A convergence analysis of SOR iterative methods for linear systems with weak H-matrices

Cheng-yi Zhang, Zichen Xue 1 , and Shuanghua Luo 2
  • 1 School of Science, Xi’an Polytechnic University, Xi’an Shaanxi 710048, China
  • 2 School of Science, Xi’an Polytechnic University, Xi’an Shaanxi 710048, China
Cheng-yi Zhang, Zichen Xue and Shuanghua Luo

Abstract

It is well known that SOR iterative methods are convergent for linear systems, whose coefficient matrices are strictly or irreducibly diagonally dominant matrices and strong H-matrices (whose comparison matrices are nonsingular M-matrices). However, the same can not be true in case of those iterative methods for linear systems with weak H-matrices (whose comparison matrices are singular M-matrices). This paper proposes some necessary and sufficient conditions such that SOR iterative methods are convergent for linear systems with weak H-matrices. Furthermore, some numerical examples are given to demonstrate the convergence results obtained in this paper.

1 Introduction

In this paper we consider the solution methods for the system of n linear equations
Ax=b,
where A = (aij) ∈ Cn×n and is nonsingular, b, xCn and x unknown.
In order to solve system (1) by iterative methods, the coefficient matrix A = (Aij) ∈ Cn×n is split into
A=MN,
where MCn×n is nonsingular and NCn×n. Then, the general form of iterative methods for (1) can be described as follows:
x(i+1)=M1Nx(i)+M1b,i=0,1,2,......

The matrix H = M−1N is called the iterative matrix of the iteration (3). It is well-known that (3) converges for any given x(0) if and only if ρ(H) < 1 (see [15]), where ρ(H) denotes the spectral radius of the matrix H. Thus, to establish the convergence results of iterative methods, we mainly study the spectral radius of the iteration matrix in iteration (3).

For simplicity, we let
A=ILU,
where I is the identity matrix, L and U are strictly lower and strictly upper triangular, respectively. According to the standard decomposition (4), the iteration matrices for SOR iterative methods of (1) are listed in the following.
The forward, backward and symmetric SOR methods (FSOR, BSOR and SSOR) iteration matrices are
HFSOR(ω)=(IωL)1[(1ω)I+ωU],
HBSOR(ω)=(IωU)1[(1ω)I+ωL],
and
HSSOR(ω)=HBSOR(ω)HFSOR(ω)
where ω ∈ (0, 1) is the overrelaxation parameter.

Recently, the class of strong H-matrices including strictly or irreducibly diagonally dominant matrices (whose comparison matrices are nonsingular M-matrices) has been extended to encompass a wider set, known as the set of general Hmatrices (whose comparison matrices are M-matrices). In a recent paper, a partition of the n × n general H−matrix set Hn was obtained in [68]. Here, we give a different partition: general H−matrix set Hn are partitioned into two mutually exclusive classes: the strong class, HnS, where the comparison matrices of all general H−matrices are nonsingular, and the weak class, HnW, where the comparison matrices of all general H−matrices are singular, in which singular and nonsingular H−matrices coexist. As is shown in [15, 927], classical iterative methods such as Jacobi, Gauss-Seidel, symmetric Gauss-Seidel, JOR and SOR(SSOR) iterative methods for linear systems, whose coefficient matrices are strong H−matrices including strictly or irreducibly diagonally dominant matrices, are convergent to the unique solution of (1) for any choice of the initial guess x(0). However, the same need not be true in case of some iterative methods for the linear systems with weak H−matrices. Let us investigate the following example.

Assume that either A or B is the coefficient matrix of linear system (1), where
A=[211121112]andB=[211121112].

It is verified that both A and B are weak Hmatrices and not strong Hmatrices, since their comparison matrices are both singular Mmatrices. How can we get the convergence on FSOR-, BSOR- and SSOR iterative methods for linear systems with this class of matrices without direct computations?

In last years, Zhang et al. in [28] and Zhang et al. in [29] studied the convergence of Jacobli, Gauss-Seidel and symmetric Gauss-Seidel iterative methods for the linear systems with nonstrictly diagonally dominant matrices and general H−matrices, and established some significant results. In this paper the convergence of FSOR-, BSOR- and SSOR- iterative method will be studied for the linear systems with weak H−matrices and some necessary and sufficient conditions are proposed, such that these iterative methods are convergent for the linear systems with this class of matrices. Then some numerical examples are given to demonstrate the convergence results obtained in this paper.

The paper is organized as follows. Some notations and preliminary results about general H-matrices are given in Section 2. The main results of this paper are given in Section 3, where we give some necessary and sufficient conditions on convergence for FSOR-, BSOR- and SSOR- iterative methods of linear systems with weak H-matrices. In Section 4, some numerical examples are given to demonstrate the convergence results obtained in this paper. Future work is given in Section 4.

2 Preliminaries

In this section we give some notations and preliminary results relating to the special matrices that are used in this paper.

m×n (ℝm×n) will be used to denote the set of all m × n complex (real) matrices. ℤ denotes the set of all integers. Let α ⊆ 〈n〉 = {1, 2,...,n} ⊂ ℤ. For nonempty index sets α, β ⊆ 〈n〉, A(α, β) is the submatrix of A ∈ ℂn×n with row indices in α and column indices in β. The submatrix A(α, α) is abbreviated to A(α). Let A ∈ ℂn×n, α ⊂ 〈nand α' = 〈n〉 − α. If A(α) is nonsingular, the matrix
A/α=A(α)A(α,α)[A(α)]1A(α,α)
is called the Schur complement with respect to A(α), indices in both α and α' are arranged with increasing order. We shall confine ourselves to the nonsingular A(α) as far as A/α is concerned.
Let A = (aij) ∈ ℂm×n and B = (bij) ∈ ℂm×n, A ° B = (aij bij) ∈ ℂm×n denotes the Hadamard product of the matrices A and B. A matrix A = (aij) ∈ ℝn×n is called nonnegative if aij ≥ 0 for all i, j ∈ 〈n〉. A matrix A = (aij) ∈ ℝn×nis called a Zmatrix if aij ≤ 0 for all ii = j. We will use Zn to denote the set of all n × n Zmatrices. A matrix A = (aij) ∈ Znis called an Mmatrix if A can be expressed in the form A = sIB, where B ≥ 0, and sρ(B), the spectral radius of B. If s > ρ(B), A is called a nonsingular Mmatrix; if s= ρ(B), A is called a singular Mmatrix. Mn, Mn and Mn0 will be used to denote the set of all n × n M−matrices, the set of all n × n nonsingular M−matrices and the set of all n × n singular M−matrices, respectively. It is easy to see that
Mn=MnMn0andMnMn0=.
The comparison matrix of a given matrix A = (aij) ∈ ℂn×n, denoted by μ(A) = (μij), is defined by
uij={|aii|,ifi=j,|aij|,ifij.
It is clear that μ(A) ∈ Zn for a matrix A ∈ ℂn×n. The set of equimodular matrices associated with A, denoted by ω(A) = {B ∈ ℂn×n: μ(B) = μ(A)}. Note that both A and μ(A) are in ω(A). A matrix A = (aij) ∈ ℂn×n is called a general Hmatrix if μ(A) ∈ Mn (See [1]). If μ(A)MnA is called a strong Hmatrix; if μ(A)Mn0, A is called a weak Hmatrix. Hn, HnS and HnW will denote the set of all n × n general H−matrices, the set of all n × n strong H−matrices and the set of all n × n weak H−matrices, respectively. Similarly to equalities (9), we have
Hn=HnSHnWandHnSHnW=.
For n2, an n × n complex matrix A is reducible if there exists an n × n permutation matrix P such that
PAPT=[A11A120A22],
where A11 is an r × r submatrix and A22 is an (nr) × (nr) submatrix, where 1r < n. If no such permutation matrix exists, then A is called irreducible. If A is a 1 × 1 complex matrix, then A is irreducible if its single entry is nonzero, and reducible otherwise.
A matrix A ∈ ℂn×nis called diagonally dominant by row if
|aii|j=1,jin|aij|
holds for all i ∈ 〈n〉. If inequality in (13) holds strictly for all i ∈ 〈n〉, A is called strictly diagonally dominant by row. If A is irreducible and the inequality in (13) holds strictly for at least one i ∈ 〈n〉, A is called irreducibly diagonally dominant by row. If (13) holds with equality for all i ∈ 〈n〉, A is called diagonally equipotent by row.

Dn(SDn, IDn) and DEn will be used to denote the sets of all n × n (strictly, irreducibly) diagonally dominant matrices and the set of all n × n diagonally equipotent matrices, respectively.

A matrix A ∈ ℂn×nis called generalized diagonally dominant if there exist positive constants αi, i ∈ 〈n〉, such that
αi|aii|j=1,jinαj|aij|
holds for all i ∈ 〈n〉. If inequality in (14) holds strictly for all i ∈ 〈n〉, A is called generalized strictly diagonally dominant. If (14) holds with equality for all i ∈ 〈n〉, A is called generalized diagonally equipotent.

We denote the sets of all n × n generalized (strictly) diagonally dominant matrices and the set of all n × n generalized diagonally equipotent matrices by GDn(GSDn) and GDEn, respectively.

(See [3032]). Let ADn(GDn). ThenAHnIif and only if A has no (generalized) diagonally equipotent principal submatrices. Furthermore, if ADn) ∩ Zn (GDnZn), thenAMnif and only if A has no (generalized) diagonally equipotent principal submatrices.

(See [31, 32]). DEnGDEnHnW and SDnIDnHnS=GSDn.

(See [6]). GDn ⊂ Hn

(See [6]). Let A ∈ ℂn×nbe irreducible. Then AHn if and only if AGDn

More importantly, under the condition of "reducibility", we have the following conclusion.

(See [6, 29]). Let A ∈ ℂn × nbe reducible. Then AHn if and only if in the Frobenius normal form of A
PAPT=[R11R12R1s0R22R2s00Rss],
each irreducible diagonal square block Rii is generalized diagonally dominant, where P is a permutation matrix, Rii = A(αi) is either 1 × 1 zero matrices or irreducible square matrices, Rij = A(αi, αj), ij, i, j = 1, 2,... ,s, further, αiαj = ∅ for ij, andi=1Sαi=n.

(See [6, 29]). A matrixAHnWif and only if in the Frobenius normal from (15) of A, each irreducible diagonal square block Rii is generalized diagonally dominant and has at least one generalized diagonally equipotent principal submatrix.

The following definitions and lemmas come from [28, 29].

Let E = (eiθrs) ∈ ℂn×n, whereeiθrs = cosrs + i sin θrs, i=1and θrs ∈ ℝ for all r, s ∈ 〈n〉. The matrix E = (eiθrs) ∈ ℂn × nis called a πray pattern matrix if

1. θrs+ θsr = 2kβ holds for all r, s ∈ 〈n〉, rs, where k ∈ ℤ;

2. θrsθrt = θts + (2k + 1)π holds for all r, s, t ∈ 〈nand rs, rt, ts, where k ∈ ℤ;

3. θrr = 0 for all r ∈ 〈n〉.

Any complex matrix A = (ars) ∈ ℂn × n has the following form:
A=ein.|A|Eiθ=(ein.|ars|eiθrs)n×n
where η ∈ ℝ, |A| = (|ars|) ∈ ℝn × nand E = (eiθrs) ∈ ℂn ×nwith θrs ∈ ℝ and θrr = 0 for r, s ∈ 〈n〉. The matrix Eiθ is called a ray pattern matrix of the matrix A. If the ray pattern matrix Eiθ of the matrix A given in (16) is a πray pattern matrix, then A is called a πray matrix.

Rnπ denote the set of all n × n π −ray matrices. Obviously, if a matrix ARnπ, then ξ.ARnπ for all ξ ∈ ℂ.

Let a matrix A = DALAUA = (ars) ∈ ℂn ×n with DA = diag(a11, a22, ... , ann). ThenARnπif and only if there exists an n × n unitary diagonal matrix D such that D−1AD = e.(|DA|−|LA|−|UA|) for η ∈ ℝ.

A matrix AHnW is singular if and only if the matrix A has at least either one zero principal submatrix or one irreducible principal submatrix A k = A(i1, i2, ... ,ik), 1 < kn, such thatDAK1AKGDEKRKπ, where DAk = diag(ai1i1,... ,aikik).

3 Main results

In numerical linear algebra, the successive overrelaxation iterative method, simultaneously introduced by Frankel (1950) [33] and Young (1950) [34], is a famous iterative method used to solve a linear system of equations. This iterative method is also called the accelerate Liebmann method by Frankel (1950) [33] and the other many subsequent researchers. Kahan (1958) [35] calls it the extrapolated Gauss-Seidel method. It is often called the method of systematic overrelaxation. Frankel showed that for the numerical solutation of the Dirichlet problem for a rectangle, successive overrelaxation iterative method gave substantially larger (by an order of magnitude) asymptotic rates of convergence than those for the point Jacobi and point Gauss-Seidel iterative methods with suitable chosen relaxation factor. Young (1950) [34] and Young (1954) [36] showed that these conclusions held more generally for matrices satisfying his definition of propertly A, and that these results could be rigorously applied to the iterative solution of matrix equations arising from discrete approximations to a large class of elliptic partial differential equations for general regions.

Later, this iterative method was developed as three iterative methods, i.e., the forward, backward and symmetric successive overrelaxation (FSOR-, BSOR- and SSOR-) iterative methods. Though these iterative methods can be applied to any matrix with non-zero elements on the diagonals, convergence is only guaranteed if the matrix is strictly or irreducibly diagonally dominant matrix, Hermitian positive definite matrix, strong H−matrix and consistently ordered p-cyclic matrix. Some classic results on convergence of SOR iterative methods are as follows:

(See [1315, 23]). Let ASDn ∪ IDn. Then ρ(HFSOR) < 1, ρ(HBSOR) < 1 and ρ(HSSOR) < 1, where HFSOR, HBSOR and HSSOR are defined in (5) , (6) and (7) , respectively, and therefore the sequence {x(i)} generated by the FSOR-, BSOR- and SSOR-scheme (3) , respectively, converges to the unique solution of (1) for any choice of the initial guess x(0).

(See [37]). LetAHnSThen the sequence {x(i)} generated by the FSOR-, BSOR- and SSOR-scheme (3) , respectively, converges to the unique solution of (1) for any choice of the initial guess x(0).

(See [4, 5, 38]). Let A ∈ ℂn × nbe a Hermitian positive definite matrix. Then the sequence {x(i)} generated by the FSOR-, BSOR- and SSOR-scheme (3) , respectively, converges to the unique solution of (1) for any choice of the initial guess x(0).

In this section, we mainly study convergence of SOR iterative methods for the linear systems with weak H-matrices.

Let A = ILU ∈ DE nbe irreducible. Then for ω ∈ (0, 1), ρ(HFSOR(ω)) < 1 and ρ(HBSOR(ω)) < 1, i.e. the sequence {x(i))} generated by the FSOR and BSOR iterative schemes (5) and (6) converges to the unique solution of (1) for any choice of the initial guess x(0)if and only if AARnπ.

The sufficiency can be proved by contradiction. We assume that there exists an eigenvalue λ of HFSOR such that |λ| ≥ 1. According to Equality (5),
det(λIHFSOR(ω))=det{λI(IωL)1[(1ω)I+ωU]}=det[(IωL)1]det{λ(IωL)[(1ω)I+ωU]}=det[(λ1+ω)IλωLωU]det(IωL)=(λ1+ω)det[Iλωλ1+ωLωλ1+ωU]det(IωL)=0.
Since |λ ≥ 1, λ − 1 + ω ≠ 0. Hence, equality (17) yields
detA(λ,ω)=det[Iλωλ1+ωLωλ1+ωU]=0
i.e.
A(λ,ω)=Iλωλ1+ωLωλ1+ωU
is singular. Set λ = μe with μ ≥ 1 and θR. Then 1 − cosθ ≥ 0, μ − 1 ≥ 0 and μ2 − 1 ≥ 0. Again, ω ∈ (0, 1) shows 1 − ω > 0. Therefore, we have
|λ1+ω|2|λω|2=|μeiθ1+ω2||μeiθω|2=μ22(1ω)μcosθ+(1ω)2μ2ω2=[μ2(1+ω)2μcosθ+1ω](1ω)=[μ2+12μcosθ+μ2ωω](1ω)[2μ2μcosθ+(μ21)ω](1ω)=[2μ(1cosθ)+(μ21)ω](1ω)0.
and
|λ1+ω|2ω2=|μeiθ1+ω|2ω2=μ22(1ω)μcosθ+(1ω)2ω2=[μ(1ω)]2ω2+2μ(1ω)2μ(1ω)cosθ=[μ1+ω]2ω2+2μ(1ω)(1cosθ)ω2ω20.
This shows that
|λωλ1+ω|1,|ωλ1+ω|1.
Since A = ILUDEn is irreducible, both L ≠ 0 and U ≠ 0. As a result, (19) and (22) indicate that A(λ, ω) ∈ Dn and irreducible. Again, since A(λ, ω) is singular and hence A(λ,ω)HnW, it follows from Lemma 2.12 that A(λ,ω)An0DEn, i.e. there exists an unitary diagonal matrix D such that
D1A(λ,ω)D=Iλωλ1+ωD1LDωλ1+ωD1UD=I|λωλ1+ω||L||ωλ1+ω||U|DEn.
Since A = ILU ∈ DE n,
μ(A)=I|L|π|U|DEn
(23) and (24) shows
|λωλ1+ω|=1,|ωλ1+ω|=1.

Because of |λ ≥ 1 and ω ∈ (0,1), the latter equality of (25) implies |λ| = 1. As a result, (25) and (19) show A(λ, ω) = ILU = A. From (23), it is easy to see A(λ,ω)=ARnπ. This contradicts ARnπ. Thus, ρ(HFSOR(ω)) < 1, i.e. FSOR-method converges.

The following will prove the necessity by contradiction. Assume that ARnπ. Then it follows from Lemma 2.11 that there exists an n × n unitary diagonal matrix D such that A = ILU = A = ID |L| D−1D |U| D−1 and
HFSOR(ω)=(IωL)1[(1ω)I+ωU]=D(Iω|L|)1[(1ω)I+ω|U|]D1.
Hence,
det(IHFSOR(ω))=det{I(IωL)1[(1ω)I+ωU]}=det{I(Iω|L|)1[(1ω)I+ω|U|]}=det[Iω|L|(1ω)Iω|U|]det(Iω|L|)=ωdet[I|U||L|]det(Iω|L|).

Sice A = ILU ∈ DE n and is irreducible, Lemma 2.4 shows that I|L||U|HnWRnπ and is irreducible. Lemma 2.12 shows that I − |L| − |U| is singular and hence det(I − |L| − |U|) = 0. Therefore, (27) yields det(IHFSOR(ω)) = 0, which shows that 1 is an eigenvalue of HFSOR(ω). Then, we have that ρ(HFSOR(ω)) ≥ 1, i.e. FSOR-method doesn’t converge. This is a contradiction. Thus, the assumption is incorrect and hence, ARnπ. This completes the necessity.

In the same way, we can prove that for ω ∈ (0,1), BSOR-method converges, i.e. ρ(HBSOR(ω)) < 1 if and only if AAn0. Here, we finish the proof. □

Let A = ILU = (aij) ∈ Dn with aii ≠ 0 for all i ∈ 〈n〉. Then for ω ∈ (0, 1), ρ(HFSOR(ω)) < 1 and ρ(HBSOR(ω) < 1, i.e. the sequence {x(i))} generated by the FSOR and BSOR iterative schemes (5) and (6) converges to the unique solution of (1) for any choice of the initial guess x(0)if and only if A is nonsingular.

The conclusion of this theorem is not difficult to be obtained form Lemma 2.1 2, Theorem 3.1 and Theorem 3.4. □

In what follows we will propose the convergence result for SSOR iterative method for linear system with weak H-matrices including nonstrictly diagonally dominant matrices. Firstly, the following lemma will be given for the convenience of the proof.

([32]). LetA=[EULF]C2n×2n, where E, F, L, UCn × nand E is nonsingular. Then A/E is nonsingular if and only if A is nonsingular, where A/E = FLE−1U is the Schur complement of A with respect to E.

Let A = ILUDEn be irreducible. Then for ω ∈ (0, 1), ρ(HSOR(ω)) < 1, i.e. the sequence {x(i))} generated by the SSOR iterative schemes (7) converges to the unique solution of (1) for any choice of the initial guess x(0)if and only if AARnπ.

The sufficiency can be proved by contradiction. We assume that there exists an eigenvalue λ of HSSOR such that |λ| ≥ 1. According to equalities (5), (6) and (7),
det(λIHSSOR(ω))=det{λI(IωU)1[(1ω)I+ωL](IωL)1[(1ω)I+ωU]}=det{λ(IωU)[(1ω)I+ωL](IωL)1[(1ω)I+ωU]}det(IωU)=λdet{(IωU)λ1[(1ω)I+ωL](IωL)1[(1ω)I+ωU]}det(IωU)=0.
Equality (28) gives
detB(λ,ω)=det{(IωU)λ1[(1ω)I+ωL](IωL)1[(1ω)I+ωU]}=0,
i.e.
B(λ,ω)=(IωU)λ1[(1ω)I+ωL](IωL)1[(1ω)I+ωU]
is singular. Let R = IωL, S = λ.IωU) , T = λ−1[(1 − ω)I + ωL], V = (1 − ω)I + ωU and
B=[RVTS]=[IωL[(1ω)I+ωU]λ1[(1ω)I+ωL]IωU].
Then, B(λ, ω) = ℬ/R is the Schur complement of with respect to the principal submatrix R. Since B(λ, ω) = ℬ/R is singular, Lemma 3.6 shows that is also singular. Again since A is irreducible, both L ≠ 0 and U ≠ 0. As a result, is also irreducible. Since A = ILU = (aij) ∈ DE n with unit diagonal entries,
1=j=1i1|aij|+j=i+1n|aij|,i=1,2,...,n.
Thus, for all ω ∈ (0, 1) and |λ ≥ 1, we have that both
1ωj=1i1|aij|(1ω)ωj=i+1n|aij|=ω(1j=1i1|aij|j=i+1n|aij|)=0
and
1ωj=1+1n|aij||λ|1[(1ω)+ωj=1i1|aij|]ω(1j=1i1|aij|j=i+1n|aij|)=0
hold for all iN = {1, 2,..., n}. Immediately, we obtain
1=ωj=1i1|aij|+(1ω)+ωj=i+1n|aij|,1ωj=i+1n|aij|+|λ|1[(1ω)+ωj=1i1|aij|],i=1,2,...,n.
(33) shows that D2n. Again, is irreducible and singular, and hence Lemma 2.4 shows that A(λ,ω)H2nW. Then, it follows from Lemma 2.12 that BDE2nB2nπ. Consequently |λ| = 1. Let λ = e with θR. Since BR2nπ, Lemma 2.12 shows that there exists an n × n unitary diagonal matrix D such that = diag(D, D) and
D~1BD~=D~1[IωL(1ω)I+ωUλ1[(1ω)I+ωL]IωU]D~1=D~1[IωL[(1ω)I+ωU]eiθ[(1ω)I+ωL]IωU]D~1=[IωD1LD[(1ω)I+ωD1UD]eiθ[(1ω)I+ωD1LD]IωD1UD]=[Iω|L|[(1ω)I+ω|U|]eiθ[(1ω)I+ω|L|]Iω|U|]=[Iω|L|[(1ω)I+ω|U|][(1ω)I+ω|L|]Iω|U|].

The latter two equalities of (34) indicate that θ = 2kβ; where k is an integer and thus λ = ei2kβ = 1, and there exists an n × n unitary diagonal matrix D such that D−1AD = I − |L| − |U|, i.e. AAn0. However, this contradicts AAn0. According to the proof above, we have that ρ(HSSOR(ω)) < 1, i.e. SSOR-method converges.

The following will prove the necessity by contradiction. Assume that AAn0. Then there exists an n × n unitary diagonal matrix D such that A = ILU = ID |L| D−1D |U| D−1 and hence
HSSOR(ω)=(IωU)1[(1ω)I+ωL](IωL)1[(1ω)I+ωU]=D{(Iω|U|)1[(1ω)I+ω|L|](1ω|L|)1[(1ω)I+ω|U|]}D1.
Thus,
det(IHSSOR(ω))=det{I(Iω|U|)1[(1ω)I+ω|L|](1ω|L|)1[(1ω)I+ω|U|]}=det{(Iω|U|)[(1ω)I+ω|L|](1ω|L|)1[(1ω)I+ω|U|]}det(Iω|U|).
Set C(ω) = (Iω |U|)− [(1−ω)I + ω|L|](Iω |L|)−1 [(1−ω)I + ω |U|] and let = Iω |L|, = Iω |U|, = (1 − ω)I + ω |L|, = (1 − ω)I + ω |U| and
B^=[R^V^T^S^]=[Iω|L|[(1ω)I+ω|U|][(1ω)I+ω|L|Iω|U|].
Then, C(ω) = ℬ̂/ is the Schur complement of ℬ̂ with respect to the principal submatrix . (33) and (37) show B^H2nWR2nπ. It comes from Lemma 2.12 that ℬ̂ is singular. Therefore, Lemma 3.6 yields that C(ω) is singular, i.e.
detC(ω)=det{(Iω|U|)[(1ω)I+ω|L|](Iω|L|)1[(1ω)I+ω|U|]}=0.
(36) gives det(IHSSOR) = 0, which shows that 1 is an eigenvalue of HSSOR(ω). Then, we have that ρ(HSSOR(ω)) ≥ 1, i.e. SSOR-method doesn’t converge. This is a contradiction. Thus, the assumption is incorrect and AAn0. This completes the proof. □

Let A = ILU = (aij) ∈ Dn with aii ≠ 0 for all i ∈ 〈n〉. Then for ω ∈ (0, 1), ρ(HSSOR(ω)) < 1, i.e. the sequence {x(i)} generated by the SSOR iterative schemes (7) converges to the unique solution of (1) for any choice of the initial guess x(0)if and only if A is nonsingular.

The conclusion of this theorem is easy to be obtained form Lemma 2.1 2, Theorem 3.1 and Theorem 3. 7. □

LetA=ILUHnWbe irreducible. Then for ω ∈ (0, 1), ρ(HFSOR(ω)) < 1 and ρ(HBSOR(ω)) < 1, i.e. the sequence {x(i)} generated by the FSOR and BSOR iterative schemes (5) and (6) converges to the unique solution of (1) for any choice of the initial guess x(0)if and only if AARnπ.

Since A=ILUHnW is irreducible, it follows form Lemma 2.3 and Lemma 2.6 that A = ILUGDEn. Then, from Definition 2. 2, there exists a positive diagonal matrix D such that = D−1AD = ID−1LDD−1UD = IDEn and is irreducible, where = D−1LD and = D−1UD. Theorem 3.4 shows that for ω ∈ (0, 1), ρ(FSOR(ω)) < 1 and ρ(BSOR(ω)) < 1, i.e. the sequence {x(i)} generated by the FSOR and BSOR iterative schemes (5) and (6) converges to the unique solution of (1) for any choice of the initial guess x(0) if and only if A^Rnπ. Again, from (5), we have
H^FSOR(ω)=(IωL^)1[(1ω)I+ωU^]=(IωD1LD)1[(1ω)I+ωD1UD]=D1{(IωL)1[(1ω)I+ωU]}D=D1HFSOR(ω)D.
In the same way, we can get
H^BSOR(ω)=D1HBSOR(ω)D.
from (6). Since A^=D1ADRnπ and D is a positive diagonal matrix, it follows from Definition 2.9 and Definition 2.10 that ARnπ. Therefore, for ω ∈ (0, 1), ρ(HFSOR(ω)) = ρ(FSOR(ω)) < 1 and ρ(HBSOR(ω)) = ρ(BSOR(ω)) < 1, i.e. the sequence {x(i)} generated by the FSOR and BSOR iterative schemes (5) and (6) converges to the unique solution of (1) for any choice of the initial guess x(0) if and only if ARnπ. This completes the proof.□

LetA=ILUHnWbe irreducible. Then for ω ∈ (0, 1), ρ(HSSOR(ω)) < 1, i.e. the sequence {x(i)} generated by the SSOR iterative schemes (7) converges to the unique solution of (1) for any choice of the initial guess x(0) if and only ifARnπ.

From (5), (6), (7), (38) and (39),
H^SSOR(ω)=D1[HBSOR(ω)HFSOR(ω)]D=D1HSSOR(ω)D.

Therefore, similarly as in the proof of Theorem 3. 9, we have with Theorem 3.7 that for ω ∈ (0, 1), ρ(HSSOR(ω)) = ρ(SSOR(ω)) < 1 i.e. the sequence {x(i)} generated by the SSOR iterative schemes (7) converges to the unique solution of (1) for any choice of the initial guess x(0) if and only if ARnπ. □

LetA=ILU=(aij)HnWwith aii ≠ 0 for all i ∈ 〈n〉. Then for ω ∈ (0, 1), ρ(HFSOR(ω)) < 1 and ρ(HBSOR(ω)) < 1, i.e. the sequence {x(i)} generated by the FSOR and BSOR iterative schemes (5) and (6) converges to the unique solution of (1) for any choice of the initial guess x(0)if and only if A is nonsingular.

If AHnW is irreducible, it follows from Theorem 3.9 that the conclusion of this theorem is true. If AHnW is reducible, since AHnW with aii ≠ 0 for all i ∈ 〈n〉, HFSOR(ω) and HBSOR(ω) exist and Theorem 2.7 shows that there exists some i ∈ 〈n〉 such that diagonal square block Rii in the Frobenius normal from (15) of A is irreducible and generalized diagonally equipotent. Let HFSORRii and HBSORRii denote the FSOR- and BSOR-iteration matrices associated with diagonal square block Rii. Direct computations give
ρ(HFSOR)=max1isρ(HFSORRii)andρ(HBSOR)=max1isρ(HBSORRii).

Since RiiGDEn is irreducible, Theorem 3.9 shows that for ω ∈ (0, 1), if ρ(HFSOR(ω))=maxiisρ(HFSORRii)<1 and ρ(HBSOR(ω))=maxiisρ(HBSORRii)<1, i.e. the sequence {x(i)} generated by the FSOR and BSOR iterative schemes (5) and (6) converges to the unique solution of (1) for any choice of the initial guess x(0), then Rii=A(αi)R|ai|π. Again, RiiGDEn but RiiR|ai|π. Lemma 2.12 shows Rii = A(αi) is nonsingular. However, it is easy to obtain that Rjj = A(αj) is nonsingular that satisfies ρ(HFSOR(ω)Rjj)=maxiisρ(HFSORRii)<1 and ρ(HBSOR(ω)Rjj)=maxiisρ(HBSORRii)<1. Thus, (15) shows that A is nonsingular. This completes the proof of the necessity.

Let us prove the sufficiency. Assume that A is nonsingular, then each diagonal square block Rii in the Frobenius normal from (15) of A is nonsingular for all i ∈ 〈n〉. Since AHnW, Theorem 2.7 shows that some diagonal square block Rii in the Frobenius normal from (15) of A is irreducible and generalized diagonally equipotent and the other diagonal square block Rjj is generalized strictly diagonally dominant or generalized irreducibly diagonally dominant. Again, each irreducible and generalized diagonally equipotent diagonal square block Rii is nonsingular. Lemma 2.12 yields that Rii=A(αi)R|αi|π. Then it follows from Theorem 3. 1, Theorem 3.2 and Theorem 3.9 that ρ(HFSOR(ω))=max1<i<sρ(HFSORRii)<1 and ρ(HBSOR(ω))=max1<i<sρ(HBSORRii)<1, i.e. the sequence x(i))} generated by the FSOR and BSOR iterative schemes (5) and (6) converges to the unique solution of (1) for any choice of the initial guess x(0). This completes the proof. □

A=ILU=(aij)HnWwithaii ≠ 0 for all i ∈ 〈n〉. Then for ω ∈ (0, 1), ρ(HSSOR(ω)) < 1, i.e. the sequence {x(i)} generated by the SSOR iterative schemes (7) converges to the unique solution of (1) for any choice of the initial guess x(0)if and only if A is nonsingular.

Similar to the proof of Theorem 3.1 1, the conclusion of this theorem is easy to be obtained form Theorem 3.1, Theorem 3.2 and Theorem 3.1 0. □

4 Numerical examples

In this section, some numerical examples are given to demonstrate the convergence results obtained in this paper.

Let the coefficient matrix A of linear system (1) be given by the following n × n matrix
An=[1100000121000001210000012000000021000001210000011].
It is easy to see that AnDEn is irreducible. Since
Dn1AnDn=|DAn||LAn|+|UAn|,
where
Dn=diag[1,1,,(1)K1,,(1)n1],
it follows from Lemma 2.11 that AnRnπ. Then, Lemma 2.12 shows that An is nonsingular. Therefore, Theorem 3.4 and Theorem 3.7 show that for ω ∈ (0, 1),
ρ(HFSOR(ω))<1,ρ(HBSOR(ω))<1andρ(HSSOR(ω))<1,
i.e. the sequence {x(i))} generated by the FSOR-, BSOR- and SSOR-iterative schemes (5), (6) and (7) converges to the unique solution of (1) for any choice of the initial guess x(0).

In what follows, the computations on the spectral radii ρ1 = ρ(HFSOR(ω)) , ρ2 = ρ(HBSOR(ω)) and ρ3 = ρ(HSSOR(ω)) of FSOR-, BSOR- and SSOR-iterative matrices for A100 were performed on PC computer with Matlab 7.0 to verify that the results above are true. The computational results are shown in Table 1.

Table 1

The comparison of spectral radii of FSOR-, BSOR- and SSOR-iterative matrices with different ω

ω00.10.20.30.40.50.60.70.80.91
ρ110.9000.8000.7020.6060.5140.4300.3580.5140.8901.284
ρ210.9000.8000.7020.6060.5140.4300.3580.5140.8901.284
ρ310.8080.6240.4490.3100.1190.3240.7801.5863.4571.640

It is shown in Table 1 and Fig. 1 that: (i) the changes of ρ(HFSOR(ω)) and ρ(HBSOR(ω)) are identical with ω increasing. They gradually decrease from 1 to 0.358 with ω increasing from 0 to 0.7 while they gradually increase from 0.358 to 1.284 with ω increasing from 0.7 to 1. This shows the optimal value of ω should be ωopt ∈ (0:50; 0:80) such that the SOR iterative method converges faster to the unique solution of (1) for any choice of the initial guess x0.

Fig. 1
Fig. 1

The change of spectral radii of FSOR-, BSOR- and SSOR-iterative matrices with different ω.

Citation: Open Mathematics 14, 1; 10.1515/math-2016-0065

(ii) ρ(HSSOR(ω)) performs better than ρ(HFSOR(ω)) and (HBSOR(ω)). It decreases quickly from 1 to 0.119 with ω increasing from 0 to 0.5 while it increases fast from 0.119 to 1.640 with ω increasing from 0.5 to 1. The optimal value of ω should be ωopt ∈ (0:40; 0:60) such that the SOR iterative method converges faster to the unique solution of (1) for any choice of the initial guess x0. It follows from Table 1 and Fig. 1 that SSOR iterative method is superior to the other two SOR iterative methods.

Let the coefficient matrix A of linear system (1) be given by the following 6 × 6 matrix
An=[511111151111115111000211000121000112].
Although ADE6 are reducible, there is not any principal submatrix Ak(k < 6) in A such that DAk1AkRkπ, Lemma 2.12 shows that A is nonsingular. Therefore, Theorem 3.5 and Theorem 3.8 show that for ω ∈ (0, 1),
ρ=ρ(HFSOR(ω))<1,ρ2=ρ(HBSOR(ω))<1andρ3=ρ(HSSOR(ω))<1,
i.e. the sequence {x(i))} generated by the FSOR-, BSOR- and SSOR-iterative schemes (5), (6) and (7) converges to the unique solution of (1) for any choice of the initial guess x(0).

The computations on Matlab 7.0 of PC yield some comparison results on the spectral radius of SSOR iterative matrices, see Table 2.

Table 2

The comparison of spectral radii of FSOR-, BSOR- and SSOR-iterative matrices with different ω

ω00.10.20.30.40.50.60.70.80.91
ρ110.9240.8460.7650.6790.6200.5730.5270.5100.7881.092
ρ210.9240.8490.7740.7000.6280.5610.4980.4490.7121.000
ρ310.8490.6970.5430.4170.3320.3510.3420.4430.5320.585

It is shown in Table 2 and Fig. 2 that (i) the change of ρ(HFSOR(ω)) is similar to the one of ρ(HBSOR(ω)). They gradually decrease to their minimal values then gradually increase from their minimal values with ω increasing from 0 to 1. This shows the optimal value of ω for FSOR- and BSOR-iterative method should be ωopt ∈ (0:70; 0:90). But ρ(HFSOR(ωopt)) > ρ(HFSOR(ωopt)) shows that the BSOR iterative method converges much faster than the FSOR does to the unique solution of (1) for any choice of the initial guess x0.

Fig. 2
Fig. 2

The change of spectral radii of FSOR-, BSOR- and SSOR-iterative matrices with different ω

Citation: Open Mathematics 14, 1; 10.1515/math-2016-0065

(ii) Similarly as in Example 1, ρ(HSSOR(ω)) performs better than ρ(HFSOR(ω)) and ρ(HBSOR(ω)). It decreases quickly from 1 to 0.332 with ω increasing from 0 to 0.5 while it quickly increases near about ω = 0.5 and then decreases gradually to 0.342 at ω = 0.7. Finally, it increases quickly from 0.342 to 0.585 with ω increasing from 0.7 to 1. The optimal value of ω should be ωopt ∈ (0.40, 0.60) such that the SOR iterative method converges faster to the unique solution of (1) for any choice of the initial guess x0. It follows from Table 2 and Fig. 2 that SSOR iterative method is superior to the other two SOR iterative methods.

5 Further work

In this paper some necessary and sufficient conditions are proposed such that SOR iterative methods, including FSOR, BSOR and SSOR iterative methods, are convergent for linear systems with weak H-matrices. The class of weak H-matrices with singular comparison matrices is a subclass of general H−matrices [29] and has some theoretical problems. In particular, the convergence problem on AOR iterative methods for this class of matrices is an open problem and is a focus of our further work.

Acknowledgement

This work is supported by the National Natural Science Foundations of China (Nos.11201362, 11601409, 11271297), the Natural Science Foundation of Shaaxi Province of China (No. 2016JM1009) and the Science Foundation of the Education Department of Shaanxi Province of China (No. 14JK1305).

References

  • [1]

    Berman A., Plemmons R.J.: Nonnegative Matrices in the Mathematical Sciences. Academic, New York, 1979.

  • [2]

    Demmel J.W.: Applied Numerical Linear Algebra. SIAM Press, 1997.

  • [3]

    Golub G.H., Van Loan C.F.: Matrix Computations, third ed. Johns Hopkins University Press, Baltimore, 1996.

  • [4]

    Saad Y.: Iterative methods for sparse linear systmes. PWS publishing Company, Boston, 1996.

  • [5]

    Varga R.S.: Matrix Iterative Analysis, Second ed. Springer-Verlag, Berlin, Heidelberg, 2000.

  • [6]

    Bru R., Corral C., Gimenez I. and Mas J.: Classes of general H-matrices. Linear Algebra Appl., 429(2008), 2358-2366.

  • [7]

    Bru R., Corral C., Gimenez I. and Mas J.: Schur coplement of general H –matrices. Numer. Linear Algebra Appl., 16(2009), 935-974.

  • [8]

    Bru R., Gimenez I. and Hadjidimos A.: Is A ∈ ℂn,n a general H –matrices. Linear Algebra Appl., 436(2012), 364-380.

  • [9]

    Cveković L.J., Herceg D.: Convergence theory for AOR method. Journal of Computational Mathematics, 8(1990), 128-134.

  • [10]

    Darvishi M.T., Hessari P.: On convergence of the generalized AOR method for linear systems with diagonally dominant coef?cient matrices. Applied Mathematics and Computation, 176(2006), 128-133.

  • [11]

    Evans D.J., Martins M.M.: On the convergence of the extrapolated AOR method. Internat. J. Computer Math, 43(1992), 161-171.

  • [12]

    Gao Z.X., Huang T.Z.: Convergence of AOR method. Applied Mathematics and Computation, 176(2006), 134-140.

  • [13]

    Hadjidimos A.: Accelerated overrelaxation method. Mathematics of Computation. 32(1978), 149-157.

  • [14]

    James K.R., Riha W.: Convergence Criteria for Successive Overrelaxation. SIAM Journal on Numerical Analysis. 12(1975), 137-143.

  • [15]

    James K.R.: Convergence of Matrix Iterations Subject to Diagonal Dominance. SIAM Journal on Numerical Analysis. 10(1973), 478-484.

  • [16]

    Li W.: On nekrasov matrices. Linear Algebra Appl., 281(1998), 87-96.

  • [17]

    Martins M.M.: On an Accelerated Overrelaxation Iterative Method for Linear Systems With Strictly Diagonally Dominant Matrix. Mathematics of Computation, 35(1980), 1269-1273.

  • [18]

    Ortega J.M., Plemmons R.J.: Extension of the Ostrowski-Reich theorem for SOR iterations. Linear Algebra Appl., 28(1979), 177-191.

  • [19]

    Plemmons R.J.: M –matrix characterization I: Nonsingular M –matrix. Linear Algebra Appl., 18(1977), 175-188.

  • [20]

    Song Y.Z.: On the convergence of the MAOR method, Journal of Computational and Applied Mathematics, 79(1997), 299-317.

  • [21]

    Song Y.Z.: On the convergence of the generalized AOR method. Linear Algebra and its Applications, 256(1997), 199-218.

  • [22]

    Tian G.X., Huang T.Z., Cui S.Y.: Convergence of generalized AOR iterative method for linear systems with strictly diagonally dominant matrices. Journal of Computational and Applied Mathematics, 213(2008), 240-247.

  • [23]

    Varga R.S.: On recurring theorems on diagonal dominance. Linear Algebra Appl., 13(1976), 1-9.

  • [24]

    Wang X.M.: Convergence for the MSOR iterative method applied to H-matrices. Applied Numerical Mathematics, 21(1996), 469-479.

  • [25]

    Wang X.M.: Convergence theory for the general GAOR type iterative method and the MSOR iterative method applied to H-matrices. Linear Algebra and its Applications, 250(1997), 1-19.

  • [26]

    Xiang S.H., Zhang S.L.: A convergence analysis of block accelerated over-relaxation iterative methods for weak block H-matrices to partition ?, Linear Algebra and its Applications. 418(2006), 20-32.

  • [27]

    Young D.M.: Iterative solution of large linear systmes. Academic Press, New York, 1971.

  • [28]

    Zhang C.Y., Xu F.M., Xu Z.B., Li J.C.: General H-matrices and their Schur complements. Frontiers of Mathematics in China, 9(2014), 1141-1168.

  • [29]

    Zhang C.Y., Ye D., Zhong C.L., Luo S.H.: Convergence on Gauss-Seidel iterative methods for linear systems with general H-matrices. Electronic Journal of Linear Algebra, 30(2015), 843-870.

  • [30]

    Zhang C.Y. and Li Y.T.: Diagonal Dominant Matrices and the Determing of H matrices and M matrices, Guangxi Sciences, 12(2005), 1161-164.

  • [31]

    Zhang C.Y., Xu C.X., Li Y.T.: The Eigenvalue Distribution on Schur Complements of H matrices, Linear Algebra Appl., 422(2007), 250-264.

  • [32]

    Zhang C.Y., Luo S.H., Xu C.X., Jiang H.Y.: Schur complements of generally diagonally dominant matrices and criterion for irreducibility of matrices. Electronic Journal of Linear Algebra, 18(2009), 69-87.

  • [33]

    Frankel S.P.: Convergence rates of iterative treatments of partial differential equations. Math. Tables Aids Comput., 4(1950), 65-75.

  • [34]

    Young D.M.: Iterative methods for solving partial differential equations of elliptic type. Doctoral Thesis, Harvard University, Cambridge, MA, 1950.

  • [35]

    Kahan W.: Gauss-Seidel methods of solving large systems of linear equations. Doctoral Thesis, University of Toronto, Toronto, Canada, 1958.

  • [36]

    Young D.M.: Iterative methods for solving partial differential equations of elliptic type. Trans. Amer. Math. Soc. 76(1954) 92-111.

  • [37]

    Neumaier A. and Varga R.S.: Exact convergence and divergence domains for the symmetric successive overrelaxation iterative (SSOR) method applied to H -matrices. Linear Algebra Appl., 58(1984), 261-272.

  • [38]

    Meurant G.: Computer Solution of Large Linear Systems. Studies in Mathematics and its Applications, Vol. 28, North-Holland Publishing Co., Amsterdam, 1999.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Berman A., Plemmons R.J.: Nonnegative Matrices in the Mathematical Sciences. Academic, New York, 1979.

  • [2]

    Demmel J.W.: Applied Numerical Linear Algebra. SIAM Press, 1997.

  • [3]

    Golub G.H., Van Loan C.F.: Matrix Computations, third ed. Johns Hopkins University Press, Baltimore, 1996.

  • [4]

    Saad Y.: Iterative methods for sparse linear systmes. PWS publishing Company, Boston, 1996.

  • [5]

    Varga R.S.: Matrix Iterative Analysis, Second ed. Springer-Verlag, Berlin, Heidelberg, 2000.

  • [6]

    Bru R., Corral C., Gimenez I. and Mas J.: Classes of general H-matrices. Linear Algebra Appl., 429(2008), 2358-2366.

  • [7]

    Bru R., Corral C., Gimenez I. and Mas J.: Schur coplement of general H –matrices. Numer. Linear Algebra Appl., 16(2009), 935-974.

  • [8]

    Bru R., Gimenez I. and Hadjidimos A.: Is A ∈ ℂn,n a general H –matrices. Linear Algebra Appl., 436(2012), 364-380.

  • [9]

    Cveković L.J., Herceg D.: Convergence theory for AOR method. Journal of Computational Mathematics, 8(1990), 128-134.

  • [10]

    Darvishi M.T., Hessari P.: On convergence of the generalized AOR method for linear systems with diagonally dominant coef?cient matrices. Applied Mathematics and Computation, 176(2006), 128-133.

  • [11]

    Evans D.J., Martins M.M.: On the convergence of the extrapolated AOR method. Internat. J. Computer Math, 43(1992), 161-171.

  • [12]

    Gao Z.X., Huang T.Z.: Convergence of AOR method. Applied Mathematics and Computation, 176(2006), 134-140.

  • [13]

    Hadjidimos A.: Accelerated overrelaxation method. Mathematics of Computation. 32(1978), 149-157.

  • [14]

    James K.R., Riha W.: Convergence Criteria for Successive Overrelaxation. SIAM Journal on Numerical Analysis. 12(1975), 137-143.

  • [15]

    James K.R.: Convergence of Matrix Iterations Subject to Diagonal Dominance. SIAM Journal on Numerical Analysis. 10(1973), 478-484.

  • [16]

    Li W.: On nekrasov matrices. Linear Algebra Appl., 281(1998), 87-96.

  • [17]

    Martins M.M.: On an Accelerated Overrelaxation Iterative Method for Linear Systems With Strictly Diagonally Dominant Matrix. Mathematics of Computation, 35(1980), 1269-1273.

  • [18]

    Ortega J.M., Plemmons R.J.: Extension of the Ostrowski-Reich theorem for SOR iterations. Linear Algebra Appl., 28(1979), 177-191.

  • [19]

    Plemmons R.J.: M –matrix characterization I: Nonsingular M –matrix. Linear Algebra Appl., 18(1977), 175-188.

  • [20]

    Song Y.Z.: On the convergence of the MAOR method, Journal of Computational and Applied Mathematics, 79(1997), 299-317.

  • [21]

    Song Y.Z.: On the convergence of the generalized AOR method. Linear Algebra and its Applications, 256(1997), 199-218.

  • [22]

    Tian G.X., Huang T.Z., Cui S.Y.: Convergence of generalized AOR iterative method for linear systems with strictly diagonally dominant matrices. Journal of Computational and Applied Mathematics, 213(2008), 240-247.

  • [23]

    Varga R.S.: On recurring theorems on diagonal dominance. Linear Algebra Appl., 13(1976), 1-9.

  • [24]

    Wang X.M.: Convergence for the MSOR iterative method applied to H-matrices. Applied Numerical Mathematics, 21(1996), 469-479.

  • [25]

    Wang X.M.: Convergence theory for the general GAOR type iterative method and the MSOR iterative method applied to H-matrices. Linear Algebra and its Applications, 250(1997), 1-19.

  • [26]

    Xiang S.H., Zhang S.L.: A convergence analysis of block accelerated over-relaxation iterative methods for weak block H-matrices to partition ?, Linear Algebra and its Applications. 418(2006), 20-32.

  • [27]

    Young D.M.: Iterative solution of large linear systmes. Academic Press, New York, 1971.

  • [28]

    Zhang C.Y., Xu F.M., Xu Z.B., Li J.C.: General H-matrices and their Schur complements. Frontiers of Mathematics in China, 9(2014), 1141-1168.

  • [29]

    Zhang C.Y., Ye D., Zhong C.L., Luo S.H.: Convergence on Gauss-Seidel iterative methods for linear systems with general H-matrices. Electronic Journal of Linear Algebra, 30(2015), 843-870.

  • [30]

    Zhang C.Y. and Li Y.T.: Diagonal Dominant Matrices and the Determing of H matrices and M matrices, Guangxi Sciences, 12(2005), 1161-164.

  • [31]

    Zhang C.Y., Xu C.X., Li Y.T.: The Eigenvalue Distribution on Schur Complements of H matrices, Linear Algebra Appl., 422(2007), 250-264.

  • [32]

    Zhang C.Y., Luo S.H., Xu C.X., Jiang H.Y.: Schur complements of generally diagonally dominant matrices and criterion for irreducibility of matrices. Electronic Journal of Linear Algebra, 18(2009), 69-87.

  • [33]

    Frankel S.P.: Convergence rates of iterative treatments of partial differential equations. Math. Tables Aids Comput., 4(1950), 65-75.

  • [34]

    Young D.M.: Iterative methods for solving partial differential equations of elliptic type. Doctoral Thesis, Harvard University, Cambridge, MA, 1950.

  • [35]

    Kahan W.: Gauss-Seidel methods of solving large systems of linear equations. Doctoral Thesis, University of Toronto, Toronto, Canada, 1958.

  • [36]

    Young D.M.: Iterative methods for solving partial differential equations of elliptic type. Trans. Amer. Math. Soc. 76(1954) 92-111.

  • [37]

    Neumaier A. and Varga R.S.: Exact convergence and divergence domains for the symmetric successive overrelaxation iterative (SSOR) method applied to H -matrices. Linear Algebra Appl., 58(1984), 261-272.

  • [38]

    Meurant G.: Computer Solution of Large Linear Systems. Studies in Mathematics and its Applications, Vol. 28, North-Holland Publishing Co., Amsterdam, 1999.

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    The change of spectral radii of FSOR-, BSOR- and SSOR-iterative matrices with different ω.

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    The change of spectral radii of FSOR-, BSOR- and SSOR-iterative matrices with different ω