The upper Jacobi and upper Gauss Seidel type iterative methods for preconditioned linear systems

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1 Applied Mathematics Letters 19 (2006) wwwelseviercom/locate/aml The upper Jacobi upper Gauss Seidel type iterative methods for preconditioned linear systems Zhuan-De Wang, Ting-Zhu Huang School of Applied Mathematics, University of Electronic Science Technology of China, Chengdu, Sichuan, , PR China Received 15 March 2004; received in revised form 21 October 2005; accepted 27 October 2005 Abstract The preconditioner for solving the linear system Ax = b introduced in [DJ Evans, MM Martins, ME Trigo, The AOR iterative method for new preconditioned linear systems, J Comput Appl Math 132 (2001) is generalized Results obtained in this paper show that the convergence rate of Jacobi Gauss Seidel type methods can be increased by using the preconditioned method when A is an M-matrix c 2005 Elsevier Ltd All rights reserved Keywords: Preconditioned iterative method; Upper Jacobi upper Gauss Seidel type method; Spectral radius 1 Introduction For solving the linear system Ax = b, many preconditioners have been proposed [1 3[7 11 In 2001 Evans et al [1proposed the preconditioner 1 0 a 1n P = I + S = showed that the preconditioned AOR method is better than the original one In this paper, we generalize the preconditioner as follows 1 0 αa 1n P(α) = I + S(α) = (11) Corresponding author address: tzhuang@uestceducn (T-Z Huang) /$ - see front matter c 2005 Elsevier Ltd All rights reserved doi:101016/jaml

2 1030 Z-D Wang, T-Z Huang / Applied Mathematics Letters 19 (2006) consider the convergence of the upper Jacobi upper Gauss Seidel type iterative methods for preconditioned linear systems 2 Convergence of the upper Jacobi upper Gauss Seidel iterative methods Consider the usual splitting of A,namely, A = D L U, (21) where D is adiagonal matrix L U are strictly lower upper triangular matrices, respectively Throughout this paper we assume that A is a nonsingular M-matrix So, without loss of generality, we replace (21) by A = I L U (22) Applying P(α) on (11) we obtain the equivalent linear system Ã(α)x = b(α) with Ã(α) = (I + S(α))A b(α) = (I + S(α))b, (23) where, if needed, we will write Ã(α) = D(α) L(α) Ũ(α) (24) where D(α) is diagonal L(α) Ũ(α) strictly lower strictly upper triangular matrices By the equalities above, we have Ã(α) = I L U + S(α) S(α)L S(α)U, with S(α)U = 0 The elements a ij (α) of Ã(α) are given by the expression: a ij, 2 i n, a ij (α) = (1 α)a 1n, i = 1, j = n, a 1 j αa 1n a nj, i = 1, j n Requesting that ã 1n (α) = (1 α)a 1n 0, if α 1, the nonpositivity of all the off-diagonal elements will be preserved so will the Z-matrix character of Ã(α), ã 11 = 1 αa 1n a n1 > 0 For α [0, 1,definethematrices D(α) := diag(αa 1n a n1, 0,,0) (25) S(α)L = (P(α) I)L := D α + U α, (26) where U α is the strictly upper triangular components of S(α)L Bythefact that S(α)U = 0 the preceding discussion, the three matrices on the right side of (24) are given by D(α) = I D α, L(α) = L, Ũ(α) = U S(α) + U α (27) The diagonal elements of D(α) are positive while those of L(α) Ũ(α) are non-negative Definition 21 Let B be any n n matrix with zero diagonal entries We call B = U + L L ω := (I ωu) 1 {ωl + (1 ω)i} the upper Jacobi upper successive overrelaxation matrix, respectively Especially, we call L 1 Gauss Seidel matrix the upper

3 Z-D Wang, T-Z Huang / Applied Mathematics Letters 19 (2006) Forthe needs of one of our main statements the following splitting will be considered: M(α) N(α) = (I + S(α)) (I + S(α))(L + U), Ã(α) = M (α) N (α) = I (U S(α) + U α + D α + L), M (α) N (α) = (I D α ) (U S(α) + U α + L) Below we define the upper Jacobi type iterative matrices associated with the above splittings: B(α) = B := M 1 (α)n(α) = U + L, B (α) := M 1 (α)n (α) = I (I + S(α))A = U S(α) + U α + D α + L, B(α) B (α) := M 1 (α)n (α) = (I D α ) 1 (I (I + S(α))A D α ) = (I D α ) 1 (U S(α) + U α + L), as well as the splittings that define the upper Gauss Seidel type matrices: M(α) N(α) = (I (U S(α))) (I + S(α))L, Ã(α) = M (α) N (α) = ((I (U S(α))) U α ) (D α + L), M (α) N (α) = ((I (U S(α))) U α D α ) L H (α) H := (I U) 1 L, H (α) := ((I (U S(α))) U α ) 1 (D α + L), H(α) H (α) := ((I (U S(α))) D α U α ) 1 L (28) (29) (210) (211) Lemma 21 Let the upper Jacobi matrix B := U + Lbeanon-negative n nmatrix with zero diagonal entries, let L 1 be the upper Gauss Seidel matrix, the special case ω = 1 for L ω Then, one only one of the following relations is valid: 1 ρ(b) = ρ(l 1 ) = <ρ(l 1 )<ρ(b) <1 3 1 = ρ(b) = ρ(l 1 ) 4 1 <ρ(b) <ρ(l 1 ) (Remark Thus, the upper Jacobi matrix B the upper Gauss Seidel matrix L 1 are either both convergent, or both divergent) Proof Similar to the proof of the Stein Rosenberg theorem in [4 6, the proof of this lemma is easy Theorem 21 Let Abe a nonsingular M-matrix (a) For any α [0, 1, wehave that: There exists y R n,with y 0,suchthat B (α)y By, (212) ρ( B(α)) ρ(b (α)) 1, (213) ρ( H(α)) ρ(h (α)) ρ(h )<1, (214) ρ( H(α)) ρ( B(α)), ρ(h (α)) ρ(b (α)), ρ(h )<ρ(b) <1; (215) (b) Suppose that A is irreducible Then: (i) For α [0, 1,provided that α 0,thematrices B(α), B (α) Bare irreducible all the inequalities in (213) (215) are strict Moreover, there holds ρ(b (α)) ρ(b); (216) (ii) For α = 1, the(n 1) (n 1) matrices B 1 (1) B 1 (1) of the top left corner of B (1) B(1) are irreducible all the inequalities in (213) (216) are strict

4 1032 Z-D Wang, T-Z Huang / Applied Mathematics Letters 19 (2006) Proof (a) For (212):Toprove (212) we need the expressions of the non-negative elements of the two Jacobi iteration matrices involved Below we give the elements for all three matrices in (29): b ii = 0, i N, b ij = a ij, i, j N, j i (217) b 11 (α) = αa 1na n1 = αb 1n b n1, b 1 j (α) = αa 1na nj a 1 j = αb 1n b nj + b 1 j, 2 j n 1, b 1n (α) = (α 1)a 1n = (1 α)b 1n, (218) b ii (α) = 0 2 i n b ij (α) = a ij = b ij, 2 i n, i j b ii (α) = 0 b 1 j (α) = αa 1na nj a 1 j = αb 1nb nj + b 1 j 2 j n 1, 1 αa 1n a n1 1 αb 1n b n1 b 1n (α) = (α 1)a 1n = (1 α)b (219) 1n, 1 αa 1n a n1 1 αb 1n b n1 b ij (α) = a ij = b ij, 2 i n, i j For the non-negative Jacobi iteration matrix B there exists a non-negative vector y such that By = ρ(b)y Equating the first row of the two vector replacing the elements b 1 j of B in terms of the elements b 1 j (α) of B (α) using (217) (218), wesuccessively obtain ρ(b)y 1 = n 1 b 1 j y j = b 1n y n + b 1 j y j = (b 1n (α) + αb n 1 1n)y n + (b 1 j (α) αb 1nb nj )y j = (b 1n (α) + αb n 1 1n)y n + (b 1 j (α) αb 1nb nj )y j + b 11 (α)y 1 b 11 (α)y 1 = b 1 j (α)y n 1 j αb 1n b nj y j + αb 1n y n αb 1n b n1 y 1 = b 1 j (α)y n 1 j αb 1n b nj y j + αb 1n y n By the fact that ρ(b)y n = n 1 b njy j replacing in (220),wehave ρ(b)y 1 = b 1 j (α)y j + αb 1n ( 1 ρ(b) 1 ) n 1 (220) b nj (α)y j (221) Since the second term on the sum in (221) is non-negative, b ij (α)y j b ij y j (222) Then, (212) follows from (222) For (213): ForaZ-matrix A the statement A is a nonsingular M-matrix is equivalent to the statement there exists a positive vector y(> 0) R n such that Ay > 0 (see Theorem 623 Condition I 27 of [6) But P(α) = I + S(α) 0, implies that Ã(α)y = P(α)Ay > 0

5 Z-D Wang, T-Z Huang / Applied Mathematics Letters 19 (2006) Consequently, Ã(α), whichisaz-matrix,is a nonsingular M-matrix So, the last two splittings in (28)are regular ones because M 1 (α) = I 1 = I 0, N (α) 0 M 1 (α) = (I D α ) 1 0, N (α) 0, so they are convergent For a Z-matrix, the statement A is a nonsingular M-matrix is equivalent to the statement all the principal minors of A are positive (see Theorem 622, condition (A 1 ) of [6) So, we have0 a 1n a n1 < 1 Thus, 0 αa 1n a n1 < 1, 0 < 1 αa 1n a n1 1M 1 (α) M 1 (α), the left inequality in (213) is true For (214): Consider the splittings (210) that define the iteration matrices in (211) Thematrix M(α) = I (U S(α)) of the first splitting isupper triangular with units on the diagonal, elements of the first row the last column (1 α)a 1n, remaining ones those of the strictly lower triangular part of ASo, all the off-diagonal elements of M(α) are nonpositive therefore M(α) is a nonsingular M-matrix which implies that M 1 (α) 0 Also, (I + S(α))L 0, so the first splitting in (210) is a regular one M (α) can be written as M (α) = M(α) U α = M(α)(I M 1 (α)u α ), setting Ū = M 1 (α)u α 0, we have M 1 (α) = (I Ū) 1 M 1 (α) = (I + Ū + Ū 2 + +Ū n 1 )M 1 (α) 0 (223) Since N (α) = D α + L 0, the second splitting in (210) is also a regular one The last splitting is a regular one since Ã(α) is a nonsingular M-matrix so is M (α) since the latter is derived from the former by setting some off-diagonal elements equal to zero N (α) = L + L α 0 The inequalities in (214) are established because we notice that N(α) = U α + D α + L N (α) = D α + L N (α) = L For (215): SinceA is a nonsingular M-matrix, the rightmost inequality is a straightforward implication of Lemma 21 as was mentioned before The other two inequalities in (215) are implied directly by the facts that Ã(α) is a nonsingular M-matrix, the last two pairs of splittings in (28) (210) Fromthefour matrices involved, H(α), B(α), H (α) B (α), areproduced, are regular ones with U S(α) + U α + L L U S(α) + U α + D α + L D α + L (b): For α [0, 1), Ã(α) is irreducible because it inherits the nonzero structure of the irreducible matrix A (i) of (b): For (213) (216): Byvirtue of the irreducibility of the corresponding matrices involved, the theorem used previously also can be applied to prove the strict inequalities in (213) (215) Similar to Theorem 22 of [7, (216) is easily proved (ii) of (b): We consider the block partitions [ [ A1 a h I1 a h A = av T, P(1) = 1 0n 1 T 1 [Ã1 (1) 0 Ã(1) = n 1 av T 1, (224) Then the associated block upper Jacobi upper Gauss Seidel iteration matrices will be [ [ B1 a h B = av T, B B (1) = 1 (1) 0 n 1 0 av T, 0 [ (225) B B(1) = 1 (1) 0 n 1 av T 0 H = [ H1 0 n 1 av T, H (1) = 0 [ H 1 (1) 0 n 1 av T 0 H(1) = [ H 1 (1) 0 n 1 a T v 0, (226)

6 1034 Z-D Wang, T-Z Huang / Applied Mathematics Letters 19 (2006) For (213) (216): Bystudying the structure of the matrices B 1, B 1 (1), B 1 (1), H 1, H 1 (1) H 1 (1), wecan find out that the associated irreducibility properties hold for these matrices So the theorems used previously also can be applied in each case to prove the strict inequalities (213) (215) Similar to Theorem 22 of [7, (216) is easily proved Lemma 22 ([8) Let A 1,A 2 R n,n A i = M i N i,i = 1, 2,beweak splittings (T i = M 1 i N i 0,i = 1, 2) If the Perron eigenvector z 2 ( 0) of T 2 satisfies T 1 z 2 T 2 z 2,thenρ(T 1 ) ρ(t 2 ) Theorem 22 Let Abe a nonsingular M-matrix Then, for 0 α α 1,wehave ρ( B(α )) ρ( B(α)) ρ( H(α )) ρ( H(α)) (227) Proof Note that the upper Jacobi upper Gauss Seidel iteration matrices associated with any A = D L U are the same as those associated with D 1 A = I D 1 L D 1 UObservethat by virtue of Lemma 21,thenature of the vector y in (212) (213), isρ( B(α)) ρ(b) By(214), ρ( H(α)) ρ(h )Therefore, the Jacobi the Gauss Seidel iterative methods associated with a preconditioned matrix Ã(α), are no worse than the corresponding ones of the unpreconditioned matrix ASince D 1 Ã has the same Jacobi Gauss Seidel iteration matrices with Ã, its elements, denoted by thesamesymbols as those of Ã, are ã ii = a ii = 1, 1 i n, ã ij = a ij, i 1, ã 1 j = a 1 j αa 1n a nj 1 αa 1n a n1, 2 j n 1, ã 1n = (1 α)a 1n 1 αa 1n a n1 Consider β which is defined by β = 0 ifα = 1, β = α α 1 α if α 1 (228) Apply to D 1 Ã the preconditioner P(β) The upper Jacobi the upper Gauss Seidel iterative methods associated with the new preconditioned matrix Ã(β) = P(β) D 1 Ã will be no worse than the ones corresponding to D 1 ÃTheelements a ij of the matrix D 1 (β) Ã(β) will be given by the sameexpressions as those in (228) where the a ij will be replaced by a ij the α by βthea ij are given by ã ii = 1, 1 i n, ã ij =ã ij, i 1, ã 1 j = ã1 j βã 1n ã nj 1 βã 1n ã n1, 2 j n 1, ã 1n = (1 β)ã 1n 1 βã 1n ã n1 Substituting in (229) the ã ij β,aftersomesimple calculation, we obtain (229) ã ii = 1, 1 i n, ã ij =ã ij, i 1, ã 1 j = a 1 j α a 1n a nj 1 α a 1n a n1, 2 j n 1, ã 1n = (1 α )a 1n 1 α, a 1n a n1 which proves (227)

7 Z-D Wang, T-Z Huang / Applied Mathematics Letters 19 (2006) Numerical examples Fig 1 Interior mesh point of five-point difference approximation Example 1 In order to obtain the numerical solution of the Laplace equation 2 u(x, y) x u(x, y) y 2 = u xx (x, y) + u yy (x, y) = 0, under a uniform square mesh of five-point difference approximations, the interior mesh points as shown in Fig 1, we can obtain the linear system Ax = b where A = If we use the preconditioned method, for α = 09, we have ρ(b(α )) = 07239, ρ(b (α )) = ρ( B(α )) = 07176

8 1036 Z-D Wang, T-Z Huang / Applied Mathematics Letters 19 (2006) Analogously, we obtain ρ(h (α )) = 05241, ρ(h (α )) = ρ( H(α )) = For α = 05, we obtain ρ( H(α )) = <ρ( H(α)) = ρ( B(α )) = <ρ( B(α)) = Example 2 Let the coefficient matrix A of (11) be 1 q r s q s 1 q r q q s s A = r, 1 q r s q s 1 q s r q s 1 where q = p/n, r = p/(n + 1) s = p/(n + 2) [11 Here, we let n = 10 p = 1 If we use the preconditioned method, for α = 09, we have ρ(b(α )) = 08227, ρ(b (α )) = ρ( B(α )) = Analogously, we obtain ρ(h (α )) = 06804, ρ(h (α )) = ρ( H(α )) = For α = 05, we obtain ρ( H(α )) = <ρ( H(α)) = ρ( B(α )) = <ρ( B(α)) = Acknowledgement The second author was supported by the NCET in universitiesofchina foundation of national key lab, Beijing Appl Phy Math Institute References [1 DJ Evans, MM Martins, ME Trigo, The AOR iterative method for new preconditioned linear systems, J Comput Appl Math 132 (2001) [2 A Hadjimos, Accelerated overelaxation method, Math Comp 32 (1978) [3 AD Gunawardena, SK Jain, L Snyder, Modified iterative methods for consistent linear systems, Linear Algebra Appl 41 (1981) [4 RS Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, New York, 2000 [5 DM Young, Iterative Solution of Large Linear Systems, Academic Press, New York, London, 1971 [6 A Berman, RJ Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979 Reprinted by SIAM, Philadelphia, PA, 1994 [7 JP Milaszewicz, Improving Jacobi Gauss Seidel iterations, Linear Algebra Appl 93 (1987) [8 A Hadjidimos, D Noutsos, M Tzoumas, More on modifications improvements of classical iterative schemes for M-matrices, Linear Algebra Appl 364 (2003) [9 W Li, W Sun, Modified Gauss Seidel type methods Jacobi type methods for Z-matrices, Linear Algebra Appl 317 (2000) [10 T Kohno, H Kotakemori, H Niki, M Usui,Improving the Gauss Seidel method for Z-matrices, Linear Algebra Appl 267 (1997) [11 M Usui, H Niki, T Kohno, Adaptive Gauss Seidel method for linear systems, Int J Comput Math 51 (1994)