A Modified Precondition in the Gauss-Seidel Method

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(1)Advances in Linear Algebra & Matrix Theory, 2012, 1, 31-37 doi:10.4236/alamt.2012.23005 Published Online September 2012 (http://www.SciRP.org/journal/alamt) A Modified Precondition in the Gauss-Seidel Method Alimohammad Nazari, Sajjad Zia Borujeni Department of Mathematics, Faculty of Science, Arak University, Arak, Iran Email: a-nazari@araku.ac.ir, z-borujeni@arshad.araku.ac.ir Received March 6, 2012; revised June 28, 2012; accepted August 21, 2012 ABSTRACT In recent years, a number of preconditioners have been applied to solve the linear systems with Gauss-Seidel method (see [1-13]). In this paper we use Sl instead of (S + Sm) and compare with M. Morimoto’s precondition [3] and H. Niki’s precondition [5] to obtain better convergence rate. A numerical example is given which shows the preference of our method. Keywords: Preconditioning; Gauss-Seidel Method; Regular Splitting; Z-Matrix; Nonnegative Matrix 1. Introduction Consider the linear system (1) Ax = b, where A   aij   R nn is a known nonsingular matrix and x, b  R n are vectors. For any splitting A = M – N with a nonsingular matrix M, the basic splitting iterative method can be expressed as x  k 1 1 = M Nx k  1  M b. k = 0,1, 2, (2) where D and E are the diagonal and strictly lower triangular parts of SL, respectively. If 1 aii 1ai 1i  11  i  n  1 , then  I  D    L  E  exists. Therefore, the preconditioned Gauss-Seidel iterative matrix TS for AS becomes TS =  I  D    L  E  1 which is referred to as the modified Gauss-Seidel iterative matrix. Gunawardena et al. proved the following inequality:  TS    T  < 1, Assume that aii  0, i = 1, 2, , n, without loss of generality we can write A = I  L U , (3) where I is the identity matrix,  L and U are strictly lower triangular and strictly upper triangular parts of A , respectively. In order to accelerate the convergence of the iterative method for solving the linear system (1), the original linear system (1) is transformed into the following preconditioned linear system PAx = Pb, (4) where P, called a preconditioner, is a nonsingular matrix. In 1991, Gunawardena et al. [2] considered the modified Gauss-Seidel method with P   I  S  , where for i = 1, 2,, n  1, j  i  1 a S   sij    ii 1 otherwise. 0 Then, the preconditioned matrix AS =  I  S  A can be written as AS = I  L  SL  U  S  SU =  I  D    L  E   U  S  SU  , Copyright © 2012 SciRes. U  S  SU  , where  T  denotes the spectral radius of the GaussSeidel iterative matrix T. Morimoto et al. [3] have proposed the following preconditioner,  Ps  m = aij s  m = I SS m . In this preconditioner, Sm is defined by  aili 1  i < n  1, i  1 < j  n,    = 0 Sm = sij m  otherwise, where li = min I i and   I i = j : aij is maximal for i  1 < j  n , for 1  i < n  1 . The preconditioned matrix As  m =  I  S  S m  A can then be written as As  m = I  L  SL  U  S  SU  Sm  Sm L  SmU   I  Ds  m    L  Es  m   U  S  Sm  SU  SmU  Fs  m  , where Ds  m , Es  m and Fs  m are the diagonal, strictly ALAMT

(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 G1 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 Sl1 exists and the preconditioned GaussSeidel iterative matrix TSl for ASl is defined by TSl = M Sl1 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 ui 1  li , where li and ui 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 A1 > O . Theorem 2.9 [16, Theorem 3.29]. Let A = MN be a regular splitting of the matrix A. Then, A is nonsingular with A1 > O if and only if  M 1 N < 1 , where   M N  = 1    A1 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 ASl1  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 s1m = [ 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 11 > M 21 and N1  N 2 , then M Sl1   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 11 N1 , T2 = M 21 N 2 . Let x  0, y  0 be such that T1 x =  T1  x and T2 y =  T2  y . If M 11  M 21 , 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 s1m  M Sl1 . 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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