# Accelerated Circulant and Skew Circulant Splitting Methods for Hermitian Positive Definite Toeplitz Systems

Authors: N. Akhondi; F. Toutounian
Journal: Advances in Numerical Analysis
 (2012)

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## Abstract

We study the CSCS method for large Hermitian positive definite Toeplitz linear systems, which first appears in Ng's paper published in (Ng, 2003), and CSCS stands for circulant and skew circulant splitting of the coefficient matrixA. In this paper, we present a new iteration method for the numerical solution of Hermitian positive definite Toeplitz systems of linear equations. The method is a two-parameter generation of the CSCS method such that when the two parameters involved are equal, it coincides with the CSCS method. We discuss the convergence property and optimal parameters of this method. Finally, we extend our method to BTTB matrices. Numerical experiments are presented to show the effectiveness of our new method.


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## Body

## 1. Introduction

We consider iterative solution of the large system of linear equations(1.1)Ax=b, where A∈ℂn×n is Hermitian positive definite Toeplitz matrix and b,x∈ℂn. An n-by-n matrix A=(ai,j)i,j=1n is said to be Toeplitz if ai,j=ai-j; that is, A is constant along its diagonals. Toeplitz systems arise in a variety of applications, especially in signal processing and control theory. Many direct methods are proposed for solving Toeplitz linear systems. A straightforward application of Gaussian elimination will lead to an algorithm with O(n3) complexity. There are a number of fast Toeplitz solvers that decrease the complexity to O(n2) operations, see for instance [1–3]. Around 1980, superfast direct Toeplitz solvers of complexity O(nlog2n), such as the one by Ammar and Gragg [4], were also developed. Recent research on using the preconditioned conjugate gradient method as an iterative method for solving Toeplitz systems has brought much attention. One of the main important results of this methodology is that the PCG method has a computational complexity proportional to O(nlogn) for a large class of problem [5] and is therefore competitive with any direct method.In [6], an iterative method based on circulant and skew circulant splitting (CSCS) of the Toeplitz matrix was given. The authors have driven an upper bound of the contraction factor of the CSCS iteration which is dependent solely on the spectra of the circulant and the skew circulant matrices involved.In [7] the authors studied the HSS iteration method for large sparse non-Hermitian positive definite Toeplitz linear systems, which first appears in [8]. They used the HSS iteration method based on a special case of the HSS splitting, where the symmetric part H=(1/2)(A+AT) is a centrosymmetric matrix and skew-symmetric part S=(1/2)(A-AT) is a skew-centrosymmetric matrix for a given Toeplitz matrix and discussed the computational complexity of the HSS and IHSS methods.In this paper we present an efficient iterative method for the numerical solution of Hermitian positive definite Toeplitz systems of linear equations. The method is a two-parameter generation of the CSCS method such that when the two parameters involved are equal, it coincides with the CSCS method. We discuss the convergence property and optimal parameters of this method. Then we extend our method to block-Toeplitz-Toeplitz-block (BTTB) matrices.For convenience, some of the terminology used in this paper will be given. The symbolℂn×n will denote the set of all n×n complex matrices. Let A,B∈ℂn×n. We use the notation A≻0(A≽0) if  A  is Hermitian positive (semi-)definite. If A and B are both Hermitian, we write A≻B(A≽B) if and only if A-B≻0(A-B≽0). For a Hermitian positive definite matrix A, we define the ∥·∥A norm of a vector z∈ℂn as ∥z∥A=z*Az. Then the induced ∥·∥A norm of a matrix B∈ℂn×n is defined as ∥B∥A=∥A1/2BA-1/2∥A.The organization of this paper is as follows. In Section2, we present accelerated circulant and skew circulant splitting (ACSCS) method for Toeplitz systems. In Section 3, we study the convergence properties and analyze the convergence rate of ACSCS iteration and derive optimal parameters. The convergence results of ACSCS method for BTTB matrices are given in Section 4. Numerical experiments are presented in Section 5 to show the effectiveness of our new method. Finally some conclusions are given in Section 6.


## 2. Accelerated Circulant and Skew Circulant Splitting Method

Let us begin by supposing that the entriesai,j=ai-j of n-by-n Toeplitz matrix An(=A) are the Fourier coefficients of the real generating function (2.1)f(θ)=∑-∞∞ake-ikθ
defined on [-π,π]. Since f is a real-valued function, a-k=a¯k for all integers and An is a Hermitian matrix. For Hermitian Toeplitz matrix An we note that it can always be split as(2.2)An=Cn+Sn,
where
(2.3)Cn=12(a0a-1+an-1a-(n-1)+a1a1+a-(n-1)a0⋱⋱⋱⋱a-1+an-1a(n-1)+a-1a0),Sn=12(a0a-1-an-1a-(n-1)-a1a1-a-(n-1)a0⋱⋱⋱⋱a-1-an-1a(n-1)-a-1a0).ClearlyCn is Hermitian circulant matrix and Sn is Hermitian skew circulant matrix. The positive definiteness of Cn and Sn is given in the following theorem. Its proof is similar to that of  Theorem 2 in [9].Theorem 2.1.
Letf be a real-valued function in the Wiener class (∑-∞∞|ak|<∞) and satisfies the condition
(2.4)f(θ)=∑-∞∞ake-ikθ≥δ>0∀θ.
Then the circulant matrix Cn and the skew circulant matrix Sn, defining by the splitting An=Cn+Sn, are uniformly positive and bounded for sufficiently large n.The subscriptn of matrices is omitted hereafter whenever there is no confusion.Based on the splitting (2.2), Ng [6] presented the CSCS iteration method: given an initial guess x(0), for k=0,1,2,…, until x(k) converges, compute(2.5)(αI+C)x(k+1/2)=(αI-S)x(k)+b,(αI+S)x(k+1)=(αI-C)x(k+1/2)+b,
where α is a given positive constant. He has also proved that if the circulant and the skew circulant splitting matrices are positive definite, then the CSCS method converges to the unique solution of the system of linear equations. Moreover, he derived an upper bound of the contraction of the CSCS iteration which is dependent solely on the spectra of the circulant and the skew circulant matrices C and S, respectively.In this paper, based on the CSCS splitting, we present a different approach to solve (1.1) with the Hermitian positive definite coefficient matrix, called the Accelerated Circulant and Skew Circulant Splitting method, shortened to the ACSCS iteration. Let us describe it as follows.The ACSCS iteration method: given an initial guessx(0), for k=0,1,2,…, until x(k) converges, compute(2.6)(αI+C)x(k+1/2)=(αI-S)x(k)+b,(βI+S)x(k+1)=(βI-S)x(k+1/2)+b,
where α is a given nonnegative constant and β is given positive constant.The ACSCS iteration alternates between the circulant matrixC and the skew circulant matrix S. Theoretical analysis shows that if the coefficient matrix A is Hermitian positive definite the ACSCS iteration (2.6) can converge to the unique solution of linear system (1.1) with any given nonnegative α, if β is restricted in an appropriate region. And the upper bound of contraction factor of the ACSCS iteration is dependent on the choice of α, β, the spectra of the circulant matrix C, and the skew circulant matrix S. The two steps at each ACSCS iterate require exact solutions with the n×n matrices αI+C and βI+S. Since circulant matrices can be diagonalized by the discrete Fourier matrix F and skew circulant matrices can be diagonalized by the diagonal times discrete Fourier F̂ [10], that is, (2.7)C=F*Λ1F,S=F̂*Λ2F̂,
where Λ1 and Λ2 are diagonal matrices holding the eigenvalues of C and S, respectively, the exact solutions with circulant matrices and skew circulant matrices can be obtained by using fast Fourier transforms (FFTs). In particular, the number of operations required for each step of the ACSCS iteration method is O(nlogn).Noting that the roles of the matricesC and S in (2.6) can be reverse, we can first solve the system of linear equation with the βI+S and then solve the system of linear equation with coefficient matrix αI+C.


## 3. Convergence Analysis of the ACSCS Iteration

In this section we study the convergence rate of the ACSCS iteration and we suppose that the entriesai,j=ai-j of A are the Fourier coefficient of the real generating function f that satisfies the conditions of Theorem 2.1. So, for sufficiently large n, the matrices A, C, and S will be Hermitian positive definite. Let us denote the eigenvalues of C and S by λi,μi, i=1,…,n, and the minimum and maximum eigenvalues of C and S by λmin,λmax and μmin,μmax, respectively. Therefore, from Theorem 2.1, for sufficiently large n we have λmin>0 and μmin>0.We first note that the ACSCS iteration method can be generalized to the two-step splitting iteration framework, and the following lemma describes a general convergence criterion for a two-step splitting iteration.Lemma 3.1.
LetA∈ℂn×n, A=Mi-Ni(i=1,2) be two splitting of the matrix A, and x(0)∈ℂn be a given initial vector. If x(k) is a two-step iteration sequence defined by
(3.1)M1x(k+1/2)=N1x(k)+b,M2x(k+1)=N2x(k+1/2)+b,k=0,1,2,…, then
(3.2)xk+1=M2-1N2M1-1N1xk+M2-1(I+N2M1-1)b,k=0,1,2,….
Moreover, if the spectral radius ρ(M2-1N2M1-1N1)<1, then the iterative sequence x(k) converges to the unique solution x*∈ℂn of the system of linear equations (1.1) for all initial vectors x(0)∈ℂn.Applying this lemma to the ACSCS iteration, we obtain the following convergence property.Theorem 3.2.
LetA∈ℂ(n×n) be a Hermitian positive definite Toeplitz matrix, and let C,S be its Hermitian positive circulant and skew circulant parts, α be a nonnegative constant, and β be a positive constant. Then the iteration matrix M(α,β) of the ACSCS method is
(3.3)M(α,β)=(βI+S)-1(βI-C)(αI+C)-1(αI-S),
and its spectral radius ρ(M(α,β)) is bounded by
(3.4)δ(α,β)=max⁡i=1n|β-λiα+λi|max⁡i=1n|α-μiβ+μi|,
where λi,μi,  i=1,…,n are eigenvalues of C, S, respectively. And for any given parameter α if
(3.5)α-2μmin⁡<β<α+2λmin⁡,
then δ(α,β)<1, that is, the ACSCS iteration converges, where λmin,μmin are the minimum eigenvalue of C and S, respectively.Proof.
Setting(3.6)M1=αI+C,N1=αI-S,M2=βI+S,N2=βI-C
in Lemma 3.1. Since αI+C and βI+S are nonsingular for any nonnegative constant α and positive β, we get (3.3).
By similarity transformation, we have(3.7)ρ(M(α,β))=ρ((βI+S)-1(βI-C)(αI+C)-1(αI-S))=ρ((βI-C)(αI+C)-1(αI-S)(βI+S)-1)≤‖(βI-C)(αI+C)-1(αI-S)(βI+S)-1‖2≤‖(βI-C)(αI+C)-1‖2‖(αI-S)(βI+S)-1‖2=max⁡i=1n|β-λiα+λi|max⁡i=1n|α-μiβ+μi|.
Then the bound for ρ(M(α,β)) is given by (3.4).
Sinceα≥0, β>0, and β satisfies the relation (3.5), the following equalities hold:
(3.8)max⁡i=1n|β-λiα+λi|=max⁡⁡(|β-λmax⁡α+λmax⁡|,|β-λmin⁡α+λmin⁡|)<1,max⁡i=1n|α-μiβ+μi|=max⁡⁡(|α-μmax⁡β+μmax⁡|,|α-μmin⁡β+μmin⁡|)<1,
so δ(α,β)<1.Theorem3.2 mainly discusses the available β for a convergent ACSCS iteration for any given nonnegative α. It also shows that the choice of β is dependent on the minimum eigenvalue of the circulant matrix C and the skew circulant matrix S and the choice of α. Notice that (3.9)(α+2λmin⁡)-(α-2μmin⁡)=2(λmin⁡+μmin⁡)>0,
then we remark that for any α the available β exists. And if λmin and μmin are large, then the restriction put on β is loose. The bound on δ(α,β) of the convergence rate depends on the spectrum of C and S and the choice of α and β. Moreover, δ(α,β) is also an upper bound of the contraction of the ACSCS iteration.Moreover, from the proof of Theorem3.2 we can simplify the bound δ(α,β) as(3.10)δ(α,β)=max⁡⁡(|β-λmax⁡α+λmax⁡|,|β-λmin⁡α+λmin⁡|)×max⁡⁡(|α-μmax⁡β+μmax⁡|,|α-μmin⁡β+μmin⁡|).In the following lemma, we list some useful relations related to the minimum and maximum eigenvalues of matricesC and S, which are essential for us to obtain the optimal parameters α and β and to describe their properties.Lemma 3.3.
LetSλ=λmin+λmax, Sμ=μmin+μmax and Pλ=λminλmax, Pμ=μminμmax, then the following relations hold:
(3.11)μmax⁡(Sμ+Sλ)-(Pμ-Pλ)=(μmax⁡+λmin⁡)(μmax⁡+λmax⁡),(3.12)μmin⁡(Sμ+Sλ)-(Pμ-Pλ)=(μmin⁡+λmin⁡)(μmin⁡+λmax⁡),(3.13)λmax⁡(Sλ+Sμ)-(Pλ-Pμ)=(λmax⁡+μmin⁡)(λmax⁡+μmax⁡),(3.14)λmin⁡(Sλ+Sμ)-(Pλ-Pμ)=(λmin⁡+μmin⁡)(λmin⁡+μmax⁡),(3.15)(Pμ-Pλ)2+(Sλ+Sμ)(SμPλ+SλPμ)=(λmax⁡+μmin⁡)(μmax⁡+λmin⁡)×(λmax⁡+μmax⁡)(λmin⁡+μmin⁡).Proof.
The equalities follow from straightforward computation.Theorem 3.4.
LetA, C, S be the matrices defined in Theorem 3.2 and Sλ, Sμ, Pλ, Pμ be defined as Lemma 3.3. Then the optimal α*,β* should be
(3.16)α*=(Pμ-Pλ)+(Pμ-Pλ)2+(Sμ+Sλ)(SμPλ+SλPμ)Sμ+Sλ,(3.17)β*=(Pλ-Pμ)+(Pμ-Pλ)2+(Sμ+Sλ)(SμPλ+SλPμ)Sμ+Sλ,
and they satisfy the relations
(3.18)μmin⁡<α*<μmax⁡,(3.19)λmin⁡<β*<λmax⁡,(3.20)α*-2μmin⁡<β*<α*+2λmin⁡.
And the optimal bound is
(3.21)δ*(α*,β*)=(λmax⁡+μmin⁡)(μmax⁡+λmin⁡)/(λmax⁡+μmax⁡)(λmin⁡+μmin⁡)-1(λmax⁡+μmin⁡)(μmax⁡+λmin⁡)/(λmax⁡+μmax⁡)(λmin⁡+μmin⁡)+1.Proof.
From Theorem3.2 and (3.8) there exist a β*∈[λmin,λmax] and α*∈[μmin,μmax] such that
(3.22)max⁡i=1n|β-λiα+λi|={λmax⁡-βλmax⁡+α,β≤β*,β-λmin⁡α+λmin⁡,β≥β*,(3.23)max⁡i=1n|α-μiβ+μi|={μmax⁡-αμmax⁡+β,α≤α*,α-μmin⁡β+μmin⁡,α≥α*,
respectively. In order to minimize the bound in (3.10), the following equalities must hold:
(3.24)β-λmin⁡α+λmin⁡=λmax⁡-βλmax⁡+α,α-μmin⁡β+μmin⁡=μmax⁡-αμmax⁡+β.
By using Sλ=λmax+λmin, Pλ=λmaxλmin, Sμ=μmax+μmin, Pμ=μmaxμmin, the above equalities can be rewritten as
(3.25)2(αβ-Pλ)=(α-β)Sλ,2(αβ-Pμ)=(β-α)Sμ.
These relations imply that
(3.26)α-β=2(Pμ-Pλ)Sλ+Sμ,αβ=SμPλ+SλPμSλ+Sμ.
By putting β′=-β, the parameters α and β′ will be the roots of the quadratic polynomial
(3.27)x2-2(Pμ-Pλ)Sμ+Sλx-SμPλ+SλPμSμ+Sλ=0.
Solving this equation we get the parameters α* and β* given by (3.16) and (3.17), respectively. These parameters α* and β* can be considered as optimal parameters if they satisfy the relations (3.18)–(3.20).
From (3.12), (3.15) and (3.11), (3.15), we have
(3.28)μmin⁡(Sλ+Sμ)-(Pμ-Pλ)≤(Pμ-Pλ)2+(Sλ+Sμ)(SμPλ+SλPμ),(3.29)μmax⁡(Sλ+Sμ)-(Pμ-Pλ)≥(Pμ-Pλ)2+(Sλ+Sμ)(SμPλ+SλPμ),
respectively. From these inequalities, by the definition of α* and simple computation, we get μmin≤α*≤μmax. By similarity computation, we can also show that λmin≤β*≤λmax. So, the parameters α* and β* satisfy the relations (3.18) and (3.19).
Moreover, for the optimal parameterα* and β*, we have
(3.30)β*-α*-2λmin⁡=2[(Pλ-Pμ)-λmin⁡(Sμ+Sλ)(Sμ+Sλ)]=-2(λmin⁡+μmin⁡)(λmin⁡+μmax⁡)(Sμ+Sλ)(from(3.14))<0.
By similarity computation, we obtain β*-α*-2μmin>0. So, the parameters α* and β* satisfy the relation (3.20).
Finally, by denoting(3.31)Δ=(Pμ-Pλ)2+(Sλ+Sμ)(SμPλ+SλPμ),
and substituting α* and β* in (3.10) and using the relations (3.11)–(3.15), we obtain the optimal bound
(3.32)δ*(α*,β*)=α*-μmin⁡β*+μmin⁡×β*-λmin⁡α*+λmin⁡=(Pμ-Pλ)+Δ-μmin⁡(Sμ+Sλ)(Pλ-Pμ)+Δ+μmin⁡(Sμ+Sλ)×(Pλ-Pμ)+Δ-λmin⁡(Sμ+Sλ)(Pμ-Pλ)+Δ+λmin⁡(Sμ+Sλ)=Δ-(μmin⁡+λmin⁡)(μmin⁡+λmax⁡)Δ+(μmin⁡+λmin⁡)(μmin⁡+λmax⁡)×Δ-(λmin⁡+μmin⁡)(λmin⁡+μmax⁡)Δ+(λmin⁡+μmin⁡)(λmin⁡+μmax⁡)=(μmax⁡+λmin⁡)(λmax⁡+μmin⁡)-Δ(μmax⁡+λmin⁡)(λmax⁡+μmin⁡)+Δ=(λmax⁡+μmin⁡)(μmax⁡+λmin⁡)/(λmax⁡+μmax⁡)(λmin⁡+μmin⁡)-1(λmax⁡+μmin⁡)(μmax⁡+λmin⁡)/(λmax⁡+μmax⁡)(λmin⁡+μmin⁡)+1.Remark 3.5.
We remark that if the eigenvalues of matricesC and D contain in Ω=[γmin,γmax], and we estimate δ(α,β), as [6], by
(3.33)δ¯(α,β)=max⁡γ∈Ω(|β-γα+γ|max⁡γ∈Ω|α-γβ+γ|),
then by taking β=α, we obtain
(3.34)α*=γmin⁡γmax⁡,(3.35)δ¯(α*)=γmin⁡+γmax⁡-2γmin⁡γmax⁡γmin⁡+γmax⁡+2γmin⁡γmax⁡,
which are the same as those given in [6] for Hermitian positive definite matrix A.


## 4. ACSCS Iteration Method for the BTTB Matrices

In this section we extend our method to block-Toeplitz-Toeplitz-block (BTTB) matrices of the form(4.1)A=(A0A1⋯Am-1A1A0⋯Am-2⋮⋱⋮Am-1Am-2⋯A0)
with(4.2)Aj=(aj,0aj,1⋯aj,n-1aj,1aj,0⋯aj,n-2⋮⋱⋮aj,n-1aj,n-2⋯aj,0).
Similar to the Toeplitz matrix, the BTTB matrix A possesses a splitting [11]:(4.3)A=Cc+Cs+Sc+Ss,
where Cc is a block-circulant-circulant-block (BCCB) matrix, Cs is a block-circulant-skew-circulant-block (BCSB) matrix, Sc is a block-skew-circulant-circulant-block (BSCB) matrix, and Ss is a block-skew-circulant-skew-circulant block (BSSB) matrix. We note that the matrices Cc, Cs, Sc, and Ss can be diagonalized by F⊗F, F⊗F̂, F̂⊗F, and F̂⊗F̂, respectively. Therefore, the systems of linear equations with coefficient matrices (α1,1I+Cc), (α1,2I+Cs), (α2,1I+Sc), and (α2,2I+Ss), where αi,j for i,j=1,2 are positive constants, can be solved efficiently using FFTs. The total number of operations required for each step of the method is O(nmlog(nm)) where nm is the size of the BTTB matrix A. Based on the splitting of A given in (4.3), the ACSCS iteration is as follows.The ACSCS iteration method for BTTB matrix: given an initial guessx(0), for k=0,1,2,…, until x(k) converges, compute(4.4)(α1,1I+Cc)x(k+1/4)=(α1,1I-Cs-Sc-Ss)x(k)+b,(α1,2I+Cs)x(k+2/4)=(α1,2I-Cc-Sc-Ss)x(k+1/4)+b,(α2,1I+Sc)x(k+3/4)=(α2,2I-Ss-Cc-Cs)x(k+2/4)+b,(α2,2I+Ss)x(k+1)=(α2,2I-Sc-Cc-Cs)x(k+3/4)+b,
where αi,j, i,j=1,2 are given positive constants.In the sequel, we need the following definition and results.Definition 4.1 (see [12]).
A splittingA=M-N is called P-regular if MH+N is Hermitian positive definite.Theorem 4.2 (see [13]).
LetA be Hermitian positive definite. Then A=M-N is a P-regular splitting if and only if ∥M-1N∥A<1.Lemma 4.3 (see [14]).
SupposeA,B∈ℂn×n be two Hermitian matrices, then
(4.5)λmax⁡(A+B)≤λmax⁡(A)+λmax⁡(B),λmin⁡(A+B)≥λmin⁡(A)+λmin⁡(B),
where λmin(X) and λmax(X) denote the minimum and the maximum eigenvalues of matrix X, respectively.Now we give the main results as follows.Theorem 4.4.
LetA be a Hermitian positive definite BTTB matrix, and Cc, Cs, Sc, and Ss be its BCCB, BCSB, BSCB, and BSSB parts, and αi,j, i,j=1,2 be positive constants. Then the iteration matrix M of the ACSCS method for BTTB matrices is
(4.6)M=(α2,2I+Ss)-1(α2,2I-Sc-Cc-Cs)(α2,1I+Sc)-1(α2,1I-Ss-Cc-Cs)×(α1,2I+Cs)-1(α1,2I-Cc-Sc-Ss)(α1,1I+Cc)-1(α1,1I-Cs-Sc-Ss).
And if
(4.7)α1,1>λmax⁡(Cs)+λmax⁡(Sc)+λmax⁡(Ss)-λmin⁡(Cc)2(≡α̃1,1),α1,2>λmax⁡(Cc)+λmax⁡(Sc)+λmax⁡(Ss)-λmin⁡(Cs)2(≡α̃1,2),α2,1>λmax⁡(Ss)+λmax⁡(Cc)+λmax⁡(Cs)-λmin⁡(Sc)2(≡α̃2,1),α2,2>λmax⁡(Sc)+λmax⁡(Cc)+λmax⁡(Cs)-λmin⁡(Ss)2(≡α̃2,2),
then the spectral radius ρ(M)<1, and the ACSCS iteration converges to the unique solution x*∈ℂn of the system of linear equations (1.1) for all initial vectors x(0)∈ℂn.Proof.
By the definitions of BCCB, BCSB, BSCB, and BSSB parts ofA, the matrices Cc, Cs, Sc, and Ss are Hermitian. Let us consider the Hermitian matrices
(4.8)M1=α1,1I+Cc,N1=α1,1I-Cs-Sc-Ss,M2=α1,2I+Cs,N2=α1,2I-Cc-Sc-Ss,M3=α2,1I+Sc,N3=α2,1I-Ss-Cc-Cs,M4=α2,2I+Ss,N4=α2,2I-Sc-Cc-Cs.
Since A is Hermitian positive definite, it follows that
(4.9)Mi-Ni≻0,fori=1,2,3,4.
By the assumptions (4.7) and Lemma 4.3, we have also
(4.10)Mi+Ni≻0,fori=1,2,3,4.
The relations (4.9) and (4.10) imply that Mi≻0, for i=1,2,3,4. So, the matrices Mi≻0, for i=1,2,3,4, are nonsingular and we get (4.6). In addition, the splittings A=Mi-Ni,i=1,2,3,4 are P-regular and by Theorem 4.2, we have
(4.11)‖Mi-1Ni‖A<1,fori=1,2,3,4.
Finally, by using (4.11), we can obtain
(4.12)‖M‖A≤‖M1-1N1‖A‖M2-1N2‖A‖M3-1N3‖A‖M4-1N4‖A<1,
which completes the proof.


## 5. Numerical Experiments

In this section, we compare the ACSCS method with CSCS and CG methods for 1D and 2D Toeplitz problems. We used the vector of all ones for the right-hand side vectorb. All tests are started from the zero vector, performed in MATLAB 7.6 with double precision, and terminated when (5.1)‖r(k)‖2‖r(0)‖2≤10-7,
or when the number of iterations is over 1000. This case is denoted by the symbol “−”. Here r(k) is the residual vector of the system of linear equation (1.1) at the current iterate x(k), and r(0) is the initial one.For 1D Toeplitz problems (Examples5.1–5.3), our comparisons are done for the number of iterations of the CG, CSCS, and ACSCS methods (denoted by “IT”) and the elapsed CPU time (denoted by “CPU”). All numerical results are performed for n=16,32,64,128,256,512,1024. The corresponding numerical results are listed in Tables 1–4. In these tables λmin and μmin represent the minimum eigenvalue of matrices C and S, respectively. Note that the CPU time in these tables does not account those for computing the iteration parameters. For ACSCS method, α* and β* are computed by (3.16) and (3.17), respectively. And for CSCS method, α* is computed by (3.34).Table 1
Numerical results of Example5.1.
nλminμminITCPUCGCSCSACSCSCGCSCSACSCS160.41830.5825835370.00770.01010.0095320.48010.51992039390.00850.01060.0100640.49510.50493740390.00980.01080.01061280.49880.50125540400.01080.01150.01282560.49970.50036740400.01380.01420.01445120.49910.50017040400.01890.02030.020510240.50000.50007140400.02830.03160.0292Table 2
Numerical results of Example5.2.
nλminμminITCPUCGCSCSACSCSCGCSCSACSCS160.44780.417712880.00740.00790.0079320.44190.424715990.00760.00870.0084640.43770.42921710100.00790.00900.00911280.43550.43151911110.00830.00940.00932560.43440.43252012120.00950.01060.01035120.43390.43302113130.01010.01110.011110240.43370.43332214140.01480.01610.0150Table 3
Numerical results of Example5.3 for β=10, γ=0.5.
nλminμminITCPUCGCSCSACSCSCGCSCSACSCS161.06210.1280820100.00720.00950.0080320.7746−0.025116—130.0077—0.0085640.6285−0.100523—150.0082—0.00901280.5169−0.137928—180.0096—0.00962560.44100.14853220150.01070.01170.01005120.38410.12073423160.01280.01390.012510240.34440.10673524170.01730.02450.0185Table 4
Numerical results of Example5.3 for β=10, γ=0.1.
nλminμminITCPUCGCSCSACSCSCGCSCSACSCS160.8963−0.07718—120.0077—0.0082320.5967−0.236716—180.0082—0.0090640.44440.11942622160.00900.00920.00931280.32820.002536—200.0102—0.01052560.24900.047347—270.0123—0.01305120.1898−0.001259—290.0170—0.017110240.14840.012568—310.0279—0.0297Example 5.1 (see [10]).
In this exampleAn is symmetric positive definite Toeplitz matrix,
(5.2)ak={4π2(-1)kk2-24(-1)kk4,k≠0,1+π55,k=0
with generating function f(θ)=θ4+1, θ∈[-π,π]. Numerical results for this example are listed in Table 1.Example 5.2 (see [15]).
LetAn be a Hermitian positive definite Toeplitz matrix,
(5.3)ak={1+-1(1+k)1.1,k>0,2,k=0,a¯-k,k<0.
The associated generating function is f(θ)=2∑k=0∞(sin(kθ)+cos(kθ))/(1+k)1.1,  θ∈[0,2π]. Numerical results of this example are presented in Table 2.Example 5.3 (see [10]).
In this exampleAn is Hermitian positive definite Toeplitz matrix,
(5.4)ak={--1(β-γ)(1+(-1)k),2πkk>0,(β+γ),k=0,a¯-k,k<0
with generating function
(5.5)f{β,γ}(θ)={β-γπθ+β,-π≤θ<0,β-γπθ+γ,0<θ≤π,
where β and γ are the maximum and minimum values of f{β,γ} on [-π,π], respectively. In Tables 3 and 4, numerical results are reported for β=10, γ=0.5 and β=10, γ=0.1, respectively.In the following, we summarize the observation from Tables1–4. In all cases, in terms of CPU time needed for convergence, the ACSCS converges at the same rate that the CG method converges. However, the number of ACSCS iterations is less than that of CG iterations required for convergence. The convergence behavior of ACSCS method, in terms of the number of iterations and CPU time needed for convergence, is similar to that of CSCS method when λmin and μmin are positive and not too small (all the cases in Tables 1 and 2). Moreover, we observe that, when λmin and μmin are too small or negative (the cases n=32,64,128 in Table 3 and the cases n≠64 in Table 4), the ACSCS method converges at the same rate that the CG converges, but the CSCS method does not converge. These results imply that the computational efficiency of the ACSCS iteration method is similar to that of the CG method and is higher than that of the CSCS iterations.For 2D Toeplitz problems, we tested three problems of the form given in (4.1) with the diagonals of the blocks Aj. The diagonals of Aj are given by the generating sequences (see [10])(a)
aj,i=1/(j+1)(|i|+1)1+.1(j+1), j≥0,i=0,±1,±2,…,(b)
aj,i=1/(j+1)1.1(|i|+1)1+.1(j+1), j≥0,i=0,±1,±2,…,(c)
aj,i=1/(j+1)2.1+(|i|+1)2.1, j≥0,i=0,±1,±2,….The generating sequences  (b) and  (c) are absolutely summable while  (a) is not. Our comparisons are done for the number of iterations of the CG, CSCS, and ACSCS methods (denoted by “IT”). All numerical results are performed forn=16,32,64,128. The corresponding numerical results are listed in Table 5. For ACSCS method, parameters αi,j=α̃i,j, i,j=1,2 are used. For CSCS method, we used α=maxi,j=12{αi,j}. Table 5 shows that, in all cases, the number of ACSCS iterations required for convergence is less than that of CSCS method and more than that of CG method. We mention that the relations (4.7) are sufficient conditions for convergence of ACSCS iteration for BTTB matrices. The numerical experiments show that the convergence behavior of ACSCS method, in terms of the number of iterations needed for convergence, is better than that of CG and CSCS methods if one of the parameters αi,j, i,j=1,2 is chosen less than the corresponding lower bound α̃i,j given in Theorem 4.2. Table 6 presents the results which are obtained for the ACSCS method with α1,1=0.5(<α̃1,1), α1,2=α̃1,2, α2,1=α̃2,1, and α2,2=α̃2,2. Table 7 presents the results obtained for CSCS method with the optimal parameter α, obtained computationally by trial and error. As we observe, from Tables 5–7, the number of ACSCS iterations required for convergence is less than that of CG and CSCS methods.Table 5
Numerical results of 2D examples.
nSequence  (a)Sequence  (b)Sequence (c)CGACSCSCSCSCGACSCSCSCSCGACSCSCSCS8153542153036101719162850572743491625283237606535515623343764456468415458304144128496366465456374649Table 6
Numerical results of 2D examples for ACSCS method withα1,1=0.5(<α̃1,1).
nSequence  (a)Sequence  (b)Sequence  (c)ITITIT8151413161515143217171564202016128222217Table 7
Numerical results of 2D examples for CSCS method with optimalα.
nSequence  (a)Sequence  (b)Sequence  (c)α*ITα*ITα*IT82.48262.31231.1815163.75353.53311.7920325.14424.65362.4125646.33445.70393.17301287.39446.74393.9334These results imply that the computational efficiency of the ACSCS iteration method is similar to that of the CG method and is higher than that of the CSCS iterations.


## 6. Conclusion

In this paper, a new iteration method for the numerical solution of Hermitian positive definite Toeplitz systems of linear equations has been described. This method, which called ACSCS method, is a two- (four-) parameter generation of the CSCS of Ng for 1D (2D) problems and is based on the circulant and the skew circulant splitting of the Toeplitz matrix. We theoretically studied the convergence properties of the ACSCS method. Moreover, the contraction factor of the ACSCS iteration and its optimal parameters are derived. Theoretical considerations and numerical examples indicate that the splitting method is extremely effective when the generation function is positive. Numerical results also showed that the computational efficiency of the ACSCS iteration method is similar to that of the CG method and is higher than that of the CSCS iteration method.



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*Source: 973407-2011-12-11.xml*