For 2 = 1.23966 we have: = 0.0227 From the numerical results, we see that this method with the preconditioner PGSl opt produces a spectral radius smaller than the recent preconditioners that above was introduced. 1 3 = 0, 0.0154, 0.0102, 0.0629 , opt 3 and TGSl AGSl 1 = 0, 0.003, 0.042, 0.733 , 3 opt AGSl 1 and TGSl 1 = 0.0509 . opt 3 = 0.0547, 0.0781, 0, 0.6609 , opt 3 TG TG = 0, 0.0021, 0.0015, 0.0093 , = 0.0779 , and TGSl 2 = 0.0483 . 1 and TG 2 ALAMT

(7) A. M. NAZARI, S. ZIA BORUJENI [7] [8] [9] M. Morimoto, H. Kotakemori, T. Kohno and H. Niki, “The Gauss-Seidel Method with Preconditioner (I + R),” Transactions of the Japan Society for Industrial and Applied Mathematics, Vol. 13, 2003, pp. 439-445. H. Kotakemori, K. Harada, M. Morimoto and H. Niki, “A Comparison Theorem for the Iterative Method with the Preconditioner (I + Smax),” Journal of Computational and Applied Mathematics, Vol. 145, No. 2, 2002, pp. 373-378. doi:10.1016/S0377-0427(01)00588-X A. Hadjidimos, D. Noutsos amd M. Tzoumas, “More on Modifications and Improvements of Classical Iterative Schemes for Z-Matrices,” Linear Algebra and Its Applications, Vol. 364, 2003, pp. 253-279. doi:10.1016/S0024-3795(02)00570-0 [10] T. Kohno, H. Kotakemori, H. Niki and M. Usui, “Improving the Gauss-Seidel Method for Z-Matrices,” Linear Algebra and Its Applications, Vol. 267, No. 1-3, 1997, pp. 113-123. [11] W. Li and W. Sun, “Modified Gauss-Seidel Type Methods and Jacobi Type Methods for Z-Matrices,” Linear Algebra and Its Applications, Vol. 317, 2000, pp. 227-240. doi:10.1016/S0024-3795(00)00140-3 [12] J. P. Milaszewicz, “On Modified Jacobi Linear Operators,” Copyright © 2012 SciRes. 37 Linear Algebra and Its Applications, Vol. 51, 1983, pp. 127-136. doi:10.1016/0024-3795(83)90153-2 [13] J. P. Milaszewicz, “Improving Jacobi and Gauss-Seidel Iterations,” Linear Algebra and Its Applications, Vol. 93, 1987, pp. 161-170. doi:10.1016/S0024-3795(87)90321-1 [14] H. Niki, K. Harada, M. Morimoto and M. Sakakihara, “The Survey of Preconditioners Used for Accelerating the Rate of Convergence in the Gauss-Seidel Method,” Journal of Computational and Applied Mathematics, Vol. 164165, 2004, pp. 587-600. doi:10.1016/j.cam.2003.11.012 [15] D. M. Young, “Iterative Solution of Large Linear Systems,” Academic Press, New York, 1971. [16] R. S. Varga, “Matrix Iterative Analysis,” Prentice-Hall, Englewood Cliffs, 1962. [17] I. Marek and D. Szyld, “Comparison Theorems for Weak Splittings of Bound Operators,” Numerische Mathematik, Vol. 58, No. 1, 1990, pp. 387-397. doi:10.1007/BF01385632 [18] W. Li, “Comparison Results for Solving Preconditioned Linear Systems,” Journal of Computational and Applied Mathematics, Vol. 176, No. 2, 2005, pp. 319-329. doi:10.1016/j.cam.2004.07.022 ALAMT

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