A relaxed block splitting preconditioner for complex symmetric indefinite linear systems

Yunying Huang; Guoliang Chen

doi:10.1515/math-2018-0051

Artikel Open Access

A relaxed block splitting preconditioner for complex symmetric indefinite linear systems

Yunying Huang und Guoliang Chen

Veröffentlicht/Copyright: 7. Juni 2018

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Abstract

In this paper, we propose a relaxed block splitting preconditioner for a class of complex symmetric indefinite linear systems to accelerate the convergence rate of the Krylov subspace iteration method and the relaxed preconditioner is much closer to the original block two-by-two coefficient matrix. We study the spectral properties and the eigenvector distributions of the corresponding preconditioned matrix. In addition, the degree of the minimal polynomial of the preconditioned matrix is also derived. Finally, some numerical experiments are presented to illustrate the effectiveness of the relaxed splitting preconditioner.

Keywords: Complex symmetric linear system; Preconditioner; Krylov subspace method; GMRES

MSC 2010: 65F10; 75F50

1 Introduction

Consider the following large sparse nonsingular complex symmetric linear systems

Au=b, A∈Cn×n and u, b∈Cn,(1)

where A = W + iT, W, T ∈ ℝ^n×n are symmetric matrices and i = −1 denotes the imaginary unit. Such complex linear system (1) arise in many areas of scientific computing and engineering applications, such as FFT-based solution of certain time-dependent PDEs [1], distributed control problems [2], diffuse optical tomography [3], PDE-constrained optimization problems [4], quantum mechanics [5] and so on; see [6,7,8] and references therein for other applications.

Let u = x + iy and b = f + ig with x, y, f, g ∈ ℝⁿ, then the complex linear system (1) can be equivalently rewritten as the real block two-by-two linear system as follows

A~u=W−TTWxy=fg=b,(2)

or the following two-by-two block real equivalent formulation

Au=T−WWTx−y=gf=b.(3)

It can be seen that the linear systems (2) and (3) both avoid using complex arithmetic, but the coefficient matrix in (2) and (3) have become doubled in size. Because it is more attractive to use iterative methods rather than direct methods for solving (1) or (2), then many efficient iterative methods as well as their numerical properties have been studied in the literature, see [9,10]. When both W and T are symmetric positive semi-definite with at least one of them being positive definite, some efficient iterative methods have been proposed. Recently, based on the Hermitian and skew-Hermitian splitting (HSS) method [11], Bai et al. [12] developed the modified HSS (MHSS) iteration method to solve the complex linear system (1), which is convergent for any positive constant α. To accelerate the convergence rate of the MHSS iteration method, Bai et al. [13] introduced the preconditioned MHSS (PMHSS) which is unconditionally convergent and shows h-independent convergence behavior. In [14], Bai further analyzed algebraic and convergence properties of the PMHSS iteration method for solving complex linear systems. Moreover, there are also some other effective iterative methods, such as the lopsided PMHSS (LPMHSS) iteration method, the accelerated PMHSS (APMHSS) iteration method, the preconditioned generalized SOR (PGSOR) iterative method, the skew-normal splitting (SNS) method, the double-step scale splitting (DSS) iteration method [15,16,17,18,19], and so on.

If the matrices W and T are symmetric positive semi-definite, some efficient preconditioners have been presented. Bai [20] proposed the rotated block triangular (RBT) preconditioners which is based on the PMHSS preconditioning matrix [21] for the linear system (2). Lang and Ren [22] established the inexact RBT (IRBT) preconditioners. For solving the linear system (2), Liang and Zhang [23] developed the symmetric SOR (SSOR) method.

However, when W ∈ ℝ^n×n is symmetric indefinite, T ∈ ℝ^n×n is symmetric positive definite, the matrices αI + W, αV + W and αW + T² are indefinite or singular which may lead to the slow convergence speeds of the MHSS method, the PMHSS method and the SNS method. To overcome these problems, in [24], the Hermitian normal splitting (HNS) method and its variant simplified HNS (SHNS) method have been introduced by Wu for solving the complex symmetric linear system (1). Zhang and Dai [25] presented a preconditioned SHNS (PSHNS) iteration method and constructed a new preconditioner. In this paper, we solve the linear system (3) with W being symmetric indefinite and T being symmetric positive definite. In order to solve real formulation (3), Zhang and Dai [26] established a new block preconditioner and also analyzed the spectral properties of the corresponding preconditioned matrix. An improved block (IB) splitting preconditioner was proposed by Zhang et al. in [27] and they gave the convergence property of the corresponding iteration method under suitable conditions.

In this paper, by using the relaxing technique, we construct a relaxed block splitting preconditioner for the real block two-by-two linear system (3). The relaxed splitting preconditioner is much closer to the original coefficient matrix than the block preconditioners in [26]. We analyze the spectral properties of the corresponding preconditioned matrix and show that the corresponding preconditioned matrix has an eigenvalue 1 with algebraic multiplicity at least n. The structure of the eigenvectors and the dimension of the Krylov subspace for the preconditioned matrix are also derived.

The organization of the paper is as follows. In Section 2, a relaxed block splitting preconditioner is presented. In Section 3, we study the distribution of eigenvalues, the form of the eigenvectors and the dimension of the Krylov subspace of the corresponding preconditioned matrix. In Section 4, some numerical examples are presented to compare the new preconditioner with the PSHNS preconditioner and the HSS preconditioner. Finally, we give some brief concluding remarks in Section 5.

2 A new relaxed splitting preconditioner

In this section, by employing the relaxation techniques, we establish a relaxed block splitting preconditioner for solving the linear system (3).

Pan et al. [28] employed the positive-definite and skew-Hermitian splitting (PSS) iteration method in [29] to develop a PSS preconditioner for saddle point problems. For the linear system (3), the coefficient matrix 𝓐 can be split as follows

A=J+K,(4)

where J=T000andK=0−WWT. Recently, based on the PSS preconditioner, Zhang and Dai presented a preconditioner in [26] as follows

P1=1α(αI+K)(αI+J)=1ααI−WWαI+TαI+T00αI=αI+T−WW(I+1αT)αI+T.(5)

Then the difference between the preconditioner 𝓟₁ and the coefficient matrix 𝓐 is given as follows

R1=P1−A=αI01αWTαI.(6)

A general criterion for an efficient preconditioner is that it should be as close as possible to the coefficient matrix 𝓐 which makes the preconditioned matrix have a clustered spectrum, and hope that it is easy to implement. Inspired by the idea of the relaxation technique in [30,31,32,33], we give a relaxed splitting preconditioner which is defined as follows

P2=1ααI−WWTT00αI=T−W1αWTT.(7)

From (3) and (7), we get the difference between the preconditioner 𝓟₂ and the coefficient matrix 𝓐

R2=P2−A=0 01αWT−W 0.(8)

It is easy to see that the (2, 1)-block matrix in (8) is different from that in (6) and the (1, 1)-block and (2, 2)-block matrix in (8) now vanish, which indicates that the preconditioner 𝓟₂ is much closer to the coefficient matrix 𝓐 than the preconditioner 𝓟₁.

From (8), we know that the preconditioner 𝓟₂ can be established by the following splitting of the coefficient matrix 𝓐

A=P2−R2=T−W1αWTT−0 01αWT−W 0.(9)

In the following section, we will analyze the spectral properties of the preconditioned matrix P2−1 𝓐.

3 Spectral analysis of the preconditioned matrix

In this section, we mainly study the spectral properties of the preconditioned matrix P2−1 𝓐 and give an upper bound of the dimension of the Krylov subspace for this preconditioned matrix. Therefore, firstly we should get the inverse of the matrix 𝓟₂ and the explicit form of the matrix P2−1 will be given in the following lemma. The proof of this lemma consists of straightforward calculations and is omitted here.

Lemma 3.1

Let

P=αI−WWTandP=T00αI.

Here 𝓟 has the following block-triangular factorization

P=I01αWIαI 00 T+1αW2I−1αW0I.

Then we obtain

P2−1=αP^−1P−1=T−1−1αT−1W(T+1αW2)−1W T−1W(T+1αW2)−1−1α(T+1αW2)−1W(T+1αW2)−1.(10)

In the following, we will analyze the eigenvalue distributions of the preconditioned matrix P2−1 𝓐.

Theorem 3.2

Assume that the coefficient matrix 𝓐 is nonsingular, W ∈ ℝ^n×nis symmetric indefinite andT ∈ ℝ^n×nis symmetric positive definite. Letαbe a real positive constant andU = WT⁻¹W. The preconditioner 𝓟₂is defined as in (7). Then the preconditioned matrixP2−1 𝓐 has eigenvalues at 1 with multiplicity at leastnand the remaining eigenvalues are λ_i, i = 1, 2, ⋯, n, where λ_i (i = 1, 2, ⋯, n) are the eigenvalues of the matrixαT⁻¹(αI + U)⁻¹(T + U).

Proof

From (9) and Lemma 3.1, we have

P2−1A=I−P2−1R2=I−T−1W(T+1αW2)−1W(1αT−I) 0(T+1αW2)−1W(1αT−I) 0=I−T−1W(T+1αW2)−1W(1αT−I) 0−(T+1αW2)−1W(1αT−I) I.

It follows from U = WT⁻¹W that

I−T−1W(T+1αW2)−1W(1αT−I)=T−1(T−W(T+1αW2)−1W(1αT−I))=T−1(T−W(W(W−1TW−1+1αI)W)−1W(1αT−I))=T−1(T−(1αI+W−1TW−1)−1(1αT−I))=T−1(T−(1αU+I)−1U(1αT−I))=αT−1(αI+U)−1(T+U).

Hence, we get

P2−1A=αT−1(αI+U)−1(T+U) 0−(T+1αW2)−1W(1αT−I) I.(11)

Then, it is obvious from (11) that the results are proved. □

The detail implementation of the preconditioning process will be described as follows. Applying the preconditioner 𝓟₂ within a Krylov subspace method, the following system should be solved at each step

P2z=1ααI−WWTT00αIz=r,(12)

where z=[z1T, z2T]T and r=[r1T, r2T]T are the current and generalized residual vectors, respectively, z₁, z₂, r₁, r₂ ∈ ℝⁿ. Based on Lemma 3.1 and (12), we have

z1z2=αT−1001αII1αW0I1αI00(T+1αW2)−1I0−1αWIr1r2.(13)

Then, we can derive the implementing process of the preconditioner 𝓟₂ as follows.

Algorithm 3.3

(The preconditioner 𝓟₂). For a given vectorr=[r1T, r2T]T , the vectorz=[z1T, z2T]Tcan be computed by (13) from the following steps:

Computet1=r2−1αWr1;
Solve(T+1αW2)z2=t1;
Computet₂ = r₁ + Wz₂;
Solve Tz₁ = t₂;
Set the generalized residual vectorz=[z1T, z2T]T

It can be seen from Algorithm 3.3 that we need to solve two sub-linear systems with coefficient matrices T + 1αW² and T at each iteration. Based on the assumptions of the matrices W and T in Section 1, we know that the two matrices T + 1αW² and T are both symmetric positive definite. Hence, we can solve the two sub-linear systems (T + 1αW²)z₂ = t₁ and Tz₁ = t₂ by applying the sparse Cholesky decomposition, the conjugate gradient (CG) method or the preconditioned CG method. In order to prove the effectiveness of the preconditioner 𝓟₂, we compare it with the PSHNS preconditioner and the HSS preconditioner which are given as follows:

PPSHNS=12α(αW+iI)(αT+I) and PHSS=1αI+T00αI+TαI−WWαI.

Then, the implementing process about the PSHNS preconditioner and the HSS preconditioner with GMRES will be described as follows, respectively.

Algorithm 3.4

(The PSHNS Preconditioner). For a given vector r, the vector z can be computed from the following steps:

Solve (αV + iW)d = 2αr;
Solve (αT + WV^–1W)z = Wd.

Algorithm 3.5

(The HSS Preconditioner). For a given vectorr=[r1T, r2T]T,thevectorz=[z1T, z2T]Tcan be computed from the following steps:

Solve (αI + T)ν₁ = r₁;
Solve (αI + T)ν₂ = r₂;
Compute μ₁ = αν₂ – Wν₁;
Solve(αI+1αW2)z2=μ1;
Compute z₁ = ν1+1αWz2;
Set the generalized residual vectorz=[z1T, z2T]T.

From Algorithm 3.4, we can see that the PSHNS preconditioner iteration involves the complex coefficient matrix αV + iW whose corresponding linear subsystem may be hard to be solved. Meanwhile, it can be seen from Algorithm 3.5 that the implementation of the HSS preconditioner need to solve three symmetric positive definite linear subsystems.

In the following, we will discuss the eigenvector distributions of the preconditioned matrix P2−1A in detail.

Theorem 3.6

Let the preconditioner 𝓟₂be defined as in (7), then the preconditioned matrixP2−1Ahas n + i + j (0 ≤ i + j ≤ n) linearly independent eigenvectors, which are described as follows:

n eigenvectors 0vk (k=1,2,⋯,n) that correspond to the eigenvalue 1, where v_k (k = 1, 2, ⋯, n) are arbitrary linearly independent vectors.
i (0 ≤ i ≤ n) eigenvectors uk1vk1 (1≤k≤i) that correspond to the eigenvalue 1, where uk1≠0,Tuk1=αuk1 and vk1 are arbitrary vectors.
j (0 ≤ j ≤ n) eigenvectors uk2vk2 (1≤k≤j) that correspond to the eigenvalues λ_k ≠ 1, where α(T+U)uk2=λk(αI+U)Tuk2,uk2≠0andvk2=11−λk(T+1αW2)−1(1αWT−W)uk2.

Proof

Let λ be an eigenvalue of the preconditioned matrix P2−1Aanduv be the corresponding eigenvector. Then

αT−1(αI+U)−1(T+U) 0−(T+1αW2)−1W(1αT−I) Iuv=λuv.

By simple calculation, we have

α(T+U)u=λ(αI+U)Tu,(T+1αW2)−1W(1αT−I)u=(1−λ)v.(14)

If λ = 1, the Eq. (14) become

α(T+U)u=(αI+U)Tu,(T+1αW2)−1W(1αT−I)u=0.(15)

Then we have the following two situations.

When u = 0, the Eq. (15) are always true. Hence, there are n linearly independent eigenvectors 0vk (k = 1, 2, ⋯, n) corresponding to the eigenvalue 1, where v_k (k = 1, 2, ⋯, n) are arbitrary linearly independent vectors.
When u ≠ 0, through a simple calculation for (15), we have Tu = αu. Then, there will be i (0 ≤ i ≤ n) linearly independent eigenvectors uk1vk1 (1≤k≤i) that correspond to the eigenvalue 1, where uk1≠0,Tuk1=αuk1andvk1 are arbitrary vectors.

Next, we consider the case λ ≠ 1. It can be seen from (14) that the results in (3) hold.

Finally, we show that the n + i + j eigenvectors are linearly independent. Let c = [c₁, c₂, ⋯, c_n]^T, c¹ = [c11,c21,⋯,ci1]Tandc2=[c12,c22,⋯,cj2]T be three vectors with 0 ≤ i, j ≤ n. Then, we have to show that

0⋯0v1⋯vnc1⋮cn+u11⋯ui1v11⋯vi1c11⋮ci1+u12⋯uj2v12⋯vj2c12⋮cj2=0⋮0(16)

holds if and only if the vectors c, c¹ and c² are all zero vectors, where the first matrix consists of the eigenvectors corresponding to the eigenvalue 1 for the case (1), the second matrix consists of those for the case (2) and the third matrix consists of the eigenvectors corresponding to λ ≠ 1 for the case (3). By multiplying matrix P2−1A from left on both sides of Eq. (16), we get

0⋯0v1⋯vnc1⋮cn+u11⋯ui1v11⋯vi1c11⋮ci1+u12⋯uj2v12⋯vj2λ1c12⋮λjcj2=0⋮0.(17)

From the difference between (17) and (16), we obtain

u12⋯uj2v12⋯vj2(λ1−1)c12⋮(λj−1)cj2=0⋮0.

Because the eigenvalues λ_k ≠ 1 and uk2vk2 (k=1,⋯,j) are linearly independent, we know that ck2 = 0 (k = 1, ⋯, j). Thus, the Eq. (16) reduces to

0⋯0v1⋯vnc1⋮cn+u11⋯ui1v11⋯vi1c11⋮ci1=0⋮0.

As the vectors uk1 (k = 1, ⋯, i) are also linearly independent, then we have ck1 = 0 (k = 1, ⋯, i). Hence, the above equation becomes

0⋯0v1⋯vnc1⋮cn=0⋮0.

Since v_k (k = 1, ⋯, n) are linearly independent, we have c_k = 0 (k = 1, ⋯, n). Therefore, the n + i + j eigenvectors are linearly independent. The proof of this theorem is completed □

Based on the Krylov subspace theory, we know that the iterative method with an optimal property (such as GMRES [34] method) will terminate at the moment when the degree of the minimal polynomial is attained. Next, we will give an upper bound of the degree of the minimal polynomial of the preconditioned matrix P2−1A.

Theorem 3.7

Assume that the conditions of Theorem 3.2 are satisfied and let the preconditioner 𝓟₂be defined as in(7). Then the degree of the minimal polynomial of the preconditioned matrixP2−1A is at most n + 1. Thus, the dimension of the Krylov subspace 𝓚( P2−1A, b) is at most n + 1.

Proof

From (11), we know that the preconditioned matrix can be expressed as

P2−1A=θ10θ2I,

where θ₁ = αT^–1(αI + U)^–1(T + U) and θ2=−(T+1αW2)−1W(1αT−I). Let μ_i (i = 1, ⋯, n) be the eigenvalues of matrix θ₁. The characteristic polynomial of the preconditioned matrix P2−1A is

(P2−1A−I)n∏i=1n(P2−1A−μiI).

Then

(P2−1A−I)∏i=1n(P2−1A−μiI)=(θ1−I)∏i=1n(θ1−μiI) 0θ2∏i=1n(θ1−μiI) 0.

Because μ_i (i = 1, ⋯, n) are the eigenvalues of θ₁, then by the Hamilton-Cayley theorem, we have

∏i=1n(θ1−μiI)=0.

Therefore, we obtain that the degree of the minimal polynomial of the preconditioned matrix P2−1A is at most n+1. It is well known that the degree of the minimal polynomial is equal to the dimension of the corresponding Krylov subspace. So the dimension of the Krylov subspace 𝓚( P2−1A, b) is also at most n + 1. Thus, the proof of this theorem is completed. □

4 Numerical experiments

In this section, some numerical experiments are presented to illustrate the effectiveness of the preconditioner 𝓟₂ for the linear system (3). We compare the performance of the preconditioner 𝓟₂ with the PSHNS preconditioner and the HSS preconditioner. We denote the number of iteration steps as ”Iter”, the elapsed CPU time in seconds as “CPU”, and the relative residual norm as “RES”. Unless otherwise specified, we use left preconditioning with the GMRES method [34]. All the computations are implemented in MATLAB on a PC computer with Intel (R) Core (TM) i5 CPU 2.50GHz, and 4.00 GB memory.

In our implementations, the zero vector is adopted as the initial vector and the iteration stops when

RES:=∥g−Txk−Wyk∥22+∥f−Wxk+Tyk∥22∥g∥22+∥f∥22<10−6.

The maximum number of iteration steps allowed is set to 3000 for the unpreconditioned GMRES method, and to 500 for the preconditioned GMRES method. It should be emphasized that the sub-linear systems arising from the application of the preconditioners are solved by direct methods. In matlab, the sub-systems of linear equations are solved through sparse Cholesky or LU factorization in combination with AMD or column AMD reordering. A symbol “-” is used to indicate that the method does not obtain the required stopping criterion before maximum iterations or out of memory. We test three values of the parameter α, that is, α = 0.001, α = 0.01, α = 1.

Example 1

We consider the following complex symmetric linear system which comes from [13, 24, 26]

[(−ω2M+K)+i(ωCv+CH)]x=b.

where M and K are the inertia and stiffness matrices, C_v and C_H are the viscous and hysteretic damping matrices, respectively, ω is the driving circular frequency, K = I ⊗ V_m + V_m ⊗ I, V_m = h^–2tridiag(–1, 2, n-1) ∈ ℝ^m×mis a tridiagonal matrix, ω = 2π, h=1m+1,Cv=12M, CH=μKwith μ = 0.02 being a damping coefficient. We choose W = h²(–ω²M + K), T = h²(ωC_v + C_H) and test M = 10I, 15I, 25I, 35I, 50I. In this example, we set m = 32 and the right-hand side b = 𝓐 * ones(2m², 1).

By choosing different α and M, we list the numerical results of the unpreconditioned GMRES method, the 𝓟₂ preconditioned GMERS method, the PSHNS preconditioned GMRES method and the HSS preconditioned GMRES method in Table 1. Table 1 indicates that applying the 𝓟₂ preconditioned GMERS method requires less iteration steps and CPU times than applying the other preconditioned GMRES methods by choosing suitable parameters. In Table 1, we find that the PSHNS preconditioner and the HSS preconditioner are both more sensitive to the parameter α than the preconditioner 𝓟₂. Figure 1 describes the eigenvalue distributions of the original coefficient matrix 𝓐, the preconditioned matrix PHSS−1A(forα=0.06), the preconditioned matrix PPSHNS−1A(forα=0.06) and the preconditioned matrix P2−1A (for α = 0.06 and α = 0.006) with M = 35I. From Figure 1, we observe that the eigenvalue distributions of the preconditioned matrix P2−1A are much better than those of the other preconditioned matrices. Moreover, the smaller parameter α generally makes the eigenvalues of the preconditioned matrix P2−1A become more and more clustered.

$Fig. 1 The eigenvalue distributions of unpreconditioned matrix, the preconditioned matrices PHSS−1A,PPSHNS−1AandP2−1A$\begin{array}{} \displaystyle \mathcal{P}^{-1}_{\rm{HSS}}\mathcal{A}, \mathcal{P}^{-1}_{\rm{PSHNS}}\mathcal{A} \,\,\text{and}\,\, \mathcal{P}^{-1}_{2}\mathcal{A} \end{array}$ for Example 1 with M = 35I.$

Fig. 1

The eigenvalue distributions of unpreconditioned matrix, the preconditioned matrices PHSS−1A,PPSHNS−1AandP2−1A for Example 1 with M = 35I.

Table 1

Numerical results of GMRES and Preconditioned GMRES methods for Example 1

Preconditioner	α	M	101	151	251	351	501
unpreconditioner		Iter	126	155	197	243	305
		CPU	0.26206	0.36578	0.56839	0.83868	1.34459
𝓟_PSHNS	α = 0.001	Iter	71	76	87	93	108
		CPU	0.16812	0.189197	0.224558	0.241821	0.300449
𝓟_HSS	α = 0.001	Iter	22	22	23	26	31
		CPU	0.052688	0.053715	0.05616	0.060838	0.077092
𝓟₂	α = 0.001	Iter	12	11	12	12	13
		CPU	0.032933	0.039289	0.041518	0.035258	0.037504
𝓟_PSHNS	α = 0.01	Iter	71	76	87	92	108
		CPU	0.04293	0.04455	0.05594	0.06078	0.007809
𝓟_HSS	α = 0.01	Iter	15	16	20	21	27
		CPU	0.00858	0.00885	0.01142	0.01187	0.01464
𝓟₂	α = 0.01	Iter	12	11	10	11	11
		CPU	0.00792	0.00776	0.00714	0.00751	0.00761
𝓟_PSHNS	α = 1	Iter	31	34	37	35	27
		CPU	0.06931	0.0775	0.08297	0.07733	0.05919
𝓟HSS	α = 1	Iter	77	73	54	40	29
		CPU	0.18827	0.18229	0.13074	0.08941	0.06585
𝓟₂	α = 1	Iter	15	15	18	16	14
		CPU	0.03641	0.03653	0.0418	0.0383	0.03443

Example 2

We consider the following complex symmetric linear system [25, 26]

[(Tm⊗Im+Im⊗Tm−k2h2(Im⊗Im))+iσ2(Im⊗Im)]x=b,

where T_m = tridiag(–1, 2, –1) is a tridiagonal matrix with order m and k denotes the wavenumber, σ₂ = 0.1, h=1m+1and b = 𝓐 * ones(2m², 1). Here, we choose W = T_m ⊗ I_m + I_m ⊗ T_m – k²h²(I_m ⊗ I_m) and T = σ₂(I_m ⊗ I_m).

Table 2 shows that these three kinds preconditioned GMRES methods present much better convergence behavior than the unpreconditioned GMRES method. Meanwhile, the 𝓟₂ preconditioned GMERS method gives a better numerical results than the PSHNS preconditioned GMRES method and the HSS preconditioned GMRES method with different parameters α in terms of iteration steps and CPU times. From Figure 2, we find that the eigenvalue distributions of the preconditioned matrix P2−1A are much better than those of the other preconditioned matrices and the eigenvalues of the preconditioned matrix P2−1A will be more gathered as α decrease.

$Fig. 2 The eigenvalue distributions of unpreconditioned matrix, the preconditioned matrices PHSS−1A,PPSHNS−1AandP2−1A$\begin{array}{} \displaystyle \mathcal{P}^{-1}_{\rm{HSS}}\mathcal{A}, \mathcal{P}^{-1}_{\rm{PSHNS}}\mathcal{A} \,\,\text{and}\,\, \mathcal{P}^{-1}_{2}\mathcal{A} \end{array}$ for Example 2 with k = 20, m = 32.$

Fig. 2

The eigenvalue distributions of unpreconditioned matrix, the preconditioned matrices PHSS−1A,PPSHNS−1AandP2−1A for Example 2 with k = 20, m = 32.

Table 2

Numerical results of GMRES and Preconditioned GMRES methods for Example 2

Preconditioner	α	k	10	20	30	40	50
		m²	16²	32²	64²	128²	256²
unpreconditioner		Iter	47	121	324	443	-
		CPU	0.02811	0.24058	5.20897	29.3303	-
𝓟_PSHNS	α = 0.001	Iter	29	68	123	116	99
		CPU	0.073258	0.154954	1.349414	3.920827	17.513853
𝓟_HSS	α = 0.001	Iter	12	24	45	103	211
		CPU	0.019048	0.047163	0.463713	7.020476	160.268749
𝓟₂	α = 0.001	Iter	5	10	16	30	38
		CPU	0.013813	0.02835	0.156661	1.394879	11.409131
𝓟_PSHNS	α = 0.01	Iter	29	68	123	115	98
		CPU	0.03101	0. 15599	1.33949	4.06219	30.38906
𝓟_HSS	α = 0.01	Iter	9	19	30	48	57
		CPU	0.00913	0.03617	0.23154	1.64472	14.81967
𝓟₂	α = 0.01	Iter	e	8	13	18	17
		CPU	0.00809	0.02318	0.12537	0.68779	3.90741
𝓟_PSHNS	α = 1	Iter	14	29	41	39	34
		CPU	0.060745	0.04931	0.235227	0.859889	4.907534
𝓟_HSS	α = 1	Iter	29	52	62	64	65
		CPU	0.033833	0.127379	0.705647	3.464846	23.253554
𝓟₂	α = 1	Iter	10	17	20	21	22
		CPU	0.019189	0.037429	0.176674	0.842907	4.844297

5 Conclusions

In this paper, we have proposed a relaxed block splitting preconditioner 𝓟₂ for the real block two-by-two linear system (3) and analyzed the eigenvalue distributions of the corresponding preconditioned matrix. Meanwhile, the structure of the eigenvectors and an upper bound for the degree of the minimal polynomial of the preconditioned matrix P2−1A were also derived. Some numerical experiments are given to illustrate the effectiveness of the preconditioner 𝓟₂ for solving the linear system (3).

Acknowledgement

The authors would like to thank the referees for their very detailed comments and suggestions which improved the presentation of this paper greatly.

This work is supported by the National Natural Science Foundation of China No. 11471122 and supported in part by Science and Technology Commission of Shanghai Municipality (No. 18dz2271000).

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Received: 2017-10-08

Accepted: 2018-03-29

Published Online: 2018-06-07

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Artikel in diesem Heft

https://doi.org/10.1515/math-2018-0051

Schlagwörter für diesen Artikel

Complex symmetric linear system; Preconditioner; Krylov subspace method; GMRES

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