On validity of m-step multisplitting preconditioners for linear systems

Applied Mathematics and Computation 126 (2002) 199–211 www.elsevier.com/locate/amc

On validity of m-step multisplitting preconditioners for linear systems q Zhi-Hao Cao *, Yang Wang Laboratory of Mathematics for Nonlinear Sciences and Department of Mathematics, Fudan University, Shanghai 200433, People’s Republic of China

Abstract Let Ax ¼ b be a linear system where A is a symmetric positive definite (spd) matrix. m-step multisplitting preconditioners, which include the preconditioners based on multisplittings obtained by incomplete Cholesky factorizations [R. Bru, C. Corral, A. Mart
inez, J. Mas, SIAM J. Matrix Anal. Appl. 16 (1995) 1210–1222], for the conjugate gradient method are studied. The validity of the proposed m-step multisplitting preconditioners when A is an spd matrix is proved. Our results improve and extend previous ones. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Iterative method; Linear system; Multisplitting; Preconditioner

1. Introduction The preconditioned conjugate gradient algorithms are powerful tools in the numerical solution of large system of linear equations Ax ¼ b

ð1:1Þ

on high performance computers, where A is an n  n spd matrix. The choice of preconditioners is still a topic of research. There exist several techniques for

q This work is supported by the Foundation of National Key Laboratory of Computational Physics and the Doctoral Point Foundation of China. * Corresponding author. E-mail address: [email protected] (Z.-H. Cao).

0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 0 ) 0 0 1 5 1 - X

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constructing the preconditioner. A usual technique consists of considering a splitting of matrix A A ¼ M  N;

ð1:2Þ

where M is nonsingular and the spectral radius qðM 1 N Þ of M 1 N is less than 1. From the splitting one constructs the preconditioner matrix Mm by using a partial sum of the power series of A1 . That is Mm ¼ MðI þ H þ    þ H m1 Þ1 ;

ð1:3Þ

1

where H ¼ M N . This is called a preconditioner based on approximating inverse [6]. From (1.3) we have m 1 M1 m ¼ ðI  H ÞA

ð1:4Þ

which implies A ¼ Mm  Mm H m  Mm  Nm ;

ð1:5Þ m

i.e., the splitting (1.5) is induced by the iterative matrix H (cf. [9, Lemma 2.3]). Thus, Mm in (1.3) is also called an m-step preconditioner [1]. For this type of preconditioners in order to obtain M1 m v with a given vector v, i.e., to solve the system Mm x ¼ v, one needs to do m steps Mxj ¼ Nxj1 þ v;

j ¼ 1; . . . ; m

ð1:6Þ

of the iteration given by the splitting (1.2) with x0 ¼ 0, i.e., xm ¼ M1 m v. If the preconditioner matrix Mm in (1.3) is spd, then the corresponding mstep preconditioner is called a validity m-step preconditioner. Recently, some m-step preconditioners based on multisplittings have been proposed, see e.g., [4,7,8]. However, the validity of all these preconditioners was proved under the assumptions that matrix A is an spd M-matrix (i.e., Stieltjes matrix) or the weighting matrices are multiples of the identity. In this paper, we will present two types of m-step multisplitting preconditioners, they are presented in [4,5], respectively, and prove the validity of these preconditioners when A is only assumed to be an spd matrix and the weighting matrices need not be multiples of identity.

2. Preliminaries We begin with some basic notation and preliminary results which we refer to later. A matrix A ¼ ðaij Þ 2 Rn;n is called a Z-matrix [3] if aij 6 0 for i 6¼ j. If A is a nonsingular Z-matrix and A1 P 0, then A is called an M-matrix. An spd Zmatrix is called a Stieltjes matrix. A symmetric M-matrix is a Stieltjes matrix and vice versa (cf. [12]).

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201

A splitting A ¼ M  N of A is called regular if M 1 P 0 and N P 0; (left) weak regular if M 1 P 0 and M 1 N P 0; two-side weak regular if it is (left) weak regular and NM 1 P 0; convergent if qðM 1 N Þ < 1. Let A be symmetric, a splitting A ¼ M  N is called symmetric if M and N are symmetric. As defined in [11] a multisplitting of A is a collection of triples of matrices ðMk ; Nk ; Ek Þ, k ¼ 1; . . . ; K, satisfying (i) A ¼ Mk  Nk , k ¼ 1; . . . ; K; (ii) Mk is nonsingular for k ¼ 1; . . . ; K; (iii) P Ek , k ¼ 1; . . . ; K, are diagonal matrices with nonnegative entries such that K k¼1 Ek ¼ I. The main tool for deriving our m-step multisplitting preconditioners is the diagonal compensation reduction [2]. Now let us turn to this conception, the description of the method of diagonal compensation reduction is slight extension of that in [2]. Let A be a symmetric positive definite matrix, let R be symmetric and nonnegative. Consider the reduced matrix B¼AR

ð2:1Þ

and (arbitrarily) select a positive (weighting) vector v, then define the diagonally compensated reduced matrix of A b ¼ D þ B; A

ð2:2Þ

where D is a diagonal matrix defined by ð2:3Þ

Dv P Rv;

D is the diagonal compensation matrix for the reduced entry matrix R. Note b v P Av, then A becomes split as that D P 0 and A b  ðD  RÞ: A¼A

ð2:4Þ

Since D P 0, R P 0 and ðD  RÞv P 0, D  R is positive semidefinite. Hence b  A is positive semidefinite, and for any eigenvalue kj ð A b 1 AÞ we have A b 1 AÞ 6 1: kj ð A

ð2:5Þ

Moreover, let b ¼MN b A

ð2:6Þ

be a symmetric splitting with a positive definite matrix M. Then it holds the following convergent splitting result. b ¼ D þ B be a Lemma 2.1 (cf. [2]). Let A be an n  n spd matrix and let A diagonally compensated reduced matrix of A. Then b Þ; 0 < kj ðM 1 AÞ 6 kj ðM 1 A

j ¼ 1; . . . ; n

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and thus, the splitting A¼M N b ¼MN b is convergent. (where N ¼ M  A) is convergent if the splitting A We note that if the reduced matrix B is a Z-matrix (e.g., all positive offdib ¼DþB agonal entries of A are reduced by the reduced entry matrix R), then A b b is a Z-matrix too. Since A  A is positive semidefinite, A is a Stieltjes matrix b is an spd Z-matrix). Hence, A b is an spd M-matrix (cf. [12]). Thus the (i.e., A problem of constructing a convergent splitting for an spd matrix, owing to the Lemma 2.1, can always be reduced to that for a Stieltjes matrix which is an spd M-matrix. Henceforth, when we construct a diagonally compensated reduced b from an spd matrix A we will always make A b a Stieltjes matrix. matrix A b is We also note that (2.2) and (2.3) imply that one can select D such that A strictly diagonally dominant. Therefore, for a spd matrix A one can always construct a strictly diagonally dominant Stieltjes diagonally compensated reb of A. duced matrix A Lemma 2.2 (cf. [1,5]). Let A be spd and the splitting A ¼ M  N of A be such that M is spd and H  M 1 N satisfies qðH Þ < 1. Then the m-step preconditioner Mm ¼ MðI þ H þ    þ H m1 Þ1 is spd. Proof. Since H ¼ I  M 1 A, we have M1 m ¼

m1 X j¼0

¼

m1 X

H j M 1 ¼

m1 X

H j ðI  H ÞA1 ¼ ðI  H m ÞA1

j¼0 j

ðI  M 1 AÞ M 1 :

ð2:7Þ

j¼0

Eq. (2.7) implies that M1 m is symmetric. Since H ¼ I  M 1 A ¼ A1=2 ðI  A1=2 M 1 A1=2 ÞA1=2 ; all eigenvalues of H are real. We now have 1=2 1=2 A1=2 M1 ¼ A1=2 ðM1 ¼ A1=2 ðI  H m ÞA1=2 : m A m AÞA

Since all eigenvalues of I  H m are positive, M1 m is spd.



From Lemma 2.2 we know that in order to prove m-step preconditioner Mm being validity one we need only to show that matrix M in the splitting (1.2) is spd and iterative matrix H  M 1 N is convergent.

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203

3. Main results Let A be an n  n spd matrix, we are going to determine the multisplitting ðMk ; Nk ; Ek Þ, k ¼ 1; . . . ; K, of A by using the diagonal compensation reduction. b of A be constructed. Let a Stieltjes diagonally compensated reduced matrix A 3.1. Block diagonal conformable multisplitting The first type of multisplittings is block diagonal conformable (cf. [4, Definition 2]). b be partitioned into q  q blocks Let A 1 0 b1 C12    C1q A C B b2 B C21    C2q C A b C B ð3:1Þ A¼B . .. C; .. .. @ .. . A . . bq Cq1 Cq2    A Pq b j , j ¼ 1; . . . ; q, are nj  nj matrices with b where A j¼1 nj ¼ n. Since A is a b j , j ¼ 1; . . . ; q, are Stieltjes matrices too. Cij ¼ C T P 0, for Stieltjes matrix, A ji j 6¼ i and i; j ¼ 1; . . . ; q. b j ¼ MjðkÞ  N b j for j ¼ 1; . . . ; q. We b jðkÞ , k ¼ 1; . . . ; K, be splittings of A Let A b in the form b k ; Ek Þ, k ¼ 1; . . . ; K, of A construct a multisplitting ðMk ; N 1 0 ðkÞ M1 C B .. C B . C B C B ðkÞ Mj ð3:2Þ Mk ¼ B C; C B .. C B A @ . ðkÞ Mq b; b k ¼ Mk  A N 0 ðkÞ d I1 B 1 .. B . B B ðkÞ d j Ij Ek ¼ B B B @

1

..

. dqðkÞ Iq

C C C C C; C C A

ð3:3Þ

PK ðkÞ ðkÞ where 0 6 dj 6 1, k¼1 dj ¼ 1, and Ij , j ¼ 1; . . . ; q; are nj  nj identity matrices. b is defined, a b k ; Ek Þ, k ¼ 1; . . . ; K, of A Obviously, when a multisplitting ðMk ; N multisplitting ðMk ; Nk ; Ek Þ, k ¼ 1; . . . ; K, of A is also defined. The difference

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b , while Nk ¼ Mk  A for b k ¼ Mk  A between these two multisplittings is that N k ¼ 1; . . . ; K. b be any Stieltjes diagonally compensated reTheorem 3.1. Let A be spd, let A b duced matrix of A and be partitioned into the form (3.1). Assume that A and A b k ; Ek Þ, k ¼ 1; . . . ; K, respectively, have multisplittings ðMk ; Nk ; Ek Þ and ðMk ; N where Mk and Ek are in (3.2) and (3.3), respectively, with splittings b j ¼ MjðkÞ  N b jðkÞ , j ¼ 1; . . . ; q; k ¼ 1; . . . ; K, being symmetric weak regular with A ðkÞ Mj being spd. Then the m-step preconditioner Mm resulting from the multisplitting ðMk ; Nk ; Ek Þ, k ¼ 1; . . . ; K, of A is spd. Proof. We have, for k ¼ 1; . . . ; K, b ¼ Mk  N bk A 0 ðkÞ M B 1 ðkÞ B M2 B B @

1 ..

. MqðkÞ

0

b 1ðkÞ N C B C B C21 CB C B .. A @ .

C12 b 2ðkÞ N .. .

  .. .

Cq1

Cq2



1 C1q C C2q C C .. C: . A b N ðkÞ

ð3:4Þ

q

Let G¼

K X

Ek Mk1 ;

b¼ b ¼ I  GA H

k¼1

K X

bk: Ek Mk1 N

ð3:5Þ

k¼1

b are symmetric weak b ¼ Mk  N b k , k ¼ 1; . . . ; K, of A Obviously, the splittings A regular with Mk being spd. b is a Stieltjes matrix, A b is monotone. It is well known that the Since A b k ; Ek Þ, multisplitting iterative method resulting from the multisplitting ðMk ; N b b k ¼ 1; . . . ; K, of A converges (cf. [11, Theorem 1(a)]), i.e., qð H Þ < 1 and G is nonsingular. Let H ¼ I  GA, then we have two single splittings A ¼ G1  G1 H  M  N ;

b ¼ G1  G1 H b MN b: A

ð3:6Þ

Eqs. (3.2), (3.3) and (3.5) imply G is symmetric positive semidefinite. Note that G is also nonsingular. Thus, G and hence M are spd. From Lemma 2.1, (3.6) b Þ < 1 we have qðH Þ < 1. Then, Lemma 2.2 implies Mm ¼ MðIþ and qð H 1 H þ    þ Hm1 Þ is spd.  b j ¼ MjðkÞ  N b jðkÞ , j ¼ 1; . . . ; q; k ¼ 1; . . . ; K, are taken as incomplete If A b j is a symmetric M-matrix, these inCholesky factorizations (ICF) (since A ðkÞ ðkÞ ðkÞT ðkÞ complete Cholesky factorizations exist [10]), then Mj ¼ Lj Lj ; where Lj is ðkÞ b j P 0. Following Theorem 3.1, we obtain a triangular lower M-matrix, and N an spd m-step multisplitting preconditioner based on incomplete Cholesky

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205

factorization. Thus, Theorem 3.1 improves and extends [4, Theorem 4] in which matrix A is assumed to be a symmetric M-matrix. Based on the above discussion it is easy to see that all the implementation techniques presented in [4] of the multisplitting preconditioning PCG iterative methods for solving symmetric M-matrix linear systems can also be used for solving symmetric positive definite systems. 3.2. Symmetric multisplitting The second type of multisplittings is symmetric which is presented in [5]. Let A be an n  n spd matrix and let ðMk ; Nk ; Ek Þ, k ¼ 1; . . . ; K, be a multisplitting of A. Then we construct a parallel symmetric multisplitting algorithm as follows (cf. [13]) x pþ1 ¼ x p þ Gr p ;

ð3:7Þ

where G¼

K 1X ðEk Mk1 þ MkT Ek Þ 2 k¼1

and

r p ¼ b  Ax p ;

ð3:8Þ

the iteration matrix of (3.7) is H ¼ I  GA: If G is nonsingular, the iteration can be induced by a single splitting of A A ¼ G1  G1 H : Now let us determine the multisplitting ðMk ; Nk ; Ek Þ, k ¼ 1; . . . ; K, by using the diagonal compensation reduction. Let a diagonally compensated (Stieltjes) b of A be constructed (see Section 2). Let A b be partitioned into K  K matrix A blocks 0 1 b1 C12    C1K A B C b2    C2K C A 21 C b¼B ; ð3:9Þ A B . .. C .. . .. @ .. . A . bK CK1 CK2    A PK b i , i ¼ 1; . . . ; K, are ni  ni matrices with b where A i¼1 ni ¼ n. Since A is a b Stieltjes matrix, A i , i ¼ 1; . . . ; K, are Stieltjes matrices too, and Cij ¼ CjiT P 0, for i 6¼ j and i; j ¼ 1; . . . ; K. b i ¼ Bi  Ci be a splitting of A b i for i ¼ 1; . . . ; K. We construct a mulLet A b in the form b tisplitting ðMk ; N k ; Ek Þ, k ¼ 1; . . . ; K, of A

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Z.-H. Cao, Y. Wang / Appl. Math. Comput. 126 (2002) 199–211

0 B B B B Mk ¼ B B B B @

B1

1 ..

C C C C C; C C C A

. Bk Ckþ1;k .. .

Bkþ1 .. .

CKk

ð3:10Þ

BK

b; b k ¼ Mk  A N 0 B B B Ek ¼ B B B @

1

ðkÞ

d 1 I1

..

C C C C; C C A

. ðkÞ

d k Ik

..

.

ð3:11Þ

ðkÞ

d K IK P ðkÞ ðkÞ where 0 6 di 6 1, Kk¼1 di ¼ 1, and Ii , i ¼ 1; . . . ; K, are ni  ni identity matrices. Obviously, let Nk ¼ Mk  A, then a multisplitting ðMk ; Nk ; Ek Þ, k ¼ 1; . . . ; K, of A is defined. From (3.10) we have 1 0 1 B1 C .. B C B . C B 1 C B B 1 k C: B ð3:12Þ Mk ¼ B 1 1 1 C Bkþ1 Ckþ1;k Bk Bkþ1 C B C B .. .. A @ . . 1 1 1 BK CKk Bk BK Some simple but tedious matrix manipulations show K 1X ðEk Mk1 þ MkT Ek Þ 2 k¼1 0 ð1Þ d 1 T ðB1 þ BT Þ 22 BT 1 1 1 C12 B2 2 B ð1Þ B d2 1 1 T B B2 C21 B1 ðB1 1 2 þ B2 Þ 2 2 ¼B B .. .. B . . @

G¼

ð1Þ

dK 2

1 B1 K CK1 B1

ð2Þ

dK 2

1 B1 K CK2 B2

ð1Þ



dK 2

 .. .

dK 2



1 ðB1 K 2

ð2Þ

T BT 1 C1K BK T BT 2 C2K BK

.. . þ BT K Þ

1 C C C C C C A

ð3:13Þ

Z.-H. Cao, Y. Wang / Appl. Math. Comput. 126 (2002) 199–211

207

and 0

b h 11 Bb h 21 b¼B b ¼ I  GA H B . @ .. b h K1

b h 12 b h 22 .. . b h K2

  .. . 

1 b h 1K b h 2K C C : .. C . A b h KK

ð3:14Þ

b be any Stieltjes diagonally compensated reTheorem 3.2. Let A be spd, let A b k ; Ek Þ, k ¼ 1; . . . ; K, is in the form duced matrix of A, its multisplitting ðMk ; N b i ¼ Bi  Ci being two-side weak regular for i ¼ 1; . . . ; K. (3.10) and (3.11) with A b Þ < 1. Then G is nonsingular and qð H b is a Stieltjes matrix and the splittings A b i ¼ Bi  Ci of A b i, Proof. Since A 1 1 T T i ¼ 1; . . . ; K, are two-side weak regular, we have Bi P 0; Bi Ci P 0; Bi Ci P 0 and Cij P 0 ði 6¼ jÞ for i; j ¼ 1; . . . ; K. It follows from (3.13) and (3.14) that GP0

and

b P 0: H

ð3:15Þ

It is also easy to know from (3.13) that each row of G has at least one nonzero b 1 P 0 and H b , it follows from [13, b ¼ I  GA component. Noting (3.15), A b Theorem 1] that G is nonsingular and qð H Þ < 1. Thus the proof is finished.  We now have two single splittings A ¼ G1  G1 H  M  N

and

b ¼ G1  G1 H b A

ð3:16Þ

b , respectively. Moreover, the latter is convergent, i.e., qð H b Þ < 1. of A and A First, we have the following result. b be any Stieltjes diagonally compensated reTheorem 3.3. Let A be spd, let A b duced matrix of A and be partitioned into the form (3.9). Assume that A and A b have multisplittings ðMk ; Nk ; Ek Þ and ðMk ; N k ; Ek Þ, k ¼ 1; . . . ; K, respectively, b i ¼ Bi  Ci where Mk and Ek are in (3.10) and (3.11), respectively, with splittings A b of A i being two-side weak regular for i ¼ 1; . . . ; K. If the matrix G in (3.13) is spd, then the m-step multisplitting preconditioner Mm ¼ G1 ðI þ H þ    þ 1 H m1 Þ resulting from the parallel symmetric multisplitting method (3.7) is spd. b ¼ G1  G1 H b b Þ < 1, i.e., the splitting A b of A Proof. Theorem 3.2 implies qð H 1 1 is convergent. It follows from Lemma 2.1 that A ¼ G  G H  M  N is a convergent splitting of A, i.e., qðH Þ < 1. Then, Lemma 2.2 implies 1 Mm ¼ MðI þ H þ    þ H m1 Þ is spd. Thus, the proof is finished.  b i ¼ Bi  Ci of A b i , i ¼ 1; . . . ; K, being symmetric, If we restrict the splittings A then the matrix G in (3.13) has the form

208

Z.-H. Cao, Y. Wang / Appl. Math. Comput. 126 (2002) 199–211 1 e 1 1 G ¼ diagðB1 1 ; . . . ; BK Þ G diagðB1 ; . . . ; BK Þ;

where 0

ð1Þ

B1

B ð1Þ B d2 B C21 2 e G¼B B . B . @ . ð1Þ

dK 2

CK1

d2 2

ð1Þ

C12



dK 2

 .. .

dK 2

C1K

ð2Þ

B2 .. . ð2Þ

dK 2

CK2



1

C C C2K C C: .. C C . A

ð3:17Þ

BK

In this case Theorem 3.3 can be rewritten as follows. b be any Stieltjes diagonally compensated Theorem 3.4. Let A be spd, let A b reduced matrix of A and be partitioned into the form (3.9). Assume that A and A b have multisplittings ðMk ; Nk ; Ek Þ and ðMk ; N k ; Ek Þ, k ¼ 1; . . . ; K, respectively, b i ¼ Bi  Ci where Mk and Ek are in (3.10) and (3.11), respectively, with splittings A b e in of A i being symmetric and weak regular for i ¼ 1; . . . ; K. If the matrix G 1 (3.17) is spd, then the m-step multisplitting preconditioner Mm ¼ G ðIþ 1 H þ    þ H m1 Þ resulting from the parallel symmetric multisplitting method (3.7) is spd. In order to get a more practically feasible parallel multisplitting algorithm we further restrict the weighting matrices Ek , k ¼ 1; . . . ; K, in (3.11) to satisfy the following conditions: 1 0 ðkÞ d 1 I1 C B .. C B . C B ðkÞ C; B ð3:18Þ Ek ¼ B d k Ik C C B . A @ .. ðkÞ d K IK where ðkÞ

0 6 dj 6 1;

K X

ðkÞ

ðkÞ

dj ¼ 1 and dj ¼ 0

k¼1

for j > k; k ¼ 1; . . . ; K; j ¼ 1; . . . ; K: In this case, the matrix G in (3.13) has the following simple form: 0 1 1 B1 þ BT 1 1B C .. G¼ @ A: . 2 T BK þ BK

Z.-H. Cao, Y. Wang / Appl. Math. Comput. 126 (2002) 199–211

If we restrict in addition symmetric, then 0 1 B1 B .. G¼@ .

209

b i ¼ Bi  Ci of A b i , i ¼ 1; . . . ; K, being the splittings A 1 C A:

ð3:19Þ

B1 K e in (3.17) or G We now give some examples of symmetric splittings such that G in (3.19) are spd, hence the m-step multisplitting preconditioner Mm ¼ MðI þ H þ    þ H m1 Þ1 resulting from the parallel symmetric multisplitting method (3.7) is spd. bi ¼ D b i g;  b bi  b b i ¼ diagf A Let A Li  b L Ti , i ¼ 1; . . . ; K, where D L i is the strictly b i. lower part of A bi ¼ B b i , where diagf B b i g and B bi  C b i g ¼ diagf A b i is a sym(i) Band splitting: A b i. metric band matrix consisting of some (symmetric) diagonals of A bi ¼ D b i  ðb (ii) Jacobi splitting: A Li þ b L Ti Þ. Obviously, this can be regarded as a special case of the band splitting. (iii) SSOR splitting: 



 1 b T bi ¼ x b i  xb b 1 1 D A Di  xb Li D L i i 2x x x h h i i x 1 1 1 T b b b b b ð1  xÞ D i þ x L i D i ð1  xÞ D i þ x L i  : 2x x x bi ¼ e b i  Mi  N b i , where e L Ti  R Li e L i is a lower triangular (iv) ICF splitting [10]: A b M-matrix and N i P 0. Theorem 3.5. Let A ¼ ðaij Þ be an n  n spd matrix, then it exists a diagonally b of A such that A b is a strictly diagonally dominant compensated reduced matrix A b Stieltjes matrix. Assume A is partitioned into (3.9). If a multisplitting b is in the form (3.10) and (3.11) with the splittings b k ; Ek Þ, k ¼ 1; . . . ; K, of A ðMk ; N b of A i , i ¼ 1; . . . ; K, being (i) band splittings or (ii) Jacobi splittings. Then the mstep multisplitting preconditioner Mm ¼ G1 ðI þ H þ    þ H m1 Þ1 resulting from the parallel symmetric multisplitting method (3.7) according to the multisplitting ðMk ; Nk ; Ek Þ, k ¼ 1; . . . ; K, of A is spd. b of A Proof. First we construct a diagonally reduced compensated matrix A b is a strictly diagonally dominant Stieltjes matrix as follows. such that A We first construct a reduced matrix B ¼ ðbij Þ such that bii ¼ aii for i ¼ 1; . . . ; n and bij ¼ aij if aij < 0, otherwise, bij ¼ 0, for i 6¼ j and i; j ¼ 1; . . . ; n. Let R ¼ A  B, then R P 0, and construct a diagonal matrix Pn D ¼ diagðdii Þ; D P 0 such that De P Re and dii > j¼1 jbij j; i ¼ 1; . . . ; n, where j6¼i

b ¼ D þ B is the demanded diagonally compensated e ¼ ½1; . . . ; 1T 2 Rn . Then A

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b is a strictly diagonally dominant Stieltjes reduced matrix of A. Obviously, A matrix. b is a Stieltjes matrix, i.e., an spd Z-matrix, the main submatrices A b i, Since A b i ¼ 1; . . . ; K, of A are all spd Z-matrices and hence are all Stieltjes matrices. b i ¼ Bi  Ci of A b i for i ¼ 1; . . . ; K, Clearly, in either case (i) or (ii) of splittings A e the corresponding matrix G in (3.17) is strictly diagonally dominant with e is spd. Moreover, all the splittings positive diagonal entries. Hence G b b A i ¼ Bi  Ci of A i , i ¼ 1; . . . ; K, in case (i) or (ii) are symmetric and regular (note that in either case Bi , i ¼ 1; . . . ; K, are Stieltjes matrices and hence are Mmatrices). Then the conclusion of this theorem follows from Theorem 3.4. Thus the proof is completed.  If matrix A with positive diagonal entries is strictly diagonally dominant or irreducibly diagonally dominant, then we have the following result. Theorem 3.6. Let A be symmetric with positive diagonal entries and be strictly diagonally dominant or irreducibly diagonally dominant. Then there exists a dib of A such that A b is a strictly diagonally agonally compensated reduced matrix A b dominant Stieltjes matrix. Assume A is partitioned into (3.9). If a multisplitting b is in the form (3.10) and (3.11) with the splittings b k ; Ek Þ, k ¼ 1; . . . ; K, of A ðMk ; N b of A i , i ¼ 1; . . . ; K, being (i) band splittings or (ii) Jacobi splittings. Then the mstep multisplitting preconditioner Mm ¼ G1 ðI þ H þ    þ H m1 Þ1 resulting from the parallel symmetric multisplitting method (3.7) according to the multisplitting ðMk ; Nk ; Ek Þ, k ¼ 1; . . . ; K, of A is spd. Proof. With either assumption on matrix A, It is well known (cf. [12]) that A is spd. Then the conclusion of the theorem follows from Theorem 3.5.  If we restrict the weighting matrices Ek , k ¼ 1; . . . ; K, in the form (3.18), then we have the following result. b be any Stieltjes diagonally compensated reTheorem 3.7. Let A be spd, let A b is partitioned into (3.9). If a multisplitting duced matrix of A. Assume A b is such that Ek are in the form (3.18), Mk are in b k ; Ek Þ, k ¼ 1; . . . ; K, of A ðMk ; N b i , i ¼ 1; . . . ; K, being (i) band splittings or the form (3.10) with the splittings of A (ii) Jacobi splittings or (iii) SSOR splittings with x 2 ð0; 1 or (iv) ICF splittings. 1 Then the m-step multisplitting preconditioner Mm ¼ G1 ðI þ H þ    þ H m1 Þ resulting from the parallel symmetric multisplitting method (3.7) according to the multisplitting ðMk ; Nk ; Ek Þ, k ¼ 1; . . . ; K, of A is spd. b i , i ¼ 1; . . . ; K, for either of the four cases (i), (ii), Proof. In all the splittings of A (iii) (with x 2 ð0; 1) and (iv) are symmetric and regular and the corresponding

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matrix G in (3.19) is spd, then by Theorem 3.4 we have 1 Mm ¼ G1 ðI þ H þ    þ H m1 Þ is spd. Thus the proof is finished. 

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