Computers and Mathematics with Applications. Convergence analysis of the preconditioned Gauss Seidel method for H-matrices

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1 Computers Mathematics with Applications 56 (2008) Contents lists available at ScienceDirect Computers Mathematics with Applications journal homepage: wwwelseviercom/locate/camwa Convergence analysis of the preconditioned Gauss Seidel method for H-matrices Qingbing Liu a,b, Guoliang Chen a,, Jing Cai a,c a Department of Mathematics, East China Normal University, Shanghai , PR China b Department of Mathematics, Zhejiang Wanli University, Ningbo , PR China c School of Science, Huzhou Teachers College, Huzhou , PR China a r t i c l e i n f o a b s t r a c t Article history: Received 8 August 2011 Received in revised form 25 February 2012 Accepted 5 March 2012 Keywords: H-matrix Preconditioner Preconditioned iterative method Gauss Seidel method H-splitting In 1997, Kohno et al [Toshiyuki Kohno, Hisashi Kotakemori, Hiroshi Niki, Improving the modified Gauss Seidel method for Z-matrices, Linear Algebra Appl 267 (1997) ] proved that the convergence rate of the preconditioned Gauss Seidel method for irreducibly diagonally dominant Z-matrices with a preconditioner I + S α is superior to that of the basic iterative method In this paper, we present a new preconditioner I + K β which is different from the preconditioner given by Kohno et al [Toshiyuki Kohno, Hisashi Kotakemori, Hiroshi Niki, Improving the modified Gauss Seidel method for Z-matrices, Linear Algebra Appl 267 (1997) ] prove the convergence theory about two preconditioned iterative methods when the coefficient matrix is an H-matrix Meanwhile, two novel sufficient conditions for guaranteeing the convergence of the preconditioned iterative methods are given Crown Copyright 2008-Published by Elsevier Ltd All rights reserved 1 Introduction We consider the following linear system Ax b, where A is a complex n n matrix, x b are n-dimensional vectors For any splitting, A M N with the nonsingular matrix M, the basic iterative method for solving the linear system (1) is as follows: x i+1 M 1 Nx i + M 1 b i 0, 1, 2, Some techniques of preconditioning which improve the rate of convergence of these iterative methods have been developed Let us consider a preconditioned system of (1) PAx Pb, where P is a nonsingular matrix The corresponding basic iterative method is given in general by x i+1 M 1 P N P x i + M 1 P Pb i 0, 1, 2,, where PA M P N P is a splitting of PA (1) (2) This project was granted financial support by the Shanghai Science Technology Committee Shanghai Priority Academic Discipline Foundation (no ) Corresponding author addresses: lqb_2008@hotmailcom (Q Liu), glchen@mathecnueducn (G Chen)

2 2049 In 1997, Kohno et al [1] proposed a general method for improving the preconditioned Gauss Seidel method with the preconditioned matrix P I + S α, if A is a nonsingular diagonally dominant Z-matrix with some conditions, where 0 α 1 a 1, α 2 a 2,3 0 S α α n1 a n1,n They showed numerically that the preconditioned Gauss Seidel method is superior to the original iterative method if the parameters α i 0 (i 1, 2,, n 1) are chosen appropriately Many other researchers have considered left preconditioners applied to linear system (1) that made the associated Jacobi Gauss Seidel methods converge faster than the original ones Such modifications or improvements based on prechosen preconditioners were considered by Milaszewicz [2] who based his ideas on previous ones (see, eg, [3]), by Gunawardena et al [4], very recently by Li Sun [5] who extended the class of matrices considered in [1] by other researchers (see, eg, [6 12]), many results for more general preconditioned iterative methods were obtained In this paper, besides the above preconditioned method, we will consider the following preconditioned linear system A β x b β, where A β (I + K β )A b β (I + K β )b with β 1 a 2, K β 0 β 2 a 3,2 0 0, 0 0 β n1 a n,n1 0 where β i 0 (i 1, 2,, n 1) Our work gives the convergence analysis of the above two preconditioned Gauss Seidel methods for the case when a coefficient matrix A is an H-matrix obtains two sufficient conditions for guaranteeing the convergence of two preconditioned iterative methods (3) 2 Preliminaries Without loss of generality, let the matrix A of the linear system (1) be A I L U, where I is an identity matrix, L U are strictly lower upper triangular matrices obtained from A, respectively We assume a i,i+1 0, considering the preconditioner P I + S α, then we have whenever A α (I + S α )A I L S α L (U S α + S α U) b α (I + Sα)b, α i a i,i+1 a i+1,i 1 for i 1, 2,, n 1, then (I L S α L) 1 exists Hence it is possible to define the Gauss Seidel iteration matrix for A α, namely T α (I L S α L) 1 (U S α + S α U) (4) Similarly, if a i,i1 0, considering the preconditioner P I + K β, then we have A β (I + K β )A I L + K β K β L (U + K β U) b β (I + K β )b, define the Gauss Seidel iteration matrix for A β, namely T β (I L + K β K β L) 1 (U + K β U) We first recall the following: A real vector x (x 1, x 2,, x n ) T is called nonnegative(positive) denoted by x 0 (x > 0), if x i 0 (x i > 0) for all i Similarly, a real matrix A (a i,j ) is called nonnegative denoted by A 0 (A > 0) if a i,j 0 (a i,j > 0) for all i, j, the absolute value of A is denoted by A ( a i,j ) Definition 21 ([13]) A real matrix A is called an M-matrix if A si B, B 0 s > ρ(b), where ρ(b) denotes the spectral radius of B (5)

3 2050 Definition 22 ([13]) A complex matrix A (a i,j ) is an H-matrix, if its comparison matrix A (ā i,j ) is an M-matrix, where ā i,j is ā i,i a i,i, ā i,j a i,j, i j Definition 23 ([14]) The splitting A M N is called an H-splitting if M N is an M-matrix Lemma 21 ([14]) Let A M N be a splitting If it is an H-splitting, then A M are H-matrices ρ(m 1 N) ρ( M 1 N ) < 1 Lemma 22 ([15]) Let A have nonpositive off-diagonal entries Then a real matrix A is an M-matrix if only if there exists some positive vector u (u 1,, u n ) T > 0 such that Au > 0 3 Convergence results Theorem 31 Let A be an H-matrix with unit diagonal elements, A α (I + S α )A M α N α, M α I L S α L N α US α +S α U Let u (u 1,, u n ) T be a positive vector such that A u > 0 Assume that a i,i+1 0 for i 1, 2,, n1, u i i1 a i,j u j n a i,j u j + a i,i+1 u i+1 α i, a i,i+1 n a i+1,j u j then α i > 1 for i 1, 2,, n 1 for 0 α i < α i, the splitting A α M α N α is an H-splitting ρ(mα 1 N α) < 1 so that the iteration (2) converges to the solution of (1) Proof By assumption, let a positive vector u > 0 satisfy A u > 0, from the definition of A, we have u i a i,j u j > 0 for i 1, 2,, n 1 Therefore, we have u i a i,j u j u i a i,j u j + a i,i+1 u i+1 a i,i+1 a i,j u j + a i,i+1 u i+1 Observe that u i n u i a i,j u j a i+1,j u j a i+1,j u j for i 1, 2,, n 1 +1 a i,j u j > 0 u i+1 n a i+1,j u j > 0, then we have +1 a i,j u j + a i,i+1 u i+1 a i,i+1 a i+1,j u j > 0, u i a i,j u j a i,j u j + a i,i+1 u i+1 > a i,i+1 a i+1,j u j > 0 for i 1, 2,, n 1 This implies u i i1 a i,j u j n a i,j u j + a i,i+1 u i+1 α i > 1 for i 1, 2,, n 1 a i,i+1 n a i+1,j u j Hence, α i > 1 for i 1, 2,, n 1

4 2051 In order to prove that ρ(mα 1 N α) < 1, we first show that M α N α is an M-matrix Let [( M α N α )u] i be the ith element in the vector ( M α N α )u for i 1, 2,, n 1 Then we have [( M α N α )u] i 1 α i a i,i+1 a i+1,i u i a i,j α i a i,i+1 a i+1,j u j a i,j α i a i,i+1 a i+1,j u j u i α i a i,i+1 a i+1,i u i a i,j u j α i a i,i+1 a i+1,j u j [( M α N α )u] n u n a i,j u j α i a i,i+1 a i+1,j u j 1 α i a i,i+1 u i+1, (6) a n,j u j > 0 (7) j n If 0 α i 1 for i 1, 2,, n 1, then we have [( M α N α )u] i u i α i a i,i+1 a i+1,i u i a i,j u j α i a i,i+1 a i+1,j u j Since u i n u i a i,j u j α i a i,i+1 a i+1,j u j (1 α i ) a i,i+1 u i+1 a i,j u j + α i a i,i+1 u i+1 α i a i,i+1 u i a i,j u j + α i a i,i+1 u i+1 a i,j u j > 0 u i+1 n a i+1,j u j > 0, we have +1 a i+1,j u j +1 a i+1,j u j (8) [( M α N α )u] i > 0 for i 1, 2,, n 1 (9) +1 If 1 < α i < α i for i 1, 2,, n 1, from (6) the definition of α i, then we have [( M α N α )u] i u i α i a i,i+1 a i+1,i u i a i,j u j α i a i,i+1 a i+1,j u j a i,j u j α i a i,i+1 a i+1,j u j (α i 1) a i,i+1 u i+1 u i a i,j u j > 0 Therefore, from (7) to (10), we have ( M α N α )u > 0 for 0 α i < α i a i,j u j + a i,i+1 u i+1 α i a i,i+1 a i+1,j u j By Lemma 22, M α N α is an M-matrix for 0 α i < α i (i 1, 2,, n 1) From Definition 23, A α M α N α is an H-splitting for 0 α i < α i (i 1, 2,, n 1) Hence, according to Lemma 21, we have that A α M α are H-matrices ρ(mα 1 N α) ρ( M α 1 N α ) < 1 for 0 α i < α i (i 1, 2,, n 1) Theorem 32 Let A be an H-matrix with unit diagonal elements, A β (I + K β )A M β N β, M β I L + K β K β L N β U + K β U Let v (v 1,, v n ) T be a positive vector such that A v > 0 Assume that a i,i1 0 for i 2,, n, v i i2 a i,j v j n a i,j v j + a i,i1 v i1 β i, a i,i1 n a i1,j v j (10)

5 2052 then β i > 1 for i 2,, n for 0 β i < β i, the splitting A β M β N β is an H-splitting ρ(m 1 β N β) < 1 so that the iteration (2) converges to the solution of (1) Proof From the conditions of the theorem, we know there exists a positive vector v > 0 satisfying A u > 0, then we have v i a i,j v j > 0 for i 2,, n Therefore, we have v i a i,j v j v i Since v i n a i,j v j + a i,i1 v i1 a i,i1 a i,j v j + a i,i1 v i1 v i a i,j v j a i1,j v j a i1,j v j for i 2,, n 1 a i,j v j > 0 v i1 n a i1,j v j > 0, we have 1 a i,j v j + a i,i1 v i1 a i,i1 So it is obvious to obtain v i a i,j v j a i,j v j + a i,i1 v i1 > a i,i1 This implies β i v i i2 a i,j v j n a i,j v j + a i,i1 v i1 a i1,j v j > 0 a i1,j v j > 0 for i 2,, n > 1 for i 2,, n a i,i1 n a i1,j v j Namely, β i > 1 for i 2,, n In order to prove that ρ(m 1 β N β) < 1, we first show that M β N β is an M-matrix, let [( M β N β )v] i be the ith element in the vector ( M β N β )v for i 2,, n Then we have [( M β N β )v] i v i β i1 a i,i1 a i1,i v i a i,j β i1 a i,i1 a i1,j v j a i,j β i1 a i,i1 a i1,j v j v i β i1 a i,i1 a i1,i v i a i,j v j β i1 a i,i1 a i1,j v j a i,j v j β i1 a i,i1 a i1,j v j 1 β i1 a i,i1 v i1, (11) [( M β N β )v] 1 v 1 a 1,j v j > 0 (12) j2 If 0 β i1 1 for i 2,, n, then we have [( M β N β )v] i v i β i1 a i,i1 a i1,i v i a i,j v j β i1 a i,i1 a i1,j v j a i,j v j β i1 a i,i1 a i1,j v j (1 β i1 ) a i,i1 v i1

6 Observe that v i n v i a i,j v j + β i1 a i,i1 v i1 β i1 a i,i1 v i a i,j v j + β i1 a i,i1 v i1 a i1,j v j a i1,j v j (13) 1 a i,j v j > 0 v i1 n a i1,j v j > 0, then we have 1 [( M β N β )v] i > 0 for i 2, 3,, n (14) If 1 < β i1 < β i1 for i 2,, n, from (11) the definition of β i1, then we have [( M β N β )v] i v i β i1 a i,i1 a i1,i v i a i,j v j β i1 a i,i1 a i1,j v j a i,j v j β i1 a i,i1 a i1,j v j (β i1 1) a i,i1 v i1 v i a i,j v j > 0 Therefore, from (12) to (15), we have a i,j v j + a i,i1 v i1 β i1 a i,i1 ( M β N β )v > 0 for 0 β i1 < β i1 a i1,j v j By Lemma 22, M β N β is an M- matrix for 0 β i1 < β i1 (i 2,, n) From Definition 23, A β M β N β is an H-splitting for 0 β i1 < β i1 (i 2,, n) Hence, from Lemma 21, we have that A β M β are H-matrices ρ(m 1 β N β) ρ( M β 1 N β ) < 1 for 0 β i1 < β i1 (i 2,, n) Remark 31 From Theorems 31 32, we observe that the two preconditioned iterative methods converge to the solution of the linear system (1) if the coefficient matrix A which is an H-matrix satisfies the conditions of the corresponding theorems Hence the two theorems provide sufficient conditions for guaranteeing the convergence of the preconditioned iterative methods Acknowledgements The authors wish to express their thanks to Professor EY Rodin the two referees for helpful suggestions comments References [1] Toshiyuki Kohno, Hisashi Kotakemori, Hiroshi Niki, Improving the modified Gauss Seidel method for Z-matrices, Linear Algebra Appl 267 (1997) [2] JP Milaszewicz, Improving Jacobi Gauss Seidel iterations, Linear Algebra Appl 93 (1987) [3] JP Milaszewicz, On modified Jacobi linear operators, Linear Algebra Appl 51 (1983) [4] A Gunawardena, SK Jain, L Snyder, Modified iterative methods for consistent linear systems, Linear Algebra Appl 149/156 (1991) [5] W Li, WW Sun, Modified Gauss Seidel type methods Jacobi type methods for Z-matrix, Linear Algebra Appl 317 (2000) [6] A Hadjidimos, D Noutsos, M Tzoumas, More on modifications improvements of classical iterative schemes for M-matrices, Linear Algebra Appl 364 (2003) [7] D Noutsos, M Tzoumas, On optimal improvements of classical iterative schemes for Z-matrix, J Comput Appl Math 188 (2006) [8] M Neumann, RJ Plemmons, Convergence of parallel multisplitting iterative methods for M-matrix, Linear Algebra Appl 88/89 (1987) [9] W Li, A note on the preconditioned Gauss Seidel(GS) method for linear systems, J Comput Appl Math 182 (2005) [10] XZ Wang, TZ Huang, YD Fu, Comparison results on preconditioned SOR-type method for Z-matrix linear systems, J Comput Appl Math 206 (2007) [11] MJ Wu, L Wang, YZ Song, Preconditioned AOR iterative method for linear systems, Appl Numer Math 57 (2007) [12] LY Sun, Some extensions of the improved modified Gauss Seidel iterative method for H-matrix, Numer Linear Algebr 13 (2006) [13] RS Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962, pp [14] A Frommer, DB Szyld, H-splitting two-stage iterative methods, Numer Math 63 (1992) [15] KY Fan, Topological proofs for certain theorems on matrices with non-negative elements, Monatsh Math 62 (1958) (15)