- Email: [email protected]
On the convergence of parallel multisplitting asynchronous GAOR method for H-matrix
Applied Mathematics and Computation 160 (2005) 477–485 www.elsevier.com/locate/amc
On the convergence of parallel multisplitting asynchronous GAOR method for H -matrix Dongjin Yuan Department of Mathematics, Yangzhou University, Jiangsu 225002, PR China
Abstract In this paper we establish several algorithms of parallel chaotic generalized AOR iterative methods for solving large nonsingular systems based on some given models. Under some different assumptions of coefficient matrix A and its multisplittings we obtain corresponding sufficient conditions of convergence for some relaxed parameters. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Multisplitting; Parallel; Chaotic; Convergence; Generalized AOR method; H -matrix
1. Introduction Parallel multisplitting iterative method for solving a large system of linear equations Ax ¼ b;
ð1Þ
where A 2 Rnn , x 2 Rn , b 2 Rn , take two basic forms, synchronous when all of the processor wait until they are updated with the results of the current iteration before they begin the next iteration or asynchronous when they act more or less independently of each other, using possibly delayed iterative values of the output of the other processors in computing their next iterate. In view of
E-mail address: [email protected] (D. Yuan). 0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.11.015
478
D. Yuan / Appl. Math. Comput. 160 (2005) 477–485
the potential time saving inherent in them, asynchronous iterative methods, or chaotic as they are often called, have attracted much attention since the early paper of Chazan and Miranker [2] introduced them in the context of point iterative schemes. Naturally, their convergence is of crucial interest and a number of convergence results (such as [1,3–6,8] etc.) have been obtained. In particular, the convergence of three relaxed chaotic parallel AOR methods have been investigated in [6,8]. In this paper, we will establish some algorithms of chaotic parallel generalized AOR method and investigate their convergence for H -matrices. Let us consider the splitting of the matrix A of (1) as follows: A ¼ D CL CU
ð2Þ
with D ¼ diag A and CL , CU strictly lower and upper triangular matrices obtained from A. The generalized accelerated overrelaxation (AOR) method, as in [7], is given by 1
xðkþ1Þ ¼ F ða; XÞxðkÞ þ ðI aXLÞ b;
k ¼ 1; 2; . . . ;
ð3Þ
where F ða; XÞ ¼ ðI aXLÞ1 ½I X þ ð1 aÞXL þ XU is an iterative matrix and L ¼ D1 CL , U ¼ D1 CU , X ¼ diagðx1 ; x2 ; . . . ; xn Þ with xi 2 Rþ and a real parameter. The matrix B¼LþU is the Jacobi iterative matrix. For a ¼ c=x and X ¼ xI, the generalized AOR method reduces to the AOR method with the iteration matrix 1
F ðc; xÞ ¼ ðI cLÞ ½ð1 xÞI þ ðx cÞL þ xU : The main purpose of this paper is to present several algorithms of relaxed parallel chaotic generalized AOR schemes for solving large nonsingular system (1), in which the coefficient matrix A is an H -matrix, and investigate the corresponding convergence of these algorithms.
2. Notation and algorithms Let us first introduce some of the notation and terminology which will be used in this paper. For x ¼ ðx1 ; x2 ; . . . ; xn ÞT 2 Rn and A ¼ ðaij Þ 2 Rnn by x P 0 we mean that xi P 0 for i ¼ 1; . . . ; n, and by A P 0 that aij P 0 for i; j ¼ 1; . . . ; n, in which case we say that x and A are nonnegative. For
D. Yuan / Appl. Math. Comput. 160 (2005) 477–485
479
A; B 2 Rnn , we write A 6 B if aij 6 bij hold for all entries of A ¼ ðaij Þ and B ¼ ðbij Þ. By jAj ¼ ðjaij jÞ we define the absolute value of A 2 Rnn , it is a nonnegative matrix satisfying jABj 6 jAj jBj. The notation hAi ¼ ðhaiij Þ represents the comparison matrix of A 2 Rnn where jaij j if i ¼ j; haiij ¼ jaij j if i 6¼ j: A matrix A ¼ ðaij Þ 2 Rnn is an M-matrix if it is nonsingular with A1 P 0 and aij 6 0 for all i 6¼ j. It is an H -matrix if hAi is an M-matrix, and an L-matrix if aii > 0 for i ¼ 1; . . . ; n, and aij 6 0 for all i 6¼ j. A splitting of a matrix A ¼ ðaij Þ 2 Rnn is a pair of matrices M; N 2 Rnn with detðMÞ 6¼ 0 such that A ¼ M N . It is called a nonnegative splitting if M 1 N P 0 and an M-splitting if M is an M-matrix and N P 0. For any s P 2 a multisplitting of A 2 Rnn is a collection of s triples ðMl ; Nl ; El Þ of n n real matrices, l ¼ 1; 2; . . . ; s, for which each El is nonnegative diagonal, each Ml is invertible and the equations A ¼ Ml N l ;
l ¼ 1; 2; . . . ; s
ð4Þ
and s X
El ¼ I
ð5Þ
l¼1
are satisfied. Finally, a sequence of sets Pi with Pi f1; . . . ; sg is admissible if every integer 1; . . . ; s appears infinitely often in the Pi , while such an admissible sequence is regulated if there exists a positive integer T such that each of the integer 1; . . . ; s appears at least once in any T consecutive sets of the sequence. Using the given models in [1,6,8] and (3) we can now describe three algorithms of relaxed parallel chaotic generalized AOR method by above notation. Algorithm 2.1. Choose xð0Þ 2 Rn arbitrarily. For k ¼ 1; 2; . . ., until convergence, perform xðkþ1Þ ¼
s X
l
El Fl l;k ða; X; xðkÞ Þ;
l¼1
Fl ða; X; xðkÞ Þ ¼ ðI aXLl Þ1 ½I X þ ð1 aÞXLl þ XUl xðkÞ þ ðI aXLl Þ1 b
with a P 0, xi > 0, and ll;k P 1.
480
D. Yuan / Appl. Math. Comput. 160 (2005) 477–485 l
where Fl l;k is the ll;k th composition of the affine mapping satisfying Fl Fl Fl ; ll;k P 1; ll;k Fl ¼ I; ll;k ¼ 0 and B ¼ Ll þ Ul ;
l ¼ 1; 2; . . . ; s:
By using a suitable positive relaxation parameter b, we then get the following relaxed algorithm which is based on Algorithm 2.1. Algorithm 2.2. Choose xð0Þ 2 Rn arbitrarily. For k ¼ 1; 2; . . ., until convergence, perform xðkþ1Þ ¼ b
s X
l
El Fl l;k ða; X; xðkÞ Þ þ ð1 bÞxðkÞ ;
l¼1
Fl ða; X; xðkÞ Þ ¼ ðI aXLl Þ1 ½I X þ ð1 aÞXLl þ XUl xðkÞ þ ðI aXLl Þ1 b
with b > 0, a P 0, xi > 0, and ll;k P 1. Next if we consider the case of relaxed chaotic generalized AOR method and assume that the index sequence fPi g is admissible and regulated, then we can get the following algorithm. Algorithm 2.3. Choose xð0Þ 2 Rn arbitrarily. For k ¼ 1; 2; . . ., until convergence, perform ! X X l ðkþ1Þ x ¼ I b El xðkÞ þ b El Fl l;k ða; X; xðkrk þ1Þ Þ; l2Pi
l2Pi 1
Fl ða; X; xðkrk þ1Þ Þ ¼ ðI aXLl Þ ½I X þ ð1 aÞXLl þ XUl xðkrk þ1Þ l
þ ðI aXLl Þ b; T ðkrð1;kÞÞ ðkrð2;kÞÞ xðkrk þ1Þ ¼ x1 ; x2 ; . . . ; xnðkrðn;kÞÞ with b > 0, a P 0, xi > 0, ll;k P 1, and ; 6¼ Pi f1; . . . ; sg. Now we can point that because the AOR method is only the special case of the generalized AOR method, the corresponding algorithms in [6,8] are also the special cases of the above Algorithms 2.1–2.3.
D. Yuan / Appl. Math. Comput. 160 (2005) 477–485
481
3. Convergence of the algorithms Before starting our convergence results concerning above algorithms we should first introduce the following two lemmas, which have been presented in [8]. Lemma 3.1. If A is an H -matrix, then 1
(a) jA1 j 6 hAi ; (b) there exists a diagonal matrix P whose diagonal entries are positive such that AP is by rows strictly diagonally dominant, i.e., hAiPe > 0
ð6Þ T
with e ¼ ð1; . . . ; 1Þ . Lemma 3.2. Let A be an M-matrix, and let the splitting A¼M N be an M-splitting. If P is the diagonal matrix defined in Lemma 3.1, then kP 1 M 1 NP k1 < 1:
ð7Þ
Using above two lemmas, now we can prove one of our main results, which is a sufficient condition for the convergence of Algorithm 2.1. Theorem 3.1. Let A 2 Rnn be an H -matrix and ðD ðCL Þl ; ðCU Þl ; El Þ, l ¼ 1; 2; . . . ; s, be a multisplitting of A. Assume that for l ¼ 1; 2; . . . ; s, we have (1) ðCL Þl is the strictly lower triangular matrices and ðCU Þl is the matrices such that the equalities A ¼ D ðCL Þl ðCU Þl hold. (2) hAi ¼ jDj jðCL Þl j jðCU Þl j ¼ jDj jB1 j, where jB1 j ¼ jðCL Þl j þ jðCU Þl j. Then the sequence fxðkÞ g generated by Algorithm 2.1 converges to the solution vector of system (1) for any starting vector xð0Þ 2 Rn if ða; xi Þ, i ¼ 1; 2; . . . ; n 2 S1 , where S1 ¼ fða; xi Þ 2 R2 : 0 6 a 6 1; 0 < xi < 2=ð1 þ qÞ; i ¼ 1; 2; . . . ; ng 1
with q ¼ qðjDj jB1 jÞ. Proof. Let us first denote Ll ¼ D1 ðCL Þl , Ul ¼ D1 ðCU Þl , l ¼ 1; 2; . . . ; s, then we can define the iterative matrix in Algorithm 2.1 H ða; XÞk ¼
s X l¼1
1
El fðI aXLl Þ ½I X þ ð1 aÞXLl þ XUl g
ll;k
:
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D. Yuan / Appl. Math. Comput. 160 (2005) 477–485
It is clear that we need to find a constant r with 0 6 r < 1 and some norm, which are independent of k, such that for k P 1, kH ða; XÞk k 6 r. Since A is an H -matrix and for l ¼ 1; 2; . . . ; s, Ll is a strictly lower triangular matrix, we see that each hI aXLl i is an M-matrix for l ¼ 1; 2; . . . ; s, and hI aXLl i
1
¼ ðI aXjLl jÞ
1
P 0:
Hence each ðI aXLl Þ is an H -matrix for l ¼ 1; 2; . . . ; s, and we have the following inequality 1
jðI aXLl Þ j 6 hI aXLl i
1
1
¼ ðI aXjLl jÞ :
From this relation it follows that 1
1
jðI aXLl Þ ½I X þ ð1 aÞXL þ XU j 6 hI aXLl i jI X 1
þ ð1 aÞXLl þ XUl j 6 ðI aXjLl jÞ ½jI Xj þ j1 ajXjLl j þ XjUl j: Case 1. xi 6 1. In this case jI Xj þ j1 ajXjLl j þ XjUl j ¼ I X þ XjLl j ajXjLl j þ XjUl j: We denote Ml ða; XÞ ¼ I aXjLl j and Nl ða; XÞ ¼ jI Xj þ ð1 aÞXjLl j þ XjUl j: Evidently, for l ¼ 1; 2; . . . ; s, we have the following relation Ml ða; XÞ Nl ða; XÞ ¼ X XjLl j XjUl j ¼ XðI jBjÞ: Since, for l ¼ 1; 2; . . . ; s, Ml ða; XÞ are M-matrices and Nl ða; XÞ P 0, the splittings Ml ða; XÞ Nl ða; XÞ are M-splittings of the matrix XðI jBjÞ, which is M-matrix. Case 2. xi > 1. Suppose x ¼ max1 6 i 6 n xi . We have jI Xj þ j1 ajXjLl j þ XjUl j 6 ðx 1ÞI þ ð1 aÞxjLl j þ xjUl j and 1
ðI aXjLl jÞ
1
6 ðI axjLl jÞ :
We also denote Ml ða; xÞ ¼ I axjLl j
D. Yuan / Appl. Math. Comput. 160 (2005) 477–485
483
and Nl ða; xÞ ¼ ðx 1ÞI þ ð1 aÞxjLl j þ xjUl j; then Ml ða; xÞ Nl ða; xÞ ¼ ð1 j1 xjÞI xjBj: It is easy to prove (see from [6,8]) that ð1 j1 xjÞI xjBj is an M-matrix. Since, for l ¼ 1; 2; . . . ; s, Ml ða; XÞ are M-matrices and Nl ða; XÞ P 0, the splittings Ml ða; XÞ Nl ða; XÞ are M-splittings of the matrix ð1 j1 xjÞI xjBj. Thus, for Cases 1 and 2, from Lemma 3.2 in the above it derives kP 1 Ml1 ða; XÞNl ða; XÞP k1 < 1;
l ¼ 1; 2; . . . ; s
and hence P 1 jH ða; XÞk jPe 6 6
s X l¼1 s X
El fP 1 Ml1 ða; XÞNl ða; XÞP g
ll;k
e
l
El kP 1 Ml1 ða; XÞNl ða; XÞP k1l;k e
l¼1
6 max kP 1 Ml1 ða; XÞNl ða; XÞP k1 e 16l6s
which implies kP 1 H ða; XÞk P k1 6 max kP 1 Ml1 ða; XÞNl ða; XÞP k1 < 1: 16l6s
Consequently jH ða; XÞk jPe ¼ P ðP 1 jH ða; XÞk jP Þe 6 P kP 1 H ða; XÞk P k1 e 6 max kP 1 Ml1 ða; XÞNl ða; XÞP k1 Pe; 16l6s
l ¼ 1; 2; . . . ; s:
Let us denote r ¼ max kP 1 Ml1 ða; XÞNl ða; XÞP k1 ; 16l6s
then kH ða; XÞk k 6 r < 1: We have completed the proof.
h
Theorem 3.2. Let A 2 Rnn be an H -matrix and ðD ðCL Þl ; ðCU Þl ; El Þ, l ¼ 1; 2; . . . ; s, be a multisplitting of A. Assume that for l ¼ 1; 2; . . . ; s, we have (1) ðCL Þl is the strictly lower triangular matrices and ðCU Þl is the matrices such that the equalities A ¼ D ðCL Þl ðCU Þl hold.
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D. Yuan / Appl. Math. Comput. 160 (2005) 477–485
(2) hAi ¼ jDj jðCL Þl j jðCU Þl j ¼ jDj jB1 j, where jB1 j ¼ jðCL Þl j þ jðCU Þl j. (3) P is the diagonal matrix defined in Lemma 3.1 and Ml ða; XÞ, Nl ða; XÞ in Theorem 3.1. Then the sequence fxðkÞ g generated by Algorithm 2.2 converges to the solution vector of system (1) for any starting vector xð0Þ 2 Rn if ða; b; xi Þ, i ¼ 1; 2; . . . ; n 2 S2 , where S2 ¼ fða; b; xi Þ 2 R3 : 0 6 a 6 1; 0 < b < 2=ð1 þ hÞ; 0 < xi < 2=ð1 þ qÞ; i ¼ 1; 2; . . . ; ng 1
with q ¼ qðjDj jB1 jÞ, h ¼ max1 6 l 6 s kP 1 Ml1 ða; XÞNl ða; XÞP k1 . Proof. Let us define the iterative matrix in Algorithm 2.2 H ða; b; XÞk ¼ bH ða; XÞk þ ð1 bÞI; where H ða; XÞk ¼
s X
El fðI aXLl Þ1 ½I X þ ð1 aÞXLl þ XUl gll;k :
l¼1
Similar to the proof of Theorem 3.1 we only need to prove that there exists a constant r with 0 6 r < 1, which is independent of k, such that kP 1 H ða; b; XÞk P k1 6 r: From the relation in the proof of Theorem 3.1 kP 1 H ða; XÞk P k1 6 max kP 1 Ml1 ða; XÞNl ða; XÞP k1 < 1; 16l6s
we obtain kP 1 H ða; b; XÞk P k1 6 bkP 1 H ða; XÞk P k1 þ j1 bj 6 b max kP 1 Ml1 ða; XÞNl ða; XÞP k1 þ j1 bj: 16l6s
Clearly, if xi (i ¼ 1; 2; . . . ; n) and b satisfy the condition of this theorem then r b max kP 1 Ml1 ða; XÞNl ða; XÞP k1 þ j1 bj < 1 16l6s
which completes the proof.
h
Using the proving process of Theorems 3.1, 3.2 and [8, Theorem 2.8] we get the following convergence of Algorithm 2.3. Theorem 3.3. Let A 2 Rnn be an H -matrix and ðD ðCL Þl ; ðCU Þl ; El Þ, l ¼ 1; 2; . . . ; s, be a multisplitting of A. Assume that for l ¼ 1; 2; . . . ; s, we have
D. Yuan / Appl. Math. Comput. 160 (2005) 477–485
485
(1) ðCL Þl is the strictly lower triangular matrices and ðCU Þl is the matrices such that the equalities A ¼ D ðCL Þl ðCU Þl hold. (2) hAi ¼ jDj jðCL Þl j jðCU Þl j ¼ jDj jB1 j, where jB1 j ¼ jðCL Þl j þ jðCU Þl j. (3) P is the diagonal matrix defined in Lemma 3.1 and Ml ða; XÞ, Nl ða; XÞ in Theorem 3.1. (4) The index sequence fPi g is admissible and regulated. Then the sequence fxðkÞ g generated by Algorithm 2.3 converges to the solution vector of system (1) for any starting vector xð0Þ 2 Rn if ða; b; xi Þ, i ¼ 1; 2; . . . ; n 2 S3 , where S3 ¼ fða; b; xi Þ 2 R3 : 0 6 a 6 1; 0 < b < 2=ð1 þ hÞ; 0 < xi < 2=ð1 þ qÞ; i ¼ 1; 2; . . . ; ng with q ¼ qðjDj1 jB1 jÞ, h ¼ max1 6 l 6 s kP 1 Ml1 ða; XÞNl ða; XÞP k1 . References [1] R. Bru, L. Elsner, M. Neumman, Models of parallel chaotic iterative methods, Linear Algebra Appl. 103 (1988) 175–192. [2] D. Chazan, W. Miranker, Chaotic relaxation, Linear Algebra Appl. 2 (1969) 199–222. [3] L. Elsner, M. Neumman, B. Vemmer, The effect the number of processors on the convergence of the parallel block Jacobi method, Linear Algebra Appl. 154–156 (1991) 311–330. [4] L. Elsner, M. Neumman, Monotonic sequences and rates of convergence of asynchronized iterative methods, Linear Algebra Appl. 180 (1993) 17–33. [5] A. Frommon, G. Mayer, Convergence of relaxed parallel multisplittings methods, Linear Algebra Appl. 119 (1989) 141–152. [6] P.E. Kloeden, D. Yuan, Convergence of relaxed chaotic parallel iterative methods, Bull. Aust. Math. Soc. 50 (1994) 167–176. [7] Y. Song, Convergence of the generalized AOR method, Linear Algebra Appl. 256 (1997) 199–218. [8] Y. Song, D. Yuan, On the convergence of relaxed parallel chaotic iterations for H-matrix, Int. J. Comput. Math. 52 (1994) 195–209.
On the convergence of parallel multisplitting asynchronous GAOR method for H -matrix Dongjin Yuan Department of Mathematics, Yangzhou University, Jiangsu 225002, PR China
Abstract In this paper we establish several algorithms of parallel chaotic generalized AOR iterative methods for solving large nonsingular systems based on some given models. Under some different assumptions of coefficient matrix A and its multisplittings we obtain corresponding sufficient conditions of convergence for some relaxed parameters. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Multisplitting; Parallel; Chaotic; Convergence; Generalized AOR method; H -matrix
1. Introduction Parallel multisplitting iterative method for solving a large system of linear equations Ax ¼ b;
ð1Þ
where A 2 Rnn , x 2 Rn , b 2 Rn , take two basic forms, synchronous when all of the processor wait until they are updated with the results of the current iteration before they begin the next iteration or asynchronous when they act more or less independently of each other, using possibly delayed iterative values of the output of the other processors in computing their next iterate. In view of
E-mail address: [email protected] (D. Yuan). 0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.11.015
478
D. Yuan / Appl. Math. Comput. 160 (2005) 477–485
the potential time saving inherent in them, asynchronous iterative methods, or chaotic as they are often called, have attracted much attention since the early paper of Chazan and Miranker [2] introduced them in the context of point iterative schemes. Naturally, their convergence is of crucial interest and a number of convergence results (such as [1,3–6,8] etc.) have been obtained. In particular, the convergence of three relaxed chaotic parallel AOR methods have been investigated in [6,8]. In this paper, we will establish some algorithms of chaotic parallel generalized AOR method and investigate their convergence for H -matrices. Let us consider the splitting of the matrix A of (1) as follows: A ¼ D CL CU
ð2Þ
with D ¼ diag A and CL , CU strictly lower and upper triangular matrices obtained from A. The generalized accelerated overrelaxation (AOR) method, as in [7], is given by 1
xðkþ1Þ ¼ F ða; XÞxðkÞ þ ðI aXLÞ b;
k ¼ 1; 2; . . . ;
ð3Þ
where F ða; XÞ ¼ ðI aXLÞ1 ½I X þ ð1 aÞXL þ XU is an iterative matrix and L ¼ D1 CL , U ¼ D1 CU , X ¼ diagðx1 ; x2 ; . . . ; xn Þ with xi 2 Rþ and a real parameter. The matrix B¼LþU is the Jacobi iterative matrix. For a ¼ c=x and X ¼ xI, the generalized AOR method reduces to the AOR method with the iteration matrix 1
F ðc; xÞ ¼ ðI cLÞ ½ð1 xÞI þ ðx cÞL þ xU : The main purpose of this paper is to present several algorithms of relaxed parallel chaotic generalized AOR schemes for solving large nonsingular system (1), in which the coefficient matrix A is an H -matrix, and investigate the corresponding convergence of these algorithms.
2. Notation and algorithms Let us first introduce some of the notation and terminology which will be used in this paper. For x ¼ ðx1 ; x2 ; . . . ; xn ÞT 2 Rn and A ¼ ðaij Þ 2 Rnn by x P 0 we mean that xi P 0 for i ¼ 1; . . . ; n, and by A P 0 that aij P 0 for i; j ¼ 1; . . . ; n, in which case we say that x and A are nonnegative. For
D. Yuan / Appl. Math. Comput. 160 (2005) 477–485
479
A; B 2 Rnn , we write A 6 B if aij 6 bij hold for all entries of A ¼ ðaij Þ and B ¼ ðbij Þ. By jAj ¼ ðjaij jÞ we define the absolute value of A 2 Rnn , it is a nonnegative matrix satisfying jABj 6 jAj jBj. The notation hAi ¼ ðhaiij Þ represents the comparison matrix of A 2 Rnn where jaij j if i ¼ j; haiij ¼ jaij j if i 6¼ j: A matrix A ¼ ðaij Þ 2 Rnn is an M-matrix if it is nonsingular with A1 P 0 and aij 6 0 for all i 6¼ j. It is an H -matrix if hAi is an M-matrix, and an L-matrix if aii > 0 for i ¼ 1; . . . ; n, and aij 6 0 for all i 6¼ j. A splitting of a matrix A ¼ ðaij Þ 2 Rnn is a pair of matrices M; N 2 Rnn with detðMÞ 6¼ 0 such that A ¼ M N . It is called a nonnegative splitting if M 1 N P 0 and an M-splitting if M is an M-matrix and N P 0. For any s P 2 a multisplitting of A 2 Rnn is a collection of s triples ðMl ; Nl ; El Þ of n n real matrices, l ¼ 1; 2; . . . ; s, for which each El is nonnegative diagonal, each Ml is invertible and the equations A ¼ Ml N l ;
l ¼ 1; 2; . . . ; s
ð4Þ
and s X
El ¼ I
ð5Þ
l¼1
are satisfied. Finally, a sequence of sets Pi with Pi f1; . . . ; sg is admissible if every integer 1; . . . ; s appears infinitely often in the Pi , while such an admissible sequence is regulated if there exists a positive integer T such that each of the integer 1; . . . ; s appears at least once in any T consecutive sets of the sequence. Using the given models in [1,6,8] and (3) we can now describe three algorithms of relaxed parallel chaotic generalized AOR method by above notation. Algorithm 2.1. Choose xð0Þ 2 Rn arbitrarily. For k ¼ 1; 2; . . ., until convergence, perform xðkþ1Þ ¼
s X
l
El Fl l;k ða; X; xðkÞ Þ;
l¼1
Fl ða; X; xðkÞ Þ ¼ ðI aXLl Þ1 ½I X þ ð1 aÞXLl þ XUl xðkÞ þ ðI aXLl Þ1 b
with a P 0, xi > 0, and ll;k P 1.
480
D. Yuan / Appl. Math. Comput. 160 (2005) 477–485 l
where Fl l;k is the ll;k th composition of the affine mapping satisfying Fl Fl Fl ; ll;k P 1; ll;k Fl ¼ I; ll;k ¼ 0 and B ¼ Ll þ Ul ;
l ¼ 1; 2; . . . ; s:
By using a suitable positive relaxation parameter b, we then get the following relaxed algorithm which is based on Algorithm 2.1. Algorithm 2.2. Choose xð0Þ 2 Rn arbitrarily. For k ¼ 1; 2; . . ., until convergence, perform xðkþ1Þ ¼ b
s X
l
El Fl l;k ða; X; xðkÞ Þ þ ð1 bÞxðkÞ ;
l¼1
Fl ða; X; xðkÞ Þ ¼ ðI aXLl Þ1 ½I X þ ð1 aÞXLl þ XUl xðkÞ þ ðI aXLl Þ1 b
with b > 0, a P 0, xi > 0, and ll;k P 1. Next if we consider the case of relaxed chaotic generalized AOR method and assume that the index sequence fPi g is admissible and regulated, then we can get the following algorithm. Algorithm 2.3. Choose xð0Þ 2 Rn arbitrarily. For k ¼ 1; 2; . . ., until convergence, perform ! X X l ðkþ1Þ x ¼ I b El xðkÞ þ b El Fl l;k ða; X; xðkrk þ1Þ Þ; l2Pi
l2Pi 1
Fl ða; X; xðkrk þ1Þ Þ ¼ ðI aXLl Þ ½I X þ ð1 aÞXLl þ XUl xðkrk þ1Þ l
þ ðI aXLl Þ b; T ðkrð1;kÞÞ ðkrð2;kÞÞ xðkrk þ1Þ ¼ x1 ; x2 ; . . . ; xnðkrðn;kÞÞ with b > 0, a P 0, xi > 0, ll;k P 1, and ; 6¼ Pi f1; . . . ; sg. Now we can point that because the AOR method is only the special case of the generalized AOR method, the corresponding algorithms in [6,8] are also the special cases of the above Algorithms 2.1–2.3.
D. Yuan / Appl. Math. Comput. 160 (2005) 477–485
481
3. Convergence of the algorithms Before starting our convergence results concerning above algorithms we should first introduce the following two lemmas, which have been presented in [8]. Lemma 3.1. If A is an H -matrix, then 1
(a) jA1 j 6 hAi ; (b) there exists a diagonal matrix P whose diagonal entries are positive such that AP is by rows strictly diagonally dominant, i.e., hAiPe > 0
ð6Þ T
with e ¼ ð1; . . . ; 1Þ . Lemma 3.2. Let A be an M-matrix, and let the splitting A¼M N be an M-splitting. If P is the diagonal matrix defined in Lemma 3.1, then kP 1 M 1 NP k1 < 1:
ð7Þ
Using above two lemmas, now we can prove one of our main results, which is a sufficient condition for the convergence of Algorithm 2.1. Theorem 3.1. Let A 2 Rnn be an H -matrix and ðD ðCL Þl ; ðCU Þl ; El Þ, l ¼ 1; 2; . . . ; s, be a multisplitting of A. Assume that for l ¼ 1; 2; . . . ; s, we have (1) ðCL Þl is the strictly lower triangular matrices and ðCU Þl is the matrices such that the equalities A ¼ D ðCL Þl ðCU Þl hold. (2) hAi ¼ jDj jðCL Þl j jðCU Þl j ¼ jDj jB1 j, where jB1 j ¼ jðCL Þl j þ jðCU Þl j. Then the sequence fxðkÞ g generated by Algorithm 2.1 converges to the solution vector of system (1) for any starting vector xð0Þ 2 Rn if ða; xi Þ, i ¼ 1; 2; . . . ; n 2 S1 , where S1 ¼ fða; xi Þ 2 R2 : 0 6 a 6 1; 0 < xi < 2=ð1 þ qÞ; i ¼ 1; 2; . . . ; ng 1
with q ¼ qðjDj jB1 jÞ. Proof. Let us first denote Ll ¼ D1 ðCL Þl , Ul ¼ D1 ðCU Þl , l ¼ 1; 2; . . . ; s, then we can define the iterative matrix in Algorithm 2.1 H ða; XÞk ¼
s X l¼1
1
El fðI aXLl Þ ½I X þ ð1 aÞXLl þ XUl g
ll;k
:
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D. Yuan / Appl. Math. Comput. 160 (2005) 477–485
It is clear that we need to find a constant r with 0 6 r < 1 and some norm, which are independent of k, such that for k P 1, kH ða; XÞk k 6 r. Since A is an H -matrix and for l ¼ 1; 2; . . . ; s, Ll is a strictly lower triangular matrix, we see that each hI aXLl i is an M-matrix for l ¼ 1; 2; . . . ; s, and hI aXLl i
1
¼ ðI aXjLl jÞ
1
P 0:
Hence each ðI aXLl Þ is an H -matrix for l ¼ 1; 2; . . . ; s, and we have the following inequality 1
jðI aXLl Þ j 6 hI aXLl i
1
1
¼ ðI aXjLl jÞ :
From this relation it follows that 1
1
jðI aXLl Þ ½I X þ ð1 aÞXL þ XU j 6 hI aXLl i jI X 1
þ ð1 aÞXLl þ XUl j 6 ðI aXjLl jÞ ½jI Xj þ j1 ajXjLl j þ XjUl j: Case 1. xi 6 1. In this case jI Xj þ j1 ajXjLl j þ XjUl j ¼ I X þ XjLl j ajXjLl j þ XjUl j: We denote Ml ða; XÞ ¼ I aXjLl j and Nl ða; XÞ ¼ jI Xj þ ð1 aÞXjLl j þ XjUl j: Evidently, for l ¼ 1; 2; . . . ; s, we have the following relation Ml ða; XÞ Nl ða; XÞ ¼ X XjLl j XjUl j ¼ XðI jBjÞ: Since, for l ¼ 1; 2; . . . ; s, Ml ða; XÞ are M-matrices and Nl ða; XÞ P 0, the splittings Ml ða; XÞ Nl ða; XÞ are M-splittings of the matrix XðI jBjÞ, which is M-matrix. Case 2. xi > 1. Suppose x ¼ max1 6 i 6 n xi . We have jI Xj þ j1 ajXjLl j þ XjUl j 6 ðx 1ÞI þ ð1 aÞxjLl j þ xjUl j and 1
ðI aXjLl jÞ
1
6 ðI axjLl jÞ :
We also denote Ml ða; xÞ ¼ I axjLl j
D. Yuan / Appl. Math. Comput. 160 (2005) 477–485
483
and Nl ða; xÞ ¼ ðx 1ÞI þ ð1 aÞxjLl j þ xjUl j; then Ml ða; xÞ Nl ða; xÞ ¼ ð1 j1 xjÞI xjBj: It is easy to prove (see from [6,8]) that ð1 j1 xjÞI xjBj is an M-matrix. Since, for l ¼ 1; 2; . . . ; s, Ml ða; XÞ are M-matrices and Nl ða; XÞ P 0, the splittings Ml ða; XÞ Nl ða; XÞ are M-splittings of the matrix ð1 j1 xjÞI xjBj. Thus, for Cases 1 and 2, from Lemma 3.2 in the above it derives kP 1 Ml1 ða; XÞNl ða; XÞP k1 < 1;
l ¼ 1; 2; . . . ; s
and hence P 1 jH ða; XÞk jPe 6 6
s X l¼1 s X
El fP 1 Ml1 ða; XÞNl ða; XÞP g
ll;k
e
l
El kP 1 Ml1 ða; XÞNl ða; XÞP k1l;k e
l¼1
6 max kP 1 Ml1 ða; XÞNl ða; XÞP k1 e 16l6s
which implies kP 1 H ða; XÞk P k1 6 max kP 1 Ml1 ða; XÞNl ða; XÞP k1 < 1: 16l6s
Consequently jH ða; XÞk jPe ¼ P ðP 1 jH ða; XÞk jP Þe 6 P kP 1 H ða; XÞk P k1 e 6 max kP 1 Ml1 ða; XÞNl ða; XÞP k1 Pe; 16l6s
l ¼ 1; 2; . . . ; s:
Let us denote r ¼ max kP 1 Ml1 ða; XÞNl ða; XÞP k1 ; 16l6s
then kH ða; XÞk k 6 r < 1: We have completed the proof.
h
Theorem 3.2. Let A 2 Rnn be an H -matrix and ðD ðCL Þl ; ðCU Þl ; El Þ, l ¼ 1; 2; . . . ; s, be a multisplitting of A. Assume that for l ¼ 1; 2; . . . ; s, we have (1) ðCL Þl is the strictly lower triangular matrices and ðCU Þl is the matrices such that the equalities A ¼ D ðCL Þl ðCU Þl hold.
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D. Yuan / Appl. Math. Comput. 160 (2005) 477–485
(2) hAi ¼ jDj jðCL Þl j jðCU Þl j ¼ jDj jB1 j, where jB1 j ¼ jðCL Þl j þ jðCU Þl j. (3) P is the diagonal matrix defined in Lemma 3.1 and Ml ða; XÞ, Nl ða; XÞ in Theorem 3.1. Then the sequence fxðkÞ g generated by Algorithm 2.2 converges to the solution vector of system (1) for any starting vector xð0Þ 2 Rn if ða; b; xi Þ, i ¼ 1; 2; . . . ; n 2 S2 , where S2 ¼ fða; b; xi Þ 2 R3 : 0 6 a 6 1; 0 < b < 2=ð1 þ hÞ; 0 < xi < 2=ð1 þ qÞ; i ¼ 1; 2; . . . ; ng 1
with q ¼ qðjDj jB1 jÞ, h ¼ max1 6 l 6 s kP 1 Ml1 ða; XÞNl ða; XÞP k1 . Proof. Let us define the iterative matrix in Algorithm 2.2 H ða; b; XÞk ¼ bH ða; XÞk þ ð1 bÞI; where H ða; XÞk ¼
s X
El fðI aXLl Þ1 ½I X þ ð1 aÞXLl þ XUl gll;k :
l¼1
Similar to the proof of Theorem 3.1 we only need to prove that there exists a constant r with 0 6 r < 1, which is independent of k, such that kP 1 H ða; b; XÞk P k1 6 r: From the relation in the proof of Theorem 3.1 kP 1 H ða; XÞk P k1 6 max kP 1 Ml1 ða; XÞNl ða; XÞP k1 < 1; 16l6s
we obtain kP 1 H ða; b; XÞk P k1 6 bkP 1 H ða; XÞk P k1 þ j1 bj 6 b max kP 1 Ml1 ða; XÞNl ða; XÞP k1 þ j1 bj: 16l6s
Clearly, if xi (i ¼ 1; 2; . . . ; n) and b satisfy the condition of this theorem then r b max kP 1 Ml1 ða; XÞNl ða; XÞP k1 þ j1 bj < 1 16l6s
which completes the proof.
h
Using the proving process of Theorems 3.1, 3.2 and [8, Theorem 2.8] we get the following convergence of Algorithm 2.3. Theorem 3.3. Let A 2 Rnn be an H -matrix and ðD ðCL Þl ; ðCU Þl ; El Þ, l ¼ 1; 2; . . . ; s, be a multisplitting of A. Assume that for l ¼ 1; 2; . . . ; s, we have
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485
(1) ðCL Þl is the strictly lower triangular matrices and ðCU Þl is the matrices such that the equalities A ¼ D ðCL Þl ðCU Þl hold. (2) hAi ¼ jDj jðCL Þl j jðCU Þl j ¼ jDj jB1 j, where jB1 j ¼ jðCL Þl j þ jðCU Þl j. (3) P is the diagonal matrix defined in Lemma 3.1 and Ml ða; XÞ, Nl ða; XÞ in Theorem 3.1. (4) The index sequence fPi g is admissible and regulated. Then the sequence fxðkÞ g generated by Algorithm 2.3 converges to the solution vector of system (1) for any starting vector xð0Þ 2 Rn if ða; b; xi Þ, i ¼ 1; 2; . . . ; n 2 S3 , where S3 ¼ fða; b; xi Þ 2 R3 : 0 6 a 6 1; 0 < b < 2=ð1 þ hÞ; 0 < xi < 2=ð1 þ qÞ; i ¼ 1; 2; . . . ; ng with q ¼ qðjDj1 jB1 jÞ, h ¼ max1 6 l 6 s kP 1 Ml1 ða; XÞNl ða; XÞP k1 . References [1] R. Bru, L. Elsner, M. Neumman, Models of parallel chaotic iterative methods, Linear Algebra Appl. 103 (1988) 175–192. [2] D. Chazan, W. Miranker, Chaotic relaxation, Linear Algebra Appl. 2 (1969) 199–222. [3] L. Elsner, M. Neumman, B. Vemmer, The effect the number of processors on the convergence of the parallel block Jacobi method, Linear Algebra Appl. 154–156 (1991) 311–330. [4] L. Elsner, M. Neumman, Monotonic sequences and rates of convergence of asynchronized iterative methods, Linear Algebra Appl. 180 (1993) 17–33. [5] A. Frommon, G. Mayer, Convergence of relaxed parallel multisplittings methods, Linear Algebra Appl. 119 (1989) 141–152. [6] P.E. Kloeden, D. Yuan, Convergence of relaxed chaotic parallel iterative methods, Bull. Aust. Math. Soc. 50 (1994) 167–176. [7] Y. Song, Convergence of the generalized AOR method, Linear Algebra Appl. 256 (1997) 199–218. [8] Y. Song, D. Yuan, On the convergence of relaxed parallel chaotic iterations for H-matrix, Int. J. Comput. Math. 52 (1994) 195–209.