# A Generalized HSS Iteration Method for Continuous Sylvester Equations

Authors: Xu Li; Yu-Jiang Wu; Ai-Li Yang; Jin-Yun Yuan
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

Based on the Hermitian and skew-Hermitian splitting (HSS) iteration technique, we establish a generalized HSS (GHSS) iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and positive definite/semidefinite matrices. The GHSS method is essentially a four-parameter iteration which not only covers the standard HSS iteration but also enables us to optimize the iterative process. An exact parameter region of convergence for the method is strictly proved and a minimum value for the upper bound of the iterative spectrum is derived. Moreover, to reduce the computational cost, we establish an inexact variant of the GHSS (IGHSS) iteration method whose convergence property is discussed. Numerical experiments illustrate the efficiency and robustness of the GHSS iteration method and its inexact variant.


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## Body

## 1. Introduction

Consider the following continuous Sylvester equation:(1)AX+XB=C,
where A∈ℂm×m, B∈ℂn×n, and C∈ℂm×n are given complex matrices. Assume that (i)
A,B, and C are large and sparse matrices;(ii)
at least one ofA and B is non-Hermitian;(iii)
bothA and B are positive semi-definite, and at least one of them is positive definite.Since under assumptions (i)–(iii) there is no common eigenvalue between A and -B, we obtain from [1, 2] that the continuous Sylvester equation (1) has a unique solution. Obviously, the continuous Lyapunov equation is a special case of the continuous Sylvester equation (1) with B=A* and C Hermitian, where A* represents the conjugate transpose of the matrix A. This continuous Sylvester equation arises in several areas of applications. For more details about the practical backgrounds of this class of problems, we refer to [2–15] and the references therein.Before giving its numerical scheme, we rewrite the continuous Sylvester equation (1) in the mathematically equivalent system of linear equations
(2)Ax=c,
where A=I⊗A+BT⊗I; the vectors x and c contain the concatenated columns of the matrices X and C, respectively, with ⊗ being the Kronecker product symbol and BT representing the transpose of the matrix B. However, it is quite expensive and ill-conditioned to use the iteration method to solve this variation of the continuous Sylvester equation (1).There is a large number of numerical methods for solving the continuous Sylvester equation (1). The Bartels-Stewart and the Hessenberg-Schur methods [16, 17] are direct algorithms, which can only be applied to problems of reasonably small sizes. When the matrices A and B become large and sparse, iterative methods are usually employed for efficiently and accurately solving the continuous Sylvester equation (1), for instance, the Smith’s method [18], the alternating direction implicit (ADI) method [19–22], and others [23–26].Recently, Bai established the Hermitian and skew-Hermitian splitting (HSS) [4] iterative method for solving the continuous Sylvester equation (1), which is based on the Hermitian and skew-Hermitian splitting of the matrices A and B. This HSS iteration method is a matrix variant of the original HSS iteration method firstly proposed by Bai et al. [27] for solving systems of linear equations; see [28–39] for more detailed descriptions about the HSS iteration method and its variants.To further improve the convergence efficiency, in this paper we present a new generalized Hermitian and skew-Hermitian splitting (GHSS) method for solving the continuous Sylvester equation (1). It is a four-parameter iteration which enables the optimization of the iterative process, thereby achieving high efficiency and robustness. Similar approaches of using the parameterized acceleration technique in the algorithmic designs of the iterative methods can be seen in [34–39].In the remainder of this paper, a matrix sequence{Y(k)}k=0∞⊆ℂm×n is said to be convergent to a matrix Y∈ℂm×n if the corresponding vector sequence {y(k)}k=0∞⊆ℂmn is convergent to the corresponding vector y∈ℂmn, where the vectors y(k) and y contain the concatenated columns of the matrices Y(k) and Y, respectively. If {Y(k)}k=0∞ is convergent, then its convergence factor and convergence rate are defined as those of {y(k)}k=0∞, correspondingly. In addition, we use sp(V), ∥V∥2 and ∥V∥F to denote the spectrum, the spectral norm, and the Frobenius norm of the matrix V∈ℂm×m, respectively. Note that ∥·∥2 is also used to represent the 2-norm of a vector.The rest of this paper is organized as follows. In Section2, we present the GHSS method for solving the continuous Sylvester equation (1), in which we use four parameters instead of two parameters in the HSS method [4]. An exact parameter region of convergence for the method is strictly proved and a minimum value for the upper bound of the iterative spectrum is derived in Section 3. In Section 4, an inexact variant of the GHSS (IGHSS) iteration method is presented and its convergence property is studied. Numerical examples are given to illustrate the theoretical results and the effectiveness of the GHSS method in Section 5. Finally, we draw our conclusions.


## 2. The GHSS Method

Here and in the sequel, we useH(V):=(1/2)(V+V*) and S(V):=(1/2)(V-V*) to denote the Hermitian part and the skew-Hermitian part of the matrix V∈ℂm×n, respectively. Obviously, the matrix V naturally possesses the Hermitian and skew-Hermitian splitting (HSS):
(3)V=H(V)+S(V);
see [4, 27, 28].Similar to the HSS method [4], we obtain the following splitting of A and B:
(4)A=(α1I+H(A))-(α1I-S(A))=(β1I+S(A))-(β1I-H(A)),B=(α2I+H(B))-(α2I-S(B))=(β2I+S(B))-(β2I-H(B)),
where αj(j=1,2) are given nonnegative constants and βj(j=1,2) are given positive constants and I is the identity matrix of suitable dimension. Then the continuous Sylvester equation (1) can be equivalently reformulated as
(5)(α1I+H(A))X+X(α2I+H(B))=(α1I-S(A))X+X(α2I-S(B))+C,(β1I+S(A))X+X(β2I+S(B))=(β1I-H(A))X+X(β2I-H(B))+C.Under assumptions (i)–(iii), we observe that there is no common eigenvalue between the matricesα1I+H(A) and -(α2I+H(B)), as well as between the matrices β1I+S(A) and -(β2I+S(B)), so that the above two fixed-point matrix equations have unique solutions for all given right-hand side matrices. This leads to the following generalized Hermitian and skew-Hermitian splitting (GHSS) iteration method for solving the continuous Sylvester equation (1).Algorithm 1 (the GHSS iteration method).
Given an initial guessX(0)∈ℂm×n, compute X(k+1)∈ℂm×n for k=0,1,2,… using the following iteration scheme until {X(k)}k=0∞ satisfies the stopping criterion:
(6)(α1I+H(A))X(k+1/2)+X(k+1/2)(α2I+H(B))=(α1I-S(A))X(k)+X(k)(α2I-S(B))+C,(β1I+S(A))X(k+1)+X(k+1)(β2I+S(B))=(β1I-H(A))X(k+1/2)+X(k+1/2)(β2I-H(B))+C,
where αj(j=1,2) are given nonnegative constants and βj(j=1,2) are given positive constants and I is the identity matrix of suitable dimension.Remark 2.
The GHSS method has the same algorithmic structure as the HSS method [4], and thus two methods have the same computational cost in each iteration step. It is easy to see that the former reduces to the latter when α1=β1 and α2=β2.Remark 3.
WhenB is a zero matrix, and X(k) and C reduce to column vectors, the GHSS iteration method becomes the one for systems of linear equations; see [34–36]. In addition, when B=A* and C is Hermitian, it leads to an GHSS iteration method for the continuous Lyapunov equations.


## 3. Convergence Analysis of the GHSS Method

LetH(A), H(B), and S(A), S(B) be the Hermitian and the skew-Hermitian parts of the matrices A and B, respectively.Denote(7)λmax(H(A))=maxλj∈sp(H(A)){λj},μmax(H(B))=maxμk∈sp(H(B)){μk},λmin(H(A))=minλj∈sp(H(A)){λj},μmin(H(B))=minμk∈sp(H(B)){μk},ξmax(S(A))=maxiξj∈sp(S(A)){|ξj|},ζmax(S(B))=maxiζk∈sp(S(B)){|ζk|},ξmin(S(A))=miniξj∈sp(S(A)){|ξj|},ζmin(S(B))=miniζk∈sp(S(B)){|ζk|},
with i=-1 and
(8)Θmax=λmax(H(A))+μmax(H(B)),Υmax=ξmax(S(A))+ζmax(S(B)),Θmin=λmin(H(A))+μmin(H(B)),Υmin=ξmin(S(A))+ζmin(S(B)).In addition, denote byA=H+S, with
(9)H=H(A)=I⊗H(A)+H(B)T⊗I,S=S(A)=I⊗S(A)+S(B)T⊗I.
Obviously, Θmax, Υmax and Θmin, Υmin are the upper and the lower bounds of the eigenvalues of the matrices H and S, respectively.By making use of Theorems 2.2 and 2.5 in [35], we can obtain the following convergence theorem about the GHSS iteration method for solving the continuous Sylvester equation (1).Theorem 4.
Assume thatA∈ℂm×m and B∈ℂn×n are positive semi-definite matrices, and at least one of them is positive definite. Let αj(j=1,2) be nonnegative constants and βj(j=1,2) be positive constants. Denote by
(10)M(α,β)=(βI+S)-1(βI-H)(αI+H)-1(αI-S),(11)α=α1+α2,β=β1+β2.
Then the convergence factor of the GHSS iteration method (6) is given by the spectral radius ρ(M(α,β)) of the matrix M(α,β), which is bounded by
(12)σ(α,β):=maxΘmin≤Θ≤Θmax|β-Θ||α+Θ|·maxΥmin≤Υ≤Υmaxα2+Υ2β2+Υ2.
And, if the parameters α and β satisfy
(13)(α,β)∈⋃ℓ=14Ωℓ,
where
(14)Ω1={(α,β)∣α≤β<β*(α)},Ω2={(α,β)∣β≥max{α,β*(α)},ϕ1(α,β)>0},Ω3={(α,β)∣β*(α)≤β<α},Ω4={(α,β)∣β<min{α,β*(α)},ϕ2(α,β)>0},
with functions ϕ1(α,β), ϕ2(α,β) and β*(α) denoted by
(15)ϕ1(α,β)=(β-α)(Θmin2-Υmax2)+2αβΘmin+2Υmax2Θmin,ϕ2(α,β)=(β-α)(Θmax2-Υmin2)+2αβΘmax+2Υmin2Θmax,β*(α)=α(Θmax+Θmin)+2ΘmaxΘmin2α+Θmax+Θmin∈[Θmin,Θmax],
then σ(α,β)<1; that is, the GHSS iteration method (6) is convergent to the exact solution X⋆∈ℂm×n of the continuous Sylvester equation (1).Proof.
By making use of the Kronecker product, we can reformulate the GHSS iteration (6) as the following matrix-vector form:
(16)(I⊗(α1I+H(A))+(α2I+H(B))T⊗I)x(k+1/2)=(I⊗(α1I-S(A))+(α2I-S(B))T⊗I)x(k)+c,(I⊗(β1I+S(A))+(β2I+S(B))T⊗I)x(k+1)=(I⊗(β1I-H(A))+(β2I-H(B))T⊗I)x(k+1/2)+c,
which can be arranged equivalently as
(17)(αI+H)x(k+1/2)=(αI-S)x(k)+c,(βI+S)x(k+1)=(βI-H)x(k+1/2)+c.
Evidently, the iteration scheme (17) is the GHSS iteration method for solving the system of linear equations (2), with A=H+S; see [34, 35]. After concrete operations, the GHSS iteration (17) can also be expressed as a stationary iteration as follows:
(18)x(k+1)=M(α,β)x(k)+N(α,β)c,
where M(α,β) is the iteration matrix defined in (10), with H, S and α, β being given in (9) and (11), respectively, and
(19)N(α,β)=(α+β)(βI+S)-1(αI+H)-1.
We can easily verify thatH is a Hermitian matrix, S is a skew-Hermitian matrix, α is a nonnegative constant, and β is a positive constant. Moreover, when either A∈ℂm×m or B∈ℂn×n is positive definite, the matrix H is Hermitian positive definite. The spectral radius of the iteration matrix M(α,β) clearly satisfies
(20)ρ(M(α,β))=ρ((βI+S)-1(βI-H)(αI+H)-1(αI-S))≤∥(βI-H)(αI+H)-1∥2∥(αI-S)(βI+S)-1∥2=maxλj∈sp(H(A))μk∈sp(H(B))|β-λj-μk||α+λj+μk|·maxiξj∈sp(S(A))iζk∈sp(S(B))α2+(ξj+ζk)2β2+(ξj+ζk)2≤maxΘmin≤Θ≤Θmax|β-Θ||α+Θ|·maxΥmin≤Υ≤Υmaxα2+Υ2β2+Υ2;
then the bound for ρ(M(α,β)) is given by (12).
Hence, by making use of Theorem 2.2 in [35] we know that
(21)ρ(M(α,β))≤σ(α,β)<1,∀(α,β)∈⋃ℓ=14Ωℓ.
Therefore we obtain that if the parameters α and β satisfy the condition (13), the GHSS iteration method (17) converges to the exact solution x⋆∈ℂmn of the system of linear equations (2). This directly shows that the GHSS iteration method (6) is convergent to the exact solution X⋆∈ℂm×n of the continuous Sylvester equation (1) when α and β satisfy the condition (13), with the convergence factor ρ(M(α,β)) being bounded by σ(α,β). This completes the proof.Remark 5.
Whenα1=β1 and α2=β2, the GHSS method reduces to the HSS method, which is unconditionally convergent due to α=β.Theorem4 gives the convergence conditions of the GHSS iteration method for the continuous Sylvester equation (1) by analyzing the upper bound σ(α,β) of the spectral radius of the iteration matrix M(α,β). Since the optimal parameters α and β that minimize the spectral radius ρ(M(α,β)) are hardly obtained, we instead give the parameters α⋆ and β⋆, which minimize the upper bound σ(α,β) of the spectral radius ρ(M(α,β)), in the following corollary.Corollary 6.
The theoretical quasi-optimal parameters that minimize the upper boundσ(α,β) are given by
(22)(α⋆,β⋆)≡argminα,β{σ(α,β)}={(γ1,β*(γ1)),ΘmaxΘmin≤Υmin2,(γ2,β*(γ2)),Υmin2<ΘmaxΘmin<Υmax2,(γ3,β*(γ3)),ΘmaxΘmin≥Υmax2,
with
(23)γ1=Υmin2-ΘmaxΘmin+(Υmin2+Θmax2)(Υmin2+Θmin2)Θmax+Θmin,γ2=ΘmaxΘmin,γ3=Υmax2-ΘmaxΘmin+(Υmax2+Θmax2)(Υmax2+Θmin2)Θmax+Θmin,
and the corresponding upper bound of the convergence factor is given by
(24)σ(α⋆,β⋆)={σ(γ1),ΘmaxΘmin≤Υmin2,σ(γ2),Υmin2<ΘmaxΘmin<Υmax2,σ(γ3),ΘmaxΘmin≥Υmax2,
where
(25)σ(γ)≔σ(γ,β*(γ))={β*(γ)-Θminγ+Θmin·γ2+Υmin2β*(γ)2+Υmin2,γ>γ2,β*(γ)-Θminγ+Θmin·γ2+Υmax2β*(γ)2+Υmax2,γ≤γ2.Proof.
It is straightforward from Theorem 2.5 of [35].Remark 7.
We observe thatα⋆=β⋆=ΘmaxΘmin when Υmin2<ΘmaxΘmin<Υmax2, which means that GHSS with the theoretical quasi-optimal parameters reduces to HSS with the theoretical quasi-optimal parameter [4] in this case. In other cases, the GHSS iteration is superior to the HSS iteration when both of them use the theoretical quasi-optimal parameters. This phenomenon is also illustrated in the numerical results of Section 5.Remark 8.
The actual iteration parametersαi and βi(i=1,2) can be chosen as αi=αi^ and βi=βi^(i=1,2) such that α1^+α2^=α⋆ and β1^+β2^=β⋆. For example, we may take α1^=α2^=(1/2)α⋆ and β1^=β2^=(1/2)β⋆.


## 4. Inexact GHSS Iteration Methods

In the process of GHSS iteration (6), two subproblems need to be solved exactly. This is a tough task which is costly and even impractical in actual implementations. To further improve computational efficiency of the GHSS iteration, we develop an inexact GHSS (IGHSS) iteration, which solves the two sub-problems iteratively [18–24]. We write the IGHSS iteration scheme in the following algorithm for solving the continuous Sylvester equation (1).Algorithm 9 (the IGHSS iteration method).
Given an initial guessX(0)∈ℂm×n, then this algorithm leads to the solution of the continuous Sylvester equation (1):k=0
;while (not convergent)R(k)=C-AX(k)-X(k)B;approximately solve (α1I+H(A))Z(k)+Z(k)(α2I+H(B))=R(k) by employing an effective iteration method, such that the residual P(k)=R(k)-(α1I+H(A))Z(k)-Z(k)(α2I+H(B)) of the iteration satisfies ∥P(k)∥F≤εk∥R(k)∥F;X(k+1/2)=X(k)+Z(k);R(k+1/2)=C-AX(k+1/2)-X(k+1/2)B;approximately solve (β1I+S(A))Z(k+1/2)+Z(k+1/2)(β2I+S(B))=R(k+1/2) by employing an effective iteration method, such that the residual Q(k+1/2)=R(k+1/2)-(β1I+S(A))Z(k+1/2)-Z(k+1/2)(β2I+S(B)) of the iteration satisfies ∥Q(k+1/2)∥F≤ηk∥R(k+1/2)∥F;X(k+1)=X(k+1/2)+Z(k+1/2);k=k+1;end.Here,{εk} and {ηk} are prescribed tolerances used to control the accuracies of the inner iterations. We remark that when α1=β1 and α2=β2, the IGHSS method reduces the inexact HSS (IHSS) method [4].The convergence properties for the two-step iteration have been carefully studied in [27, 31]. By making use of Theorem 3.1 in [27], we can demonstrate the following convergence result about the above IGHSS iteration method.Theorem 10.
Let the conditions of Theorem4 be satisfied. If {X(k)}k=0∞⊆ℂm×n is an iteration sequence generated by the IGHSS iteration method and if X⋆∈ℂm×n is the exact solution of the continuous Sylvester equation (1), then it holds that
(26)∥X(k+1)-X⋆∥S≤(σ(α,β)+μθεk+θ(ρ+θνεk)ηk)·∥X(k)-X⋆∥S,k=0,1,2,…,
where the norm ∥·∥S is defined as
(27)∥Y∥S=∥(β1I+S(A))Y+Y(β2I+S(B))∥F
for any matrix Y∈ℂm×n, and the constants μ, θ, ρ, and ν are given by
(28)μ=∥(βI-H)(αI+H)-1∥2,θ=∥A(βI+S)-1∥2,ρ=∥(βI+S)(αI+H)-1(αI-S)(βI+S)-1∥2,ν=∥(βI+S)(αI+H)-1∥2,
with the matrices H and S being defined in (9) and the constants α and β being defined in (11). In particular, when
(29)σ(α,β)+μθεmax+θ(ρ+θνεmax)ηmax<1,
the iteration sequence {X(k)}k=0∞⊆ℂm×n converges to X⋆∈ℂm×n, where εmax=maxk{εk} and ηmax=maxk{ηk}.Proof.
By making use of the Kronecker product and the notations introduced in Theorem4, we can reformulate the above-described IGHSS iteration as the following matrix-vector form:
(30)(αI+H)z(k)=r(k),x(k+1/2)=x(k)+z(k),(βI+S)z(k+1/2)=r(k+1/2),x(k+1)=x(k+1/2)+z(k+1/2),
with r(k)=c-Ax(k) and r(k+1/2)=c-Ax(k+1/2), where z(k) is such that the residual
(31)p(k)=r(k)-(αI+H)z(k)
satisfies ∥p(k)∥2≤εk∥r(k)∥2, and z(k+1/2) is such that the residual
(32)q(k+1/2)=r(k+1/2)-(βI+S)z(k+1/2)
satisfies ∥q(k+1/2)∥2≤ηk∥r(k+1/2)∥2.
Evidently, the iteration scheme (30) is the inexact GHSS iteration method for solving the system of linear equations (2), with A=H+S; see [34, 36]. Hence, by making use of Theorem 3.1 in [27] we can obtain the estimate
(33)|∥x(k+1)-x⋆∥|≤(σ(α,β)+μθεk+θ(ρ+θνεk)ηk)·|∥x(k)-x⋆∥|,k=0,1,2,…,
where the norm |∥·∥| is defined as follows: for a vector y∈ℂmn, |∥y∥|=∥(βI+S)y∥2; and for a matrix Y∈ℂmn×mn, |∥Y∥|=∥(βI+S)Y(βI+S)-1∥2 is the correspondingly induced matrix norm. Note that
(34)|∥y∥|=∥(βI+S)y∥2=∥(β1I+S(A))Y+Y(β2I+S(B))∥F=∥Y∥S.
Hence, we can equivalently rewrite the estimate (33) as
(35)∥X(k+1)-X⋆∥S≤(σ(α,β)+μθεk+θ(ρ+θνεk)ηk)·∥X(k)-X⋆∥S,k=0,1,2,….
This proves the theorem.We remark that Theorem10 gives the choices of the tolerances {εk} and {ηk} for convergence. In general, Theorem 10 shows that in order to guarantee the convergence of the IGHSS iteration, it is not necessary for {εk} and {ηk} to approach to zero as k is increasing. All we need is that the condition (29) be satisfied. However, the theoretical optimal tolerances {εk} and {ηk} are difficult to be analyzed.


## 5. Numerical Results

In this section, we perform numerical tests to exhibit the superiority of GHSS and IGHSS to HSS and IHSS when they are used as solvers for solving the continuous Sylvester equation (1), in terms of iteration numbers (denoted as IT) and CPU times (in seconds, denoted as CPU).In our implementations, the initial guess is chosen to be the zero matrix, and the iteration is terminated once the current iterateX(k) satisfies
(36)∥C-AX(k)-X(k)B∥F∥C∥F≤10-6.
In addition, all sub-problems involved in each step of the HSS and GHSS iteration methods are solved exactly by the method in [16]. In HSS and IGHSS iteration methods, we set εk=ηk=0.01, k=0,1,2,…, and use the Smith’s method [18] as the inner iteration scheme.We consider the continuous Sylvester equation (1) with m=n and the matrices
(37)A=B=M+qN+100(n+1)2I,
where M,N∈ℝn×n are the tridiagonal matrices given by
(38)M=tridiag(-1,2,-1),N=tridiag(0.5,0,-0.5);
see also [4–6, 27, 40, 41].From [4] we known that the HSS iteration method considerably outperforms the SOR iteration method in both iteration step and CPU time, so here we just solve this continuous Sylvester equation by the GHSS and the HSS iteration methods and their inexact variants.In Table1, numerical results for GHSS and HSS with the experimental optimal iteration parameters are listed, while α1exp (with α2exp=α1exp), β1exp (with β2exp=β1exp), and αexp (with βexp=αexp) represent the experimentally found optimal values of the iteration parameters used for the GHSS and the HSS iterations, respectively.Table 1
Numerical results for GHSS and HSS with the experimental optimal iteration parameters.
q
n
Method
GHSS
HSS
α
1exp
β
1exp
IT
CPU
α
exp
IT
CPU
q
=
0.01
n
=
10
0.01
0.83
2
0.0006
1.66
12
0.0038
n
=
20
0.01
0.90
3
0.0018
0.80
23
0.0148
n
=
40
0.01
0.77
4
0.0141
0.42
43
0.1018
n
=
80
0.01
0.06
8
0.1038
0.24
85
0.9943
n
=
160
0.01
0.03
21
1.5376
0.14
169
11.605
q
=
0.1
n
=
10
0.03
1.00
4
0.0012
1.70
12
0.0036
n
=
20
0.01
0.95
4
0.0025
0.84
24
0.0151
n
=
40
0.01
0.61
6
0.0140
0.48
47
0.1102
n
=
80
0.01
0.21
11
0.1265
0.26
93
1.0787
n
=
160
0.01
0.10
24
1.6567
0.16
181
12.488
q
=
1
n
=
10
0.67
1.00
8
0.0025
1.82
13
0.0039
n
=
20
0.29
0.96
13
0.0081
0.98
23
0.0148
n
=
40
0.29
0.59
24
0.0563
0.62
35
0.0821
n
=
80
0.34
0.43
41
0.4750
0.46
50
0.5885
n
=
160
0.31
0.33
66
4.5142
0.34
70
4.7983
q
=
10
n
=
10
6.70
2.00
11
0.0034
1.88
12
0.0040
n
=
20
8.80
1.90
15
0.0097
1.16
21
0.0135
n
=
40
4.90
1.50
20
0.0477
0.68
37
0.0898
n
=
80
3.60
1.30
29
0.3395
0.90
57
0.6780
n
=
160
2.00
1.00
40
2.7869
0.68
83
5.8003
q
=
100
n
=
10
70.5
2.58
7
0.0021
1.72
12
0.0036
n
=
20
49.0
2.70
8
0.0052
0.90
20
0.0130
n
=
40
16.0
2.45
10
0.0243
0.48
36
0.0885
n
=
80
6.00
1.70
15
0.1782
0.28
64
0.7664
n
=
160
1.70
1.05
32
2.2333
0.16
110
7.6504In Table2, numerical results for GHSS and HSS with the theoretical quasi-optimal iteration parameters are listed, while α1⋆ (with α2⋆=α1⋆), β1⋆ (with β2⋆=β1⋆) and α⋆ (with β⋆=α⋆) represent the theoretical quasi-optimal iteration parameters used for the GHSS and the HSS iterations, respectively.Table 2
Numerical results for GHSS and HSS with the theoretical quasi-optimal iteration parameters.
q
n
Method
GHSS
HSS
α
1
*
β
1
*
IT
CPU
α
*
IT
CPU
q
=
0.01
n
=
10
0.0001
1.5236
2
0.0007
2.0752
15
0.0282
n
=
20
0.0002
0.4705
3
0.0019
1.0234
27
0.0261
n
=
40
0.0007
0.1294
4
0.0094
0.5147
50
0.1343
n
=
80
0.0028
0.0361
8
0.0938
0.2593
91
1.0901
n
=
160
0.0066
0.0151
21
1.4448
0.1303
169
11.704
q
=
0.1
n
=
10
0.0060
1.5263
4
0.0013
2.0752
15
0.0060
n
=
20
0.0201
0.4861
6
0.0038
1.0234
27
0.0263
n
=
40
0.0555
0.1793
15
0.0353
0.5147
49
0.1291
n
=
80
0.0867
0.1151
47
0.5458
0.2593
93
1.0875
n
=
160
0.0983
0.1017
161
11.069
0.1303
198
13.644
q
=
1
n
=
10
0.5322
1.7300
8
0.0026
2.0752
14
0.0045
n
=
20
0.9733
1.0046
22
0.0142
1.0234
23
0.0342
n
=
40
0.5147
0.5147
41
0.0967
0.5147
41
0.1106
n
=
80
0.2593
0.2593
81
0.9405
0.2593
81
0.9638
n
=
160
0.1303
0.1303
170
11.867
0.1303
170
11.872
q
=
10
n
=
10
2.0752
2.0752
12
0.0039
2.0752
12
0.0040
n
=
20
1.0234
1.0234
23
0.0259
1.0234
23
0.0598
n
=
40
0.5147
0.5147
44
0.1073
0.5147
44
0.1473
n
=
80
0.2593
0.2593
85
1.0074
0.2593
85
1.0354
n
=
160
0.1303
0.1303
169
11.913
0.1303
169
11.928
q
=
100
n
=
10
72.911
2.7778
7
0.0023
2.0752
12
0.0039
n
=
20
26.701
2.0916
9
0.0060
1.0234
20
0.0823
n
=
40
8.6843
1.6894
14
0.0342
0.5147
36
0.1035
n
=
80
3.0610
1.2284
24
0.2858
0.2593
66
0.8051
n
=
160
1.2364
0.7699
44
3.0805
0.1303
126
8.9284In Table3, numerical results for IGHSS and IHSS are listed; here we adopt the iteration parameters in Table 1 for convenience and not the experimental optimal parameters.Table 3
Numerical results for IGHSS and IHSS.
q
n
Method
IGHSS
IHSS
IT
CPU
IT
CPU
q
=
0.01
n
=
10
2
0.0006
11
0.0031
n
=
20
3
0.0015
21
0.0106
n
=
40
4
0.0120
38
0.0818
n
=
80
6
0.1005
75
0.8043
n
=
160
18
0.9076
130
6.9022
q
=
0.1
n
=
10
4
0.0010
11
0.0032
n
=
20
4
0.0019
21
0.0105
n
=
40
5
0.0090
41
0.0905
n
=
80
10
0.1065
79
0.9082
n
=
160
20
0.8568
156
7.0812
q
=
1
n
=
10
7
0.0021
12
0.0032
n
=
20
12
0.0075
20
0.0120
n
=
40
21
0.0462
31
0.0691
n
=
80
35
0.4036
45
0.5082
n
=
160
52
2.8140
60
3.5928
q
=
10
n
=
10
10
0.0029
11
0.0035
n
=
20
13
0.0081
20
0.0120
n
=
40
17
0.0401
34
0.0806
n
=
80
24
0.3059
50
0.5889
n
=
160
30
1.5861
72
4.5002
q
=
100
n
=
10
6
0.0019
11
0.0032
n
=
20
7
0.0045
18
0.0111
n
=
40
8
0.0192
32
0.0832
n
=
80
13
0.1579
55
0.6968
n
=
160
27
1.1335
98
5.8509From Tables1–3 we observe that GHSS and IGHSS methods performs better than HSS and IHSS methods in terms of iteration numbers and CPU times. Therefore, the GHSS and IGHSS methods proposed in this work are two powerful and attractive iterative approaches for solving large sparse continuous Sylvester equations.


## 6. Conclusions

As a strategy for accelerating convergence of iteration for solving a broad class of continuous Sylvester equations, we have proposed a four-parameter generalized HSS (GHSS) method. This is obviously a type of generalization of the classical HSS method [4]. When we take α1=β1 and α2=β2, we shall return to the HSS method. In our work we demonstrate that the iterative series produced by the GHSS method converge to the unique solution of the continuous Sylvester equation when the parameters satisfy some moderate conditions. The GHSS method takes HSS method as a special case. We also give a possible optimal upper bound for the iterative spectral radius. Moreover, to reduce the computational cost, an inexact variant of the GHSS (IGHSS) iteration method is developed and its convergence property is analyzed. Numerical results display that the new GHSS method and its inexact variant are typically more flexible than HSS and IHSS methods.



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*Source: 578102-2014-01-12.xml*