Convergence of TTS Iterative Method for Non-Hermitian Positive Definite Linear Systems
7
Research Article Convergence of TTS Iterative Method for Non-Hermitian Positive Definite Linear Systems Cheng-Yi Zhang, 1 Yu-Qian Yang, 2 and Qiang Sun 3 1 School of Science, Xi’an Polytechnic University, Xi’an, Shaanxi 710048, China 2 School of Mathematics and Statistics, Xidian University, Xi’an, Shaanxi 710075, China 3 Department of Electronic Engineering, Xi’an University of Technology, Xi’an, Shaanxi 710048, China Correspondence should be addressed to Cheng-Yi Zhang; [email protected] Received 26 February 2014; Accepted 28 April 2014; Published 19 May 2014 Academic Editor: Suh-Yuh Yang Copyright © 2014 Cheng-Yi Zhang et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e TTS iterative method is proposed to solve non-Hermitian positive definite linear systems and some convergence conditions are presented. Subsequently, these convergence conditions are applied to the ALUS method proposed by Xiang et al. in 2012 and comparison of some convergence theorems is made. Furthermore, an example is given to demonstrate the results obtained in this paper. 1. Introduction Many problems in scientific computing give rise to a system of linear equations in unknowns, = , = ( )∈ C × nonsingular, , ∈ C , (1) where is a large and sparse non-Hermitian matrix. In this paper we consider the important case where is positive def- inite; that is, the Hermitian part = ( + ∗ )/2 is Hermitian positive definite, where ∗ denotes the conjugate transpose of the matrix . Large and sparse systems of this type arise in many applications, including discretizations of convection- diffusion problems [1], regularized weighted least-squares problems [2], real-valued formulations of certain complex symmetric systems [3]. ere have been several studies on the convergence of splitting iterative methods for non-Hermitian positive defi- nite linear systems. In [4, pages 190–193], some convergence conditions for the splitting of non-Hermitian positive definite matrices have been established. More recently, [5, 6] give some conditions for the convergence of splittings for this class of linear systems. Recently, there has been considerable interest in the HSS (Hermitian and skew-Hermitian splitting) method intro- duced by Bai et al. in order to solve non-Hermitian positive definite linear systems; see [7]; we further note the generaliza- tions and extensions of this basic method proposed in [8–13]. Furthermore, these methods and their convergence theories have been shown to apply to (generalized) saddle point problems, either directly or indirectly (as a preconditioner); see [5, 6, 8, 9, 11–16]. Continuing in this direction, in this paper we establish new results on splitting methods for solving system (1) iteratively, focusing on a particular class of splittings—the Triangular and Triangular splitting (TTS). According to the idea of HSS method, we construct another alternative iterative method—the TTS method to solve non-Hermitian positive definite linear systems; furthermore, we will prove the convergence of this alternative method. 2. The TTS Method Let = ( )∈ C × be non-Hermitian positive definite with the Hermitian part = ( + ∗ )/2. We split as =− −, where = diag( 11 , 22 ,..., ), is strictly lower Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 328901, 6 pages http://dx.doi.org/10.1155/2014/328901
Transcript of Convergence of TTS Iterative Method for Non-Hermitian Positive Definite Linear Systems
Research ArticleConvergence of TTS Iterative Method forNon-Hermitian Positive Definite Linear Systems
Cheng-Yi Zhang1 Yu-Qian Yang2 and Qiang Sun3
1 School of Science Xirsquoan Polytechnic University Xirsquoan Shaanxi 710048 China2 School of Mathematics and Statistics Xidian University Xirsquoan Shaanxi 710075 China3Department of Electronic Engineering Xirsquoan University of Technology Xirsquoan Shaanxi 710048 China
Correspondence should be addressed to Cheng-Yi Zhang cyzhang08126com
Received 26 February 2014 Accepted 28 April 2014 Published 19 May 2014
Academic Editor Suh-Yuh Yang
Copyright copy 2014 Cheng-Yi Zhang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
The TTS iterative method is proposed to solve non-Hermitian positive definite linear systems and some convergence conditionsare presented Subsequently these convergence conditions are applied to the ALUS method proposed by Xiang et al in 2012 andcomparison of some convergence theorems is made Furthermore an example is given to demonstrate the results obtained in thispaper
1 Introduction
Many problems in scientific computing give rise to a systemof 119899 linear equations in 119899 unknowns
119860119909 = 119887 119860 = (119886119894119895) isin C119899times119899 nonsingular 119887 119909 isin C
119899 (1)
where 119860 is a large and sparse non-Hermitian matrix In thispaper we consider the important case where119860 is positive def-inite that is the Hermitian part119867 = (119860+119860lowast)2 is Hermitianpositive definite where 119860lowast denotes the conjugate transposeof the matrix119860 Large and sparse systems of this type arise inmany applications including discretizations of convection-diffusion problems [1] regularized weighted least-squaresproblems [2] real-valued formulations of certain complexsymmetric systems [3]
There have been several studies on the convergence ofsplitting iterative methods for non-Hermitian positive defi-nite linear systems In [4 pages 190ndash193] some convergenceconditions for the splitting of non-Hermitian positive definitematrices have been established More recently [5 6] givesome conditions for the convergence of splittings for this classof linear systems
Recently there has been considerable interest in the HSS(Hermitian and skew-Hermitian splitting) method intro-duced by Bai et al in order to solve non-Hermitian positivedefinite linear systems see [7] we further note the generaliza-tions and extensions of this basic method proposed in [8ndash13]Furthermore these methods and their convergence theorieshave been shown to apply to (generalized) saddle pointproblems either directly or indirectly (as a preconditioner)see [5 6 8 9 11ndash16]
Continuing in this direction in this paper we establishnew results on splitting methods for solving system (1)iteratively focusing on a particular class of splittingsmdashtheTriangular and Triangular splitting (TTS) According tothe idea of HSS method we construct another alternativeiterative methodmdashthe TTS method to solve non-Hermitianpositive definite linear systems furthermore we will provethe convergence of this alternative method
2 The TTS Method
Let 119860 = (119886119894119895) isin C119899times119899 be non-Hermitian positive definite with
the Hermitian part119867 = (119860 + 119860lowast)2 We split 119860 as 119860 = 119863 minus
119871 minus 119880 where 119863 = diag(11988611 11988622 119886
119899119899) 119871 is strictly lower
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 328901 6 pageshttpdxdoiorg1011552014328901
2 Mathematical Problems in Engineering
triangular matrix and 119880 is strictly upper triangular matrixLetL = 1198632 minus 119871 and letU = 1198632 minus 119880 Then the splitting
119860 =L +U (2)
is called the Triangular and Triangular splitting (TTS)Subsequently we give the TTS method
TheTTSMethodGiven an initial guess119909(0) for 119896 = 0 1 2 until 119909(119896) converges compute
(120572119868 +L) 119909(119896+12)
= (120572119868 minusU) 119909(119896)+ 119887
(120572119868 +U) 119909(119896+1)
= (120572119868 minusL) 119909(119896+12)
+ 119887
(3)
where 120572 is a given positive constantEliminating 119909(119894+12) from (3) we obtain the iterative
process
119909(119896+1)
= 119879120572119909(119896)+ 119865120572119887 119894 = 0 1 2 (4)
where now119879120572= (120572119868+U)
minus1(120572119868minusL)(120572119868+L)
minus1(120572119868minusU) is the
iteration matrix of the iterative scheme (4) and 119865120572= [(120572119868 +
L)(120572119868+U)]minus1[(120572+1)119868+L] It is easy to see that the iterative
scheme (3) is convergent if and only if the iterative scheme (4)is convergent Thus we only consider the convergence of theiterative scheme (4) and consequently investigate the spectralradius 120588(119879
120572) of the iterative matrix 119879
120572
3 Convergence Condition for TTS Method
In this section a convergence condition for TTS methodis presented to solve non-Hermitian positive definite linearsystems The following lemma will be used in this section
Lemma 1 Let119860 isin C119899times119899 be non-HermitianThen the iterationmatrix of the iterative scheme (4) is
119879120572= (1205722119868 + 120572119860 +LU)
minus1
(1205722119868 minus 120572119860 +LU) (5)
Proof Since 119860 =L +U
119879120572= (120572119868 +U)
minus1(120572119868 minusL) (120572119868 +L)
minus1(120572119868 minusU)
= 2120572(120572119868 +U)minus1(120572119868 +L)
minus1(120572119868 minusU)
minus (120572119868 +U)minus1(120572119868 minusU)
= 2120572 [(120572119868 +U)minus1(120572119868 +L)
minus1(120572119868 minusU) minus (120572119868 +U)
minus1] + 119868
= 119868 minus 2120572(120572119868 +U)minus1(120572119868 +L)
minus1(L +U)
= (120572119868 +U)minus1(120572119868 +L)
minus1
times [(120572119868 +L) (120572119868 +U) minus 2120572 (L +U)]
= [(120572119868 +L) (120572119868 +U)]minus1[1205722119868 minus 120572L minus 120572U +LU)
= (1205722119868 + 120572119860 +LU)
minus1
(1205722119868 minus 120572119860 +LU)
(6)
which completes the proof
Theorem 2 Let 119860 isin C119899times119899 be non-Hermitian positive definitewith the TTS as in (2) Then the TTS method converges to theunique solution of (1) for any choice of the initial guess 119909(0) ifand only if 1205722 + 119909lowast(120583S +H)119909 gt 0 where 120583 = 119909lowast119878119909119909lowast119867119909119867 = (119860
lowast+119860)2 119878 = (119860minus119860lowast)2119894H = [(LU)
lowast+LU]2S =
[LU minus (LU)lowast]2119894 and 119909 isin 119909 isin C119899 119879
120572119909 = 120582119909 119909
lowast119909 =
1 and |120582| = 120588(119879120572)
Proof Since the iterative scheme (3) is convergent if and onlyif the iterative scheme (4) is convergent it follows from [17]that (4) converges for any given 119909(0) if and only if 120588(119879
120572) lt
1 where 120588(119879120572) denotes the spectral radius of the matrix 119879
120572
Assume 119909 isin 119909 isin C119899 119879120572119909 = 120582119909 119909
lowast119909 = 1 and |120582| = 120588(119879
120572)
Then 119879120572119909 = 120582119909 It follows from Lemma 1 that
(1205722119868 minus 120572119860 +LU) 119909 = 120582 (120572
2119868 + 120572119860 +LU) 119909 (7)
Multiplying (7) on the left side by 119909lowast
119909lowast(1205722119868 minus 120572119860 +LU) 119909 = 120582119909
lowast(1205722119868 + 120572119860 +LU) 119909 (8)
We assert 120601 = 119909lowast(1205722119868 + 120572119860 + LU)119909 = 0 Otherwise 120601 =119909lowast(1205722119868 + 120572119860 +LU)119909 = 0 and consequently 120593 = 119909lowast(1205722119868 minus
120572119860 +LU)119909 = 0 As a result
Re (120601) = 119909lowast (1205722119868 + 120572119867 +H) 119909 = 0
Re (120593) = 119909lowast (1205722119868 minus 120572119867 +H) 119909 = 0(9)
Equations (9) yield that
Re (120601) minus Re (120593) = 2120572119909lowast119867119909 = 0 (10)
Equation (10) shows that 119867 = (119860lowast+ 119860)2 is not Hermitian
positive definite that is 119860 is not non-Hermitian positivedefinite Therefore a contradiction appears to indicate 120601 =119909lowast(1205722119868 + 120572119860 +LU)119909 = 0 Thus
120588 (119879120572) = |120582| =
10038161003816100381610038161003816119909lowast(1205722119868 minus 120572119860 +LU) 119909
10038161003816100381610038161003816
1003816100381610038161003816119909lowast(1205722119868 + 120572119860 +LU) 119909
1003816100381610038161003816
=
radicRe2 (120593) + Im2 (120593)
radicRe2 (120601) + Im2 (120601)
= radic
Re2 (120593) + Im2 (120593)Re2 (120601) + Im2 (120601)
(11)
which indicates that 120588(119879120572) lt 1 if and only if
[Re2 (120601) + Im2 (120601)] minus [Re2 (120593) + Im2 (120593)]
= 4120572119909lowast119867119909[120572
2+ 119909lowast(H + 120583S) 119909]
gt 0
(12)
where 120583 = 119909lowast119878119909119909lowast119867119909 119867 = (119860lowast+ 119860)2 119878 = (119860 minus 119860lowast)2119894
H = [(LU)lowast+LU]2 and S = [LU minus (LU)
lowast]2119894 Since
120572 gt 0 and 119909lowast119867119909 gt 0 (12) holds if and only if 1205722 + 119909lowast(H +
120583S)119909 gt 0 This completes the proof
Mathematical Problems in Engineering 3
Theorem 2 presents a convergence condition for the TTSmethod But in fact this condition is difficult to be appliedIt is necessary to give a practical condition It follows fromTheorem 2 that we can get the following conclusion
Theorem 3 Let 119860 isin C119899times119899 be non-Hermitian positive definitewith the TTS as in (2) and let
120591 = min 120582min (119867minus1119878) 120582max (S) 120582max (119867
minus1119878) 120582min (S)
(13)
where 120582min(119862) and 120582max(119862) denote the minimal and maximaleigenvalues of the matrix 119862 respectively119867 = (119860lowast + 119860)2 119878 =(119860 minus 119860
lowast)2119894 H = [(LU)
lowast+ LU]2 and S = [LU minus
(LU)lowast]2119894 If 1205722 + 120582min(H) + 120591 gt 0 then the TTS method
converges to the unique solution of (1) for any choice of theinitial guess 119909(0)
Although the TTS method does not always convergefor non-Hermitian positive definite linear systems it alwaysconverges for Hermitian positive definite linear systems
Theorem 4 Let 119860 isin C119899times119899 be Hermitian positive definite withthe TTS as in (2)Then for all120572 gt 0 the TTSmethod convergesto the unique solution of (1) for any choice of the initial guess119909(0) Furthermore it holds that
120572opt = argmin120572gt0
120588 (119879120572) = radic119909
lowastLLlowast119909 isin [120590119899 1205901]
120588 (119879120572opt) =
2radic119909lowastLLlowast119909 minus 119909lowast119860119909
2radic119909lowastLLlowast119909 + 119909lowast119860119909
(14)
where 119909 isin 119909 isin C119899 119879120572119909 = 120582119909 119909
lowast119909 = 1 and |120582| = 120588(119879
120572)
and 120590119899and 1205901are the minimal and maximal singular values of
L respectively
Proof Since 119860 is Hermitian positive definite 119860 = L + U =
L + Llowast and LU = LLlowast is Hermitian positive definiteFollowing (11)
120588 (119879120572) = 120582 =
119909lowast(1205722119868 minus 120572119860 +LU) 119909
119909lowast(1205722119868 + 120572119860 +LU) 119909
=
1205722minus 120572119909lowast119860119909 + 119909
lowastLLlowast119909
1205722+ 120572119909lowast119860119909 + 119909
lowastLLlowast119909
lt 1
(15)
which shows the convergence of the TTS method Following(15)
120588 (119879120572) =
1205722minus 120572119909lowast119860119909 + 119909
lowastLLlowast119909
1205722+ 120572119909lowast119860119909 + 119909
lowastLLlowast119909
= 1 minus
2120572119909lowast119860119909
1205722+ 120572119909lowast119860119909 + 119909
lowastLLlowast119909
= 1 minus 119891 (120572)
(16)
where119891(120572) = 2120572119909lowast119860119909(1205722+120572119909lowast119860119909+119909lowastLLlowast119909) As a result
min120572gt0
120588 (119879120572) = 1 minusmax
120572gt0
119891 (120572) (17)
Since
1198911015840(120572) =
2119909lowast119860119909 (119909
lowastLLlowast119909 minus 1205722)
(1205722+ 120572119909lowast119860119909 + 119909
lowastLLlowast119909)2 (18)
119891(120572) is gradually increasing if 120572 isin (0 radic119909lowastLLlowast119909) 119891(120572)
is gradually decreasing if 120572 isin (radic119909lowastLLlowast119909infin) and
consequently when 120572 = radic119909lowastLLlowast119909 119891(120572) gets its maximummax120572gt0119891(120572) = 2120572119909
lowast119860119909(119909
lowast119860119909 + 2radic119909
lowastLLlowast119909) Thereforewhen
120572opt = argmin120572gt0
120588 (119879120572) = radic119909
lowastLLlowast119909 isin [120590119899 1205901]
120588 (119879120572opt) =
2radic119909lowastLLlowast119909 minus 119909lowast119860119909
2radic119909lowastLLlowast119909 + 119909lowast119860119909
(19)
which shows that we completed the proof
4 Remark on and Comparison ofConvergence Theorems
In fact the TTS method is a special case of the generalizedassymmetric SOR iteration method with parameter matriceswhen specifically choosing the parameter matrices Thisscheme is called the ALUS method in [18 19]
The ALUS Method Given an initial guess 119909(0) for 119896 =
0 1 2 until 119909(119896) converges compute
(120572119868 +L) 119909(119896+12)
= (120572119868 minusU) 119909(119896)+ 119887
(120572119868 +U) 119909(119896+1)
= (120572119868 minusL) 119909(119896+12)
+ 119887
(20)
where 120572 is a given positive constantL = 1198631+119871 U = 119863
2+119880
1198631+ 1198632= 119863 and 119871 119880 and119863 are defined in (3)
We easily generalize the convergence theorems on TTSmethod to ALUS method
Theorem 5 Let 119860 isin C119899times119899 be non-Hermitian positive definitewith the ALUS as in (20) Then the ALUS method converges tothe unique solution of (1) for any choice of the initial guess 119909(0)if and only if 1205722 + 119909lowast(120583S +H)119909 gt 0 where 120583 = 119909lowast119878119909119909lowast119867119909119867 = (119860
lowast+119860)2 119878 = (119860minus119860lowast)2119894H = [(
LU)lowast+LU]2S =
[LU minus (
LU)lowast]2119894 and 119909 isin 119909 isin C119899 119879
120572119909 = 120582119909 119909
lowast119909 =
1 and |120582| = 120588(119879120572)
Theorem 6 Let 119860 isin C119899times119899 be non-Hermitian positive definitewith the ALUS as in (20) and let
120591 = min 120582min (119867minus1119878) 120582max (S) 120582max (119867
minus1119878) 120582min (S)
(21)
where 120582min(119862) and 120582max(119862) denote the minimal and maximaleigenvalues of the matrix 119862 respectively119867 = (119860lowast + 119860)2 119878 =(119860 minus 119860
lowast)2119894 H = [(
LU)lowast+LU]2 and S = [
LU minus
(LU)lowast]2119894 If 1205722 + 120582min(H) + 120591 gt 0 then the ALUS method
converges to the unique solution of (1) for any choice of theinitial guess 119909(0)
4 Mathematical Problems in Engineering
The proofs of Theorems 5 and 6 directly result from theproofs of Theorems 2 and 3
Like Theorem 4 in [18] or Theorem 6 in [19] Theorem5 has only theoretical significance since it is difficult to beapplied However Theorem 6 along with Theorem 3 in [18]or Theorem 4 in [19] proposes a practical condition onconvergence of ALUSmethod But the condition inTheorem6 is wider than Theorem 3 in [18] or Theorem 4 in [19] Thefollowing will give an example to demonstrate this fact
Example 7 The coefficient matrix 119860 of linear system (1) isgiven as
119860 = [
1 2
minus2 1] (22)
Now we consider solving this system by ALUS methodLet1198631= [119909 0
0 119910 ]Then1198632= 119863minus119863
1= [1minus119909 0
0 1minus119910 ] L = 119863
1+119871 =
[119909 0
minus2 119910 ] and U = 1198632+ 119880 = [
1minus119909 2
0 1minus119910 ] In order that the ALUSmethod converges to the unique solution of (1) for any choiceof the initial guess119909(0)Theorem 3 in [18] orTheorem 4 in [19]shows that L +
Llowast = [ 2119909 minus2minus2 2119910 ] and U +Ulowast = [
2(2minus119909) 2
2 2(1minus119910)]
are both positive definite for all 119909 isin (0 1) and 119910 isin (0 1) As aresult the set
(119909 119910) isin R timesR
119909119910 gt 1
119909119910 gt 119909 + 119910
119909 isin (0 1) 119910 isin (0 1)
(23)
is not empty In fact the set (23) is empty since the inequality119909119910 gt 1 fails to hold when 119909 isin (0 1) and 119910 isin (0 1) ThereforeTheorem 3 in [18] or Theorem4 in [19] does not give theconvergence of theALUSmethod if we solve the linear system(1) with the coefficient matrix 119860 in (22)
Using Theorem 6 set 119909 = 119910 = 12 and compute 120591 = minus2and 120582min(H) = minus154 Then when 120572 gt radic232 the ALUSmethod converges to the unique solution of (1) for any choiceof the initial guess 119909(0)
Example 8 Linear system (1) is shifted skew-Hermitianlinear system (see [7ndash14]) whose coefficient matrix119860 is givenas
119860 = 120574119868 + 119878 (24)
where 120574 is a positive constant 119868 is the 119899 times 119899 identity matrixand
119878 = 119871 minus 119871lowast (25)
is a skew-Hermitian matrix with 119871 a lower triangular matrix
The shifted skew-Hermitian linear system arising in theHSS iterative method can be much more problematic insome cases this solution is as difficult as that of the originallinear system [10] Since HSS method fails to solve this linearsystem we consider the ALUS method and TTS method
We assert that if 120574 le 120582max(119866) minus 120582min(119866) with 119866 = 119871 + 119871lowast
and 120582max(119866) and 120582min(119866) being themaximum andminimum
eigenvalues of thematrix119866 respectivelyTheorem 3 in [18] orTheorem4 in [19] fails to give the convergence of the ALUSmethod when solving the skew-Hermitian linear system (1)
Let1198631= 119909119868Then119863
2= 119863minus119863
1= (120574minus119909)119868L = 119909119868+119871 and
U = (120574 minus 119909)119868 minus 119871lowast In order that the ALUS method converges
to the unique solution of (1) for any choice of the initial guess119909(0)Theorem 3 in [18] or Theorem4 in [19] shows that L +
Llowast = 2119909119868 + 119866 and U + Ulowast = 2(120574 minus 119909)119868 minus 119866 are both positivedefinite for all 119909 isin (0 120574) Thus it is easy to get
2119909 + 120582min (119866) gt 0 2 (120574 minus 119909) minus 120582max (119866) gt 0 (26)
It follows from (26) that 120574 gt 120582max(119866) minus 120582min(119866) which showsthat when 120574 le 120582max(119866) minus 120582min(119866) Theorem 3 in [18] orTheorem 4 in [19] fails to give the convergence of the ALUSmethod
We consider using Theorem 6 Compute 120591 =
120582max(minus119894119878)120582min(minus119894119878)2 Theorem 6 shows that if 119891(119909) = 1205722+119909(120574 minus 119909) minus 120582max(119871119871
lowast) + (1205742 minus 119909)120582min(119866) gt 0 the ALUS
method converges to the unique solution of (1) for any choiceof the initial guess 119909(0) Furthermore
max119909isin(0120574)
119891 (119909) = 1205722+
(1205742+ 1205822
min (119866))
4
minus 120582max (119871119871lowast) (27)
when 119909 = (120574 minus 120582min(119866))2 Thus if 120574 ge radic411987122minus 1205822
min(119866)Theorem 6 shows that the ALUS method converges to theunique solution of (1) for any choice of the initial guess 119909(0)and for all 120572 gt 0 otherwise it follows from Theorem 6that the ALUS method converges to the unique solution of(1) for any choice of the initial guess 119909(0) and for all 120572 gt
radic1198712
2minus (1205742+ 1205822
min(119866))4In particular if 120574 ge 2119871
2 Theorem 3 yields that the
TTS method converges to the unique solution of (1) for anychoice of the initial guess 119909(0) and for all 120572 gt 0 otherwiseit follows fromTheorem 3 that the TTS method converges tothe unique solution of (1) for any choice of the initial guess119909(0) and for all 120572 gt radic1198712
2minus 12057424
5 Numerical Examples
In this section we describe the results of a numerical simpleexample with the TTS method on a set of linear systemsarising from a finite element discretization of a convection-diffusion equation in two dimensions
Example 1 The coefficient matrix 119860 of linear system (1) isgiven as
119860 =
[
[
[
[
1 11 12 1
minus05 1 14 065
minus02 minus06 1 155
02 025 minus045 1
]
]
]
]
(28)
Now we investigate convergence of TTS method forlinear system (1) by Theorem 3 It is known that 119860 is non-Hermitian positive definite By Matlab computations we
Mathematical Problems in Engineering 5
Table 1 The comparison results of 120588(119879120572) with different parameter pairs 120572
120572 10minus5
10minus4
10minus3
10minus2
01 05 1 11120588(119879120572) 10000 09999 09994 09940 09435 08476 07163 06622
120572 12 13 14 15 16 17 18 19120588(119879120572) 07795 08983 09422 09455 09199 08698 07921 06381
120572 2 225 25 3 305 31 315 32120588(119879120572) 06193 05784 05498 05283 05281 05282 05285 05290
120572 33 34 35 4 45 5 10 15120588(119879120572) 05307 05330 05358 05550 05779 06012 07580 08292
120572 20 30 50 100 150 103 104 105
120588(119879120572) 08694 09099 09449 09720 09813 09972 09997 10000
get 120582min(H) + 120591 = minus30471 As a result when 120572 isin
(17456 +infin) the TTS method theoretically converges to theunique solution of (1) for any choice of the initial guess 119909(0)But when 120572 rarr +infin this method either converges veryslowly or fails to converge since 120588(119879
120572) rarr 1
By numerical experiments on Matlab program one hasTable 1
Table 1 shows that for given matrix 119860 the TTS methodconverges for 120572 isin (10
minus5 105) Further 120588(119879
120572) is gradually
decreasing when 120572 isin (0 11) and 120572 isin (15 305) while it isgradually increasing when 120572 isin (11 15) and 120572 isin (305 +infin)Thus 120588(119879
120572) has two minimal values 120588(119879
11) = 06622 and
120588(119879305) = 05281 However 120588(119879
305) = 05281 is the minimal
value As a result 120572opt = 305 It follows from Table 1 thatTheorem 3 is true Furthermore the interval obtained inTheorem 3 includes the optimal point 120572opt = 305
6 Conclusions
This paper studies convergence of TTS and ALUS iterativemethods for non-Hermitian positive definite linear systemsSome sufficient and necessary conditions for convergenceare proposed But these conditions are only theoreticallysignificant and difficult to apply to practical computations Inwhat follows several conditions are presented such that theTTSmethod and ALUSmethod converge for non-Hermitianpositive definite linear systems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Cheng-Yi Zhang was partly supported by the Science Foun-dation of the Education Department of Shaanxi Provinceof China (2013JK0593) the Scientific Research Foundation(BS1014) and the Education Reform Foundation (2012JG40)of Xirsquoan Polytechnic University and the National NaturalScience Foundations of China (11201362 and 11271297) Yu-Qian Yang was partly supported by the National NaturalScience Foundation of China (no 61201297) Qiang Sun waspartly supported by the National Natural Science Foundationof China (no 61001140) and the Scientific Research Program
Funded by Shaanxi Provincial Education Department (no12JK0544)
References
[1] H Elman D Silvester and A Wathen Finite Elements andFast Iterative Solvers with Applications in Incompressible FluidDynamics Numerical Mathematics and Scientific Computa-tion Oxford University Press Oxford UK 2005
[2] M Benzi and M K Ng ldquoPreconditioned iterative methodsfor weighted toeplitz least squares problemsrdquo SIAM Journal onMatrix Analysis and Applications vol 27 no 4 pp 1106ndash11242006
[3] M Benzi and D Bertaccini ldquoBlock preconditioning of real-valued iterative algorithms for complex linear systemsrdquo IMAJournal of Numerical Analysis vol 28 no 3 pp 598ndash618 2008
[4] A Berman and R J Plemmons Nonnegative Matrices in theMathematical Sciences Academic Press New York NY USA1979 Reprinted by SIAM Philadelphia Pa USA 1994
[5] C-L Wang and Z-Z Bai ldquoSufficient conditions for the con-vergent splittings of non-Hermitian positive definite matricesrdquoLinear Algebra and Its Applications vol 330 no 1ndash3 pp 215ndash2182001
[6] L Wang and Z-Z Bai ldquoConvergence conditions for splittingiteration methods for non-Hermitian linear systemsrdquo LinearAlgebra and Its Applications vol 428 no 2-3 pp 453ndash468 2008
[7] Z-Z Bai G H Golub and M K Ng ldquoHermitian andSkew-Hermitian splitting methods for non-Hermitian positivedefinite linear systemsrdquo SIAM Journal on Matrix Analysis andApplications vol 24 no 3 pp 603ndash626 2003
[8] Z-Z Bai G H Golub and M K Ng ldquoOn successive-over-relaxation acceleration of the Hermitian and Skew-Hermitiansplitting iterationsrdquoNumerical Linear Algebra with Applicationsvol 14 no 4 pp 319ndash335 2007
[9] Z-Z Bai G H Golub and J-Y Pan ldquoPreconditioned Hermi-tian and Skew-Hermitian splitting methods for non-Hermitianpositive semidefinite linear systemsrdquo Numerische Mathematikvol 98 no 1 pp 1ndash32 2004
[10] M Benzi ldquoA Generalization of the he rmitian and Skew-hermitian splitting iterationrdquo SIAM Journal on Matrix Analysisand Applications vol 31 no 2 pp 360ndash374 2009
[11] Z-Z Bai G H Golub L-Z Lu and J-F Yin ldquoBlock triangularand Skew-Hermitian splitting methods for positive-definitelinear systemsrdquo SIAM Journal on Scientific Computing vol 26no 3 pp 844ndash863 2005
[12] Z-Z Bai GHGolub andMKNg ldquoOn inexact hermitian andSkew-Hermitian splitting methods for non-Hermitian positive
6 Mathematical Problems in Engineering
definite linear systemsrdquo Linear Algebra and Its Applications vol428 no 2-3 pp 413ndash440 2008
[13] L Li T -Z Huang and X -P Liu ldquoModified Hermitian andSkew-Hermitian splittingmethods for non-Hermitian positive-definite linear systemsrdquoNumerical Linear AlgebraWith Applica-tions vol 14 no 3 pp 217ndash235 2007
[14] Z-Z Bai and G H Golub ldquoAccelerated Hermitian and Skew-Hermitian splitting iteration methods for saddle-point prob-lemsrdquo IMA Journal of Numerical Analysis vol 27 no 1 pp 1ndash232007
[15] M Benzi M J Gander and G H Golub ldquoOptimization of theHermitian and Skew-Hermitian splitting iteration for saddle-point problemsrdquo BIT Numerical Mathematics vol 43 no 5 pp881ndash900 2003
[16] M Benzi and G H Golub ldquoA preconditioner for generalizedsaddle point problemsrdquo SIAM Journal on Matrix Analysis andApplications vol 26 no 1 pp 20ndash41 2005
[17] R S VargaMatrix Iterative Analysis Springer Berlin Germany2nd edition 2000
[18] Q Xiang and S-P Wu ldquoA modified alternating directionmethod for positive definite systemsrdquo in Information Engneeringand Applications R-B Zhu and Y Ma Eds vol 154 of LectureNotes in Electrical Engineering pp 437ndash444 2012
[19] Q Xiang S -P Wu and Y Xu ldquoAlternating lower-upper split-ting iterativemethod and its applicationrdquo Journal of Beijing Uni-versity of Aeronautics and Astronautics vol 38 pp 953ndash9562012 (Chinese)
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014
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Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
triangular matrix and 119880 is strictly upper triangular matrixLetL = 1198632 minus 119871 and letU = 1198632 minus 119880 Then the splitting
119860 =L +U (2)
is called the Triangular and Triangular splitting (TTS)Subsequently we give the TTS method
TheTTSMethodGiven an initial guess119909(0) for 119896 = 0 1 2 until 119909(119896) converges compute
(120572119868 +L) 119909(119896+12)
= (120572119868 minusU) 119909(119896)+ 119887
(120572119868 +U) 119909(119896+1)
= (120572119868 minusL) 119909(119896+12)
+ 119887
(3)
where 120572 is a given positive constantEliminating 119909(119894+12) from (3) we obtain the iterative
process
119909(119896+1)
= 119879120572119909(119896)+ 119865120572119887 119894 = 0 1 2 (4)
where now119879120572= (120572119868+U)
minus1(120572119868minusL)(120572119868+L)
minus1(120572119868minusU) is the
iteration matrix of the iterative scheme (4) and 119865120572= [(120572119868 +
L)(120572119868+U)]minus1[(120572+1)119868+L] It is easy to see that the iterative
scheme (3) is convergent if and only if the iterative scheme (4)is convergent Thus we only consider the convergence of theiterative scheme (4) and consequently investigate the spectralradius 120588(119879
120572) of the iterative matrix 119879
120572
3 Convergence Condition for TTS Method
In this section a convergence condition for TTS methodis presented to solve non-Hermitian positive definite linearsystems The following lemma will be used in this section
Lemma 1 Let119860 isin C119899times119899 be non-HermitianThen the iterationmatrix of the iterative scheme (4) is
119879120572= (1205722119868 + 120572119860 +LU)
minus1
(1205722119868 minus 120572119860 +LU) (5)
Proof Since 119860 =L +U
119879120572= (120572119868 +U)
minus1(120572119868 minusL) (120572119868 +L)
minus1(120572119868 minusU)
= 2120572(120572119868 +U)minus1(120572119868 +L)
minus1(120572119868 minusU)
minus (120572119868 +U)minus1(120572119868 minusU)
= 2120572 [(120572119868 +U)minus1(120572119868 +L)
minus1(120572119868 minusU) minus (120572119868 +U)
minus1] + 119868
= 119868 minus 2120572(120572119868 +U)minus1(120572119868 +L)
minus1(L +U)
= (120572119868 +U)minus1(120572119868 +L)
minus1
times [(120572119868 +L) (120572119868 +U) minus 2120572 (L +U)]
= [(120572119868 +L) (120572119868 +U)]minus1[1205722119868 minus 120572L minus 120572U +LU)
= (1205722119868 + 120572119860 +LU)
minus1
(1205722119868 minus 120572119860 +LU)
(6)
which completes the proof
Theorem 2 Let 119860 isin C119899times119899 be non-Hermitian positive definitewith the TTS as in (2) Then the TTS method converges to theunique solution of (1) for any choice of the initial guess 119909(0) ifand only if 1205722 + 119909lowast(120583S +H)119909 gt 0 where 120583 = 119909lowast119878119909119909lowast119867119909119867 = (119860
lowast+119860)2 119878 = (119860minus119860lowast)2119894H = [(LU)
lowast+LU]2S =
[LU minus (LU)lowast]2119894 and 119909 isin 119909 isin C119899 119879
120572119909 = 120582119909 119909
lowast119909 =
1 and |120582| = 120588(119879120572)
Proof Since the iterative scheme (3) is convergent if and onlyif the iterative scheme (4) is convergent it follows from [17]that (4) converges for any given 119909(0) if and only if 120588(119879
120572) lt
1 where 120588(119879120572) denotes the spectral radius of the matrix 119879
120572
Assume 119909 isin 119909 isin C119899 119879120572119909 = 120582119909 119909
lowast119909 = 1 and |120582| = 120588(119879
120572)
Then 119879120572119909 = 120582119909 It follows from Lemma 1 that
(1205722119868 minus 120572119860 +LU) 119909 = 120582 (120572
2119868 + 120572119860 +LU) 119909 (7)
Multiplying (7) on the left side by 119909lowast
119909lowast(1205722119868 minus 120572119860 +LU) 119909 = 120582119909
lowast(1205722119868 + 120572119860 +LU) 119909 (8)
We assert 120601 = 119909lowast(1205722119868 + 120572119860 + LU)119909 = 0 Otherwise 120601 =119909lowast(1205722119868 + 120572119860 +LU)119909 = 0 and consequently 120593 = 119909lowast(1205722119868 minus
120572119860 +LU)119909 = 0 As a result
Re (120601) = 119909lowast (1205722119868 + 120572119867 +H) 119909 = 0
Re (120593) = 119909lowast (1205722119868 minus 120572119867 +H) 119909 = 0(9)
Equations (9) yield that
Re (120601) minus Re (120593) = 2120572119909lowast119867119909 = 0 (10)
Equation (10) shows that 119867 = (119860lowast+ 119860)2 is not Hermitian
positive definite that is 119860 is not non-Hermitian positivedefinite Therefore a contradiction appears to indicate 120601 =119909lowast(1205722119868 + 120572119860 +LU)119909 = 0 Thus
120588 (119879120572) = |120582| =
10038161003816100381610038161003816119909lowast(1205722119868 minus 120572119860 +LU) 119909
10038161003816100381610038161003816
1003816100381610038161003816119909lowast(1205722119868 + 120572119860 +LU) 119909
1003816100381610038161003816
=
radicRe2 (120593) + Im2 (120593)
radicRe2 (120601) + Im2 (120601)
= radic
Re2 (120593) + Im2 (120593)Re2 (120601) + Im2 (120601)
(11)
which indicates that 120588(119879120572) lt 1 if and only if
[Re2 (120601) + Im2 (120601)] minus [Re2 (120593) + Im2 (120593)]
= 4120572119909lowast119867119909[120572
2+ 119909lowast(H + 120583S) 119909]
gt 0
(12)
where 120583 = 119909lowast119878119909119909lowast119867119909 119867 = (119860lowast+ 119860)2 119878 = (119860 minus 119860lowast)2119894
H = [(LU)lowast+LU]2 and S = [LU minus (LU)
lowast]2119894 Since
120572 gt 0 and 119909lowast119867119909 gt 0 (12) holds if and only if 1205722 + 119909lowast(H +
120583S)119909 gt 0 This completes the proof
Mathematical Problems in Engineering 3
Theorem 2 presents a convergence condition for the TTSmethod But in fact this condition is difficult to be appliedIt is necessary to give a practical condition It follows fromTheorem 2 that we can get the following conclusion
Theorem 3 Let 119860 isin C119899times119899 be non-Hermitian positive definitewith the TTS as in (2) and let
120591 = min 120582min (119867minus1119878) 120582max (S) 120582max (119867
minus1119878) 120582min (S)
(13)
where 120582min(119862) and 120582max(119862) denote the minimal and maximaleigenvalues of the matrix 119862 respectively119867 = (119860lowast + 119860)2 119878 =(119860 minus 119860
lowast)2119894 H = [(LU)
lowast+ LU]2 and S = [LU minus
(LU)lowast]2119894 If 1205722 + 120582min(H) + 120591 gt 0 then the TTS method
converges to the unique solution of (1) for any choice of theinitial guess 119909(0)
Although the TTS method does not always convergefor non-Hermitian positive definite linear systems it alwaysconverges for Hermitian positive definite linear systems
Theorem 4 Let 119860 isin C119899times119899 be Hermitian positive definite withthe TTS as in (2)Then for all120572 gt 0 the TTSmethod convergesto the unique solution of (1) for any choice of the initial guess119909(0) Furthermore it holds that
120572opt = argmin120572gt0
120588 (119879120572) = radic119909
lowastLLlowast119909 isin [120590119899 1205901]
120588 (119879120572opt) =
2radic119909lowastLLlowast119909 minus 119909lowast119860119909
2radic119909lowastLLlowast119909 + 119909lowast119860119909
(14)
where 119909 isin 119909 isin C119899 119879120572119909 = 120582119909 119909
lowast119909 = 1 and |120582| = 120588(119879
120572)
and 120590119899and 1205901are the minimal and maximal singular values of
L respectively
Proof Since 119860 is Hermitian positive definite 119860 = L + U =
L + Llowast and LU = LLlowast is Hermitian positive definiteFollowing (11)
120588 (119879120572) = 120582 =
119909lowast(1205722119868 minus 120572119860 +LU) 119909
119909lowast(1205722119868 + 120572119860 +LU) 119909
=
1205722minus 120572119909lowast119860119909 + 119909
lowastLLlowast119909
1205722+ 120572119909lowast119860119909 + 119909
lowastLLlowast119909
lt 1
(15)
which shows the convergence of the TTS method Following(15)
120588 (119879120572) =
1205722minus 120572119909lowast119860119909 + 119909
lowastLLlowast119909
1205722+ 120572119909lowast119860119909 + 119909
lowastLLlowast119909
= 1 minus
2120572119909lowast119860119909
1205722+ 120572119909lowast119860119909 + 119909
lowastLLlowast119909
= 1 minus 119891 (120572)
(16)
where119891(120572) = 2120572119909lowast119860119909(1205722+120572119909lowast119860119909+119909lowastLLlowast119909) As a result
min120572gt0
120588 (119879120572) = 1 minusmax
120572gt0
119891 (120572) (17)
Since
1198911015840(120572) =
2119909lowast119860119909 (119909
lowastLLlowast119909 minus 1205722)
(1205722+ 120572119909lowast119860119909 + 119909
lowastLLlowast119909)2 (18)
119891(120572) is gradually increasing if 120572 isin (0 radic119909lowastLLlowast119909) 119891(120572)
is gradually decreasing if 120572 isin (radic119909lowastLLlowast119909infin) and
consequently when 120572 = radic119909lowastLLlowast119909 119891(120572) gets its maximummax120572gt0119891(120572) = 2120572119909
lowast119860119909(119909
lowast119860119909 + 2radic119909
lowastLLlowast119909) Thereforewhen
120572opt = argmin120572gt0
120588 (119879120572) = radic119909
lowastLLlowast119909 isin [120590119899 1205901]
120588 (119879120572opt) =
2radic119909lowastLLlowast119909 minus 119909lowast119860119909
2radic119909lowastLLlowast119909 + 119909lowast119860119909
(19)
which shows that we completed the proof
4 Remark on and Comparison ofConvergence Theorems
In fact the TTS method is a special case of the generalizedassymmetric SOR iteration method with parameter matriceswhen specifically choosing the parameter matrices Thisscheme is called the ALUS method in [18 19]
The ALUS Method Given an initial guess 119909(0) for 119896 =
0 1 2 until 119909(119896) converges compute
(120572119868 +L) 119909(119896+12)
= (120572119868 minusU) 119909(119896)+ 119887
(120572119868 +U) 119909(119896+1)
= (120572119868 minusL) 119909(119896+12)
+ 119887
(20)
where 120572 is a given positive constantL = 1198631+119871 U = 119863
2+119880
1198631+ 1198632= 119863 and 119871 119880 and119863 are defined in (3)
We easily generalize the convergence theorems on TTSmethod to ALUS method
Theorem 5 Let 119860 isin C119899times119899 be non-Hermitian positive definitewith the ALUS as in (20) Then the ALUS method converges tothe unique solution of (1) for any choice of the initial guess 119909(0)if and only if 1205722 + 119909lowast(120583S +H)119909 gt 0 where 120583 = 119909lowast119878119909119909lowast119867119909119867 = (119860
lowast+119860)2 119878 = (119860minus119860lowast)2119894H = [(
LU)lowast+LU]2S =
[LU minus (
LU)lowast]2119894 and 119909 isin 119909 isin C119899 119879
120572119909 = 120582119909 119909
lowast119909 =
1 and |120582| = 120588(119879120572)
Theorem 6 Let 119860 isin C119899times119899 be non-Hermitian positive definitewith the ALUS as in (20) and let
120591 = min 120582min (119867minus1119878) 120582max (S) 120582max (119867
minus1119878) 120582min (S)
(21)
where 120582min(119862) and 120582max(119862) denote the minimal and maximaleigenvalues of the matrix 119862 respectively119867 = (119860lowast + 119860)2 119878 =(119860 minus 119860
lowast)2119894 H = [(
LU)lowast+LU]2 and S = [
LU minus
(LU)lowast]2119894 If 1205722 + 120582min(H) + 120591 gt 0 then the ALUS method
converges to the unique solution of (1) for any choice of theinitial guess 119909(0)
4 Mathematical Problems in Engineering
The proofs of Theorems 5 and 6 directly result from theproofs of Theorems 2 and 3
Like Theorem 4 in [18] or Theorem 6 in [19] Theorem5 has only theoretical significance since it is difficult to beapplied However Theorem 6 along with Theorem 3 in [18]or Theorem 4 in [19] proposes a practical condition onconvergence of ALUSmethod But the condition inTheorem6 is wider than Theorem 3 in [18] or Theorem 4 in [19] Thefollowing will give an example to demonstrate this fact
Example 7 The coefficient matrix 119860 of linear system (1) isgiven as
119860 = [
1 2
minus2 1] (22)
Now we consider solving this system by ALUS methodLet1198631= [119909 0
0 119910 ]Then1198632= 119863minus119863
1= [1minus119909 0
0 1minus119910 ] L = 119863
1+119871 =
[119909 0
minus2 119910 ] and U = 1198632+ 119880 = [
1minus119909 2
0 1minus119910 ] In order that the ALUSmethod converges to the unique solution of (1) for any choiceof the initial guess119909(0)Theorem 3 in [18] orTheorem 4 in [19]shows that L +
Llowast = [ 2119909 minus2minus2 2119910 ] and U +Ulowast = [
2(2minus119909) 2
2 2(1minus119910)]
are both positive definite for all 119909 isin (0 1) and 119910 isin (0 1) As aresult the set
(119909 119910) isin R timesR
119909119910 gt 1
119909119910 gt 119909 + 119910
119909 isin (0 1) 119910 isin (0 1)
(23)
is not empty In fact the set (23) is empty since the inequality119909119910 gt 1 fails to hold when 119909 isin (0 1) and 119910 isin (0 1) ThereforeTheorem 3 in [18] or Theorem4 in [19] does not give theconvergence of theALUSmethod if we solve the linear system(1) with the coefficient matrix 119860 in (22)
Using Theorem 6 set 119909 = 119910 = 12 and compute 120591 = minus2and 120582min(H) = minus154 Then when 120572 gt radic232 the ALUSmethod converges to the unique solution of (1) for any choiceof the initial guess 119909(0)
Example 8 Linear system (1) is shifted skew-Hermitianlinear system (see [7ndash14]) whose coefficient matrix119860 is givenas
119860 = 120574119868 + 119878 (24)
where 120574 is a positive constant 119868 is the 119899 times 119899 identity matrixand
119878 = 119871 minus 119871lowast (25)
is a skew-Hermitian matrix with 119871 a lower triangular matrix
The shifted skew-Hermitian linear system arising in theHSS iterative method can be much more problematic insome cases this solution is as difficult as that of the originallinear system [10] Since HSS method fails to solve this linearsystem we consider the ALUS method and TTS method
We assert that if 120574 le 120582max(119866) minus 120582min(119866) with 119866 = 119871 + 119871lowast
and 120582max(119866) and 120582min(119866) being themaximum andminimum
eigenvalues of thematrix119866 respectivelyTheorem 3 in [18] orTheorem4 in [19] fails to give the convergence of the ALUSmethod when solving the skew-Hermitian linear system (1)
Let1198631= 119909119868Then119863
2= 119863minus119863
1= (120574minus119909)119868L = 119909119868+119871 and
U = (120574 minus 119909)119868 minus 119871lowast In order that the ALUS method converges
to the unique solution of (1) for any choice of the initial guess119909(0)Theorem 3 in [18] or Theorem4 in [19] shows that L +
Llowast = 2119909119868 + 119866 and U + Ulowast = 2(120574 minus 119909)119868 minus 119866 are both positivedefinite for all 119909 isin (0 120574) Thus it is easy to get
2119909 + 120582min (119866) gt 0 2 (120574 minus 119909) minus 120582max (119866) gt 0 (26)
It follows from (26) that 120574 gt 120582max(119866) minus 120582min(119866) which showsthat when 120574 le 120582max(119866) minus 120582min(119866) Theorem 3 in [18] orTheorem 4 in [19] fails to give the convergence of the ALUSmethod
We consider using Theorem 6 Compute 120591 =
120582max(minus119894119878)120582min(minus119894119878)2 Theorem 6 shows that if 119891(119909) = 1205722+119909(120574 minus 119909) minus 120582max(119871119871
lowast) + (1205742 minus 119909)120582min(119866) gt 0 the ALUS
method converges to the unique solution of (1) for any choiceof the initial guess 119909(0) Furthermore
max119909isin(0120574)
119891 (119909) = 1205722+
(1205742+ 1205822
min (119866))
4
minus 120582max (119871119871lowast) (27)
when 119909 = (120574 minus 120582min(119866))2 Thus if 120574 ge radic411987122minus 1205822
min(119866)Theorem 6 shows that the ALUS method converges to theunique solution of (1) for any choice of the initial guess 119909(0)and for all 120572 gt 0 otherwise it follows from Theorem 6that the ALUS method converges to the unique solution of(1) for any choice of the initial guess 119909(0) and for all 120572 gt
radic1198712
2minus (1205742+ 1205822
min(119866))4In particular if 120574 ge 2119871
2 Theorem 3 yields that the
TTS method converges to the unique solution of (1) for anychoice of the initial guess 119909(0) and for all 120572 gt 0 otherwiseit follows fromTheorem 3 that the TTS method converges tothe unique solution of (1) for any choice of the initial guess119909(0) and for all 120572 gt radic1198712
2minus 12057424
5 Numerical Examples
In this section we describe the results of a numerical simpleexample with the TTS method on a set of linear systemsarising from a finite element discretization of a convection-diffusion equation in two dimensions
Example 1 The coefficient matrix 119860 of linear system (1) isgiven as
119860 =
[
[
[
[
1 11 12 1
minus05 1 14 065
minus02 minus06 1 155
02 025 minus045 1
]
]
]
]
(28)
Now we investigate convergence of TTS method forlinear system (1) by Theorem 3 It is known that 119860 is non-Hermitian positive definite By Matlab computations we
Mathematical Problems in Engineering 5
Table 1 The comparison results of 120588(119879120572) with different parameter pairs 120572
120572 10minus5
10minus4
10minus3
10minus2
01 05 1 11120588(119879120572) 10000 09999 09994 09940 09435 08476 07163 06622
120572 12 13 14 15 16 17 18 19120588(119879120572) 07795 08983 09422 09455 09199 08698 07921 06381
120572 2 225 25 3 305 31 315 32120588(119879120572) 06193 05784 05498 05283 05281 05282 05285 05290
120572 33 34 35 4 45 5 10 15120588(119879120572) 05307 05330 05358 05550 05779 06012 07580 08292
120572 20 30 50 100 150 103 104 105
120588(119879120572) 08694 09099 09449 09720 09813 09972 09997 10000
get 120582min(H) + 120591 = minus30471 As a result when 120572 isin
(17456 +infin) the TTS method theoretically converges to theunique solution of (1) for any choice of the initial guess 119909(0)But when 120572 rarr +infin this method either converges veryslowly or fails to converge since 120588(119879
120572) rarr 1
By numerical experiments on Matlab program one hasTable 1
Table 1 shows that for given matrix 119860 the TTS methodconverges for 120572 isin (10
minus5 105) Further 120588(119879
120572) is gradually
decreasing when 120572 isin (0 11) and 120572 isin (15 305) while it isgradually increasing when 120572 isin (11 15) and 120572 isin (305 +infin)Thus 120588(119879
120572) has two minimal values 120588(119879
11) = 06622 and
120588(119879305) = 05281 However 120588(119879
305) = 05281 is the minimal
value As a result 120572opt = 305 It follows from Table 1 thatTheorem 3 is true Furthermore the interval obtained inTheorem 3 includes the optimal point 120572opt = 305
6 Conclusions
This paper studies convergence of TTS and ALUS iterativemethods for non-Hermitian positive definite linear systemsSome sufficient and necessary conditions for convergenceare proposed But these conditions are only theoreticallysignificant and difficult to apply to practical computations Inwhat follows several conditions are presented such that theTTSmethod and ALUSmethod converge for non-Hermitianpositive definite linear systems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Cheng-Yi Zhang was partly supported by the Science Foun-dation of the Education Department of Shaanxi Provinceof China (2013JK0593) the Scientific Research Foundation(BS1014) and the Education Reform Foundation (2012JG40)of Xirsquoan Polytechnic University and the National NaturalScience Foundations of China (11201362 and 11271297) Yu-Qian Yang was partly supported by the National NaturalScience Foundation of China (no 61201297) Qiang Sun waspartly supported by the National Natural Science Foundationof China (no 61001140) and the Scientific Research Program
Funded by Shaanxi Provincial Education Department (no12JK0544)
References
[1] H Elman D Silvester and A Wathen Finite Elements andFast Iterative Solvers with Applications in Incompressible FluidDynamics Numerical Mathematics and Scientific Computa-tion Oxford University Press Oxford UK 2005
[2] M Benzi and M K Ng ldquoPreconditioned iterative methodsfor weighted toeplitz least squares problemsrdquo SIAM Journal onMatrix Analysis and Applications vol 27 no 4 pp 1106ndash11242006
[3] M Benzi and D Bertaccini ldquoBlock preconditioning of real-valued iterative algorithms for complex linear systemsrdquo IMAJournal of Numerical Analysis vol 28 no 3 pp 598ndash618 2008
[4] A Berman and R J Plemmons Nonnegative Matrices in theMathematical Sciences Academic Press New York NY USA1979 Reprinted by SIAM Philadelphia Pa USA 1994
[5] C-L Wang and Z-Z Bai ldquoSufficient conditions for the con-vergent splittings of non-Hermitian positive definite matricesrdquoLinear Algebra and Its Applications vol 330 no 1ndash3 pp 215ndash2182001
[6] L Wang and Z-Z Bai ldquoConvergence conditions for splittingiteration methods for non-Hermitian linear systemsrdquo LinearAlgebra and Its Applications vol 428 no 2-3 pp 453ndash468 2008
[7] Z-Z Bai G H Golub and M K Ng ldquoHermitian andSkew-Hermitian splitting methods for non-Hermitian positivedefinite linear systemsrdquo SIAM Journal on Matrix Analysis andApplications vol 24 no 3 pp 603ndash626 2003
[8] Z-Z Bai G H Golub and M K Ng ldquoOn successive-over-relaxation acceleration of the Hermitian and Skew-Hermitiansplitting iterationsrdquoNumerical Linear Algebra with Applicationsvol 14 no 4 pp 319ndash335 2007
[9] Z-Z Bai G H Golub and J-Y Pan ldquoPreconditioned Hermi-tian and Skew-Hermitian splitting methods for non-Hermitianpositive semidefinite linear systemsrdquo Numerische Mathematikvol 98 no 1 pp 1ndash32 2004
[10] M Benzi ldquoA Generalization of the he rmitian and Skew-hermitian splitting iterationrdquo SIAM Journal on Matrix Analysisand Applications vol 31 no 2 pp 360ndash374 2009
[11] Z-Z Bai G H Golub L-Z Lu and J-F Yin ldquoBlock triangularand Skew-Hermitian splitting methods for positive-definitelinear systemsrdquo SIAM Journal on Scientific Computing vol 26no 3 pp 844ndash863 2005
[12] Z-Z Bai GHGolub andMKNg ldquoOn inexact hermitian andSkew-Hermitian splitting methods for non-Hermitian positive
6 Mathematical Problems in Engineering
definite linear systemsrdquo Linear Algebra and Its Applications vol428 no 2-3 pp 413ndash440 2008
[13] L Li T -Z Huang and X -P Liu ldquoModified Hermitian andSkew-Hermitian splittingmethods for non-Hermitian positive-definite linear systemsrdquoNumerical Linear AlgebraWith Applica-tions vol 14 no 3 pp 217ndash235 2007
[14] Z-Z Bai and G H Golub ldquoAccelerated Hermitian and Skew-Hermitian splitting iteration methods for saddle-point prob-lemsrdquo IMA Journal of Numerical Analysis vol 27 no 1 pp 1ndash232007
[15] M Benzi M J Gander and G H Golub ldquoOptimization of theHermitian and Skew-Hermitian splitting iteration for saddle-point problemsrdquo BIT Numerical Mathematics vol 43 no 5 pp881ndash900 2003
[16] M Benzi and G H Golub ldquoA preconditioner for generalizedsaddle point problemsrdquo SIAM Journal on Matrix Analysis andApplications vol 26 no 1 pp 20ndash41 2005
[17] R S VargaMatrix Iterative Analysis Springer Berlin Germany2nd edition 2000
[18] Q Xiang and S-P Wu ldquoA modified alternating directionmethod for positive definite systemsrdquo in Information Engneeringand Applications R-B Zhu and Y Ma Eds vol 154 of LectureNotes in Electrical Engineering pp 437ndash444 2012
[19] Q Xiang S -P Wu and Y Xu ldquoAlternating lower-upper split-ting iterativemethod and its applicationrdquo Journal of Beijing Uni-versity of Aeronautics and Astronautics vol 38 pp 953ndash9562012 (Chinese)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Theorem 2 presents a convergence condition for the TTSmethod But in fact this condition is difficult to be appliedIt is necessary to give a practical condition It follows fromTheorem 2 that we can get the following conclusion
Theorem 3 Let 119860 isin C119899times119899 be non-Hermitian positive definitewith the TTS as in (2) and let
120591 = min 120582min (119867minus1119878) 120582max (S) 120582max (119867
minus1119878) 120582min (S)
(13)
where 120582min(119862) and 120582max(119862) denote the minimal and maximaleigenvalues of the matrix 119862 respectively119867 = (119860lowast + 119860)2 119878 =(119860 minus 119860
lowast)2119894 H = [(LU)
lowast+ LU]2 and S = [LU minus
(LU)lowast]2119894 If 1205722 + 120582min(H) + 120591 gt 0 then the TTS method
converges to the unique solution of (1) for any choice of theinitial guess 119909(0)
Although the TTS method does not always convergefor non-Hermitian positive definite linear systems it alwaysconverges for Hermitian positive definite linear systems
Theorem 4 Let 119860 isin C119899times119899 be Hermitian positive definite withthe TTS as in (2)Then for all120572 gt 0 the TTSmethod convergesto the unique solution of (1) for any choice of the initial guess119909(0) Furthermore it holds that
120572opt = argmin120572gt0
120588 (119879120572) = radic119909
lowastLLlowast119909 isin [120590119899 1205901]
120588 (119879120572opt) =
2radic119909lowastLLlowast119909 minus 119909lowast119860119909
2radic119909lowastLLlowast119909 + 119909lowast119860119909
(14)
where 119909 isin 119909 isin C119899 119879120572119909 = 120582119909 119909
lowast119909 = 1 and |120582| = 120588(119879
120572)
and 120590119899and 1205901are the minimal and maximal singular values of
L respectively
Proof Since 119860 is Hermitian positive definite 119860 = L + U =
L + Llowast and LU = LLlowast is Hermitian positive definiteFollowing (11)
120588 (119879120572) = 120582 =
119909lowast(1205722119868 minus 120572119860 +LU) 119909
119909lowast(1205722119868 + 120572119860 +LU) 119909
=
1205722minus 120572119909lowast119860119909 + 119909
lowastLLlowast119909
1205722+ 120572119909lowast119860119909 + 119909
lowastLLlowast119909
lt 1
(15)
which shows the convergence of the TTS method Following(15)
120588 (119879120572) =
1205722minus 120572119909lowast119860119909 + 119909
lowastLLlowast119909
1205722+ 120572119909lowast119860119909 + 119909
lowastLLlowast119909
= 1 minus
2120572119909lowast119860119909
1205722+ 120572119909lowast119860119909 + 119909
lowastLLlowast119909
= 1 minus 119891 (120572)
(16)
where119891(120572) = 2120572119909lowast119860119909(1205722+120572119909lowast119860119909+119909lowastLLlowast119909) As a result
min120572gt0
120588 (119879120572) = 1 minusmax
120572gt0
119891 (120572) (17)
Since
1198911015840(120572) =
2119909lowast119860119909 (119909
lowastLLlowast119909 minus 1205722)
(1205722+ 120572119909lowast119860119909 + 119909
lowastLLlowast119909)2 (18)
119891(120572) is gradually increasing if 120572 isin (0 radic119909lowastLLlowast119909) 119891(120572)
is gradually decreasing if 120572 isin (radic119909lowastLLlowast119909infin) and
consequently when 120572 = radic119909lowastLLlowast119909 119891(120572) gets its maximummax120572gt0119891(120572) = 2120572119909
lowast119860119909(119909
lowast119860119909 + 2radic119909
lowastLLlowast119909) Thereforewhen
120572opt = argmin120572gt0
120588 (119879120572) = radic119909
lowastLLlowast119909 isin [120590119899 1205901]
120588 (119879120572opt) =
2radic119909lowastLLlowast119909 minus 119909lowast119860119909
2radic119909lowastLLlowast119909 + 119909lowast119860119909
(19)
which shows that we completed the proof
4 Remark on and Comparison ofConvergence Theorems
In fact the TTS method is a special case of the generalizedassymmetric SOR iteration method with parameter matriceswhen specifically choosing the parameter matrices Thisscheme is called the ALUS method in [18 19]
The ALUS Method Given an initial guess 119909(0) for 119896 =
0 1 2 until 119909(119896) converges compute
(120572119868 +L) 119909(119896+12)
= (120572119868 minusU) 119909(119896)+ 119887
(120572119868 +U) 119909(119896+1)
= (120572119868 minusL) 119909(119896+12)
+ 119887
(20)
where 120572 is a given positive constantL = 1198631+119871 U = 119863
2+119880
1198631+ 1198632= 119863 and 119871 119880 and119863 are defined in (3)
We easily generalize the convergence theorems on TTSmethod to ALUS method
Theorem 5 Let 119860 isin C119899times119899 be non-Hermitian positive definitewith the ALUS as in (20) Then the ALUS method converges tothe unique solution of (1) for any choice of the initial guess 119909(0)if and only if 1205722 + 119909lowast(120583S +H)119909 gt 0 where 120583 = 119909lowast119878119909119909lowast119867119909119867 = (119860
lowast+119860)2 119878 = (119860minus119860lowast)2119894H = [(
LU)lowast+LU]2S =
[LU minus (
LU)lowast]2119894 and 119909 isin 119909 isin C119899 119879
120572119909 = 120582119909 119909
lowast119909 =
1 and |120582| = 120588(119879120572)
Theorem 6 Let 119860 isin C119899times119899 be non-Hermitian positive definitewith the ALUS as in (20) and let
120591 = min 120582min (119867minus1119878) 120582max (S) 120582max (119867
minus1119878) 120582min (S)
(21)
where 120582min(119862) and 120582max(119862) denote the minimal and maximaleigenvalues of the matrix 119862 respectively119867 = (119860lowast + 119860)2 119878 =(119860 minus 119860
lowast)2119894 H = [(
LU)lowast+LU]2 and S = [
LU minus
(LU)lowast]2119894 If 1205722 + 120582min(H) + 120591 gt 0 then the ALUS method
converges to the unique solution of (1) for any choice of theinitial guess 119909(0)
4 Mathematical Problems in Engineering
The proofs of Theorems 5 and 6 directly result from theproofs of Theorems 2 and 3
Like Theorem 4 in [18] or Theorem 6 in [19] Theorem5 has only theoretical significance since it is difficult to beapplied However Theorem 6 along with Theorem 3 in [18]or Theorem 4 in [19] proposes a practical condition onconvergence of ALUSmethod But the condition inTheorem6 is wider than Theorem 3 in [18] or Theorem 4 in [19] Thefollowing will give an example to demonstrate this fact
Example 7 The coefficient matrix 119860 of linear system (1) isgiven as
119860 = [
1 2
minus2 1] (22)
Now we consider solving this system by ALUS methodLet1198631= [119909 0
0 119910 ]Then1198632= 119863minus119863
1= [1minus119909 0
0 1minus119910 ] L = 119863
1+119871 =
[119909 0
minus2 119910 ] and U = 1198632+ 119880 = [
1minus119909 2
0 1minus119910 ] In order that the ALUSmethod converges to the unique solution of (1) for any choiceof the initial guess119909(0)Theorem 3 in [18] orTheorem 4 in [19]shows that L +
Llowast = [ 2119909 minus2minus2 2119910 ] and U +Ulowast = [
2(2minus119909) 2
2 2(1minus119910)]
are both positive definite for all 119909 isin (0 1) and 119910 isin (0 1) As aresult the set
(119909 119910) isin R timesR
119909119910 gt 1
119909119910 gt 119909 + 119910
119909 isin (0 1) 119910 isin (0 1)
(23)
is not empty In fact the set (23) is empty since the inequality119909119910 gt 1 fails to hold when 119909 isin (0 1) and 119910 isin (0 1) ThereforeTheorem 3 in [18] or Theorem4 in [19] does not give theconvergence of theALUSmethod if we solve the linear system(1) with the coefficient matrix 119860 in (22)
Using Theorem 6 set 119909 = 119910 = 12 and compute 120591 = minus2and 120582min(H) = minus154 Then when 120572 gt radic232 the ALUSmethod converges to the unique solution of (1) for any choiceof the initial guess 119909(0)
Example 8 Linear system (1) is shifted skew-Hermitianlinear system (see [7ndash14]) whose coefficient matrix119860 is givenas
119860 = 120574119868 + 119878 (24)
where 120574 is a positive constant 119868 is the 119899 times 119899 identity matrixand
119878 = 119871 minus 119871lowast (25)
is a skew-Hermitian matrix with 119871 a lower triangular matrix
The shifted skew-Hermitian linear system arising in theHSS iterative method can be much more problematic insome cases this solution is as difficult as that of the originallinear system [10] Since HSS method fails to solve this linearsystem we consider the ALUS method and TTS method
We assert that if 120574 le 120582max(119866) minus 120582min(119866) with 119866 = 119871 + 119871lowast
and 120582max(119866) and 120582min(119866) being themaximum andminimum
eigenvalues of thematrix119866 respectivelyTheorem 3 in [18] orTheorem4 in [19] fails to give the convergence of the ALUSmethod when solving the skew-Hermitian linear system (1)
Let1198631= 119909119868Then119863
2= 119863minus119863
1= (120574minus119909)119868L = 119909119868+119871 and
U = (120574 minus 119909)119868 minus 119871lowast In order that the ALUS method converges
to the unique solution of (1) for any choice of the initial guess119909(0)Theorem 3 in [18] or Theorem4 in [19] shows that L +
Llowast = 2119909119868 + 119866 and U + Ulowast = 2(120574 minus 119909)119868 minus 119866 are both positivedefinite for all 119909 isin (0 120574) Thus it is easy to get
2119909 + 120582min (119866) gt 0 2 (120574 minus 119909) minus 120582max (119866) gt 0 (26)
It follows from (26) that 120574 gt 120582max(119866) minus 120582min(119866) which showsthat when 120574 le 120582max(119866) minus 120582min(119866) Theorem 3 in [18] orTheorem 4 in [19] fails to give the convergence of the ALUSmethod
We consider using Theorem 6 Compute 120591 =
120582max(minus119894119878)120582min(minus119894119878)2 Theorem 6 shows that if 119891(119909) = 1205722+119909(120574 minus 119909) minus 120582max(119871119871
lowast) + (1205742 minus 119909)120582min(119866) gt 0 the ALUS
method converges to the unique solution of (1) for any choiceof the initial guess 119909(0) Furthermore
max119909isin(0120574)
119891 (119909) = 1205722+
(1205742+ 1205822
min (119866))
4
minus 120582max (119871119871lowast) (27)
when 119909 = (120574 minus 120582min(119866))2 Thus if 120574 ge radic411987122minus 1205822
min(119866)Theorem 6 shows that the ALUS method converges to theunique solution of (1) for any choice of the initial guess 119909(0)and for all 120572 gt 0 otherwise it follows from Theorem 6that the ALUS method converges to the unique solution of(1) for any choice of the initial guess 119909(0) and for all 120572 gt
radic1198712
2minus (1205742+ 1205822
min(119866))4In particular if 120574 ge 2119871
2 Theorem 3 yields that the
TTS method converges to the unique solution of (1) for anychoice of the initial guess 119909(0) and for all 120572 gt 0 otherwiseit follows fromTheorem 3 that the TTS method converges tothe unique solution of (1) for any choice of the initial guess119909(0) and for all 120572 gt radic1198712
2minus 12057424
5 Numerical Examples
In this section we describe the results of a numerical simpleexample with the TTS method on a set of linear systemsarising from a finite element discretization of a convection-diffusion equation in two dimensions
Example 1 The coefficient matrix 119860 of linear system (1) isgiven as
119860 =
[
[
[
[
1 11 12 1
minus05 1 14 065
minus02 minus06 1 155
02 025 minus045 1
]
]
]
]
(28)
Now we investigate convergence of TTS method forlinear system (1) by Theorem 3 It is known that 119860 is non-Hermitian positive definite By Matlab computations we
Mathematical Problems in Engineering 5
Table 1 The comparison results of 120588(119879120572) with different parameter pairs 120572
120572 10minus5
10minus4
10minus3
10minus2
01 05 1 11120588(119879120572) 10000 09999 09994 09940 09435 08476 07163 06622
120572 12 13 14 15 16 17 18 19120588(119879120572) 07795 08983 09422 09455 09199 08698 07921 06381
120572 2 225 25 3 305 31 315 32120588(119879120572) 06193 05784 05498 05283 05281 05282 05285 05290
120572 33 34 35 4 45 5 10 15120588(119879120572) 05307 05330 05358 05550 05779 06012 07580 08292
120572 20 30 50 100 150 103 104 105
120588(119879120572) 08694 09099 09449 09720 09813 09972 09997 10000
get 120582min(H) + 120591 = minus30471 As a result when 120572 isin
(17456 +infin) the TTS method theoretically converges to theunique solution of (1) for any choice of the initial guess 119909(0)But when 120572 rarr +infin this method either converges veryslowly or fails to converge since 120588(119879
120572) rarr 1
By numerical experiments on Matlab program one hasTable 1
Table 1 shows that for given matrix 119860 the TTS methodconverges for 120572 isin (10
minus5 105) Further 120588(119879
120572) is gradually
decreasing when 120572 isin (0 11) and 120572 isin (15 305) while it isgradually increasing when 120572 isin (11 15) and 120572 isin (305 +infin)Thus 120588(119879
120572) has two minimal values 120588(119879
11) = 06622 and
120588(119879305) = 05281 However 120588(119879
305) = 05281 is the minimal
value As a result 120572opt = 305 It follows from Table 1 thatTheorem 3 is true Furthermore the interval obtained inTheorem 3 includes the optimal point 120572opt = 305
6 Conclusions
This paper studies convergence of TTS and ALUS iterativemethods for non-Hermitian positive definite linear systemsSome sufficient and necessary conditions for convergenceare proposed But these conditions are only theoreticallysignificant and difficult to apply to practical computations Inwhat follows several conditions are presented such that theTTSmethod and ALUSmethod converge for non-Hermitianpositive definite linear systems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Cheng-Yi Zhang was partly supported by the Science Foun-dation of the Education Department of Shaanxi Provinceof China (2013JK0593) the Scientific Research Foundation(BS1014) and the Education Reform Foundation (2012JG40)of Xirsquoan Polytechnic University and the National NaturalScience Foundations of China (11201362 and 11271297) Yu-Qian Yang was partly supported by the National NaturalScience Foundation of China (no 61201297) Qiang Sun waspartly supported by the National Natural Science Foundationof China (no 61001140) and the Scientific Research Program
Funded by Shaanxi Provincial Education Department (no12JK0544)
References
[1] H Elman D Silvester and A Wathen Finite Elements andFast Iterative Solvers with Applications in Incompressible FluidDynamics Numerical Mathematics and Scientific Computa-tion Oxford University Press Oxford UK 2005
[2] M Benzi and M K Ng ldquoPreconditioned iterative methodsfor weighted toeplitz least squares problemsrdquo SIAM Journal onMatrix Analysis and Applications vol 27 no 4 pp 1106ndash11242006
[3] M Benzi and D Bertaccini ldquoBlock preconditioning of real-valued iterative algorithms for complex linear systemsrdquo IMAJournal of Numerical Analysis vol 28 no 3 pp 598ndash618 2008
[4] A Berman and R J Plemmons Nonnegative Matrices in theMathematical Sciences Academic Press New York NY USA1979 Reprinted by SIAM Philadelphia Pa USA 1994
[5] C-L Wang and Z-Z Bai ldquoSufficient conditions for the con-vergent splittings of non-Hermitian positive definite matricesrdquoLinear Algebra and Its Applications vol 330 no 1ndash3 pp 215ndash2182001
[6] L Wang and Z-Z Bai ldquoConvergence conditions for splittingiteration methods for non-Hermitian linear systemsrdquo LinearAlgebra and Its Applications vol 428 no 2-3 pp 453ndash468 2008
[7] Z-Z Bai G H Golub and M K Ng ldquoHermitian andSkew-Hermitian splitting methods for non-Hermitian positivedefinite linear systemsrdquo SIAM Journal on Matrix Analysis andApplications vol 24 no 3 pp 603ndash626 2003
[8] Z-Z Bai G H Golub and M K Ng ldquoOn successive-over-relaxation acceleration of the Hermitian and Skew-Hermitiansplitting iterationsrdquoNumerical Linear Algebra with Applicationsvol 14 no 4 pp 319ndash335 2007
[9] Z-Z Bai G H Golub and J-Y Pan ldquoPreconditioned Hermi-tian and Skew-Hermitian splitting methods for non-Hermitianpositive semidefinite linear systemsrdquo Numerische Mathematikvol 98 no 1 pp 1ndash32 2004
[10] M Benzi ldquoA Generalization of the he rmitian and Skew-hermitian splitting iterationrdquo SIAM Journal on Matrix Analysisand Applications vol 31 no 2 pp 360ndash374 2009
[11] Z-Z Bai G H Golub L-Z Lu and J-F Yin ldquoBlock triangularand Skew-Hermitian splitting methods for positive-definitelinear systemsrdquo SIAM Journal on Scientific Computing vol 26no 3 pp 844ndash863 2005
[12] Z-Z Bai GHGolub andMKNg ldquoOn inexact hermitian andSkew-Hermitian splitting methods for non-Hermitian positive
6 Mathematical Problems in Engineering
definite linear systemsrdquo Linear Algebra and Its Applications vol428 no 2-3 pp 413ndash440 2008
[13] L Li T -Z Huang and X -P Liu ldquoModified Hermitian andSkew-Hermitian splittingmethods for non-Hermitian positive-definite linear systemsrdquoNumerical Linear AlgebraWith Applica-tions vol 14 no 3 pp 217ndash235 2007
[14] Z-Z Bai and G H Golub ldquoAccelerated Hermitian and Skew-Hermitian splitting iteration methods for saddle-point prob-lemsrdquo IMA Journal of Numerical Analysis vol 27 no 1 pp 1ndash232007
[15] M Benzi M J Gander and G H Golub ldquoOptimization of theHermitian and Skew-Hermitian splitting iteration for saddle-point problemsrdquo BIT Numerical Mathematics vol 43 no 5 pp881ndash900 2003
[16] M Benzi and G H Golub ldquoA preconditioner for generalizedsaddle point problemsrdquo SIAM Journal on Matrix Analysis andApplications vol 26 no 1 pp 20ndash41 2005
[17] R S VargaMatrix Iterative Analysis Springer Berlin Germany2nd edition 2000
[18] Q Xiang and S-P Wu ldquoA modified alternating directionmethod for positive definite systemsrdquo in Information Engneeringand Applications R-B Zhu and Y Ma Eds vol 154 of LectureNotes in Electrical Engineering pp 437ndash444 2012
[19] Q Xiang S -P Wu and Y Xu ldquoAlternating lower-upper split-ting iterativemethod and its applicationrdquo Journal of Beijing Uni-versity of Aeronautics and Astronautics vol 38 pp 953ndash9562012 (Chinese)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
The proofs of Theorems 5 and 6 directly result from theproofs of Theorems 2 and 3
Like Theorem 4 in [18] or Theorem 6 in [19] Theorem5 has only theoretical significance since it is difficult to beapplied However Theorem 6 along with Theorem 3 in [18]or Theorem 4 in [19] proposes a practical condition onconvergence of ALUSmethod But the condition inTheorem6 is wider than Theorem 3 in [18] or Theorem 4 in [19] Thefollowing will give an example to demonstrate this fact
Example 7 The coefficient matrix 119860 of linear system (1) isgiven as
119860 = [
1 2
minus2 1] (22)
Now we consider solving this system by ALUS methodLet1198631= [119909 0
0 119910 ]Then1198632= 119863minus119863
1= [1minus119909 0
0 1minus119910 ] L = 119863
1+119871 =
[119909 0
minus2 119910 ] and U = 1198632+ 119880 = [
1minus119909 2
0 1minus119910 ] In order that the ALUSmethod converges to the unique solution of (1) for any choiceof the initial guess119909(0)Theorem 3 in [18] orTheorem 4 in [19]shows that L +
Llowast = [ 2119909 minus2minus2 2119910 ] and U +Ulowast = [
2(2minus119909) 2
2 2(1minus119910)]
are both positive definite for all 119909 isin (0 1) and 119910 isin (0 1) As aresult the set
(119909 119910) isin R timesR
119909119910 gt 1
119909119910 gt 119909 + 119910
119909 isin (0 1) 119910 isin (0 1)
(23)
is not empty In fact the set (23) is empty since the inequality119909119910 gt 1 fails to hold when 119909 isin (0 1) and 119910 isin (0 1) ThereforeTheorem 3 in [18] or Theorem4 in [19] does not give theconvergence of theALUSmethod if we solve the linear system(1) with the coefficient matrix 119860 in (22)
Using Theorem 6 set 119909 = 119910 = 12 and compute 120591 = minus2and 120582min(H) = minus154 Then when 120572 gt radic232 the ALUSmethod converges to the unique solution of (1) for any choiceof the initial guess 119909(0)
Example 8 Linear system (1) is shifted skew-Hermitianlinear system (see [7ndash14]) whose coefficient matrix119860 is givenas
119860 = 120574119868 + 119878 (24)
where 120574 is a positive constant 119868 is the 119899 times 119899 identity matrixand
119878 = 119871 minus 119871lowast (25)
is a skew-Hermitian matrix with 119871 a lower triangular matrix
The shifted skew-Hermitian linear system arising in theHSS iterative method can be much more problematic insome cases this solution is as difficult as that of the originallinear system [10] Since HSS method fails to solve this linearsystem we consider the ALUS method and TTS method
We assert that if 120574 le 120582max(119866) minus 120582min(119866) with 119866 = 119871 + 119871lowast
and 120582max(119866) and 120582min(119866) being themaximum andminimum
eigenvalues of thematrix119866 respectivelyTheorem 3 in [18] orTheorem4 in [19] fails to give the convergence of the ALUSmethod when solving the skew-Hermitian linear system (1)
Let1198631= 119909119868Then119863
2= 119863minus119863
1= (120574minus119909)119868L = 119909119868+119871 and
U = (120574 minus 119909)119868 minus 119871lowast In order that the ALUS method converges
to the unique solution of (1) for any choice of the initial guess119909(0)Theorem 3 in [18] or Theorem4 in [19] shows that L +
Llowast = 2119909119868 + 119866 and U + Ulowast = 2(120574 minus 119909)119868 minus 119866 are both positivedefinite for all 119909 isin (0 120574) Thus it is easy to get
2119909 + 120582min (119866) gt 0 2 (120574 minus 119909) minus 120582max (119866) gt 0 (26)
It follows from (26) that 120574 gt 120582max(119866) minus 120582min(119866) which showsthat when 120574 le 120582max(119866) minus 120582min(119866) Theorem 3 in [18] orTheorem 4 in [19] fails to give the convergence of the ALUSmethod
We consider using Theorem 6 Compute 120591 =
120582max(minus119894119878)120582min(minus119894119878)2 Theorem 6 shows that if 119891(119909) = 1205722+119909(120574 minus 119909) minus 120582max(119871119871
lowast) + (1205742 minus 119909)120582min(119866) gt 0 the ALUS
method converges to the unique solution of (1) for any choiceof the initial guess 119909(0) Furthermore
max119909isin(0120574)
119891 (119909) = 1205722+
(1205742+ 1205822
min (119866))
4
minus 120582max (119871119871lowast) (27)
when 119909 = (120574 minus 120582min(119866))2 Thus if 120574 ge radic411987122minus 1205822
min(119866)Theorem 6 shows that the ALUS method converges to theunique solution of (1) for any choice of the initial guess 119909(0)and for all 120572 gt 0 otherwise it follows from Theorem 6that the ALUS method converges to the unique solution of(1) for any choice of the initial guess 119909(0) and for all 120572 gt
radic1198712
2minus (1205742+ 1205822
min(119866))4In particular if 120574 ge 2119871
2 Theorem 3 yields that the
TTS method converges to the unique solution of (1) for anychoice of the initial guess 119909(0) and for all 120572 gt 0 otherwiseit follows fromTheorem 3 that the TTS method converges tothe unique solution of (1) for any choice of the initial guess119909(0) and for all 120572 gt radic1198712
2minus 12057424
5 Numerical Examples
In this section we describe the results of a numerical simpleexample with the TTS method on a set of linear systemsarising from a finite element discretization of a convection-diffusion equation in two dimensions
Example 1 The coefficient matrix 119860 of linear system (1) isgiven as
119860 =
[
[
[
[
1 11 12 1
minus05 1 14 065
minus02 minus06 1 155
02 025 minus045 1
]
]
]
]
(28)
Now we investigate convergence of TTS method forlinear system (1) by Theorem 3 It is known that 119860 is non-Hermitian positive definite By Matlab computations we
Mathematical Problems in Engineering 5
Table 1 The comparison results of 120588(119879120572) with different parameter pairs 120572
120572 10minus5
10minus4
10minus3
10minus2
01 05 1 11120588(119879120572) 10000 09999 09994 09940 09435 08476 07163 06622
120572 12 13 14 15 16 17 18 19120588(119879120572) 07795 08983 09422 09455 09199 08698 07921 06381
120572 2 225 25 3 305 31 315 32120588(119879120572) 06193 05784 05498 05283 05281 05282 05285 05290
120572 33 34 35 4 45 5 10 15120588(119879120572) 05307 05330 05358 05550 05779 06012 07580 08292
120572 20 30 50 100 150 103 104 105
120588(119879120572) 08694 09099 09449 09720 09813 09972 09997 10000
get 120582min(H) + 120591 = minus30471 As a result when 120572 isin
(17456 +infin) the TTS method theoretically converges to theunique solution of (1) for any choice of the initial guess 119909(0)But when 120572 rarr +infin this method either converges veryslowly or fails to converge since 120588(119879
120572) rarr 1
By numerical experiments on Matlab program one hasTable 1
Table 1 shows that for given matrix 119860 the TTS methodconverges for 120572 isin (10
minus5 105) Further 120588(119879
120572) is gradually
decreasing when 120572 isin (0 11) and 120572 isin (15 305) while it isgradually increasing when 120572 isin (11 15) and 120572 isin (305 +infin)Thus 120588(119879
120572) has two minimal values 120588(119879
11) = 06622 and
120588(119879305) = 05281 However 120588(119879
305) = 05281 is the minimal
value As a result 120572opt = 305 It follows from Table 1 thatTheorem 3 is true Furthermore the interval obtained inTheorem 3 includes the optimal point 120572opt = 305
6 Conclusions
This paper studies convergence of TTS and ALUS iterativemethods for non-Hermitian positive definite linear systemsSome sufficient and necessary conditions for convergenceare proposed But these conditions are only theoreticallysignificant and difficult to apply to practical computations Inwhat follows several conditions are presented such that theTTSmethod and ALUSmethod converge for non-Hermitianpositive definite linear systems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Cheng-Yi Zhang was partly supported by the Science Foun-dation of the Education Department of Shaanxi Provinceof China (2013JK0593) the Scientific Research Foundation(BS1014) and the Education Reform Foundation (2012JG40)of Xirsquoan Polytechnic University and the National NaturalScience Foundations of China (11201362 and 11271297) Yu-Qian Yang was partly supported by the National NaturalScience Foundation of China (no 61201297) Qiang Sun waspartly supported by the National Natural Science Foundationof China (no 61001140) and the Scientific Research Program
Funded by Shaanxi Provincial Education Department (no12JK0544)
References
[1] H Elman D Silvester and A Wathen Finite Elements andFast Iterative Solvers with Applications in Incompressible FluidDynamics Numerical Mathematics and Scientific Computa-tion Oxford University Press Oxford UK 2005
[2] M Benzi and M K Ng ldquoPreconditioned iterative methodsfor weighted toeplitz least squares problemsrdquo SIAM Journal onMatrix Analysis and Applications vol 27 no 4 pp 1106ndash11242006
[3] M Benzi and D Bertaccini ldquoBlock preconditioning of real-valued iterative algorithms for complex linear systemsrdquo IMAJournal of Numerical Analysis vol 28 no 3 pp 598ndash618 2008
[4] A Berman and R J Plemmons Nonnegative Matrices in theMathematical Sciences Academic Press New York NY USA1979 Reprinted by SIAM Philadelphia Pa USA 1994
[5] C-L Wang and Z-Z Bai ldquoSufficient conditions for the con-vergent splittings of non-Hermitian positive definite matricesrdquoLinear Algebra and Its Applications vol 330 no 1ndash3 pp 215ndash2182001
[6] L Wang and Z-Z Bai ldquoConvergence conditions for splittingiteration methods for non-Hermitian linear systemsrdquo LinearAlgebra and Its Applications vol 428 no 2-3 pp 453ndash468 2008
[7] Z-Z Bai G H Golub and M K Ng ldquoHermitian andSkew-Hermitian splitting methods for non-Hermitian positivedefinite linear systemsrdquo SIAM Journal on Matrix Analysis andApplications vol 24 no 3 pp 603ndash626 2003
[8] Z-Z Bai G H Golub and M K Ng ldquoOn successive-over-relaxation acceleration of the Hermitian and Skew-Hermitiansplitting iterationsrdquoNumerical Linear Algebra with Applicationsvol 14 no 4 pp 319ndash335 2007
[9] Z-Z Bai G H Golub and J-Y Pan ldquoPreconditioned Hermi-tian and Skew-Hermitian splitting methods for non-Hermitianpositive semidefinite linear systemsrdquo Numerische Mathematikvol 98 no 1 pp 1ndash32 2004
[10] M Benzi ldquoA Generalization of the he rmitian and Skew-hermitian splitting iterationrdquo SIAM Journal on Matrix Analysisand Applications vol 31 no 2 pp 360ndash374 2009
[11] Z-Z Bai G H Golub L-Z Lu and J-F Yin ldquoBlock triangularand Skew-Hermitian splitting methods for positive-definitelinear systemsrdquo SIAM Journal on Scientific Computing vol 26no 3 pp 844ndash863 2005
[12] Z-Z Bai GHGolub andMKNg ldquoOn inexact hermitian andSkew-Hermitian splitting methods for non-Hermitian positive
6 Mathematical Problems in Engineering
definite linear systemsrdquo Linear Algebra and Its Applications vol428 no 2-3 pp 413ndash440 2008
[13] L Li T -Z Huang and X -P Liu ldquoModified Hermitian andSkew-Hermitian splittingmethods for non-Hermitian positive-definite linear systemsrdquoNumerical Linear AlgebraWith Applica-tions vol 14 no 3 pp 217ndash235 2007
[14] Z-Z Bai and G H Golub ldquoAccelerated Hermitian and Skew-Hermitian splitting iteration methods for saddle-point prob-lemsrdquo IMA Journal of Numerical Analysis vol 27 no 1 pp 1ndash232007
[15] M Benzi M J Gander and G H Golub ldquoOptimization of theHermitian and Skew-Hermitian splitting iteration for saddle-point problemsrdquo BIT Numerical Mathematics vol 43 no 5 pp881ndash900 2003
[16] M Benzi and G H Golub ldquoA preconditioner for generalizedsaddle point problemsrdquo SIAM Journal on Matrix Analysis andApplications vol 26 no 1 pp 20ndash41 2005
[17] R S VargaMatrix Iterative Analysis Springer Berlin Germany2nd edition 2000
[18] Q Xiang and S-P Wu ldquoA modified alternating directionmethod for positive definite systemsrdquo in Information Engneeringand Applications R-B Zhu and Y Ma Eds vol 154 of LectureNotes in Electrical Engineering pp 437ndash444 2012
[19] Q Xiang S -P Wu and Y Xu ldquoAlternating lower-upper split-ting iterativemethod and its applicationrdquo Journal of Beijing Uni-versity of Aeronautics and Astronautics vol 38 pp 953ndash9562012 (Chinese)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Table 1 The comparison results of 120588(119879120572) with different parameter pairs 120572
120572 10minus5
10minus4
10minus3
10minus2
01 05 1 11120588(119879120572) 10000 09999 09994 09940 09435 08476 07163 06622
120572 12 13 14 15 16 17 18 19120588(119879120572) 07795 08983 09422 09455 09199 08698 07921 06381
120572 2 225 25 3 305 31 315 32120588(119879120572) 06193 05784 05498 05283 05281 05282 05285 05290
120572 33 34 35 4 45 5 10 15120588(119879120572) 05307 05330 05358 05550 05779 06012 07580 08292
120572 20 30 50 100 150 103 104 105
120588(119879120572) 08694 09099 09449 09720 09813 09972 09997 10000
get 120582min(H) + 120591 = minus30471 As a result when 120572 isin
(17456 +infin) the TTS method theoretically converges to theunique solution of (1) for any choice of the initial guess 119909(0)But when 120572 rarr +infin this method either converges veryslowly or fails to converge since 120588(119879
120572) rarr 1
By numerical experiments on Matlab program one hasTable 1
Table 1 shows that for given matrix 119860 the TTS methodconverges for 120572 isin (10
minus5 105) Further 120588(119879
120572) is gradually
decreasing when 120572 isin (0 11) and 120572 isin (15 305) while it isgradually increasing when 120572 isin (11 15) and 120572 isin (305 +infin)Thus 120588(119879
120572) has two minimal values 120588(119879
11) = 06622 and
120588(119879305) = 05281 However 120588(119879
305) = 05281 is the minimal
value As a result 120572opt = 305 It follows from Table 1 thatTheorem 3 is true Furthermore the interval obtained inTheorem 3 includes the optimal point 120572opt = 305
6 Conclusions
This paper studies convergence of TTS and ALUS iterativemethods for non-Hermitian positive definite linear systemsSome sufficient and necessary conditions for convergenceare proposed But these conditions are only theoreticallysignificant and difficult to apply to practical computations Inwhat follows several conditions are presented such that theTTSmethod and ALUSmethod converge for non-Hermitianpositive definite linear systems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Cheng-Yi Zhang was partly supported by the Science Foun-dation of the Education Department of Shaanxi Provinceof China (2013JK0593) the Scientific Research Foundation(BS1014) and the Education Reform Foundation (2012JG40)of Xirsquoan Polytechnic University and the National NaturalScience Foundations of China (11201362 and 11271297) Yu-Qian Yang was partly supported by the National NaturalScience Foundation of China (no 61201297) Qiang Sun waspartly supported by the National Natural Science Foundationof China (no 61001140) and the Scientific Research Program
Funded by Shaanxi Provincial Education Department (no12JK0544)
References
[1] H Elman D Silvester and A Wathen Finite Elements andFast Iterative Solvers with Applications in Incompressible FluidDynamics Numerical Mathematics and Scientific Computa-tion Oxford University Press Oxford UK 2005
[2] M Benzi and M K Ng ldquoPreconditioned iterative methodsfor weighted toeplitz least squares problemsrdquo SIAM Journal onMatrix Analysis and Applications vol 27 no 4 pp 1106ndash11242006
[3] M Benzi and D Bertaccini ldquoBlock preconditioning of real-valued iterative algorithms for complex linear systemsrdquo IMAJournal of Numerical Analysis vol 28 no 3 pp 598ndash618 2008
[4] A Berman and R J Plemmons Nonnegative Matrices in theMathematical Sciences Academic Press New York NY USA1979 Reprinted by SIAM Philadelphia Pa USA 1994
[5] C-L Wang and Z-Z Bai ldquoSufficient conditions for the con-vergent splittings of non-Hermitian positive definite matricesrdquoLinear Algebra and Its Applications vol 330 no 1ndash3 pp 215ndash2182001
[6] L Wang and Z-Z Bai ldquoConvergence conditions for splittingiteration methods for non-Hermitian linear systemsrdquo LinearAlgebra and Its Applications vol 428 no 2-3 pp 453ndash468 2008
[7] Z-Z Bai G H Golub and M K Ng ldquoHermitian andSkew-Hermitian splitting methods for non-Hermitian positivedefinite linear systemsrdquo SIAM Journal on Matrix Analysis andApplications vol 24 no 3 pp 603ndash626 2003
[8] Z-Z Bai G H Golub and M K Ng ldquoOn successive-over-relaxation acceleration of the Hermitian and Skew-Hermitiansplitting iterationsrdquoNumerical Linear Algebra with Applicationsvol 14 no 4 pp 319ndash335 2007
[9] Z-Z Bai G H Golub and J-Y Pan ldquoPreconditioned Hermi-tian and Skew-Hermitian splitting methods for non-Hermitianpositive semidefinite linear systemsrdquo Numerische Mathematikvol 98 no 1 pp 1ndash32 2004
[10] M Benzi ldquoA Generalization of the he rmitian and Skew-hermitian splitting iterationrdquo SIAM Journal on Matrix Analysisand Applications vol 31 no 2 pp 360ndash374 2009
[11] Z-Z Bai G H Golub L-Z Lu and J-F Yin ldquoBlock triangularand Skew-Hermitian splitting methods for positive-definitelinear systemsrdquo SIAM Journal on Scientific Computing vol 26no 3 pp 844ndash863 2005
[12] Z-Z Bai GHGolub andMKNg ldquoOn inexact hermitian andSkew-Hermitian splitting methods for non-Hermitian positive
6 Mathematical Problems in Engineering
definite linear systemsrdquo Linear Algebra and Its Applications vol428 no 2-3 pp 413ndash440 2008
[13] L Li T -Z Huang and X -P Liu ldquoModified Hermitian andSkew-Hermitian splittingmethods for non-Hermitian positive-definite linear systemsrdquoNumerical Linear AlgebraWith Applica-tions vol 14 no 3 pp 217ndash235 2007
[14] Z-Z Bai and G H Golub ldquoAccelerated Hermitian and Skew-Hermitian splitting iteration methods for saddle-point prob-lemsrdquo IMA Journal of Numerical Analysis vol 27 no 1 pp 1ndash232007
[15] M Benzi M J Gander and G H Golub ldquoOptimization of theHermitian and Skew-Hermitian splitting iteration for saddle-point problemsrdquo BIT Numerical Mathematics vol 43 no 5 pp881ndash900 2003
[16] M Benzi and G H Golub ldquoA preconditioner for generalizedsaddle point problemsrdquo SIAM Journal on Matrix Analysis andApplications vol 26 no 1 pp 20ndash41 2005
[17] R S VargaMatrix Iterative Analysis Springer Berlin Germany2nd edition 2000
[18] Q Xiang and S-P Wu ldquoA modified alternating directionmethod for positive definite systemsrdquo in Information Engneeringand Applications R-B Zhu and Y Ma Eds vol 154 of LectureNotes in Electrical Engineering pp 437ndash444 2012
[19] Q Xiang S -P Wu and Y Xu ldquoAlternating lower-upper split-ting iterativemethod and its applicationrdquo Journal of Beijing Uni-versity of Aeronautics and Astronautics vol 38 pp 953ndash9562012 (Chinese)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
definite linear systemsrdquo Linear Algebra and Its Applications vol428 no 2-3 pp 413ndash440 2008
[13] L Li T -Z Huang and X -P Liu ldquoModified Hermitian andSkew-Hermitian splittingmethods for non-Hermitian positive-definite linear systemsrdquoNumerical Linear AlgebraWith Applica-tions vol 14 no 3 pp 217ndash235 2007
[14] Z-Z Bai and G H Golub ldquoAccelerated Hermitian and Skew-Hermitian splitting iteration methods for saddle-point prob-lemsrdquo IMA Journal of Numerical Analysis vol 27 no 1 pp 1ndash232007
[15] M Benzi M J Gander and G H Golub ldquoOptimization of theHermitian and Skew-Hermitian splitting iteration for saddle-point problemsrdquo BIT Numerical Mathematics vol 43 no 5 pp881ndash900 2003
[16] M Benzi and G H Golub ldquoA preconditioner for generalizedsaddle point problemsrdquo SIAM Journal on Matrix Analysis andApplications vol 26 no 1 pp 20ndash41 2005
[17] R S VargaMatrix Iterative Analysis Springer Berlin Germany2nd edition 2000
[18] Q Xiang and S-P Wu ldquoA modified alternating directionmethod for positive definite systemsrdquo in Information Engneeringand Applications R-B Zhu and Y Ma Eds vol 154 of LectureNotes in Electrical Engineering pp 437ndash444 2012
[19] Q Xiang S -P Wu and Y Xu ldquoAlternating lower-upper split-ting iterativemethod and its applicationrdquo Journal of Beijing Uni-versity of Aeronautics and Astronautics vol 38 pp 953ndash9562012 (Chinese)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![コプト語の複数定冠詞（ハンドアウト）[Coptic Plural Definite Article (handout)]](https://static.fdokumen.com/doc/165x107/6321a6dc64690856e108cc3b/coptic-plural-definite-article.jpg)
