A note on the parallel GSAOR method for block diagonally dominant matrices

1 Introduction

For the linear system

where A 𝜖 ℝ^n×n is nonsingular, and x, b 𝜖 ℝⁿ are n-dimensional vectors. Let us consider the splitting of the matrix A of linear system (1.1):

where D = diag(A), −L and −U are strictly lower and strictly upper triangular parts of A, respectively. James [2] presented a generalized accelerated overrelaxation (GAOR) method given by

or

where

and

are iterative matrices and Ω = diag(ω₁, ω₂ …, ω_n) with ω_iϵR⁺ and r 𝜖 [0, 1].

Then generalized symmetric AOR method (GSAOR) can be defined as follows [8].

where T = U₁(r, Ω)L₁(r,Ω) and C = U₁(r,Ω)(D − rΩL)⁻¹Ω + (D − rΩU)⁻¹Ω.

In the following, we recall the mathematical descriptions of the block linear system and the BMM introduced in [5, 6].

Let s(≤ n) and n_i(≤ n), i = 1, 2, ..., s, be given positive integers satisfying ∑i=1sn_i = n, and denote

for the convenience, we will simply use 𝕃_n for 𝕃_n(n₁, ..., n_s) and V_n for V_n(n₁, ..., n_s). Then, the block linear system to be solved can be expressed as the form

where A ϵ 𝕃_n is nonsingular and b ϵ V_n are general known coefficient matrix and right-hand vector, respectively, and x ϵ V_n is the unknown vector.

If block matrices M_k, N_k, E_k ϵ 𝕃_n, k = 1, 2, ..., 𝛼, satisfy:

• 1) A = M_k − N_k, M_k nonsingular, k = 1, 2, ..., α;

• 2) E_k = diag(E11(k),...,Ess(k)),, k = 1,2,…,α

• 3)∑k=1α∥Eii(k)∥=1,i = 1, 2, ..., s;

then we call the collection of triples (M_k, N_k, E_k), k = 1, 2, ..., 𝛼, is a block matrices multisplitting (BMM) of the block matrix A ϵ 𝕃_n where || · || denotes the consistent matrix norm.

O’Leary and White [4] invented the matrix multisplitting method in 1985 for solving parallely the large sparse linear systems on the multiprocessor systems and was further studied by many authors [1,5–14,17–citenum19].

Suppose that we have a multiprocessor with α processors connected to a host processor, that is, the same number of processors as splittings, and that all processors have the last update vector x^k, then the kth processor only computes those entries of the vector

which correspond to the block diagonal entries Eii(k) of the block matrix E_k. The processor then scales these entries so as to be able to deliver the vector

to the host processor, performing the parallel multisplitting scheme

In this paper, we investigate the domain of convergence of block GSAOR multisplittings methods for solving linear system (1.1). When the coe_cient matrix A is a block H-matrix or a block strictly diagonally dominant matrix.

2 Parallel multisplitting GSAOR methods

Given a positive integer 𝛼 (𝛼 ≤ s), we separate the number set {1, 2,…, s} into a nonempty subsets J_k, k = 1, 2,…,α such that

Note that there may be overlappings among the subsets J₁, J₂,…, J_𝛼. Corresponding to this separation, we introduce matrices

Obviously, D is a block diagonal matrix, L_k, k = 1, 2,…, 𝛼, are block strictly lower triangular matrices, U_k, k = 1, 2,…, 𝛼, are general block matrices, and E_k, k = 1, 2,…, 𝛼, are block diagonal matrices. If they satisfy:

1. D is nonsingular;

2. A = D − L_k − U_k, k = 1, 2,…, 𝛼;

3. ∑k=1αE_k = I;

then the collection of triples (D−U_k, L_k, E_k) and (D−L_k, U_k, E_k), k = 1, 2, ...,α, are BMM of the block matrix A ϵ 𝕃_n. Here, I denotes the identity matrix of order n × n.

Let (M_k, N_k, E_k), k = 1, 2,…, 𝛼, is a BMM of the block matrix A 𝜖 𝕃_n .We will definite local parallel multisplittings blockwise relaxation generalized SAOR method (LMBGSAOR) and global parallel multisplittings blockwise relaxation generalized SAOR method (GMBGSAOR).

Algorithm 2.1(local parallel multisplittings blockwise relaxation method).

Given the initial vector.

For m = 0, 1, 2, ... repeat (I) and (II), until convergence.

• (I) For k = 1, 2,…, 𝛼, (parallel) solving y_k:

  Mkyk=Nk𝜒m+b.

• (II) Computing

  𝜒m+1=∑k=1αEkyk.

Algorithm 2.1 associated with LMBGSAOR method can be written as

where

By using a suitable positive relaxation parameter 𝛽, we will establish global parallel multisplitting blockwise relaxation GSAOR method which is based on Algorithm 2.1.

Algorithm 2.2(global parallel multisplittings blockwise relaxation method).

Given the initial vector.

For m = 0, 1, 2, ... repeat (I) and (II), until convergence.

• (I) For k = 1, 2,…, α, (parallel) solving y_k:

  Mkyk=Nk𝜒m+b.

• (II) Computing

Algorithm 2.2 associated with GMBGSAOR method can be written as

where H_GMBGSAOR = 𝛽H_GMBGSAOR + (1 − 𝛽)I.

3 Preliminaries

We shall use the following notations and Lemmas. A matrix A = (a_ij) is called a Z-matrix if for any i ≠ j, a_ij ≤ 0. A Z-matrix is a nonsingular M-matrix if A is nonsingular and A⁻¹ ≥ 0. Additionally, we denote the spectral radius of A by ρ(A). It is well-known that if A ≥ 0 and there exists a vector x > 0 such that Ax < 𝛼x, then 𝜌(A) < 𝛼 [3]. Let

We will review the concepts of strictly block diagonally dominant matrix and block H-matrix.

Definition 3.1([1, 20]). Let A ϵ 𝕃_n,I, (I) block comparison matrix ℳ(A) = ((ℳ(A))_ij) ϵℝ^s×s and (II) block comparison matrix 𝒩(A) = ((𝒩(A))_ij) ϵℝ ^s×s are defined respectively as follows:

where ∥·∥ is a consistent matrix norm such that ∥I∥ = 1.

For block matrices A ϵ 𝕃_n,I, we define D(A) = diag(A₁₁, A₂₂,…, A_ss), B(A) = D(A) − A, J(A) = D(A)⁻¹B(A),μ₁(A) = 𝜌(Jℳ(A)), μ₂(A) = 𝜌(I − 𝒩(A)). In [1], Liu et al. show thatℳ (I − J(M)) = 𝒩 (I − J(M)), μ₂(A) ≤ μ₁(A).

Definition 3.2([15, 16]). Let A ϵ𝕃_n,I . A matrix A is said to be a strictly (I) block diagonally dominant matrix, if

A matrix A is said to be a strictly (II) block diagonally dominant matrix, if

Remark 3.1. From Definition 3.2, we know a strictly (I) block diagonally dominant matrix must be a strictly (II) block diagonally dominant matrix, but not conversely.

Definition 3.3([1, 15, 16]). Let A ϵ𝕃_n,I. A matrix A is said to be a (I) block H-matrix (HB(I)(P,Q)-matrix) with respect to nonsingular matrices P and Q if there are two matrices P, Q∈Ln,Id such that ℳ(PAQ) is an M-matrix; A matrix A is said to be a (II) block H-matrix (HB(II)(P,Q)-matrix) with respect to nonsingular matrices P and Q if 𝒩(PAQ) is an M-matrix.

Remark 3.2([1]). From Definition 3.3, we obtain a (I) block H-matrix (HB(I)(P,Q)-matrix) must be a (II) block H-matrix (HB(II)(P,Q)-matrix), but conversely, it is not true.

Combining Remarks 3.1 and 3.2, we have

Remark 3.3. A strictly (I) block diagonally dominant matrix must be (HB(I)(P,Q)-matrix; A strictly (II) block diagonally dominant matrix must be a (II) block H-matrix (HB(II)(P,Q)-matrix.

Definition 3.4([15]). If there exists a block diagonal matrix X such that AX is a strictly block diagonally dominant matrix, then A is said to be a block H-matrix (HB(I)(P,Q)-matrix and (HB(II)(P,Q)-matrix).

Definition 3.5([1, 5, 6]). Let Aϵ𝕃_n.We call [A] = (∥A_ij∥) ϵℝ^N×N the block absolute value of the block matrix M. The block absolute value [x] ∈ R^N of a block vector x ϵ V_n is defined in an analogous way.

These kinds of block absolute values have the following important properties.

Lemma 3.1([1, 5, 6]). Let L, M 𝜖 𝕃_n, x, y 𝜖 V_n and r 𝜖 ℝ¹. Then

1) |[L] − [M]| ≤ [L + M] ≤ [L] + [M] (|[x] − [y]| ≤ [x + y] ≤ [x] + [y]);

2) [LM] ≤ [L][M] ([xy] ≤ [x][y]);

3) [rM] ≤ |r|[M] ([rx] ≤ |r|[x]);

4)ρ(M) ≤ ρ(|M|) ≤ρ ([M]) (here, ∥·∥ is either ∥·∥_∞or ∥·∥₁ ).

Lemma 3.2([5, 6]). Let A 𝜖 𝕃_n.I be a strictly block diagonally dominant matrix, then

1) A is nonsingular;

2) [(A)⁻¹] ≤ℳ(A)⁻¹;

3)ρ(J(ℳ(A))) < 1.

Let

denote respectively the set of (I) and (II) matrices such that the absolute values of whose elements are equal to absolute values of corresponding elements of matrix A.

4 Main results

For the present Algorithms 2.1 and 2.2, we give convergence theorems for block diagonally dominant matrices and block H-matrices.

Theorem 4.1. Let M ϵ 𝕃_n,I(n₁, n₂,…, n_s) be a strictly (I) block diagonally dominant matrix, A∈ΩBI(PMQ) and the collection of triples (D − U_k, L_k, E_k) and (D − L_k, U_k, E_k), k = 1, 2,…, 𝛼, are BMM of the block matrix A ϵ 𝕃_n,I (n₁, n₂,…, n_s). Assume that

then LMBGSAOR method converges for any initial vector x⁰ϵ V_s.

Proof. By Lemma 3.1, we know

and then, the iteration (2.1) converges for any initial vector x⁰ ∈ V_N if and only if

Since A∈ΩBI (PMQ), thenℳ(A) = ℳ(PMQ). Because

is a strictly (I) block diagonally dominant matrix, so A 𝜖 𝕃_n,I (n₁, n₂,…, n_s) is a strictly (I) block diagonally dominant matrix and μ₁ = 𝜌(J(ℳ(A))) = 𝜌 (J(ℳ(PMQ))) = μ₁(PMQ), and thus, we have μ1 = 𝜌 (J(ℳ(PMQ))) =μ₁(PMQ) < 1, it follows from Lemma 3.2. Let B = L_k + U_k, by (2.2), we know that [B] = [L_k] + [U_k], k =1, 2,…, 𝛼, clearly, D − rΩL_k, k = 1, 2,…, 𝛼 are strictly (I) block diagonally dominant matrix and ℳ(D) −r[Ω][B] are strictly diagonally dominant matrix for 0 < ω_i < 2/(1 + μ₁(PMQ)), i = 1, 2,…, n, and r ϵ [0, 1] which follow from A is a strictly (I) block diagonally dominant matrix. Since

for 0 < ω_i < 2/(1 + μ₁(PMQ)), i = 1, 2,…, n, r ϵ [0, 1], k = 1, 2,…, 𝛼, and ℳ(A) is a strictly diagonally dominant matrix, we have ℳ(D)−r[Ω][B] and ℳ(D) are strictly diagonally dominantM-matrices, for 0 < ω_i < 2/(1 + μ₁(PMQ)), i = 1, 2,…, n, r ϵ[0, 1]. Therefore, ℳ(D) − r[Ω][U_k] are strictly diagonally dominant M-matrices, and then D − rΩU_k, k = 1, 2,…, 𝛼, are strictly (I) block diagonally dominant matrices, for 0 < ω_i < 2/(1 + μ₁(PMQ)), i = 1, 2,…, n, and r ϵ [0, 1].

Let L¯k=D−1Lk and, U¯k=D−1Ukthen I−rΩL¯k and I−rΩU¯k are also strictly (I) block diagonally dominant matrices, for 0 < ω_i < 2/(1 + μ₁(PMQ)), i = 1, 2,…, n, r ϵ [0, 1], k = 1, 2,…, 𝛼. Thus, by Lemma 3.1, we have

From (4.1), we have

Since L¯k=D−1Lk and U¯k=D−1Uk we have [L¯k] ≤ ℳ(D)⁻¹[L_k] and [U¯k] ≤ ℳ(D)⁻¹[U_k] which follow from Lemma 3.1 and Lemma 3.2, and then

Therefore, we have

where T([Ω]) = [I −Ω ] + [Ω]J(ℳ(A)). Note that(I−r[Ω][U¯k(k)])−1≥I, k = 1, 2,…, 𝛼, and then

Similar to the above proving process, we have

Let ϑ₁ = max{ω_i}, ϑ₂ = min{ ω_i } and f(t) = |1−t|ϑ+tJ(ℳ(A)), where t > 0. Obviously, f(t) is nonincreasing for 0 < t ≤ 1 and f(t) is nondecreasing for t≤ 1. Therefore, we have

Let e denotes the vector e = (1, 1,…, 1)^T ϵ V_s and J_𝜀(ℳ(A)) = J(ℳ(A)) + 𝜀ee^T, since J(ℳ(A)) is nonnegative, the matrix J_𝜀(ℳ(A)) has only positive entries and irreducible for any 𝜀 > 0. By the Perron-Frobenius theorem [3] for any 𝜀 > 0, there exists a vector x_𝜀 > 0 such that

where 𝜌_𝜀 = 𝜌 (J_𝜀 (ℳ (A))). Moreover, if 𝜀 > 0 is small enough, we have 𝜌_𝜀 < 1 by continuity of the spectral radius. Thus, we can get ϑ₁ − 1 + ϑ₁𝜌_𝜀 < 1 and 1 − ϑ₂ + ϑ₂𝜌_𝜀 < 1. And then

Step 1. For 0 < ω_i≤ 1, i = 1, 2,…, n,

Step 2. For 1 < ω_i < 2/(1 + μ₁(PMQ)), i = 1, 2,…, n,

Then, we have [H_LMBGSAOR]x_∈ < x_∈ and ρ([H_LMBGSAOR]) < 1. □

Theorem 4.2. Let M ϵ 𝕃_n,I(n₁, n₂,…, n_s) be an HB(I)(P,Q) -matrix, A∈ΩBI(PMQ), and the collection of triples (D − L_k, U_k, E_k) and (D − U_k, L_k, E_k), k = 1, 2,…, 𝛼, are BMM of the block matrix A ∈ 𝕃_n,I(n₁, n₂,…, n_s). Assume that

If

then LMBGSAOR method converges for any initial vector x⁰ ϵV_s.

Proof. Since A∈ΩBI(PMQ), thenℳ(A) = ℳ(PMQ). Because M ∈ 𝕃_n,I(n₁, n₂,…, n_s) is an (HB(I)(P,Q)-matrix,so A ∈ 𝕃_n,I (n₁, n₂,…, n_s) is an block (HB(I)(I,I)-matrix. From Definition 3.5, there exists a block diagonal matrix X such that AX is a strictly (I) block diagonally dominant matrix. Let H_LMBGSAOR(A) and H_LMBGSAOR(AX) denote the iterative matrices of L_MBGSAOR methods for block matrix A and AX, respectively. By simple calculation, we have H_LMBGSAOR(A) and H_LMBGSAOR(AX) are similar. Since similar matrices have the same eigenvalues, it follows that (H_LMBGSAOR(A)) = 𝜌(H_LMBGSAOR(AX)) < 1. □

Theorem 4.3. Let M ∈ 𝕃_n,I (n₁, n₂,…, n_s) be a strictly (II) block diagonally dominant matrix, A∈ΩBII(PMQ)and the collection of triples (D − U_k, L_k, E_k) and (D − L_k, U_k, E_k), k = 1, 2,…, 𝛼 are BMM of the block matrixA ∈ 𝕃_n,I (n₁, n₂,…, n_s). Assume that

If

then LMBGSAOR method converges for any initial vector x⁰ ∈V_s.

Proof. The proof goes along the same lines as in the Theorem 4.1 except that strictly (I) block diagonally dominant matrix and ΩBI(PMQ) play the roles of strictly (II) block diagonally dominant matrix and ΩBII(PMQ), respectively, which completes the proof. □

Theorem 4.4. Let M ∈ 𝕃_n,I (n₁, n₂,…, n_s) be an HB(II)(P,Q)-matrix, A∈ΩBII(PMQ), and the collection of triples (D − L_k, U_k, E_k) and (D − U_k, L_k, E_k), k = 1, 2,…, α, are BMM of the block matrix A ∈ 𝕃_n,I (n₁, n₂,…, n_s). Assume that

If

then LMBGSAOR method converges for any initial vector x⁰ ∈V_s.

Proof. The proof is similar to that given in Theorem 4.2, so omitted. □

Using GMBGSAOR method, we can also get the following convergence results.

Theorem 4.5. Let M ∈ 𝕃_n,I (n₁, n₂,…, n_s) be a strictly (I) block diagonally dominant matrix, A∈ΩBI(PMQ)and the collection of triples (D − U_k, L_k, E_k) and (D − L_k, U_k, E_k), k = 1, 2,…, 𝛼, are BMM of the block matrix A ∈ 𝕃_n,I (n₁, n₂,…, n_s). Assume that

(a) if 0 < ω_i ≤ 1, i = 1, 2,…, n, and0  <β<  2/(1  +h22), h₂ = 1 − I₂ + I₂μ₁(PMQ);

(b) if 1 ≤ ω_i < 2/(1 + μ₁(PMQ)), i = 1, 2,…, n, and0  <β<  2/(1  +h12), h₁ = I₁ − 1 + I₁μ₁(PMQ).

Then GMBGSAOR method converges for any initial vector x⁰ ∈V_s, where ϑ₁ = max{ω_i}, ϑ₂ = min{ω_i}.

Proof. Since ρ(H_GMBGSAOR) ≤ ρ(|H_GMBGSAOR|) ≤ ρ([H_GMBGSAOR]), the iteration (2.3) converges for any initial vector x⁰ ∈V_s if and only if

Similar to the proof of Theorem 4.1, we have

and there exists 𝜀 > 0 such that J_𝜀(ℳ(A)) = J(ℳ(A)) + 𝜀ee^T has only positive entries and irreducible for any 𝜀 > 0. By the Perron-Frobenius theorem for any 𝜀 > 0, there exists a vector x_𝜀 > 0 such that

where 𝜌_𝜀 =𝜌(J_𝜀(ℳ(A))). Moreover, if 𝜀 > 0 is small enough, we have ρ_𝜀 < 1, by continuity of the spectral radius. Under the condition of Theorem 4.5, we not only get h₁ = ϑ₁ −1+ ϑ₁ρ_𝜀 < 1 and h2 = 1− ϑ₂ + ϑ₂ρ_𝜀 < 1, but also βh12+|1+β|<1 and βh22+|1+β|<1 then

Then [H_GMBGSAOR]x_𝜀 < x_𝜀 and ρ([H_GMBGSAOR]) < 1. □

Theorem 4.6. Let M ∈ 𝕃_n,I (n₁, n₂,…, n_s) be an HB(I)(P,Q)-matrix, A∈ΩBI(PMQ), and the collection of triples (D − L_k, U_k, E_k) and (D − U_k, L_k, E_k), k = 1, 2,…, 𝛼, are BMM of the block matrix A ∈ Ln,I(n1, n2,…, ns). Assume that

(a) if 0 < ω_i ≤ 1, i = 1, 2,…, n, and0  <β<  2/(1  +h22), h₂ = 1 − ϑ₂ + ϑ₂μ₁(PMQ);

(b) if 1 ≤ ω_i < 2/(1 + μ₁(PMQ)), i = 1, 2,…, n, and0  <β<  2/(1  +h12), h₁ = ϑ₁ − 1 + ϑ₁μ₁(PMQ).

Then GMBGSAOR method converges for any initial vector x⁰ ∈V_s, where ϑ₁ = max{ω_i}, ϑ₂ = min{ω_i}.

Theorem 4.7. Let M ∈ 𝕃_n,I (n₁, n₂,…, n_s) be a strictly (II) block diagonally dominant matrix, A∈ΩBII(PMQ)and the collection of triples (D − U_k, L_k, E_k) and (D − L_k, U_k, E_k), k = 1, 2, ...,α are BMM of the block matrix A ∈ 𝕃_n,I (n₁, n₂,…, n_s). Assume that

(a) if 0 < ω_i ≤ 1, i = 1, 2,…, n, and0  <β<  2/(1  +h22), h₂ = 1 − ϑ₂ + ϑ₂μ₁(PMQ);

(b) if 1 ≤ ω_i < 2/(1 + μ₂(PMQ)), i = 1, 2,…, n, and0  <β<  2/(1  +h12), h₁ = ϑ₁ − 1 + ϑ₁μ₂(PMQ).

Then GMBGSAOR method converges for any initial vector x⁰ ∈ V_s, where ϑ₁ = max{ω_i}, ϑ₂ = min{ω_i}.

Theorem 4.8. Let M ∈ 𝕃_n,I (n₁, n₂,…, n_s) be an HB(II)(P,Q)-matrix), A∈ΩBII(PMQ) and the collection of triples (D − L_k, U_k, E_k) and (D − U_k, L_k, E_k), k = 1, 2,…, 𝛼, are BMM of the block matrix A ∈ 𝕃_n,I (n₁, n₂,…, n_s). Assume that

(a) if 0 < ω_i ≤ 1, i = 1, 2,…, n, and0  <β<  2/(1  +h22), h₂ = 1 − ϑ₂ + ϑ₂μ₂(PMQ);

(b) if 1 ≤ ω_i < 2/(1 + μ₂(PMQ)), i = 1, 2,…, n, and0  <β<  2/(1  +h12), h₁ = ϑ₁ − 1 + ϑ₁μ₂(PMQ).

Then GMBGSAOR method converges for any initial vector x⁰ ∈ V_s, where ϑ₁ = max{ω_i}, ϑ2 = min{ω_i}.

Proof. Similar to the proof of Theorem 4.2, Theorem 4.3, and Theorem 4.4, we can prove Theorem 4.6, Theorem 4.7, and Theorem 4.8, respectively. So omitted. □

5 Conclusion

Liu et al. [1] consider the convergence of block parallel multisplittings GSAOR iterative methods for 1 ≤ ω_i < 2/(1 + μ₁(PMQ)) or 1 ≤ω_i < 2/(1 + μ₂(PMQ)), i = 1, 2,…, n. In this paper, we extend interval of ω_i, i = 1, 2,…, n, to (0, 2/(1 + μ₁(PMQ))) or (0, 2/(1 + μ₂(PMQ))) for block parallel multisplittings GSAOR iterative methods.