Fast parameterized inexact Uzawa method for complex symmetric linear systems

Fast parameterized inexact Uzawa method for complex symmetric linear systems

q

Qing-Qing Zheng,Chang-Feng Ma ?

School of Mathematics and Computer Science,Fujian Normal University,Fuzhou 350007,PR China

a r t i c l e i n f o Keywords:

Complex symmetric linear system Iterative methods Correction technique The PIU method

Convergence analysis Numerical experiments

a b s t r a c t

In previous years,Bai and Wang presented a class of parameterized inexact Uzawa (PIU)methods for solving the generalized saddle point problems.In this paper,we consider the same method for iteratively solving the complex symmetric linear systems.Our main contribution is accelerating the convergence of the parameterized inexact Uzawa method by correction technique.First,the corrected model for the PIU method is established and the corrected PIU method is presented.Then we study the convergence property of the cor-rected PIU method.In fact,the corrected PIU method can converge faster than some Uza-wa-type and HSS-like methods.Finally,numerical experiments on a few model problems are presented to illustrate the theoretical results and examine the numerical effectiveness of the new method.

ó2015Elsevier Inc.All rights reserved.

1.Introduction

Let n be a positive integer.We consider the iterative solution of systems of linear equations of the form

Ax ?b ;

A 2C n ?n and x ;b 2C n ;

e1:1T

where A 2C n ?n is a complex symmetric matrix of the form

A ?W ti T e1:2T

and W ;T 2R n ?n are real,symmetric,and positive de?nite matrices.Here and in the sequel,we use i ????????

à1p to denote the

imaginary unit.

Complex symmetric linear systems of this kind arise in many important problems in scienti?c computing and engineering applications.For example,FFT-based solution of certain time-dependent PDEs [1],diffuse optical tomography [2],algebraic eigenvalue problems [3,4],molecular scattering [5],numerical solutions of the complex Helmholtz equation and numerical computations in eddy current problems.For more examples,we refer to [6–12]and the references therein.Hence,there is a strong need for the fast solution of complex symmetric linear systems.

For solving the complex symmetric linear system (1.1)ef?ciently,van derVorst and Mellissen [9]proposed the conjugate orthogonal conjugate gradient (COCG)method,which is regarded as an extension of the Conjugate Gradient (CG)method [13].Relatively complicated but robust methods such as QMR [14],CSYM [15],and Bi-CGCR [16]are also useful.QMR is

https://www.360docs.net/doc/bd19005842.html,/10.1016/j.amc.2015.01.023

0096-3003/ó2015Elsevier Inc.All rights reserved.

q

The Project supported by National Natural Science Foundation of China (Grant Nos.11071041,11201074),Fujian Natural Science Foundation (Grant No.2013J01006),The University Special Fund Project of Fujian (Grant No.JK2013060)and R&D of Key Instruments and Technologies for Deep Resources Prospecting (the National R&D Projects for Key Scienti?c Instruments)under Grant No.ZDYZ2012-1-02-04.?Corresponding author.

E-mail address:macf88@https://www.360docs.net/doc/bd19005842.html, (C.-F.Ma).

derived from the complex symmetric Lanczos method,CSYM is obtained from the idea of QMR and tridiagonalization of A by Householder re?ections,and Bi-CGCR is derived from a particular case in Bi-CG [17]for solving non-Hermitian linear sys-tems.In [18]Sogabe and Zhang extended the conjugate residual (CR)method described in [19,20]to complex symmetric linear systems based on an observation of deriving CG,CR,and COCG.Moreover,Based on the Hermitian and skew-Hermi-tian splitting (HSS)

A ?H tS ;

of the matrix A 2C n ?n ,with

H ?

12eA tA ?Tand S ?1

2

eA àA ?Tbeing the Hermitian and skew-Hermitian parts and A ?being the conjugate transpose of the matrix A 2C n ?n ,we can apply the

HSS iteration method [21]or its preconditioned variant PHSS (i.e.,the preconditioned HSS,see [22])which were proposed by Bai and his co-authors to compute an approximate solution of the linear system (1.1).In addition,the convergence properties of the PHSS method can be found in [23].In [24],Bai,Golub and Ng further generalized the technique for constructing HSS iteration method for solving large sparse non-Hermitian positive de?nite system of linear equations to the normal/skew-Hermitian (NS)splitting obtaining a class of normal/skew-Hermitian splitting (NSS)iteration methods.Theoretical analyses shown that the NSS iteration method converges unconditionally to the exact solution of the system of linear Eq.(1.1).A potential dif?culty with the HSS iteration approach is the need to solve the shifted skew-Hermitian sub-system of linear equations at each iteration step.Hence,Bai,Benzi and Chen presented a modi?cation of the HSS iteration scheme in [25]and some of its basic properties are studied.In [26],the authors proposed a preconditioned variant of the modi?ed HSS (PMHSS)iteration method for solving the complex symmetric systems of linear equations.Moreover,the authors in [27]pre-sented analytical and extensive numerical comparisons of some available numerical solution methods for the complex val-ued linear algebraic systems (1.1).

In this paper,we study the corrected parameterized inexact Uzawa method for solving the complex symmetric linear sys-tem (1.1).We call this new method CPIU method for the sake of simplicity.We also introduce the overall reduction coef?-cient a ei Tto measure the effectiveness of the correction process.If the overall reduction coef?cient a ei T<1is satis?ed in each iteration step,then the corrected PIU method will converge faster than the PIU method.Moreover,suf?cient conditions for the convergence of the CPIU method for the linear system (1.1)are also provided in the paper.

The paper is organized as follows.In Section 2,we introduce the parameterized inexact Uzawa method and establish the corrected PIU model for the Eq.(1.1),then the CPIU method is presented.In Section 3,The corrected PIU model is solved.Moreover,we analyze the convergence property of the CPIU method.In Section 4,some numerical experiments are given to show the ef?ciency of the CPIU method.Finally,some conclusion remarks are proposed in Section 5.

The following notations will be used throughout this paper.We denote the identity matrix and the 0-matrix by I and O ,respectively.?u ;v denote the inner product of vectors u and v .For a matrix A ,we denote the spectral norm of A by k A k 2.And denote the range and the null spaces of A by R eA Tand N eA T,respectively.The conjugate transpose of A is denoted by A H .More-over,the Moore–Penrose inverse of A is de?ned as the unique matrix A twhich satis?es the following four matrix equations:

AA tA ?A ;A tAA t?A t;eAA tTH

?AA t;

eA tA TH

?A tA :

2.The corrected PIU method

Let x ?y tiz and b ?p tiq ,then from (1.1)and (1.2),we can get eW tiT Tey tiz T?p tiq ,which implies that we can obtain the following block two-by-two systems of linear equation

DX ?

W àT T

W

y

z

?

p q

?g :e2:1T

Conversely,from the linear Eq.(2.1),we can get the complex symmetric linear system (1.1).So the complex symmetric linear system (1.1)is formally identical to the above block two-by-two systems of linear Eq.(2.1).Based on the PMHSS precondi-tioning matrix,Bai in [28]constructed a class of rotated block triangular preconditioners for linear system (2.1),and analyze the eigen-properties of the corresponding preconditioned matrices.Moreover,the block two-by-two systems of linear Eq.(2.1)can be formally regarded as a special case of the generalized saddle point problem [29,30].It frequently arises from ?nite element discretizations of elliptic partial differential equation (PDE)-constrained optimization problems such as dis-tributed control problems [31–35]and so on.

Based on the parameterized inexact Uzawa (PIU)iteration method for solving the following generalized saddle point problem

A B B >àC

e y e z

?e p e q

;

e2:2T

12Q.-Q.Zheng,C.-F.Ma /Applied Mathematics and Computation 256(2015)11–19

in this section we derive a corrected PIU iteration method for solving the block two-by-two systems of linear Eq.(2.1).To this end,we ?rst introduce the PIU iteration method proposed in Bai and Wang [36]for the Eq.(2.2).This PIU iteration method is methodically described in the following.

Method 2.1.Given initial vectors e y e0T2R n and e

z e0T2R n and two relaxation factors x ;s with x ;s –0.For k ?0;1;2;...,until the iteration sequence fee y ek T>;e

z ek T>T>

g converges to the exact solution of the saddle point problem (2.2),compute e y ek t1T?e y ek Ttx P à1ee p àA e y ek TàB e z ek TT;e z ek t1T?e z ek Tts Q à1eB >e y ek t1TàC e

z ek Tàe q T;(

where P 2R n ?n and Q 2R n ?n are prescribed symmetric positive de?nite matrices.

If P –A and x ?s ?1then the PIU iteration Method 2.1yields inexact Uzawa method [37–39]for solving the saddle point problems.Here,in this paper we consider the case that P ?Q ?A and x ?1.Based on the above Method 2.1,we can get the following iteration method.

Method 2.2.Given initial vectors y e0T2R n ;z e0T2R n and the relaxation factor s with s –0.For k ?0;1;2;...,until the iter-ation sequence fey ek T>;z ek T>T>

g converges to the exact solution of the saddle point problem (2.1),compute

y ek t1T?W à1ep tTz ek TT;

z ek t1T?z ek Tàs W à1eT >y ek t1TtWz ek Ttq T:

(

Denote X ek T

?

y ek T

z ek T

,then the PIU Method 2.2can be rewritten as X ek t1T?HX ek TtM à1g ;

e2:3T

where

H ?

O

W à1T

O I às W à1eC tTW à1T T

!

and M ?

W O T

1s W

!:

Let N ?M àD ?

O T O

1s à1àáW

,then

D ?M àN e2:4T

de?nes a splitting of the coef?cient matrix D of the block two-by-two systems of linear Eq.(2.1),and the PIU Method 2.2can also be induced by the matrix splitting (2.4).Easily,we see that H ?M à1N is the iteration matrix of the PIU Method 2.2.Therefore,the PIU Method 2.2is convergent if and only if the spectral radius of the iteration matrix H is less than one,i.e.,q eH T<1.See [40–42].

Assume X 1;X 2;...;X m are approximate solutions of Eq.(2.1),where m >1is a positive integer and DX i –g ei ?1;2;...;m T.

The basic idea of the correction model for the parameterized inexact Uzawa Method 2.2is:construct a vector b X

which is a better approximate solution of Eq.(2.1)than X i ei ?1;2;...;m T.That is,

k g àD b X k 2

k g àDX i k 2

:e2:5T

The vector b X

that satis?es (2.5)is called the corrected solution of X 1;X 2;...;X m ,and the progress to determine the vector X is called CPIU process.The corrected model for the PIU is presented as follows.The CPIU model.

b X

?a 1X 1ta 2X 2táááta c X m ;a 1ta 2táááta m ?1;

k g àD b X

k 2

?min :e2:6T

With the above CPIU model we can obtain the following detailed methodic description of the CPIU method for the block

two-by-two system of linear Eq.(2.1).The CPIU method.

Give an initial vector X e0T2R 2n ,let k ?0. While

X ek T

1?X ek à1T

For ej ?2;3;...;m T

X ek Tj

?HX ek T

j à1tM à1g

end For

X ek T

?a ek T

1X ek T

1ta ek T

2X ek T

2táááta ek T

m X ek T

m k ?k t1End While

Q.-Q.Zheng,C.-F.Ma /Applied Mathematics and Computation 256(2015)11–1913

Here aekT

i

ei?1;2;...;mTis determined by the k th CPIU progress.

How to solve the above CPIU model will be illustrated detailedly in Section3.From the Eq.(2.5),we can see that if the iteration(2.3)is convergent,i.e.,qeHT<1,then the sequence f XekTg will converge.

3.Analysis for the CPIU method

In this section,we will solve the CPIU model(2.6).Moreover,the suf?cient conditions for the convergence of the CPIU method are also provided.

When the linear system EX?r is consistent,then the vector X02C2n that satis?es k X0k

2?min EX?r k X k

2

is called the min-

imum norm solution of linear system EX?r.When the linear system EX?r is not consistent,then the vector X02C2n that

satis?es k X0k

2?min min k EXàr k

2

k X k

2

is called the minimal norm least squares solution of linear system EX?r.

Lemma3.1.If the linear system EX?r is consistent,then the minimum norm solution X0of EX?r is unique and X0?Etr.

Proof.Firstly,we prove X02ReE HTholds true.If X0R ReE HT,then we can obtain the decomposition X0?X1tX2;X12ReE HT;X22NeET??ReE HT ?;X2–0:

Note X1?X2,we obtain

k X0k2

2?k X1k2

2

tk X2k2

2

>k X1k2

2

:

Because EX1?EX1tEX2?EX0?r,then we get X0is not the minimum norm solution of EX?r.This is a contradiction,so X02ReE HT.

Then we illustrate the uniqueness of minimum norm solution X0.If Z0is another minimum norm solution of EX?r,then Z02ReE HT.So we have

X0àZ02ReE HT??NeDT ?:

Moreover,because EeX0àZ0T?0,then we have X0àZ02NeET.So X0àZ0?0,which implies X0?Z0.

Now we illustrate X0?Etr.Obviously,the general solution of the equation EX?r is

X?EtrteIàEtETy;ey2C nT:

Take y?0,then we obtain that X0?Etr is a solution of EX?r.Because

?eIàEtETy;X0 ?y HeIàEtETH X0?y HeIàEtETEtr?y HeEtàEtEEtTr?0;

then we have

k X k2

2?k X0k2

2

tkeIàEtETy k2

2

P k X0k2

2

;

which implies that X0?Etr is the minimum norm solution of linear system EX?r.The proof is completed.h

Lemma3.2.If the linear system EX?r is not consistent,then the minimal norm least squares solution X0of EX?r is unique and X0?Etr.

Proof.From Lemma3.1,we can demonstrate that the result holds true.h

Together with Lemmas3.1and3.2we prove the following result which show how to solve the CPIU model(2.6).

Theorem3.1.If X1;X2;...;X m are approximate solutions of Eq.(2.1),for k?1;2;3;...,let r k?gàDX k.Denote

b i?r iàr1ei?2;3;...;mT;Q?eb2;b3;...;b mT;e3:1T

y 1?ea2;a3;...;a mT>?àQtr1;a1?1à

X m

i?2

a i:e3:2T

Then b X?a1X1ta2X2táááta m X m is the corrected solution of X1;X2;...;X m.

Proof.Assume b X?b1X1tb2X2tááátb m X m is the corrected solution of X1;X2;...;X m of CPIU model(2.6),where b1tb2tááátb m?1.Then we have

b X?X1tX m

i?2b ieX iàX1T;

14Q.-Q.Zheng,C.-F.Ma/Applied Mathematics and Computation256(2015)11–19

which implies

reb XT?gàD b X?r1tX m

i?2

b ier iàr1T:

From(3.1)and(3.2),we obtain that the above equation can be rewritten as reb XT?r1tQeb2;b3;...;b mT>.This implies that?nding a vector b X which can minimize k reb XTk2is equivalent to?nding y2C mà1which can minimize k r1tQ y k2.

By making use of Lemmas 3.1and 3.2,the minimum norm solution satis?es k y1k2?min k r

1tQ y k2

k y k2is

y1?ea2;a3;...;a mT>?Qtr1.Let a1?1àP m

i?2

a i,then we can obtain that

b X?a1X1ta2X2táááta m X m is the corrected

solution of X1;X2;...;X m of the CPIU model(2.6).

The proof is completed.h

Remark3.1.Take j2f1;2;...;m g,and denote

b

i

?r iàr jei–jT;Q j?eb1;...;b jà1;b jt1;...;b mT;

y j ?ea1;...;a jà1;a jt1;...;a mT>?àQt

j

r j;a1?1à

X m

i?2

a i:

Then from the proof of Theorem 3.1,we can obtain that b X?a1X1ta2X2táááta m X m is the corrected solution of X1;X2;...;X m of the CPIU model(2.6).

Lemma3.3.Let A2R n?n;P2R n?n and Q2R n?n be symmetric positive de?nite,and B2R n?n is nonsingular.When C?d Q with d>0a real constant,then the PIU Method2.1for the generalized saddle point problem(2.2)is convergent,provided that x sat-is?es0

max

and s satis?es

0

d

and s?2dtel maxàd g maxTx <2e2àg max xT;e3:3T

where g max is the largest eigenvalue of the matrix Pà1A,and l max is the largest eigenvalue of the matrix J?Qà1B>Pà1B.

Proof.See Theorem2.2of[36].h

Theorem3.2.If W2R n?n and T2R n?n are symmetric positive de?nite,then the PIU Method2.2for solving the block two-by-two systems of linear Eq.(2.1)is convergent,provided that s satis?es

0Wà1T.

Proof.Let A?C?W;B?àT;P?Q?A and x?1,then we can obtain d?1;Pà1A?I and J?Qà1B>Pà1B?Wà1T>Wà1T. Also from Pà1A?I we have g max?1,then from Lemma3.3,we can get the result directly.h

In order to measure the effect of the CPIU correction process,we introduce the overall reduction coef?cient a?k reXTk2

min16i6m k r i k

2

:e3:5T

From(2.5)we have06a<1.If a?1,then the CPIU process(2.6)is failed.If06a<1,then the CPIU process is successful. And the smaller a is,the better of the effect of the CPIU progress is.In particular,if a?0,then the CPIU process is?nished, i.e.,the corrected solution is the exact solution of Eq.(2.1).

Denote the overall reduction coef?cient of the k th correction progress as aekT,then

aekT?k reX ekTTk

2

min16j6m k rekT

j k 2

;

where rekT

j ?gàDXekT

j

ej?1;2;...;mTand reXekTT?gàDXekT,with XekT

i

ei?1;2;...;mTis the approximate solutions of k th

CPIU progress and XekTis the corrected solution of XekT

1;XekT

2

;...;XekT

m

.

With the overall reduction coef?cient of the k th correction progress aekT,we can get the following result.

Theorem3.3.If W2R n?n and T2R n?n are symmetric positive de?nite,then the CPIU method for solving the block two-by-two systems of linear Eq.(2.1)is convergent,provided that s satis?es

Q.-Q.Zheng,C.-F.Ma/Applied Mathematics and Computation256(2015)11–1915

0Wà1T.Moreover,if aekT<1is satis?ed,then the CPIU method will con-verge faster than the PIU method.

Proof.If0

4.Numerical experiments

In this section,we perform some numerical examples to illustrate the theoretical results and show the effectiveness of the CPIU iteration method for solving the complex symmetric linear system(1.1)in terms of both iteration steps(denoted as IT) and computing time(in s,denoted as CPU),and the norm of the residual(denoted as‘‘RES’’)de?ned by

RES?

??????????????????????????????????????????????????????????????????????????????????????????

k pàWyekTtTzekTk2

2

tk qtTyekTtWzekTk2

2 q

:

In actual computations,the iteration schemes are started from the zero vector and terminated if the current iterations satisfy ERR610à6or the number of the prescribed iteration steps k?500are exceeded,where

ERR?

??????????????????????????????????????????????????????????????????????????????????????????

k pàWyekTtTzekTk2

2

tk qtTyekTtWzekTk2

2 q

??????????????????????????????????????????????????????????????????????????????????????????

k pàWye0TtTze0Tk2

2

tk qtTye0TtWze0Tk2

2 q:

All experiments are performed in MATLAB(version7.4.0.336(R2010b))with machine precision10à16,and all experiments are implemented on a personal computer with2.20GHz central processing unit(Intel(R)Core(TM)i3–2310M),2.00G mem-ory and Win7operating system.

The CPIU iteration method is compared with the preconditioned Uzawa(PU)method[39]and PMHSS[26]methods.For the tests reported in this section we used the optimal values of the parameter s(denoted by s opt)for the CPIU,PU and PMHSS iteration methods.The experimentally found optimal parameters s opt are the ones resulting in the least numbers of iterations for the three methods for each of the numerical examples and for each choice of the spatial mesh-sizes.

Example4.1(See[10,25,26]).The system of linear Eq.(1.1)is of the form

Kt3à

???

3

p

s I

!

ti Kt

3t

???

3

p

s I

!

"#

x?b;e4:1Twhere s is the time step-size and K is the?ve-point centered difference matrix approximating the negative Laplacian oper-ator L?àD with homogeneous Dirichlet boundary conditions,on a uniform mesh in the unit square?0;1 ??0;1 with the mesh-size h?1.The matrix K2R n?n possesses the tensor-product form K?I V ltV l I,with V l?hà2tridiag eà1;2;à1T2R l?l.Hence,K is an n?n block-tridiagonal matrix,with n?l2.We take

W?Kt3à

???

3

p

s I;and T?Kt

3t

???

3

p

s I

and the right-hand side vector b with its j th entry?b

j

being given by

?b

j ?

e1àiTj

sejt1T2

;j?1;2;...;n:

In our tests,we take s?h.Furthermore,we normalize coef?cient matrix and right-hand side by multiplying both by h2.For more details,we refer to[10].

With respect to different sizes of the coef?cient matrix,we list IT,CPU and RES about the PU,PMHSS and CPIU(m?8) methods in Table1for Example4.1.By comparing the results in Table1,we observe that the CPIU iteration method outper-forms the PU and PMHSS methods,as it requires much less time and iteration steps to achieve the stopping criterion.

Moreover,we also take m?2;3;4;5;6;7in this numerical experiment to illustrate the special property of the CPIU method.The results for different parameters m of the CPIU iteration method for Example4.1are presented in Table2.From the results of Table2we observe that the iteration steps of the CPIU iteration method become smaller when the parameter m becomes larger which means that the CPIU iteration method converges faster if the parameter m becomes larger.Moreover, the solution that our CPIU iteration method found is more accurate than the other two methods,because the norm of the residual(RES)is more smaller than the PU and PMHSS methods.So we can get a good approximate solutions by taking a proper parameter m in our actual calculation.That illustrates that CPIU iteration method is a very good method for solving the complex symmetric linear systems.

16Q.-Q.Zheng,C.-F.Ma/Applied Mathematics and Computation256(2015)11–19

Example 4.2(See [11,25,26]).The system of linear Eqs.(1.1)is of the form

?eàx 2M tK Tti ex C V tC H T x ?b ;

e4:2T

where M and K are the inertia and the stiffness matrices,C V and C H are the viscous and the hysteretic damping matrices,respectively,and x is the driving circular frequency.We take C H ?l K with l a damping coef?cient,M ?I ;C V ?10I ,and K the ?ve-point centered difference matrix approximating the negative Laplacian operator with homogeneous Dirichlet

boundary conditions,on a uniform mesh in the unit square [0,1]?[0,1]with the mesh-size h ?1

.The matrix K 2R n ?n pos-sesses the tensor-product form K ?I B l tB l I ,with B l ?h à2

tridiag eà1;2;à1T2R l ?l .Hence,K is an n ?n block-tridiagonal

matrix,with n ?l 2

.In addition,we set x ?p ;l ?0:02,and the right-hand side vector b to be b ?e1ti TA 1,with 1being the

vector of all entries equal to 1.As before,we normalize the system by multiplying both sides through by h 2

.In fact,this com-plex symmetric system of linear equation arises in direct frequency domain analysis of an n -degree-of-freedom (n -DOF)lin-ear system.For more details,we refer to [10,40].

With respect to different sizes of the coef?cient matrix,we list IT,CPU and RES about the PU,PMHSS and CPIU (m ?8)methods in Table 3for Example 4.2.By comparing the results in Tables 2,we observe that the CPIU iteration method out-performs the PU and PMHSS methods,as it requires much less time and iteration steps to achieve the stopping criterion.Moreover,we also take m ?2;3;4;5;6;7in this numerical experiment.The results for different parameters m of the CPIU iteration method for Example 4.2are presented in Table 4.As we can see from the results of Table 4,we can always get a good approximate solutions by taking a proper parameter m in our actual calculation.Example 4.3(See [25,26]).The system of linear Eq.(1.1)is of the form eW ti T Tx ?b ,with

T ?I V tV I

and W ?10eI V c tV c I Tt9ee 1e >l te l e >

1T I ;

where V ?tridiag eà1;2;à1T2R l ?l ;V c ?V àe 1e >l àe l e >

12R l ?l ;e 1and e l are the ?rst and the last unit vectors in R l ,respec-tively.We take the right-hand side vector b to be b ?e1ti TA 1,with 1being the vector of all entries equal to 1.

Here T and W correspond to the ?ve-point centered difference matrices approximat-ing the negative Laplacian operator with homogeneous Dirichlet boundary conditions and periodic boundary conditions,respectively,on a uniform mesh in the

unit square ?0;1 ??0;1 with the mesh-size h ?1

.This problem is an arti?cially constructed one,but it is quite challenging for iterative solvers.

Table 1

IT,CPU and RES for PU,PMHSS and CPIU methods for Example 4.1.Method l

8162432PU

s opt

0.1960.2160.2260.210IT 252384164CPU 0.0080.085 1.1677.485RES

5.09e à8 3.94e-8 3.15e à8 2.30e à8PMHSS

s opt

1.091 1.356 1.350 1.120IT 21212121CPU 0.0090.104 1.245 3.099RES

4.33e à8 2.02e à8 3.04e à8 3.20e à7CPIU

s opt

0.1960.2160.2260.210IT 2

2

2

2

CPU 0.0070.0790.056 2.771RES

1.53e à13

3.29e à11

3.57e à10

1.29e à9

Table 2

Numerical results for different parameters m of CPIU method for Example 4.1.l IT CPU RES IT CPU RES IT CPU RES m ?2m ?3m ?48190.0947.80e à870.0598.18e à840.042 1.50e à816230.126 3.84e à890.095 1.71e à850.087 3.85e à924240.875 2.94e à890.633 1.83e à860.606 1.92e à93231 4.007 1.82e à810 3.103 1.12e à86 2.9867.79e à9m ?5m ?6m ?7830.030 1.63e à920.007 3.44e à920.021 1.97e à111630.079 2.68e à930.094 1.46e à1020.089 1.93e à92440.655 1.71e à930.6519.59e à1020.561 1.74e à932

4

2.952

6.34e à9

3

2.869

3.36e à9

3

3.034

3.28e à11

Q.-Q.Zheng,C.-F.Ma /Applied Mathematics and Computation 256(2015)11–19

17

18Q.-Q.Zheng,C.-F.Ma/Applied Mathematics and Computation256(2015)11–19

Table3

IT,CPU and RES for PU,PMHSS and CPIU methods for Example4.2.

Method l8162432 PU s opt0.180.160.450.18 IT215234184164

CPU0.0610.285 5.10418.171

RES 6.09eà87.95eà8 3.32eà8 2.51eà8 PMHSS s opt0.045 1.341 1.0010.150 IT121486331

CPU0.1090.101 2.241 3.145

RES 4.33eà6 2.23eà5 2.04eà5 5.34eà6 CPIU s opt0.180.160.450.18 IT1212

CPU0.0040.0800.056 2.812

RES 6.28eà14 1.02eà13 6.49eà10 6.37eà15

Table4

Numerical results for different parameters m of CPIU method for Example4.2.

l IT CPU RES IT CPU RES IT CPU RES

m?2m?3m?4 8490.0279.82eà8150.0157.53eà850.026 1.79eà8 16560.225 3.65eà8140.122 2.60eà840.086 2.53eà8 24330.976 1.81eà890.676 1.53eà840.568 3.86eà10 3274 6.619 1.46eà812 3.4247.06eà94 2.911 2.40eà9

m?5m?6m?7 820.017 4.85eà820.012 6.71eà1210.0048.15eà12 1620.075 1.68eà820.080 1.73eà1210.068 2.89eà12 2420.519 2.41eà1320.552 2.36eà1310.4917.46eà13 322 2.639 2.44eà92 2.661 2.37eà131 2.493 5.21eà10

Table5

IT,CPU and RES for PU,PMHSS and CPIU methods for Example4.3.

Method l8162432 PU s opt 1.730 2.231 1.7320.510 IT125153284132

CPU0.0640.719 5.133 2.992

RES 1.02eà4 3.45eà5 6.15eà47.30eà4 PMHSS s opt0.1240.156 1.356 1.302 IT23121201135

CPU0.0090.104 1.245 3.099

RES 4.33eà5 2.02eà4 3.04eà6 3.20eà6 CPIU s opt 1.730 2.231 1.7320.510 IT1122

CPU0.0070.0790.0518 2.630

RES7.72eà6 4.77eà7 3.38eà8 3.75eà7

Table6

Numerical results for different parameters m of CPIU method for Example4.3.

l IT CPU RES IT CPU RES IT CPU RES

m?2m?3m?4 8110.009 4.69eà550.008 3.11eà620.007 4.46eà6 16210.125 4.31eà570.99 3.06eà530.083 2.91eà6 24210.747 6.88eà570.623 6.38eà530.548 1.41eà5 3214 3.1787.72eà55 2.787 6.39eà63 2.717 2.04eà6

m?5m?6m?7 820.010 1.69eà1010.005 1.10eà710.007 1.13eà8 1620.079 1.83eà710.072 1.73eà510.076 6.37eà8 2420.526 1.85eà620.542 2.36eà1010.5098.63eà7 322 2.6647.26eà62 2.740 2.08eà91 2.633 4.50eà9

Q.-Q.Zheng,C.-F.Ma/Applied Mathematics and Computation256(2015)11–1919 With respect to different sizes of the coef?cient matrix,we list IT,CPU and RES about the PU,PMHSS and CPIU(m?8) methods in Table5for Example4.3.In addition,The results for different parameters m of the CPIU iteration method are pre-sented in Table6.From the results of Table5we observe that the CPIU iteration method outperforms the PU and PMHSS methods in both iteration steps and computing time.In Table6,it is clearly that we can get a good approximate solutions by taking a proper parameter m in our actual calculation,which further illustrates that CPIU iteration method is a very good method for solving the complex symmetric linear systems.

5.Conclusions

Based on the parameterized inexact Uzawa method for solving the saddle point problems and the correction technique, we have established and analyzed a class of correction parameterized inexact Uzawa method iteration methods for solving an important class of complex symmetric linear systems.Numerical experiments have shown that these methods may yield satisfactory results when applied to linear systems of practical interest.How to choosing the optimal parameters is an open problem and needs further research.

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