on the convergence of a new splitting iterative method for non-hermitian positive definite linear...

Applied Mathematics and Computation 248 (2014) 118–130

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

On the convergence of a new splitting iterative method fornon-Hermitian positive definite linear systems q

http://dx.doi.org/10.1016/j.amc.2014.09.0850096-3003/� 2014 Elsevier Inc. All rights reserved.

q This work is supported by the NSF of China (11371275), the NSF of Shanxi Province (2012011015-6) and STIP of Higher Education Institutions i(2013144).⇑ Corresponding author.

E-mail address: [email protected] (R.-P. Wen).

Rui-Ping Wen ⇑, Xi-Hong Yan, Chuan-Long WangHigher Education Key Laboratory of Engineering and Scientific Computing in Shanxi Province, Taiyuan Normal University, Taiyuan 030012, Shanxi Province,PR China

a r t i c l e i n f o a b s t r a c t

Keywords:ConvergenceSplitting iterative methodAccelerated algorithmNon-Hermitian positive definite matrixLinear systems

In this paper we present a new splitting method for solving a linear systems with non-Her-mitian positive definite coefficient matrix. This splitting overcomes the computation com-plexity of HSS. The spectral radius and some norm properties of the iteration matrix arediscussed. With the results obtained, we study the reasonable choices of the parameterand introduce a preconditioner. Moreover, an accelerated algorithm is proposed. Finally,the numerical examples show the new method is much more efficient than the HSS (orthe NSS and the PSS) iteration method.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction and preliminaries

For solving a large sparse system of linear equations

Ax ¼ b; A 2 Cn�n nonsingular and b 2 Cn; ð1:1Þ

a usual scheme is the so-called splitting iterative method. When A is a monotone matrix, or an H-matrix, or a Hermitianpositive definite matrix, the splitting A ¼ M � N is convergent if it is a weak regular splitting, or an H-compatible splitting,or a P-regular splitting respectively. The main idea behind the technique to establish an iterative method is the exploitationof the special structures of the coefficient matrix. Numerous details derivations on the problem can be refer to the articles[1,2,5–8,11,16] and the references given therein.

When the coefficient matrix A 2 Cn�n of the system (1.1) is only positive definite and non-Hermitian, Wang and Bai [12]presented several sufficient conditions for guaranteeing the convergence of the splitting iterative methods and we would liketo address that Bai et al. [3,4] also discussed two alternative methods, called HSS and PSS iteration methods, which convergeunconditionally to the unique solution of the system of linear equations (1.1). But a common drawback of those methods is aHermitian and a skew-Hermitian system of linear equations need to be solved at each iteration step. The research into askew-Hermitian system of linear equations is also conducted in [9,10,14]. Some iterative methods for solving a non-Hermi-tian linear system are confined to be only theoretically acceptable, their convergence conditions cannot be verified easily[15]. Recently, Wang et al. [13] introduced a practical splitting iterative method for solving a non-Hermitian positive definitelinear system. More generally, in this paper we present a practical convergent splitting iterative method for solving a

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R.-P. Wen et al. / Applied Mathematics and Computation 248 (2014) 118–130 119

non-Hermitian positive definite linear system, which is a generalized scheme presented that in [13]. The spectral radius andnorm properties of the new splitting method are discussed. Based on these results, we propose an accelerated algorithm,which will find the optimal solution in hyperplane generated by fxk; . . . ; xkþmg. The numerical examples show it is effective.Furthermore, a preconditioner generated by the splitting is proposed, the condition number of preconditioned matrix isdiscussed.

Here are some essential notations and preliminaries. Cn�nðRn�nÞ is used to denote the n� n complex (real) matrix set, andCnðRnÞ the n-dimensional complex (real) vector set. A� denotes the conjugate transpose of the matrix A, and x� denotes theconjugate transpose of the vector x.

A matrix A 2 Cn�n is called Hermitian positive definite (or semidefinite), denoted by A � 0 (or � 0Þ, if it is Hermitian andfor all x 2 Cn; x – 0, it holds that x�Ax > 0 (or x�Ax P 0Þ. A matrix A 2 Cn�n is called positive definite, if for all x 2 Cn; x – 0, itholds that Reðx�AxÞ > 0. Here, Reð:Þ and Imð:Þ denote the real part and imaginary part of the corresponding complex number,respectively. The spectral radius of the matrix A is denoted by qðAÞ.

A ¼ M � N is called a splitting of the matrix A if M 2 Cn�n is nonsingular. This splitting is called a convergent splitting ifqðM�1NÞ < 1; it is called a P-regular splitting if M� þ N is positive definite; a Hermitian splitting if both M and N are Hermi-tian; and a Hermitian P-regular splitting if M is Hermitian positive definite and M þ N is positive definite; a strong HermitianP-regular splitting if M is Hermitian positive definite and N is Hermitian positive semidefinite.

In this study, we first give introduction and preliminaries in Section 1, and then a new iterative method and an acceler-ated algorithm are presented for solving the system of linear equations (1.1) in Section 2. Section 3 We focus on the spectralradius and norm properties of the iteration matrix for this new iterative method. Based on these results, convergence the-ories of the new iterative method and the accelerated algorithm are obtained. We present a preconditioner and discuss itscondition number in Section 4. Finally, we compare our algorithms with HSS by some examples in Section 5.

2. Algorithms

In this section, a new splitting iterative method and its accelerated scheme are discussed.Let

A ¼ H1 þ H2; ð2:1Þ

where H1 is a Hermitian positive definite matrix, H2 is a non-Hermitian positive semi-definite matrix.Let

H1 ¼ M � N;

P ¼ M þ aH2; ð2:2ÞQ ¼ N þ ða� 1ÞH2: ð2:3Þ

Then,

A ¼ P � Q ; a 2 X;

where X ¼ fajP is nonsingular; 8a 2 Rg. The corresponding iteration matrix

T ¼ P�1Q :

Algorithm I

Step 0. Given an initial point xð0Þ, the precision � > 0, for k ¼ 0;1;2; . . . until the process converges.Step 1. Solving the system of linear equations

PxðkÞ ¼ Qxðk�1Þ þ b

p 2. If kAxðkÞ � bk < �, stop; Otherwise, k( kþ 1, goto step 1.
Ste
Next, the accelerated algorithm is presented as follows.

Algorithm II

Step 0. Given an initial point xð0;0Þ, the precision e > 0, for k ¼ 0;1;2; . . . until the process converges.Step 1. For l ¼ 0;1;2; . . . ;m, computing

Pxðk;lþ1Þ ¼ Qxðk;lÞ þ b:

120 R.-P. Wen et al. / Applied Mathematics and Computation 248 (2014) 118–130

p 2. Let
Ste
rðk;lÞ ¼ Axðk;lÞ � b;

r ¼Xm

l¼1

aðkÞl rðk;lÞ;

mina

r�H�11 r ð2:4Þ

s:t:Xm

l¼1

aðkÞl ¼ 1:

p 3.
Ste
xðkþ1;0Þ ¼Xm

l¼1

aðkÞl xðk;lÞ:

p 4. If jjrjj2 < e, stop; Otherwise, goto Step 1.
Ste
Remark. If let

H ¼ 12ðAþ A�Þ; S ¼ 1

2ðA� A�Þ;

then

A ¼ H þ S: ð2:5Þ
Hence, the Hermitian and skew-Hermitian splitting (2.5) is a special case of the splitting (2.1). Furthermore, the Algo-
rithm I becomes that splitting iterative method proposed in [13]. Therefore, the Method 2.1 in [13] is a special case ofour Algorithm I here.

3. Convergence analysis

In this section, we study the spectral radius and norm properties of the iteration matrix, and discuss convergence theoriesof Algorithm I and II.

In order to analyze the convergence, a simple lemma is listed first.

Lemma 3.1. Let a; b; x; y be positive real numbers. Then the inequality is holds that

minfa;bg 6 axþ byxþ y

6maxfa;bg:

Theorem 3.2. Let H1 ¼ M � N be a Hermitian P-regular splitting of the Hermitian positive definite matrix H1. Then

qðTÞ 6 max qðM�1NÞ; ja� 1jjaj

� �: ð3:1Þ

Furthermore, if a > 12, then

qðTÞ < 1:

Proof. Assume that k 2 C is an eigenvalue of the matrix T and x 2 Cn; x – 0 is the corresponding eigenvector, then it holdsthat

Qx ¼ kPx; ð3:2Þ

we rewrite (3.2) as the following form since (2.2) and (2.3)

ðN þ ða� 1ÞH2Þx ¼ kðM þ aH2Þx:

Thus, we have

x�ðN þ ða� 1ÞH2Þx ¼ kx�ðM þ aH2Þx;


Thereby,

jkj ¼ x�Nxþ ða� 1Þx�H2xx�Mxþ ax�H2x

��: ð3:3Þ

From the assumption that H1 ¼ M � N is a Hermitian P-regular splitting, we see that x�Mx and x�Nx are real numbers, andx�Mx > 0.

Hence, (3.3) implies

jkj2 ¼ ðx�Nxþ ða� 1ÞReðx�H2xÞÞ2 þ ða� 1Þ2jImðx�H2xÞj2

ðx�Mxþ aReðx�H2xÞÞ2 þ a2jImðx�H2xÞj26 max

y2Cn

jy�Nyþ ða� 1ÞReðy�H2yÞj2 þ ðða� 1ÞjImðy�H2yÞjÞ2

jy�Myþ aReðy�H2yÞj2 þ ðajImðy�H2yÞjÞ2:

For 8y 2 Cn,

maxy2Cn

y�Nyy�My

¼ maxjjyjj2¼1

y�M�12NM�1

2y ¼ q M�12NM�1

2

� �¼ qðM�1NÞ:

Combining the above result and Reðy�H2yÞP 0, we get

jkj2 6 maxy2Cn

jqðM�1NÞy�Myþ ða� 1ÞReðy�H2yÞj2 þ ðjða� 1ÞjImðy�H2yÞjÞ2

jy�Myþ aReðy�H2yÞj2 þ jaImðy�H2yÞj2

6 max q2ðM�1NÞ; ja� 1j2

jaj2

!: ðby Lemma 3:1Þ

Thereby,

jkj 6max qðM�1NÞ; ja� 1jjaj

� �:

From the assumption we see that qðM�1NÞ < 1, and ja� 1j < jaj holds when a > 12, it follows that qðTÞ < 1 holds. h

Remark. From (3.3) we can derived that qðTÞ < 1 holds when a ¼ 12. Furthermore, the spectral radius of iteration matrix T is

not more than qðM�1NÞ when a is chosen suitably.

Theorem 3.3. Let H1 ¼ M � N be a splitting of the Hermitian positive definite matrix H1. Then H�1

21 QP�1H

121

�� 2< 1 if and only if the

matrix

H1 þ N þ N� þ aðH2 þ H�2Þ þ N�H�11 H2 þ H�2H�1

1 N þ ð2a� 1ÞH�2H�11 H2

is Hermitian positive definite.

Proof. From (2.2) and (2.3), it holds that

QP�1 ¼ ðN þ ða� 1ÞH2ÞðM þ aH2Þ�1:

Hence, we have

H�1

21 QP�1H

121 ¼ H

�12

1 ðN þ ða� 1ÞH2ÞðM þ aH2Þ�1H121 ¼ H

�12

1 ðN þ ða� 1ÞH2ÞðH1 þ N þ aH2Þ�1H121

¼ ðN þ ða� 1ÞH2ÞðI þ N þ aH2Þ�1;

where N ¼ H�1

21 NH

�12

1 ; H2 ¼ H�1

21 H2H

�12

1 .

Thus, H�1

21 QP�1H

121

�� 2< 1 is equivalent to

ðN þ ða� 1ÞH2ÞðI þ N þ aH2Þ�1

�� 2< 1: ð3:4Þ

Minus the matrix

ðN� þ ða� 1ÞH�2ÞðN þ ða� 1ÞH2Þ ¼ N�N þ ða� 1ÞH�2N þ ða� 1ÞN�H2 þ ða� 1Þ2H�2H2

from the matrix

ðI þ N� þ aH�2ÞðI þ N þ aH2Þ ¼ I þ N þ N� þ aðH2�þ H2Þ þ N�N þ aN�H2 þ aH�2N þ a2H�2H2;


the resulted matrix can be written as

I þ N þ N� þ aðH�2 þ H2Þ þ N�H2 þ H�2N þ ð2a� 1ÞH�2H2: ð3:5Þ

Hence, (3.4) holds if and only if the above matrix (3.5) is Hermitian positive definite. From the definitions of N and H2,(3.4) holds if and only if the matrix

H1 þ N þ N� þ aðH�2 þ H2Þ þ N�H�11 H2 þ H�2H�1

1 N þ ð2a� 1ÞH�2H�11 H2

is Hermitian positive definite. Thus, this theorem is proved. h

Corollary 3.4. Let H1 ¼ M � N be Hermitian P-regular splitting of the Hermitian positive definite matrix H1. Assume there exist

two numbers r > � 12 and a P 1

2 such that rH1 � N � ðr þ 2a� 1ÞH1, then H�12

1 QP�1H121

�� 2< 1.

Proof. From Theorem 3.3, H�1

21 QP�1H

121

�� 2< 1, if and only if

F ¼ H1 þ N þ N� þ aðH�2 þ H2Þ þ N�H�11 H2 þ H�2H�1

1 N þ ð2a� 1ÞH�2H�11 H2

is a Hermitian positive definite matrix, which is the following sample by simple manipulation

F ¼ M þ ðI þ H�2H�11 ÞðN � rH1ÞðI � H�1

1 H2Þ þ rH1 þ ð2a� 1þ rÞH�2H�11 H2 � H�2H�1

1 NH�11 H2:

We have M þ rH1 � 0 since H1 ¼ M � N is a Hermitian P-regular splitting, and F � 0 from assumptions of N. Thus, we haveproved this corollary. h

Theorem 3.5. Let H1 ¼ M � N be a splitting of the Hermitian positive definite matrix H1. Then H121TH

�12

1

�� 2< 1 if and only if the

matrix

H1 þ N þ N� þ aðH2 þ H�2Þ þ NH�11 H�2 þ H2H�1

1 N� þ ð2a� 1ÞH2H�11 H�2

is Hermitian positive definite.

Proof. This theorem can be obtained by an analogous proof to that of Theorem 3.3. h

Corollary 3.6. Let H1 ¼ M � N be Hermitian P-regular splitting of Hermitian positive definite matrix H1. Assume there exist two

numbers r > � 12 and a P 1

2 such that rH1 � N � ðr þ 2a� 1ÞH1, then H121TH

�12

1

�� 2< 1.

Proof. This corollary can be obtained by an analogous proof to that of Corollary 3.4.We observe the iteration matrix

T ¼ ðM þ aH2Þ�1ðN þ ða� 1ÞH2Þ ¼ M�12 I þ aM�1

2H2M�12

� ��1M�1

2NM�12 þ ða� 1ÞM�1

2H2M�12

� �M

12

¼ M�12ðI þ aH2Þ

�1N þ ða� 1ÞH2

� �M

12;

where H2 ¼ M�12H2M�1

2; N ¼ M�12NM�1

2.Let

T ¼ ðI þ aH2Þ�1ðN þ ða� 1ÞH2Þ: ð3:6Þ

Then the iteration matrix T is similar to the matrix T . h

Theorem 3.7. Let H1 ¼ M � N be Hermitian P-regular splitting. Let k1 6 k2 6 . . . 6 kn be the eigenvalues of matrix N and l the

smallest singular value of the matrix H2. If 1þ l26

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ l2

pþ qðNÞl2ffiffiffiffiffiffiffiffiffi

1þl2p and l – 0, then the reasonable parameter satisfies

a� ¼ 1;a� 1

aP

k1 þ kn

2;

a� ¼min2

2� k1 � kn; a

;

a� 1a6

k1 þ kn

2;


where a is the root of the following equation

1þ a2l2 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ a2l2

q� ðqðNÞ � aþ 1Þ a2l2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ a2l2p ¼ 0:

Proof. From (3.6), we have

T ¼ a� 1a

I þ ðI þ aH2Þ�1

N � a� 1a

I� �

:

Let k be an eigenvalue of the matrix ðI þ aH2Þ�1

N � a�1a I

� �and x the corresponding eigenvector. Then

kðI þ aH2Þx ¼ N � a� 1a

I� �

x;

which implies

jkj ðI þ aH2Þx��

2¼ N � a� 1

aI

� �x

��

2

and thereby

jkj 6q N � a�1

a

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ a2l2

p :

So it holds that

qðTÞ 6 a� 1a

��þ q N � a�1

a


p : ð3:7Þ

Case I. When a P 1.

Let � �
gðaÞ ¼ a� 1
aþ

q N � a�1affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ a2l2p :

If a�1a 6

k1þkn2 , then

gðaÞ ¼ a� 1aþ

qðNÞ � a�1affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ a2l2p :

It follows from straightforward operation that

g0ðaÞ ¼1þ a2l2 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ a2l2

p� ðqðNÞ � aþ 1Þ a2l2ffiffiffiffiffiffiffiffiffiffiffiffi

1þa2l2p

a2ð1þ a2l2Þ :

It is clear that g0ðaÞ > 0 when a is large enough. And, we obtain by the assumption of this theorem

g0ð1Þ ¼1þ l2 �


p� qðNÞ l2ffiffiffiffiffiffiffiffiffi

1þl2p

1þ l2 6 0

and

g0ðaÞ ¼ 1þ a2l2 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ a2l2

q� ðqðNÞ � aþ 1Þ a2l2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ a2l2p ¼ 0:

Hence, gðaÞ is the monotone decreasing in ð1; aÞ since g0ðaÞ < 0 when 1 6 a < a; it is the monotone increasing whena > a. So a is the local minimum point of gðaÞ.

Thus, we have the reasonable parameter a� ¼min 22�k1�kn

; an o

.

If a�1a P k1þkn

2 , then

gðaÞ ¼ a� 1aþ

a�1a � k1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ a2l2

p :

Also, it follows from straightforward operation that


g0ðaÞ ¼1þ a2l2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ a2l2

p� ða� 1� ak1Þ a2l2ffiffiffiffiffiffiffiffiffiffiffiffi

1þa2l2p

a2ð1þ a2l2Þ :

Thereby

g0ð1Þ ¼1þ l2 þ


pþ k1

l2ffiffiffiffiffiffiffiffiffi1þl2p

ð1þ l2Þ > 0:

Hence, the reasonable parameter a� ¼ 1 since g0ðaÞ > 0 when a is large enough.

Case II. When a < 1.

From (3.7), we know that

qðTÞ 6 1� aaþ

q N � a�1a


p :

Let

hðaÞ ¼ 1� aaþ

q N � a�1a


p :

We analyze hðaÞ with the same technique of analyzing gðaÞ, it is obtained that the reasonable parameter a� ¼ 1. Hence,we have completed the proof of the theorem. h

Theorem 3.8. Let H1 ¼ M � N be Hermitian P-regular splitting. Assume there exist two numbers r > � 12 and a P 1

2 such thatrH1 � N � ðr þ 2a� 1ÞH1, then fxðk;0Þg generated by Algorithm II converges to the unique solution xH of the system of linearequations (1.1).

Proof. Let

eðk;lÞ ¼ xðk;lÞ � xH:

For quadratic programming (2.4) and Corollary 3.4, we have

H�1

21 rðkþ1;0Þ

�� 2¼ H

�12

1

Xm

l¼1

aðkÞl rðk;lÞ !��

��2

6 H�1

21 rðk;1Þ

�� 2¼ H

�12

1 Aeðk;1Þ��

2¼ H

�12

1 ATXm

l¼1

aðkÞl eðk�1;lÞ

!��

2

6 H�1

21 ATeðk�1;1Þ

�� 2

¼ H�1

21 ATA�1H

121H�1

21 Aeðk�1;1Þ

�� 26 H

�12

1 ATA�1H121

�� 2

H�1

21 Aeðk�1;1Þ

�� 2¼ H

�12

1 QP�1H121

�� 2

H�1

21 rðk�1;1Þ

�� 2

6 b H�1

21 rðk�1;1Þ

�� 2

. . . 6 bk H�1

21 rð0;1Þ

�� 2;

where b ¼ H�1

21 QP�1H

121

�� 2.

Hence,

limk!1ðrðkþ1;0ÞÞT H�1

1 rðkþ1;0Þ ¼ 0:

Since H�11 is Hermitian positive definite, we obtain that

limk!1

rðkþ1;0Þ ¼ 0:

This theorem is proved. h

4. Preconditioner

In this section, we give a preconditioner and discuss the preconditioned condition number.For the splitting A ¼ P � Q , let P be a new preconditioner. Next, we discuss the preconditioned condition number.Let

R ¼ P�1A:

Then

R ¼ ðM þ aH2Þ�1ðH1 þ H2Þ ¼ M�12ðI þ aH2Þ

�1ðH1 þ H2ÞM

12;


where H2 ¼ M�12H2M�1

2; H1 ¼ M�12H1M�1

2.Let

R ¼ ðI þ aH2Þ�1ðH1 þ H2Þ; a > 0:

Then the preconditioned matrix R is similar to the matrix R.For x 2 Cn, the numerical field of the matrix H2 is ½u1;un�, then there exist two vectors x1; xn such that

x�1H2x1

x�1x1

�� ¼ u1;

x�nH2xn

x�nxn

�� ¼ un:

Assumption. Let x�1H2x1 ¼ c1 þ d1i; x�nH2xn ¼ cn þ dni with c1; d1; cn; dn are all real numbers. We assume that

u21

u2n6

c1

cn6 1: ð4:1Þ

Theorem 4.1. Assume that H1 ¼ M � N is a strong Hermitian P-regular splitting of the Hermitian positive definite matrixH1. Assume that r1 6 . . . 6 rn are the eigenvalues of the matrix H1. If Assumption (4.1) holds, then the reasonable parametersatisfies

a� 2 1rn;

1r1

� �: ð4:2Þ

Proof. Let k be an eigenvalue of the matrix R and x be the corresponding eigenvector of the eigenvalue k, let jjxjj2 ¼ 1. Then

ðH1 þ H2Þx ¼ kðI þ aH2Þx:

So we have

k ¼ x�H1xþ x�H2x

1þ ax�H2x:

Let x�H2x ¼ c þ di; k ¼ aþ bi, where a; b; c; d are all real numbers. Then, we have

a2 þ b2 ¼ ðx�H1xÞ

2þ 2x�H1xc þ u2

1þ 2ac þ a2u2 ;

where u2 ¼ c2 þ d2.If u ¼ 0; jkj ¼ x�H1x, which implies

condðRÞ 6 condðHÞ:

If u – 0, let r ¼ x�H1x, and

gðr;uÞ ¼ r2 þ 2rc þ u2

1þ 2ac þ a2u2

where u 2 ½u1;un�; r1 6 r 6 rn.

We determine a such that max gðr;uÞmin gðr;uÞ arrives at the minimum. From the assumption of H2, we see c P 0. Thus, we make use of

the comparisons of r2; ra ;

1a2

�to discuss gðr;uÞ as follows:

(1) If arn 6 1, then

max gðr;uÞ ¼ r2n þ 2rncn þ u2

n

1þ 2acn þ a2u2n;

min gðr;uÞ ¼ r21 þ 2r1c1 þ u2

1

1þ 2ac1 þ a2u21

and thereby

max gðr;uÞmin gðr;uÞ ¼

r2n þ 2rncn þ u2

n

r21 þ 2r1c1 þ u2

1

� 1þ 2ac1 þ a2u21

1þ 2acn þ a2u2n: ð4:3Þ


Let

gðaÞ ¼ 1þ 2ac1 þ a2u21

1þ 2acn þ a2u2n:

It follows from straightforward operation that

g0ðaÞ ¼ 2ððcnu21 � c1u2

nÞa2 þ ðu21 � u2

nÞaþ c1 � cnÞð1þ 2acn þ a2u2

nÞ2 :

From the assumptions, we see that g0ðaÞ 6 0. Hence, (4.3) reaches the minimum at a ¼ 1rn

. Further,


r2n þ 2rnc1 þ u2

1

r21 þ 2r1c1 þ u2

1

:

If ar1 6 1; arn > 1, then
(2)
max gðr;uÞ ¼ r2n þ 2rnc1 þ u2

1

1þ 2ac1 þ a2u21

;

min gðr;uÞ ¼ r21 þ 2r1c1 þ u2

1

1þ 2ac1 þ a2u21

;

thus


r2n þ 2rnc1 þ u2

1

r21 þ 2r1c1 þ u2

1

:

If ar1 > 1, then
(3)
max gðr;uÞ ¼ r2n þ 2rnc1 þ u2

1

1þ 2ac1 þ a2u21

;

min gðr;uÞ ¼ r21 þ 2r1cn þ u2

n

1þ 2acn þ a2u2n:

Thereby


r2n þ 2rnc1 þ u2

1

r21 þ 2r1cn þ u2

n

� 1þ 2acn þ a2u2n

1þ 2ac1 þ a2u21

: ð4:4Þ

It follows from an analogous demonstration to the proof of (4.3), (4.4) reaches the minimum at a ¼ 1r1

. Further,


r2n þ 2rnc1 þ u2

1

r21 þ 2r1c1 þ u2

1

:

Hence, when a 2 1rn; 1

r1

h i, it holds


r2n þ 2rnc1 þ u2

1

r21 þ 2r1c1 þ u2

1

6 condðHÞ:

We have completed the proof of the theorem. h

5. Numerical experiments

In this section, we give some preliminary computational results. All our tests are started from zero vector, and terminatedwhen the current iterate satisfied krðkÞk2 < 10�5, where rðkÞ is the residual of the current, say kth iteration or the number ofiteration step is up to 5000. For the latter the iteration is failing.

Example 5.1. Consider the system of linear equations, for which Ax ¼ b is defined as follows:

A ¼W FX

�FT N

� �;

where W 2 Rq�q; N; X 2 Rðn�qÞ�ðn�qÞ with 2q > n. We define the matrices W ¼ ðwk;jÞ; N ¼ ðnk;jÞ:F ¼ ðf k;jÞ andX ¼ diagðx1; . . . ;xn�qÞ as follows:

Table 5The com

n

100

400

800

1600

2000

3000


wk;j ¼kþ 1; for j ¼ k;

1; for jk� jj ¼ 1; k; j ¼ 1;2; . . . ; q;

0; otherwise;

8><>:

nk;j ¼kþ 1; for j ¼ k;

1; for jk� jj ¼ 1; k; j ¼ 1;2; . . . ; n� q;

0; otherwise;

8><>:

f k;j ¼j; for k ¼ jþ 2q� n; k; j ¼ 1;2; . . . ;n� q; j ¼ 1;2; . . . ;n� q;

0; otherwise

and

xk ¼1k; k ¼ 1;2; . . . ;n� q;

the right-hand side b ¼ ð1;1; . . . ;1ÞT .Let

A ¼ H1 þ H2;

where

H1 ¼~Wq�q 0

0 ~Nðn�qÞ�ðn�qÞ

!; H2 ¼ A� H1;

~Nðn�qÞ�ðn�qÞ ¼

2 1

1 2 . ..

. .. . .

.1

1 2

0BBBBB@

1CCCCCA;

~Wq�q ¼

2 1

1 2 . ..

. .. . .

.1

1 2

0BBBBB@

1CCCCCA:

Let

H1 ¼ B� C;

where

B ¼�W aF

aFT �N

!; C ¼ B� H1:

�W ¼ 5I þ ~W; �N ¼ 5I þ ~N:

Thus

A ¼ P � Q ¼ ðBþ aH2Þ � ðC � ða� 1ÞH2Þ:

In this test, the parameters of our Algorithm II are a ¼ 0:8; q ¼ 910 n.

.1parisons of computational results.

Algorithm II Algorithm II HSSm ¼ 3 m ¼ 2

IT 6 13 49CPU(s) 0.0156 0.0312 0.0312IT 8 14 99CPU(s) 0.0312 0.0468 0.1560IT 9 18 142CPU(s) 0.2028 0.4524 0.9516IT 9 20 205CPU(s) 0.8736 1.2480 4.8048IT 10 20 230CPU(s) 1.2168 2.0904 8.4865IT 8 20 285CPU(s) 2.4336 5.0076 25.0850

TC

TC


The results given in Table 5.1 indicate the Algorithm II is much more efficient than the HSS method in the senses of theiteration step (denoted as IT) and the total CPU time (denoted as CPU) in second.

Example 5.2. We consider the two-dimensional advection–diffusion equations as follows,

Table 5Compu

b ¼ 0

b ¼ 0

able 5.3omputa

mineimaxeicond

able 5.4omputa

mineimaxeicond

�mMuþ b ruþ ru ¼ f in X;

u ¼ 0 on X:

In all experiments here, we consider this equations on X ¼ ½0;1� � ½0;1� with a body force fðxÞ such that the true solutionis u ¼ ðu;vÞT ,

u ¼ sinðpxÞ sinðpyÞ;v ¼ ðx2 � xÞðy2 � yÞ;

the convection field b ¼ ð1;1ÞT and with the parameters m ¼ 1; r ¼ 1. Let Th be a convention decomposition of X into uni-form rectangular K. All the numerical experiments have been performed using the conforming Q 1 finite element Vh,

Vh ¼ vh 2 H1ðXÞ2jvhjK 2 Q 1ðKÞ2; 8K 2 Th

n o:

We employ the standard finite element method, then the variational formulation of this equations is: find uh 2 Vh suchthat

mðuh;vhÞ þ ðb ruh;vhÞ þ rðuh;vhÞ ¼ ðf;vhÞ; 8vh 2 Vh:

The stepsizes along both x and y directions are the same, i.e., h ¼ 132 or 1

64

� �At first, we implement our Algorithm I to solve the above system of linear equations. Let

A ¼ H1 þ H2; H1 ¼ M � N;

where H1 ¼ ðEþET Þ2 � 0:7diagðAÞ is symmetric positive definite, and

E ¼ ðeijÞn�n ¼aij; jj� ij 6 2;0; jj� ij > 2;

M ¼ ðbI þ H1Þ=2; N ¼ ðbI � H1Þ=2

.2tational results of the spectral radius.

n qðP�1QÞ qðTHSSÞ

a ¼ 0:5 a ¼ 0:6 a ¼ 0:8 a ¼ 1 a ¼ 1:1 a ¼ 1:2 a ¼ 1:5

:3 32 0.8710 0.8609 0.8351 0.7977 0.7717 0.7382 0.5312 0.845464 0.9660 0.9629 0.9545 0.9411 0.9310 0.9167 0.7795 0.9581

:2782 32 0.8655 0.8546 0.8263 0.7842 0.7544 0.7151 0.4515 0.836164 0.9646 0.9611 0.9518 0.9365 0.9246 0.9072 0.6970 0.9553

tional results of the condition number n ¼ 32 and b ¼ 0:3.

P�1A PHSS A

a ¼ 0:5 a ¼ 0:6 a ¼ 0:8 a ¼ 1 a ¼ 1:1 a ¼ 1:2 a ¼ 1:5

g 0.1290 0.1391 0.1649 0.2023 0.2283 0.2618 0.4688 0.1546 0.0207g 1.8530 1.6614 1.4453 1.2781 1.2082 1.1456 0.9914 1.3181 3.9870

14.3643 11.9439 8.7647 6.3178 5.2945 4.3759 2.1148 8.5259 192.7036

tional results of condition number with n ¼ 64 and b ¼ 0:3.

P�1A PHSS A

a ¼ 0:5 a ¼ 0:6 a ¼ 0:8 a ¼ 1 a ¼ 1:1 a ¼ 1:2 a ¼ 1:5

g 0.0340 0.0371 0.0455 0.0589 0.0690 0.0833 0.2205 0.0419 0.0052g 1.8530 1.6614 1.4460 1.2794 1.2096 1.1471 0.9931 1.3264 3.9967

54.5 44.7817 31.7802 21.7216 17.5304 13.7707 4.5039 31.6563 771.2581

Table 5.5Computational results of condition number with n ¼ 32 and b ¼ 0:2872.

P�1A PHSS A

a ¼ 0:5 a ¼ 0:6 a ¼ 0:8 a ¼ 1 a ¼ 1:1 a ¼ 1:2 a ¼ 1:5

mineig 0.1344 0.1454 0.1737 0.2158 0.2456 0.2849 0.5485 0.1639 0.0207maxeig 1.8653 1.6667 1.4503 1.2820 1.2117 1.1487 0.9937 1.3041 3.9870cond 13.8787 11.4629 8.3495 5.9407 4.9336 4.0320 1.8117 7.9567 192.7036

Table 5.6Computational results of condition number with n ¼ 64 and b ¼ 0:2872.

P�1A PHSS A

a ¼ 0:5 a ¼ 0:6 a ¼ 0:8 a ¼ 1 a ¼ 1:1 a ¼ 1:2 a ¼ 1:5

mineig 0.0354 0.0389 0.0482 0.0635 0.0754 0.0928 0.3030 0.0447 0.0052maxeig 1.8633 1.6667 1.4511 1.2833 1.2132 1.1503 0.9955 1.3124 3.9967cond 52.6356 42.8458 30.1058 20.2094 16.0902 12.3955 3.2855 29.3602 771.2581

Table 5.7Computational results of iteration and CPU time.

n Algorithm I HSS

a ¼ 0:5 a ¼ 0:6 a ¼ 0:8 a ¼ 1 a ¼ 1:1 a ¼ 1:2 a ¼ 1:5

32 IT 76 70 58 47 41 35 17 72CPU 0.1094 0.0938 0.0625 0.0469 0.0469 0.0496 0.0313 1.5444

64 IT 282 258 210 161 137 113 40 221CPU 2.7031 2.4688 2.0652 1.6406 1.4688 1.2500 0.6094 21.8643


and

P ¼ M þ aH2; Q ¼ N þ ða� 1ÞH2:

In our test, the parameters are b ¼ 0:3 and b ¼ 0:2872, which is the best parameter for HSS iteration. In order to examinethe effectiveness of our algorithm, we also implement our algorithm with different a and HSS iteration when n ¼ 32; 64,respectively.

The Tables 5.2–5.6 show the computational results. Also, THSS denote the iteration matrix of HSS method in Table 5.2; PHSS

denote the HSS preconditioned matrix in Tables 5.3–5.6; mineig, maxeig and cond are minimum eigenvalue, maximumeigenvalue and condition number, respectively.

From the Tables 5.2–5.6, the spectral radius and the condition numbers of the preconditioned matrix is much less thanthese of HSS preconditioned matrix. Also, for solving the large sparse system of linear equations, the new splitting iterationmethod is much more practical and efficient than the HSS iteration method.

Furthermore, in order to show the effectiveness of our method. Let the parameter b ¼ 0:2728, which is the best param-eter. We compare Algorithm I with HSS method.

The Table 5.7 shows the computational results.The speed-up is defined as follows (see [4])

speed-up ¼ CPU of HSS iterationCPU of this new method

:

The Example 5.1 shows that the speed-up is up to 6.97 and 10.31 when n ¼ 2000 and n ¼ 3000, and the Example 5.2shows that the speed-up is up to 49.3 and 35.9 when n ¼ 2� 32� 32 and n ¼ 2� 64� 64. These show that our algorithmsis much more efficient from another angle.

Acknowledgments

The authors are very much indebted to the anonymous referees for their helpful comments and suggestions which greatlyimproved the original manuscript of this paper.

References

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