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Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 850986, 7 pages http://dx.doi.org/10.1155/2013/850986 Research Article Modified Preconditioned GAOR Methods for Systems of Linear Equations Xue-Feng Zhang, Qun-Fa Cui, and Shi-Liang Wu School of Mathematics and Statistics, Anyang Normal University, Anyang 455000, China Correspondence should be addressed to Xue-Feng Zhang; [email protected] Received 30 January 2013; Accepted 13 May 2013 Academic Editor: Giuseppe Marino Copyright © 2013 Xue-Feng Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Three kinds of preconditioners are proposed to accelerate the generalized AOR (GAOR) method for the linear system from the generalized least squares problem. The convergence and comparison results are obtained. The comparison results show that the convergence rate of the preconditioned generalized AOR (PGAOR) methods is better than that of the original GAOR methods. Finally, some numerical results are reported to confirm the validity of the proposed methods. 1. Introduction Consider the generalized least squares problem min(𝐴𝑥 − 𝑏)𝑇 𝑊−1 (𝐴𝑥 − 𝑏) , 𝑥∈R𝑛 (1) where 𝑥 ∈ R𝑛 , 𝐴 ∈ R𝑛×𝑛 , 𝑏 ∈ R𝑛 , and the variance-covariance matrix 𝑊 ∈ R𝑛×𝑛 is a known symmetric and positive-definite matrix. This problem has many scientific applications and one of the applications is a parameter estimation in mathematical model [1, 2]. In order to solve the problem simply, one has to solve a linear system of the equivalent form as follows: 𝐹𝑦 = 𝑓, (2) 𝐼−𝐵 𝐻 𝐹=( ), 𝐾 𝐼−𝐶 (3) In order to get the approximate solutions of the linear system (2), a lot of iterative methods such as Jacobi, GaussSeidel (GS), successive over relaxation (SOR), and accelerated over relaxation (AOR) have been studied by many authors [3– 8]. These iterative methods have very good results, but have a serious drawback because of computing the inverses of 𝐼 − 𝐵 and 𝐼−𝐶 in (3). To avoid this drawback, Darvishi and Hessari [9] proposed the generalized convergence of the generalized AOR (GAOR) method when the coefficient matrix 𝐹 is a diagonally dominant matrix. The GAOR method [10, 11] can be defined as follows: 𝑦𝑘+1 = T𝛾𝜔 𝑦𝑘 + 𝜔𝑔, 𝑘 = 0, 1, 2, . . . , (5) where with 𝐵 ∈ R𝑝×𝑝 , 𝐶 ∈ R𝑞×𝑞 , and 𝑝 + 𝑞 = 𝑛. Without loss of generality, we assume that 𝐹 = I − L − U, where I is the identity matrix, and L and U are strictly lower and upper triangular matrices obtained from 𝐹, respectively. So we can pretty easily get that 𝐼 0 I=( ), 0 𝐼 0 0 L=( ), −𝐾 0 𝐵 −𝐻 U=( ). 0 𝐶 (4) where T𝛾𝜔 = ( 𝐼 0 ) 𝛾𝐾 𝐼 −1 × ((1 − 𝜔) 𝐼 + (𝜔 − 𝛾) ( 0 0 𝐵 −𝐻 ) + 𝜔( )) , −𝐾 0 0 𝐶 𝐼 0 𝑔=( ) 𝑓. −𝛾𝐾 𝐼 (6) 2 Journal of Applied Mathematics Here, 𝜔 and 𝛾 are real parameters with 𝜔 ≠ 0. The iteration matrix is rewritten briefly as −𝜔𝐻 (1 − 𝜔) 𝐼 + 𝜔𝐵 ). T𝛾𝜔 = ( 𝜔 (𝛾 − 1) 𝐾 − 𝛾𝜔𝐾𝐵 (1 − 𝜔) 𝐼 + 𝜔𝐶 + 𝑤𝛾𝐾𝐻 (7) To improve the convergence rate of the GAOR iterative method, a preconditioner should be applied. Now we can transform the original linear system (2) into the preconditioned linear system 𝑃𝐹𝑦 = 𝑃𝑓, (8) (a) 𝐴 has a positive eigenvalue equal to its spectral radius. (b) 𝐴 has an eigenvector 𝑥 > 0 corresponding to 𝜌(𝐴). (c) 𝜌(𝐴) is a simple eigenvalue of 𝐴. Lemma 2 (see [12]). Let 𝐴 be a nonnegative matrix. Then, (i) if 𝛼𝑥 ≤ 𝐴𝑥 for some nonnegative vector 𝑥, 𝑥 ≠ 0, then 𝛼 ≤ 𝜌(𝐴); (ii) if 𝐴𝑥 ≤ 𝛽𝑥 for some positive vector 𝑥, then 𝜌(𝐴) ≤ 𝛽. Moreover, if 𝐴 is irreducible and if 0 ≠ 𝛼𝑥 ≤ 𝐴𝑥 ≤ 𝛽𝑥 for some nonnegative vector 𝑥, then 𝛼 ≤ 𝜌(𝐴) ≤ 𝛽 and 𝑥 is a positive vector. 3. Preconditioned GAOR Methods where 𝑃 is the preconditioner. 𝑃𝐹 can be expressed as 𝐼 − 𝐵∗ 𝐻∗ ). 𝑃𝐹 = ( ∗ 𝐾 𝐼 − 𝐶∗ Lemma 1 (see [7]). Let 𝐴 ≥ 0 be an 𝑖𝑟𝑟𝑒𝑑𝑢𝑐𝑖𝑏𝑙𝑒 matrix. Then, (9) To solve the linear system (2) with the coefficient matrix 𝐹 in (3), we consider the preconditioners as follows: 𝐼 0 ), 𝑆𝑖 𝐼 𝑃𝑖 = 𝐼 + 𝑆 = ( Meanwhile, the PGAOR method for solving the preconditioned linear system (8) is defined by 𝑦𝑘+1 = T∗𝛾𝜔 𝑦𝑘 + 𝑤𝑔∗ , 𝑘 = 0, 1, 2, . . . , 0 0 ⋅ ⋅ ⋅ −𝑘1𝑝 (10) 0 0 𝑆1 = ( 0 d (11) In this paper, we propose three new types of preconditioners and study the convergence rate of the preconditioned GAOR methods for solving the linear system (2). This paper is organized as follows. In Section 2, some notations, definitions, and preliminary results are presented. In Section 3, three new types of preconditioners are proposed and compared with that of the original GAOR methods. Lastly, a numerical example is provided to confirm the theoretical results studied in Section 4. 0 ( 0 0 ( 0 ( 𝑆3 = ( ( 0 ( For vector 𝑥 ∈ R , 𝑥 ≥ 0 (𝑥 > 0) denotes that all components of 𝑥 are nonnegative (positive). For two vectors 𝑥, 𝑦 ∈ R𝑛 , 𝑥 ≥ 𝑦 (𝑥 > 𝑦) means that 𝑥 − 𝑦 ≥ 0 (𝑥 − 𝑦 > 0). These definitions are carried immediately over to matrices. A matrix 𝐴 is said to be irreducible if the directed graph of 𝐴 is strongly connected. 𝜌(𝐴) denotes the spectral radius of 𝐴. Some useful results are provided as follows. .. . −𝑘12 ⋅⋅⋅ −𝑘21 0 −𝑘23 ( ( 𝑆2 = ( ( 0 −𝑘32 d ( .. .. . . ⋅⋅⋅ 2. Preliminaries 𝑛 ⋅ ⋅ ⋅ −𝑘2𝑝 ), (0 0 ⋅ ⋅ ⋅ −𝑘𝑞𝑝 ) −𝜔𝐻∗ (1 − 𝜔) 𝐼 + 𝜔𝐵∗ ), ∗ ∗ ∗ 𝜔 (𝛾 − 1) 𝐾 − 𝛾𝜔𝐾 𝐵 (1 − 𝜔) 𝐼 + 𝜔𝐶∗ + 𝛾𝜔𝐾∗ 𝐻∗ 𝐼 0 𝑔 = (−𝛾𝐾∗ 𝐼) 𝑃𝑓. (12) where where T∗𝛾𝜔 = ( 𝑖 = 1, 2, 3, − 𝑘𝑞1 𝛼 0 0 0 ⋅⋅⋅ 0 .. . .. . 0 −𝑘𝑞−1,𝑝 ⋅ ⋅ ⋅ −𝑘𝑞,𝑝−1 −𝑘12 ⋅ ⋅ ⋅ .. . d .. . 0 −𝑘𝑞−1,𝑝 0 (13) ) 0 0 ... 0 ) ) ), ) ) 0 ) ) ), ) (𝛼 > 0) . ) The preconditioned coefficient matrix 𝑃𝑖 𝐹 can be expressed as 𝐼−𝐵 𝐻 𝑃𝑖 𝐹 = ( ), 𝐾 + 𝑆𝑖 (𝐼 − 𝐵) 𝑆𝑖 𝐻 + 𝐼 − 𝐶 where 𝑖 = 1, 2, 3, (14) Journal of Applied Mathematics 3 𝑘11 𝑘12 ⋅ ⋅ ⋅ 𝑘1𝑝 𝑏𝑝𝑝 𝐾 + 𝑆1 (𝐼 − 𝐵) = (𝑘21 𝑘22 ⋅ ⋅ ⋅ 𝑘2𝑝 𝑏𝑝𝑝 ) , 𝑘𝑞1 𝑘𝑞2 ⋅ ⋅ ⋅ 𝑘𝑞𝑝 𝑏𝑝𝑝 ( 𝐾 + 𝑆2 (𝐼 − 𝐵) = ( 𝑘11 𝑘21 𝑏11 + 𝑘23 𝑏31 𝑘12 𝑏22 𝑘22 ⋅⋅⋅ 𝑘21 𝑏13 + 𝑘23 𝑏33 𝑘31 .. . 𝑘32 𝑏22 + 𝑘34 𝑏42 .. . d 𝑘𝑞1 𝑘𝑞2 ( ⋅⋅⋅ ⋅⋅⋅ 𝑘1,𝑝−1 ⋅⋅⋅ .. . 𝑃𝑖 𝐹 = I − L𝑖 − U𝑖 , 𝑖 = 1, 2, 3. (16) ⋅⋅⋅ d 𝐼 0 I=( ), 0 𝐼 0 0 ), L𝑖 = ( −𝐾 − 𝑆𝑖 (𝐼 − 𝐵) 0 𝐵 −𝐻 U𝑖 = ( ), 0 𝐶 − 𝑆𝑖 𝐻 . . . 𝑘𝑞−1,𝑝 𝑏𝑝𝑝 ) . ⋅⋅⋅ 𝑘𝑞𝑝 Next, we will study the convergence condition of the PGAOR methods. For simplicity, without loss of generality, we can assume that 0 < 𝜔1 ≤ 1, (17) 𝑖 = 1, 2, 3. The preconditioned GAOR methods for solving 𝑃𝑖 𝐹𝑦 = 𝑃𝑖 𝑓 are defined by 𝑦𝑘+1 = T∗𝛾𝜔𝑖 𝑦𝑘 + 𝜔𝑔𝑖∗ , 𝑘 = 0, 1, 2, . . . , 𝑖 = 1, 2, 3, (18) 𝐾 ≤ 0, 𝐵 ≥ 0, 0 < 𝜔2 ≤ 1, 𝐶 ≥ 0, 0 < 𝛾2 ≤ 𝜔2 . 𝜔1 −1 𝜔𝑔𝑖∗ = (I − ΓL𝑖 ) Ω𝑃𝑖 𝑓, 𝑖 = 1, 2, 3, (19) (23) Then, we have the following theorem. Theorem 3. Let T𝛾𝜔 and T∗𝛾𝜔1 be the iteration matrices of the GAOR method and the PGAOR method corresponding to problem (2), which are defined by (7) and (22), respectively. If matrix 𝐹 in (3) is an irreducible matrix then it holds that 𝜌(T𝛾𝜔 ) ≠ 1, 𝜌 (T∗𝛾𝜔1 ) > 𝜌 (T𝛾𝜔 ) ≠ 1 if 𝜌 (T𝛾𝜔 ) > 1, where (15) 𝑘1𝑝 𝑘2𝑝 𝐻 ≤ 0, Similarly, ) ), 0 𝑘𝑞−1,𝑝−2 𝑏𝑞−2,𝑝 + 𝑘𝑞−1,𝑝 𝑏𝑞𝑝 𝑘𝑞,𝑝−1 𝑏𝑞−1,𝑝−1 𝑘𝑞𝑝 ) 𝑘12 𝑏22 𝑘11 𝑘21 𝑘22 .. .. 𝐾 + 𝑆3 (𝐼 − 𝐵) = ( . . 1 − 𝑏11 𝑘𝑞1 (1 − ) 𝑘𝑞2 𝛼 Based on the discussed above, 𝑃𝑖 𝐹 can be spitted as 𝑘1𝑝 𝑘2𝑝 .. . 𝜌 (T∗𝛾𝜔1 ) < 𝜌 (T𝛾𝜔 ) ≠ 1 if 𝜌 (T𝛾𝜔 ) < 1. (24) Proof. By some simple calculations on (7), one can get with −1 T∗𝛾𝜔𝑖 = (𝐼 − ΓL𝑖 ) (𝐼 − Ω + (Ω − Γ) L𝑖 + ΩU𝑖 ) , 𝑖 = 1, 2, 3, (20) 0 0 + 𝜔1 𝛾2 ( ). −𝐾𝐵 𝐾𝐻 where 𝜔𝐼 0 Ω=( 1 ), 0 𝜔2 𝐼, 𝛾𝐼 0 Γ=( 1 ). 0 𝛾2 𝐼 −𝜔1 𝐻 (1 − 𝜔1 𝐼) + 𝜔1 𝐵 T𝛾𝜔 = ( ) (𝜔1 𝛾2 − 𝜔2 ) 𝐾 (1 − 𝜔2 𝐼 + 𝜔2 ) 𝐶 (21) For 𝑖 = 1, 2, 3, we have −𝜔1 𝐻 (1 − 𝜔1 ) 𝐼 + 𝜔1 𝐵 T∗𝛾𝜔𝑖 = ((𝜔1 𝛾2 − 𝜔2 ) [𝐾 + 𝑆𝑖 (𝐼 − 𝐵)] (1 − 𝛾2 ) 𝐼 + 𝛾2 𝐶 − 𝛾2 𝑆𝑖 𝐻 ) . −𝜔1 𝛾2 [𝐾 + 𝑆𝑖 (𝐼 − 𝐵)] 𝐵 +𝜔1 𝛾2 [𝐾 + 𝑆𝑖 (𝐼 − 𝐵)] 𝐻 (22) (25) Since here 𝐹 is irreducible, one can pretty easily obtain that T𝛾𝜔 is nonnegative and irreducible by the above assumptions. And so on, one can also easily prove that T∗𝛾𝜔1 is nonnegative and irreducible. By Lemma 1, there exists a positive vector 𝑥 > 0 such that T𝛾𝜔 𝑥 = 𝜆𝑥, where 𝜆 = 𝜌(T𝛾𝜔 ). (26) 4 Journal of Applied Mathematics One can easily have By far, we can easily get [I − Ω + (Ω − Γ) L + ΩU] 𝑥 = 𝜆 (I − ΓL) 𝑥. (27) −1 T∗𝛾𝜔1 𝑥 − 𝜆𝑥 = (𝜆 − 1) (I − ΓL1 ) That is, (I − Ω) 𝑥 = 𝜆 (I − ΓL𝑥 − (Ω − Γ) L𝑥 − ΩU𝑥) . 0 0 0 0 ] × [( 𝜔2 𝑆 0) + ( ) 𝑥 −𝛾2 𝑆1 (𝐼 − 𝐵) 0 1 𝜔 1 [ ] (28) With the same vector 𝑥 > 0, it holds 𝐼 0 ) −𝛾2 [𝐾 + 𝑆1 (𝐼 − 𝐵)] 𝐼 −1 = (𝜆 − 1) ( T∗𝛾𝜔1 𝑥 − 𝜆𝑥 = (I − ΓL1 ) × [I − Ω + (Ω − Γ) L1 − 𝜆 (I − ΓL1 )] 𝑥. (29) 0 0 )𝑥 × ( 𝜔2 ( − 𝛾2 ) 𝑆1 + 𝛾2 𝑆1 𝐵 0 𝜔1 Using (22), (26), and (28), we can obtain 0 0 = (𝜆 − 1) ( 𝜔2 ) 𝑥. ( − 𝛾2 ) 𝑆1 + 𝛾2 𝑆1 𝐵 0 𝜔1 −1 T∗𝛾𝜔1 𝑥 − 𝜆𝑥 = (I − ΓL1 ) × [(Ω − Γ) (L1 − L) +Ω (U1 − U) + 𝜆Γ (L1 − L)] 𝑥 (32) In view of the abovementioned assumptions, we have that −1 = (I − ΓL1 ) [Ω (U1 − U + L1 − L) 0 0 ( 𝜔2 ) 𝑥 > 0. ( − 𝛾2 ) 𝑆1 + 𝛾2 𝑆1 𝐵 0 𝜔1 + (𝜆 − 1) Γ (L1 − L)] 𝑥 −1 = (I − ΓL1 ) Ω (U1 − U + L1 − L) 𝑥 Then, if 𝜆 = 𝜌(T𝑤𝑟 ) > 1, then −1 + (𝜆 − 1) ((I − ΓL1 ) ) Γ (L1 − L) 𝑥 0 0 = (I − ΓL1 ) ( )𝑥 −𝜔2 𝑆1 (𝐼 − 𝐵) −𝜔2 𝑆1 𝐻 T∗𝛾𝜔1 − 𝜆𝑥 > 0, −1 T∗𝛾𝜔1 − 𝜆𝑥 ≠ 0. −1 𝜌 (T∗𝛾𝜔1 ) > 𝜌 (T𝑤𝑟 ) > 1. 0 0 ×( ) 𝑥. −𝛾2 𝑆1 (𝐼 − 𝐵) 0 (34) From Lemma 2, we can easily get + (𝜆 − 1) ((I − ΓL1 ) ) (35) Similarly, if 𝜆 = 𝜌(T𝑤𝑟 ) < 1, then T∗𝛾𝜔1 − 𝜆𝑥 < 0, (30) Meanwhile, we have T∗𝛾𝜔1 − 𝜆𝑥 ≠ 0. (36) So we have 0 0 (I − ΓL1 ) ( )𝑥 −𝜔2 𝑆1 (𝐼 − 𝐵) −𝜔2 𝑆1 𝐻 −1 𝜌 (T∗𝛾𝜔1 ) < 𝜌 (T𝛾𝜔 ) < 1. 0 0 −𝜔 𝐼 + 𝜔1 𝐵 −𝜔1 𝐻 −1 = (I − ΓL1 ) ( 𝜔2 𝑆 0) ( 1 )𝑥 0 0 1 𝜔1 0 0 −1 = (I − ΓL1 ) ( 𝜔2 𝑆 0) 𝜔1 1 −𝜔1 𝐼 + 𝜔1 𝐵 −𝜔1 𝐻 ×( )𝑥 (𝜔1 𝛾2 𝐾 − 𝜔1 𝛾2 𝐾𝐵) −𝜔2 𝐼 + 𝜔2 𝐶 + 𝜔1 𝛾2 𝐾𝐻 0 0 −1 = (I − ΓL1 ) ( 𝜔2 𝑆 0) (T𝑤𝑟 − I) 𝑥 𝜔1 1 0 0 −1 = (𝜆 − 1) (I − ΓL1 ) ( 𝜔2 𝑆 0) 𝑥. 𝜔1 1 (33) (37) If 𝜆 = 𝜌(T𝛾𝜔 ) = 1, then we may get that 𝐹𝑦 = 𝑓 but 𝑦 ≠ 0, which is contradictory to the fact of nonsingular matrix 𝐹 by assumptions; this completes the conclusion of the theorem. Theorem 4. Let T𝛾𝜔 and T∗𝛾𝜔2 be the iteration matrices of the GAOR method and the PGAOR method corresponding to problem (2), which are defined by (7) and (22), respectively. If the matrix 𝐹 in (3) is an irreducible matrix satisfying 𝑏11 > 0, 𝑘𝑖𝑗 ≠ 0 for 𝑖 − 𝑗 = 1, (38) then it holds that 𝜌(𝑇𝑤𝑟 ) ≠ 0, 𝜌 (T∗𝛾𝜔2 ) > 𝜌 (T𝛾𝜔 ) ≠ 1 if 𝜌 (T𝛾𝜔 ) > 1, (31) 𝜌 (T∗𝛾𝜔2 ) < 𝜌 (T𝛾𝜔 ) ≠ 1 if 𝜌 (T𝛾𝜔 ) < 1. (39) Journal of Applied Mathematics 5 Proof. One can easily prove this theorem by using similar arguments of Theorem 3. The coefficient matrix F is spitted as 𝐼−𝐵 𝐻 F=( ), 𝐾 𝐼−𝐶 Similarly, we have the following theorem. Theorem 5. Let T𝛾𝜔 and T∗𝛾𝜔3 be the iteration matrices of the GAOR method and the PGAOR method corresponding to problem (2), which are defined by (7) and (22), respectively. If the matrix 𝐹 in (3) is an irreducible matrix satisfying 𝑘𝑞1 < 0, 𝛼 > 1, 𝑏11 > 0, 𝑘𝑖,𝑖+1 < 0 for 𝑖 = 1, 2, . . . , 𝑞 − 1, (40) then it holds that 𝜌(T𝛾𝜔 ) ≠ 1, 𝜌 (T∗𝛾𝜔3 ) > 𝜌 (T𝛾𝜔 ) ≠ 1 if 𝜌 (T𝛾𝜔 ) > 1, 𝜌 (T∗𝛾𝜔3 ) < 𝜌 (T𝛾𝜔 ) ≠ 1 if 𝜌 (T𝛾𝜔 ) < 1. (41) 4. Numerical Examples In this section, we give numerical examples to demonstrate the conclusions drawn above. The numerical experiments were done by using MATLAB 7.0. Example 1. Consider the following Laplace equation: 𝜕2 𝑢 (𝑥, 𝑦) 𝜕2 𝑢 (𝑥, 𝑦) + = 0. 𝜕𝑥2 𝜕𝑦2 F𝑥 = 𝑓, (43) where 1 2 0 0 0 0 0 0 1 ( 0 0 0 0 0 ( 0 ( 2 ( ( 1 1 ( 0 0 0 0 0 − ( 2 8 ( ( ( 1 1 ( 0 0 0 0 0 − ( 2 8 ( ( ( 1 ( 0 0 0 0 0 0 ( 2 ( ( 1 ( F=( 0 0 0 0 0 0 ( 2 ( ( 1 1 1 ( 0 0 − − 0 0 ( 8 8 2 ( ( ( 1 (− − 1 − 1 − 1 0 0 0 ( 8 8 8 8 ( ( ( 1 1 ( 0 0 − 0 − 0 0 ( 8 8 ( ( 1 1 1 ( 1 (− 0 − 0 − − 0 8 8 8 8 1 1 − 0 0 0 0 − 0 ( 8 8 where 1 2 (0 ( (0 ( 𝐵=( ( (0 ( ( 0 0 0 0 0 1 0 0 0 2 1 0 0 0 2 1 0 0 0 2 1 0 0 0 2 0 0 0 0 0 ( 1 1 1 0 − − 8 8 8 1 ) − 0 0 0 ) ) 8 ) ) 1 1 1 − − − 0 ) ) 8 8 8 ) ) ) 1 − 0 0 0 ) ) 8 ) ) ) 1 1 0 − − 0 ) ) 8 8 ) 1 1) ) 0 0 − − ). 8 8) ) ) 0 0 0 0 ) ) ) ) ) 1 0 0 0 ) ) 2 ) ) ) 1 0 0 0 ) ) 2 ) ) 1 ) 0 0 0 ) 2 1 0 0 0 2 ) − (44) 0 0) ) 0) ) ), ) 0) ) ) 0 1 2) 1 2 (0 ( ( 𝐶 = (0 ( ( 0 0 0 0 0 1 0 0 0) 2 ) 1 ) 0 0) , 0 ) 2 1 ) 0 0 0 2 1 0 0 0 0 ( 2) (42) Under a uniform square domain, applying the five-point finite difference method with the uniform mesh size, we can get the following linear system: (45) 1 1 1 0 − − 8 8 8 1 − 0 0 0 ) ) 8 ) ) ) 1 1 1 − − − 0 ) ) 8 8 8 ), ) 1 − 0 0 0 ) ) 8 ) ) ) 1 1 0 − − 0 ) 8 8 1 1 0 0 − − 8 8) 0 − ( 0 ( ( ( ( 1 (− ( 8 𝐻=( ( 1 (− ( ( 8 ( ( ( 0 ( 0 1 1 − 0 0 8 8 (− 1 − 1 − 1 − 1 0 0 ) ) ( 8 8 8 8 ) ( ) ( 1 1 ) ( 𝐾=( 0 0 − 0 − 0 ). ) ( 8 8 ) ( ) ( 1 1 1 1 (− 0 − 0 − − ) 8 8 8 8 1 1 − 0 0 0 0 − ( 8 8) 0 0 − (46) Table 1 reveals the spectral radii of the GAOR methods and the PGAOR methods. It tells that the spectral radii of the preconditioned PGAOR methods are smaller than those of the GAOR methods, so we can get that the proposed three 6 Journal of Applied Mathematics Table 1: Spectral radii of GAOR method and PGAOR method. Preconditioner 𝑆1 𝑆2 𝛼 𝑆3 1.5 2 𝜔1 0.8912 0.8912 0.8912 0.8912 𝜔2 0.9654 0.9654 0.9654 0.9654 𝛾2 0.8865 0.8865 0.8865 0.8865 𝜌GAOR 0.8478 0.8478 0.8478 0.8478 𝜌PGAOR 0.8457 0.8376 0.8418 0.8420 𝜌GAOR 0.7505 0.7505 0.7505 0.7505 𝜌PGAOR 0.6618 0.7340 0.7298 0.7328 Table 2: Spectral radii of GAOR method and PGAOR method. 𝛼 Preconditioner 𝑆1 𝑆2 𝜔1 0.8 0.8 0.8 0.8 0.1 0.5 𝑆3 𝜔2 = 𝜔 0.6 0.6 0.6 0.6 preconditioners can accelerate the speed rate of the GAOR method for the linear systems (2). The results in Table 1 are in accordance with Theorems 3–5. Example 2. The coefficient matrix 𝐹 in (3) is given by 𝐼−𝐵 𝐻 𝐹=( ), 𝐾 𝐼−𝐶 (47) where 𝐵 = (𝑏𝑖𝑗 )𝑝×𝑝 , 𝐶 = (𝑐𝑖𝑗 )𝑞×𝑞 , and 𝑝 + 𝑞 = 𝑛, 𝐻 = (ℎ𝑖𝑗 )𝑝×𝑞 , 𝐾 = (𝑘𝑖𝑗 )𝑞×𝑝 with 𝑏𝑖𝑖 = 𝑏𝑖𝑗 = 1 1 , − 30 (30𝑗) + 𝑖 𝑏𝑖𝑗 = 1 , 𝑖+1 𝑖 = 1, . . . 𝑝, 𝑗 > 𝑖, 𝑖 = 1, . . . 𝑝 − 1, 𝑗 = 2, . . . 𝑝, 𝑖 > 𝑗, 𝑖 = 2, . . . 𝑝, 𝑗 = 1, . . . 𝑝 − 1, 1 , 𝑖 = 1, . . . 𝑞, 𝑛+𝑖+1 1 1 𝑐𝑖𝑗 = , − 30 30 (𝑛 + 𝑗) + 𝑛 + 𝑖 𝑐𝑖𝑖 = 𝑗 > 𝑖, 𝑖 = 1, . . . , 𝑞 − 1, 𝑗 = 2, . . . , 𝑞, 1 1 , − 30 30 (𝑖 − 𝑗 + 1) + 𝑛 + 𝑖 𝑖 > 𝑗, 𝑖 = 2, . . . , 𝑞, 𝑗 = 1, . . . 𝑞 − 1, 𝑘𝑖𝑗 = 1 1 − , 30 (𝑛 + 𝑖 − 𝑗 + 1) + 𝑛 + 𝑖 30 𝑖 = 1, . . . , 𝑞, 𝑗 = 1, . . . , 𝑝, ℎ𝑖𝑗 = 1 1 − , 30 (𝑛 + 𝑗) + 𝑖 30 Obviously, 𝐹 is irreducible. Table 2 shows the spectral radii of the corresponding iteration matrices with 𝑛 = 8 and 𝑚 = 6. Similarly, in Table 2, we get that the results are in concord with Theorems 3–5. Acknowledgments The authors would like to thank the anonymous referees for their helpful suggestions, which greatly improved the paper. This research of this author was supported by NSFC Tianyuan Mathematics Youth Fund (no. 11026040), by Science and Technology Development Plan of Henan Province (no. 122300410316), and by Natural Science Foundations of Henan Province (no. 13A110022). References 1 1 , − 30 30 ((10 (𝑖 − 𝑗 + 1)) + 𝑖) 𝑐𝑖𝑗 = 𝛾2 = 𝛾 0.7 0.7 0.7 0.7 𝑖 = 1, . . . , 𝑝, 𝑗 = 1, . . . , 𝑞. (48) [1] J. Y. Yuan, “Numerical methods for generalized least squares problem,” Journal of Computational and Applied Mathematics, vol. 66, pp. 571–584, 1996. [2] J.-Y. Yuan and X.-Q. Jin, “Convergence of the generalized AOR method,” Applied Mathematics and Computation, vol. 99, no. 1, pp. 35–46, 1999. [3] A. D. Gunawardena, S. K. Jain, and L. Snyder, “Modified iterative methods for consistent linear systems,” Linear Algebra and Its Applications, vol. 154/156, pp. 123–143, 1991. [4] A. Hadjidimos, “Accelerated overrelaxation method,” Mathematics of Computation, vol. 32, no. 141, pp. 149–157, 1978. [5] Y. T. Li, C. X. Li, and S. 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