0 Introduction

Consider the linear complementarity problem abbreviated as LCP(q, A), for finding a pair of real vectors r and z∈Rⁿ such that

$ \mathit{\boldsymbol{r}}: = \mathit{\boldsymbol{Az}} + \mathit{\boldsymbol{q}} \ge 0,\mathit{\boldsymbol{z}} \ge 0\;{\rm{and}}\;{\mathit{\boldsymbol{z}}^{\rm{T}}}\left( {\mathit{\boldsymbol{Az}} + \mathit{\boldsymbol{q}}} \right) = 0, $

(1)

where A=(a_ij)R^n×n is a given large, sparse and real matrix and q=(q₁, q₂, …, q_n)^T∈Rⁿ is a given real vector. Here, z^T and ≥ denote the transpose of the vector z and the componentwise defined partial ordering between two vectors, respectively.

Many problems in scientific computing and engineering applications may lead to solutions of LCPs of the form (1). For example, the linear complementarity problem may arise from application problems such as the convex quadratic programming, the Nash equilibrium point of the bimatrix game, the free boundary problems of fiuid dynamics etc^[1-2]. Some solvers for LCP(q, A) with a special matrix A were proposed ^[3-12]. Recently, many researches have focused on the solver of LCP(q, A) with an algebra equation ^[8-22]. In particular, BAI et al^[8-9]proposed a modulus-based matrix multisplitting iterative method for solving LCP(q, A) and presented convergence analysis for the proposed methods. ZHANG et al^[13]extended the condition of a compatible H splitting to that of an H-splitting. LI ^[23] extended the modulus-based matrix splitting iteration method to more general cases. BAI ^[24] presented parallel matrix block multisplitting relaxation iterative methods, and established the convergence theory of these new methods in a thorough manner. LI et al ^[25] studied synchronous block multisplitting iterative methods. ZHANG et al^[15] generalized the methods of BAI et al^[26] and studied modulus-based synchronous multisplitting multi-parameters methods for linear complemen-tarity problem.

In this paper, we generalize the corresponding methods of BAI et al ^[26] and ZHANG et al^[15] from point form to block form according to the modulus-based synchronous multisplitting iteration methods and consider relaxed modulus-based synchronous multisplitting multi-parameter method based on block splitting for solving LCP(q, A). Moreover, we give some theoretical analysis and improve some existing convergence results in [25-26].

The rest of this paper is organized as follows: In section 1, we give some notations and lemmas. In section 2, we propose relaxed modulus-based synchronous block multisplitting multi-parameters method for solving LCP(q, A). In section 3, we give the convergence analysis for the proposed method.

1 Notations and lemmas

In order to study mudulus-based synchronous block multisplitting iteration methods for solving LCP(q, A), let us introduce some definitions and lemmas.

A matrix A=(a_ij) is called an M-matrix if a_ij≤0 for i≠j and A^-1≥0. The comparison matrix 〈A〉=(α_ij) of matrix A=(a_ij) is defined by: α_ij=|a_ij|, if i=j; α_ij=-|a_ij|, if i≠j. A is called an H-matrix if 〈A〉 is an M-matrix and is called an H₊-matrix if it is an H-matrix with positive diagonal entries^[20]. Let ρ(A) denote the spectral radius of A. A representatio n A=M-N is called a splitting of A when M is nonsingular. Let A and B be M-matrix. If A≤B, then A^-1≥B^-1. Finally, we define R₊ⁿ={x|x≥0, x∈Rⁿ}.

Definition 1^[24]    Define the set:

(1) L_{n, I}(n₁, n₂, …, n_p)={A=A_ij∈L_n(n₁, n₂, …, n_p)|A_ii∈L(Rⁿⁱ) is nonsingular (i=1, 2, …, p)};

(2) L_{n, I}^d(n₁, n₂, …, n_p)={A=diag(A₁₁, A₂₂, …, A_pp)|A_ii∈L(Rⁿⁱ) is nonsingular (i=1, 2, …, p)}.

Definition 2^[27]    Let A∈L_{n, I}(n₁, n₂, …, n_p), and (Ⅰ)-type block comparison matrix 〈M〉=(〈M〉_ij)∈L(Rⁿ) and (Ⅱ)-type block comparison matrix 〈〈M〉〉=(〈〈M〉〉_ij)∈L(Rⁿ) are defined as

$ {\left\langle \mathit{\boldsymbol{M}} \right\rangle _{ij}} = \left\{ \begin{array}{l} \left\| {M_{ii}^{ - 1}} \right\| - 1,\;\;\;i = j\\ - \left\| {{M_{ij}}} \right\|,\;\;\;i \ne j \end{array} \right.i,j = 1,2, \cdots ,p, $

$ {\left\langle {\left\langle \mathit{\boldsymbol{M}} \right\rangle } \right\rangle _{ij}} = \left\{ \begin{array}{l} 1,\;\;\;i = j\\ - \left\| {M_{ii}^{ - 1}{M_{ij}}} \right\|,i \ne j \end{array} \right.i,j = 1,2, \cdots ,p, $

respectively.

Moreover, based on block matrix A∈L_{n, I}(n₁, n₂, …, n_p), and L∈L_{n, I}(n₁, n₂, …, n_p), let D(L)=diag (L₁₁, L₂₂, …, L_pp), B(L)=D(L)-L, J(A)=D(A)^-1B(A), μ₁(A)=ρ(J_〈A〉), μ₂(A)=ρ(I-〈〈A〉〉), using definition 2, we easily verify

$ \begin{array}{*{20}{c}} {\left\langle {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{J}}\left( \mathit{\boldsymbol{A}} \right)} \right\rangle = \left\langle {\left\langle {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{J}}\left( \mathit{\boldsymbol{A}} \right)} \right\rangle } \right\rangle = \left\langle {\left\langle \mathit{\boldsymbol{A}} \right\rangle } \right\rangle ,}\\ {{\mu _2}\left( \mathit{\boldsymbol{A}} \right) \le {\mu _2}\left( \mathit{\boldsymbol{A}} \right).} \end{array} $

Definition 3^[27]    Let A∈L_{n, I}(n₁, n₂, …, n_p), if there exist P, Q∈L_{n, I}^d(n₁, n₂, …, n_p), such that 〈PAQ〉 is M-matrix, then A is called (Ⅰ)-type block H-matrix (H_B^(Ⅰ)(P, Q)-matrix) about nonsingular block matrices P, Q; Such that〈〈PAQ〉〉 is M-matrix, then A is called (Ⅱ)-type block H-matrix (H_B^(Ⅱ)(P, Q)-matrix) about nonsingular block matrices P, Q.

Definition 4^[27]    Let A∈L_{n, I}(n₁, n₂, …, n_p), then [A]=(‖M_ij‖)∈L(Rⁿ) is called block absolute value of block matrix A. Similarly, we may define block absolute value of block vector x∈V_n(n₁, n₂, …, n_p) as [x]∈Rⁿ.

Lemma 1^[24]    Let A, B∈L_{n, I}(n₁, n₂, …, n_p), x, y∈V_n(n₁, n₂, …, n_p), γ∈R¹, then

(1) |[A]-[B]|≤[A+B]≤[A]+[B](|[x]-[y]|)≤[x+y]≤[x]+[y];

(2) [AB]≤[A][B]([Ax]≤[A][x]);

(3) [γA]≤|γ|[A]([γx]≤|γ|[x]), q∈Rⁿ;

(4) ρ(A)≤ρ(|A|)≤ρ([A]).

Lemma 2^[24]    Let A, B∈L_{n, I}(n₁, n₂, …, n_p) is H_B^Ⅰ(P, Q)-matrix, then

(1) A is nonsingular;

(2) [(PAQ)^-1]≤〈PAQ〉^-1;

(3) μ₁(PAQ) < 1.

Lemma 3^[24]    Let A, B∈L_{n, I}(n₁, n₂, …, n_p) is H^Ⅱ_B(P, Q)-matrix, then

(1) A is nonsingular;

(2) [(PAQ)^-1]≤〈〈PAQ〉〉^-1[D(PAQ)^-1];

(3) μ₂(PAQ) < 1.

Definition 5^[24]    Define the set:

(1) Ω_B^I(M)={F=(F_ij)∈L_{n, I}(n₁, n₂, …, n_p)|‖F_ii^-1‖=‖M_ii^-1‖, ‖F_ij‖=‖M_ij‖(i≠j), i, j=1, 2, …, p};

(2) Ω_B^Ⅱ(M)={F=(F_ij)∈L_{n, I}(n₁, n₂, …, n_p)|‖F_ii^-1F_ij‖=‖M_ii^-1M_ij‖, i, j=1, 2, …, p}.

Express the same mode set of (Ⅰ)-type and (Ⅱ)-type associated with the matrix M=(M_ij)∈L_{n, I}(n₁, n₂, …, n_p), respectively.

Lemma 4^[28]     Let A be an H-matrix. Then A is nonsingular, and |A^-1|≤〈A〉^-1.

Lemma 5^[29]    Let A=(a_ij)∈Z^n×n which has all positive diagonal entries. A is an M-matrix if and only if ρ(B) < 1, where B=D^-1C, D=diag(A), A=D-C.

Lemma 6^[5]    Let A∈R^n×n be an H₊-matrix. Then LCP(q, A) has a unique solution for any q∈Rⁿ.

Lemma 7^[8]    Let A=M-N be a splitting of the matrix A∈R^n×n, Ω be a positive diagonal matrix, and γ be a positive constant. Then, for the LCP(q, A), the following statements hold true:

(ⅰ) If (z, r) is a solution of the LCP(q, A), then $x = \frac{1}{2}\gamma (z - {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^{ - 1}}r)$ satisfies the implicit fixed-point equation

$ \left( {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \mathit{\boldsymbol{M}}} \right)\mathit{\boldsymbol{x}} = \mathit{\boldsymbol{Nx}} + \left( {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} - \mathit{\boldsymbol{A}}} \right)\left| \mathit{\boldsymbol{x}} \right| - \gamma q; $

(2)

(ⅱ) If x satisfies the implicit fixed-point equation (2), then

$ \mathit{\boldsymbol{z}} = {\gamma ^{ - 1}}\left( {\left| \mathit{\boldsymbol{x}} \right| + \mathit{\boldsymbol{x}}} \right),r = {\gamma ^{ - 1}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}\left( {\left| \mathit{\boldsymbol{x}} \right| - \mathit{\boldsymbol{x}}} \right) $

(3)

is a solution of the LCP(q, A).

2 RMS-BMMTOR methods

Firstly, we introduce the concept of multisplitting method and the detailed process of parallel iterative method.

{M_k, N_k, E_k}_k=1^l is a multisplitting of block matrix A if

1) A=M_k-N_K, det(M_k)≠0 is a splitting for k=1, 2, …, l;

2) E_k=diag(E₁₁^k, E₂₂^k, …, E_pp^k), k=1, 2, …, l, and $\sum\limits_{k = 1}^l {\left\| {\mathit{\boldsymbol{E}}_{ii}^{(k)}} \right\|} = 1$, i=1, 2, …, p, where the block matrices M_k, N_k, E_k∈L_n(n₁, n₂, …, n_p), and ‖·‖ expressing consistent matrix norm satisfying ‖I‖=1(I∈L(R^m) is an unit matrix).

Given a positive diagonal matrix Ω and a positive constant γ, form lemma 7, we know that if x satisfies either of the implicit fixed-point equations

$ \begin{array}{l} \left( {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + {\mathit{\boldsymbol{M}}_k}} \right)\mathit{\boldsymbol{x}} = {\mathit{\boldsymbol{N}}_k}\mathit{\boldsymbol{x}} + \left( {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} - \mathit{\boldsymbol{A}}} \right)\left| {\mathit{\boldsymbol{x}} - \gamma \mathit{\boldsymbol{q}}} \right.,\\ \;\;\;\;\;k = 1,2, \cdots ,l, \end{array} $

(4)

then

$ \mathit{\boldsymbol{z}} = {\gamma ^{ - 1}}\left( {\left| \mathit{\boldsymbol{x}} \right| + \mathit{\boldsymbol{x}}} \right),\mathit{\boldsymbol{r}} = {\gamma ^{ - 1}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}\left( {\left| \mathit{\boldsymbol{x}} \right| - \mathit{\boldsymbol{x}}} \right) $

(5)

is a solution of the LCP(q, A).

Based on block matrix A ∈ R^m×m, the corresponding block diagonal matrix is D= diag(A₁₁, A₂₂, …, A_pp), and L_k, F_k is block strictly triangular matrices, U_k =D-L_k-F_k A, then (D-L_k-F_k, U_k, E_k) is a multisplitting of block matrix A∈R^m×m. With the equivalent reformulations (4), (5) and TOR method^[16] of the LCP(q, A), we can establish the following relaxed modulus-based synchronous block multisplitting multi-parameters TOR method (RMSBMMTOR), which is similar to method 3.1 in [2] and method 3.1 in [15].

Method 1(The RMS-BMMTOR method for LCP(q, A))

Let {M_k, N_k, E_k}_k=1^l be a multisplitting of the system matrix A∈R^n×n. Given an initial vector x⁽⁰⁾∈Rⁿ for m=0, 1, …, until the iteration sequence {z^(m)}_m=0^∞⊂R₊ⁿ is convergent, compute z^(m+1)∈R₊ⁿ and x^(m+1)∈R₊ⁿ by

$ {\mathit{\boldsymbol{z}}^{\left( {m + 1} \right)}} = \frac{1}{\gamma }\left( {\left| {{\mathit{\boldsymbol{x}}^{\left( {m + 1} \right)}}} \right| + {\mathit{\boldsymbol{x}}^{\left( {m + 1} \right)}}} \right) $

and x^{(m, k)}∈Rⁿ according to

$ {\mathit{\boldsymbol{x}}^{\left( {m + 1} \right)}} = \omega \sum\limits_{k = 1}^l {{\mathit{\boldsymbol{E}}_k}{\mathit{\boldsymbol{x}}^{\left( {m,k} \right)}}} + \left( {1 - \mathit{\boldsymbol{\omega }}} \right){\mathit{\boldsymbol{x}}^{\left( m \right)}}, $

where x^{(m, k)}, k=1, 2, …, l are obtained by solving the linear systems

$ \begin{array}{l} \left( {{\alpha _k}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \mathit{\boldsymbol{D}} - \left( {{\beta _k}{\mathit{\boldsymbol{L}}_k} + {\gamma _k}{\mathit{\boldsymbol{F}}_k}} \right)} \right){x^{\left( {m,k} \right)}} = \left[ {\left( {1 - {\alpha _k}} \right)\mathit{\boldsymbol{D + }}} \right.\\ \;\;\;\;\;\;\;\;\left. {\left( {{\alpha _k} - {\beta _k}} \right){\mathit{\boldsymbol{L}}_k} + \left( {{\alpha _k} - {\gamma _k}} \right){\mathit{\boldsymbol{F}}_k} + {\alpha _k}{\mathit{\boldsymbol{U}}_k}} \right]{x^{\left( m \right)}} + \\ \;\;\;\;\;\;\;\;{\alpha _k}\left[ {\left( {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} - \mathit{\boldsymbol{A}}} \right)\left| {{x^{\left( m \right)}}} \right| - \gamma q} \right],\;\;\;\;\;\;k = 1,2, \cdots ,l, \end{array} $

(6)

respectively.

Remark 1    In method 1, when the coefficient matrix A is point form and α_k = α, β_k = β, γ_k = γ, ω =1, the RMS-BMMTOR method reduces to the MSMTOR method; When the coefficient matrix A is point form and α_k = α, β_k = β = γ_k, ω = 1, the RMS-BMMTOR method reduces to the MSMAOR method^[26]; When the coefficient matrix A is point form and ω=1, the RMS-BMMTOR method reduces to the MSMMTOR method; When the coefficient matrix A is point form and γ_k = β_k, ω =1, the RMS-BMMTOR method reduces to the MSMMAOR method ^[15]; When ω = 1, the RMS-BMMTOR method reduces to the MSBMMTOR method; When γ_k = β_k, ω =1, the RMS-BMMTOR method reduces to the MSBMMAOR method ^[25]; When the parameters (α_k, β_k, γ_k, ω) = (α_k, α_k, α_k, 1), (1, 1, 1, 1) and (1, 0, 0, 1), the RMS-BMMTOR method reduces to the MSBMMSOR, MSBMGS and MSBMJ methods, respectively; When the parameters (α_k, β_k, ω) = (α_k, α_k, α_k, ω), (1, 1, 1, ω) and (1, 0, 0, ω), the RMS-BMMTOR method reduces to the RMS-BMMSOR, RMSBMGS and RMSBMJ methods, respectively.

3 Convergence analysis

Based on the modulus-based synchronous multisplitting AOR method, BAI et al^[26] obtained the following results in 2013.

Theorem 1    Let A∈R^n×n be an H₊-matrix, with D=diag(A) and B=D-A, and let (M_k, N_k, E_k)(k=1, 2, …, l) and (D-L_k, U_k, E_k)(k=1, 2, …, l) be a multisplitting and a triangular multisplitting of the matrix A, respectively. Assume that γ>0 and the positive diagonal matrix Ω satisfies Ω≥D. If A=D-L_k-U_k(k=1, 2, …, l) satisfies 〈A〉=D-|L_k|-|U_k|(k=1, 2, …, l), then the iteration sequence {z^(m)}_m=0^∞ generated by the MSMAOR iteration method converges to the unique solution z_* of LCP(q, A) for any initial vector z⁽⁰⁾∈R₊ⁿ, provided relaxation parameters α and β satisfy

$ 0 < \beta \le \alpha < \frac{1}{{\rho \left( {{D^{ - 1}}\left| B \right|} \right)}}. $

Based on the modulus-based synchronous block multisplitting AOR method, LI et al^[25] analyzed the following results in 2013.

Theorem 2^[25]    Let A∈L_{n, I}(n₁, n₂, …, n_p) be a block H_B^I(P, Q)-matrix, with H∈Ω_B^I(PAQ), and let (M_k, N_k, E_k)(k=1, 2, …, l) and (D-L_k, U_k, E_k)(k=1, 2, …, l) be a block multisplitting and a block triangular multisplitting of block H-matrix, respectively. Assume that γ>0 and the positive matrix Ω satisfies Ω≥D(H) and diag(Ω)=diag(D(H)). If H=D-L_k-U_k(k=1, 2, …, l) satisfies 〈H〉=〈D〉-[L_k]-[U_k]=D_〈H〉-B_〈H〉(k=1, 2, …, l), then the iteration sequence {z_m=0^(m)∞} generated by the MSBMAOR iteration method converges to the unique solution z_* of LCP(q, A) for any initial vector z⁽⁰⁾∈R₊ⁿ, provided the relaxation parameters α_k and β_k satisfy

$ 0 < \beta \le \alpha < \frac{1}{{{\mu _1}\left( {\mathit{\boldsymbol{PAQ}}} \right)}}. $

Based on the modulus-based synchronous multisplitting multi-parameters AOR method, ZHANG et al^[15] gave the following results in 2014.

Theorem 3^[15]    Let A∈R^n×n be an H₊-matrix, with D=diag(A) and B=D-A, and let (M_k, N_k, E_k)(k=1, 2, …, l) and (D-L_k, U_k, E_k)(k=1, 2, …, l) be a multisplitting and a triangular multisplitting of the matrix A, respectively. Assume that γ>0 and the positive diagonal matrix Ω satisfies Ω≥D. If A=D-L_k-U_k(k=1, 2, …, l) satisfies 〈A〉=D-|L_k|-|U_k|(k=1, 2, …, l), then the iteration sequence {z^(m)}_m=0^∞ generated by the MSMMAOR iteration method converges to the unique solution z_* of LCP(q, A) for any initial vector z⁽⁰⁾∈R₊ⁿ, provided relaxation parameters α_k and β_k satisfy

$ \begin{array}{*{20}{c}} {0 < {\beta _k} \le {\alpha _k} \le 1\;{\rm{or}}\;{\rm{0}} < {\beta _k} < \frac{1}{{\rho \left( {{\mathit{\boldsymbol{D}}^{ - 1}}\left| \mathit{\boldsymbol{B}} \right|} \right)}},}\\ {1 < {\alpha _k} < \frac{1}{{\rho \left( {{\mathit{\boldsymbol{D}}^{ - 1}}\left| \mathit{\boldsymbol{B}} \right|} \right)}}.} \end{array} $

$ 0 \le {\beta _k} \le {\gamma _k},0 \le {\alpha _k} \le {\gamma _k},0 < {\gamma _k} < \frac{1}{\rho },0 < \omega < \frac{1}{{{\rho _{{\gamma _k}}}}}, $

Based on the global relaxed non-stationary multisplitting multi-parameter TOR method (GRNMMTOR) for the large sparse linear systems, ZHANG et al ^[16] obtained the following convergence theorem in 2008.

Theorem 4^[16]    Let A∈R^n×n be an H-matrix, and for k=1, 2, …, l, L_k and F_k be strictly lower triangular matrices. Define the matrix U_k, k=1, 2, …, l, such that A=D-L_k-F_k-U_k. Assume that we have 〈A〉=|D|-|L_k|-|F_k|-|U_k|=|D|-|B|. If

$ 0 \le {\beta _k} \le {\gamma _k},0 \le {\alpha _k} \le {\gamma _k},0 < {\gamma _k} < \frac{1}{\rho },0 < \omega < \frac{1}{{{\rho _{{\gamma _k}}}}}, $

then GRNMMTOR method converges for any initial vector x⁽⁰⁾, where ρ=ρ(J), J=|D|^-1×|B|, ρ_γk=$\mathop {\max }\limits_{1 \le k \le \alpha } \{ |1 - {\gamma _k}| + {\gamma _k}{\rho _\varepsilon }\} $, q(m, k)≥1, m=0, 1, …, k=1, 2, …, l.

Based on relaxed modulus-based synchronous block multisplitting multi-parameters TOR method, we will present a convergence results of the multisplitting methods for the linear complementarity problem when the system matrix is a block H₊-matrix, which is as follows:

Theorem 5    Let A∈L_{n, I}(n₁, n₂, …, n_p) be a block H_B^I(P, Q)-matrix with H∈Ω_B^I(PAQ), and let (M_k, N_k, E_k)(k=1, 2, …, l) and (D-L_k, U_k, E_k)(k=1, 2, …, l) be a block multisplitting and a block triangular multisplitting of block H-matrix, respectively. Assume that γ>0 and the positive matrix Ω satisfy Ω≥D(H) and diag(Ω)=diag(D(H)). If H=D-L_k-U_k(k=1, 2, …, l) satisfies 〈H〉=〈D-〉-[L_k]-[U_k]=D_〈H〉-B_〈H〉(k=1, 2, …, l), then the iteration sequence {z_m=0^(m)∞} generated by the RMSBMMTOR iteration method converges to the unique solution z_* of LCP(q, A) for any initial vector z⁽⁰⁾∈R₊ⁿ, prcers α_k and β_k satisfy

$ \begin{array}{l} 0 < {\beta _k},{\gamma _k} \le {\alpha _k} \le 1,0 < \omega < \frac{2}{{1 + \rho '}}{\rm{or}}\\ 0 < {\beta _k},{\gamma _k} < \frac{1}{{{\mu _1}\left( {\mathit{\boldsymbol{PAQ}}} \right)}},1 < {\alpha _k} < \frac{1}{{{\mu _1}\left( {\mathit{\boldsymbol{PAQ}}} \right)}},\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0 < \omega < \frac{2}{{1 + \rho '}}, \end{array} $

(7)

where μ₁(PAQ)=ρ(D_〈H〉^-1B_〈H〉)=ρ(J_〈H〉), ρ′=$\mathop {\max }\limits_{1 \le k \le l} ${1-2α_k+2α_kρ_ε, 2δ_kρ_ε-1, 2α_kρ_ε-1}, δ_k=max{β_k, γ_k}. Moreover, β_k, γ_k should be greater or less than α_k at once.

Proof    From lemma 6 and (6), for the RMS-BMMTOR method, it holds that

$ \begin{array}{l} \left( {{\alpha _k}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \mathit{\boldsymbol{\bar D}} - \left( {{\beta _k}{{\mathit{\boldsymbol{\bar L}}}_k} + {\gamma _k}{{\mathit{\boldsymbol{\bar F}}}_k}} \right)} \right){x_ * } = \left[ {\left( {1 - {\alpha _k}} \right)\mathit{\boldsymbol{\bar D + }}} \right.\\ \;\;\;\;\;\;\left. {\left( {{\alpha _k} - {\beta _k}} \right){{\mathit{\boldsymbol{\bar L}}}_k} + \left( {{\alpha _k} - {\gamma _k}} \right){{\mathit{\boldsymbol{\bar F}}}_k} + {\alpha _k}{{\mathit{\boldsymbol{\bar U}}}_k}} \right]{x_ * } + \\ \;\;\;\;\;\;{\alpha _k}\left[ {\left( {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} - \mathit{\boldsymbol{H}}} \right)\left| {{x_ * }} \right| - \gamma q} \right],k = 1,2, \cdots ,l. \end{array} $

(8)

By subtracting (8) and (6), we have

$ \begin{array}{l} {x^{\left( {m + 1} \right)}} - {x_ * } = \omega \sum\limits_{k = 1}^l {{\mathit{\boldsymbol{E}}_k}{{\left( {{\alpha _k}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \mathit{\boldsymbol{\bar D}} - \left( {{\beta _k}{{\mathit{\boldsymbol{\bar L}}}_k} + {\gamma _k}{{\mathit{\boldsymbol{\bar F}}}_k}} \right)} \right)}^{ - 1}}} \times \\ \;\;\;\;\;\;\left[ {\left( {1 - {\alpha _k}} \right)\mathit{\boldsymbol{\bar D}} + \left( {{\alpha _k} - {\beta _k}} \right){{\mathit{\boldsymbol{\bar L}}}_k} + \left( {{\alpha _k} - {\gamma _k}} \right){{\mathit{\boldsymbol{\bar F}}}_k} + } \right.\\ \;\;\;\;\;\;\left. {{\alpha _k}{{\mathit{\boldsymbol{\bar U}}}_k}} \right]\left( {{x^{\left( m \right)}} - {x_ * }} \right) + \omega \sum\limits_{k = 1}^l {{E_k}\left( {{\alpha _k}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \mathit{\boldsymbol{\bar D}} - \left( {{\beta _k}{{\mathit{\boldsymbol{\bar L}}}_k} + } \right.} \right.} \\ \;\;\;\;\;\;\left. {\left. {{\gamma _k}{{\mathit{\boldsymbol{\bar F}}}_k}} \right)} \right) - 1{\alpha _k}\left( {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} - \mathit{\boldsymbol{H}}} \right)\left( {\left| {{x^{\left( m \right)}}} \right| - \left| {{x_ * }} \right|} \right) + \\ \;\;\;\;\;\;\left( {1 - \omega } \right)\left( {{x^{\left( m \right)}} - {x_ * }} \right). \end{array} $

(9)

By taking absolute values on both sides of the equality (9), estimating |[x^(m)]-[x_*]|≤[x^(m)-x_*] and amplifying, we may obtain

$ \begin{array}{l} \left[ {{x^{\left( m \right)}} - {x_ * }} \right] \le \omega \sum\limits_{k = 1}^l {\left[ {{\mathit{\boldsymbol{E}}_k}} \right]\left[ {\left( {{\alpha _k}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \mathit{\boldsymbol{\bar D}} - \left( {{\mathit{\boldsymbol{\beta }}_k}{{\mathit{\boldsymbol{\bar L}}}_k} + } \right.} \right.} \right.} \\ \;\;\;\;\;\;\left. {{{\left. {\left. {{\gamma _k}{{\mathit{\boldsymbol{\bar F}}}_k}} \right)} \right)}^{ - 1}}} \right]\left[ {\left| {1 - {\alpha _k}} \right|\left[ {\mathit{\boldsymbol{\bar D}}} \right] + \left| {{\alpha _k} - {\beta _k}} \right|\left[ {{{\mathit{\boldsymbol{\bar L}}}_k}} \right] + } \right.\\ \;\;\;\;\;\;\left. {\left| {{\alpha _k} - {\gamma _k}} \right|\left[ {{{\mathit{\boldsymbol{\bar F}}}_k}} \right] + {\alpha _k}\left[ {{{\mathit{\boldsymbol{\bar U}}}_k}} \right]} \right]\left[ {{x^{\left( m \right)}} - {x_ * }} \right] + \\ \;\;\;\;\;\;\omega \sum\limits_{k = 1}^l {\left[ {{\mathit{\boldsymbol{E}}_k}} \right]\left[ {{{\left( {{\alpha _k}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \mathit{\boldsymbol{\bar D}} - \left( {{\beta _k}{{\mathit{\boldsymbol{\bar L}}}_k} + {\gamma _k}{{\mathit{\boldsymbol{\bar F}}}_k}} \right)} \right)}^{ - 1}}} \right]} \times \\ \;\;\;\;\;\;{\alpha _k}\left[ {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} - \mathit{\boldsymbol{H}}} \right]\left[ {{x^{\left( m \right)}} - {x_ * }} \right] + \left| {1 - \omega } \right|\left[ {{x^{\left( m \right)}} - {x_ * }} \right]. \end{array} $

Since[Ω-H]=〈Ω〉-(D_〈H〉-B_〈H〉) and B_〈H〉=[L_k]+[F_k]+[U_k], [D]≥D_〈H〉, we have

$ \begin{array}{l} \left[ {{x^{\left( m \right)}} - {x_ * }} \right] \le \omega \sum\limits_{k = 1}^l {\left[ {{E_k}} \right]\left[ {\left( {{\alpha _k}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \mathit{\boldsymbol{\bar D}} - } \right.} \right.} \\ \;\;\;\;\;\left. {{{\left. {\left( {\left| {{\alpha _k} - {\gamma _k}} \right| + {\alpha _k}} \right)\left[ {{{\mathit{\boldsymbol{\bar L}}}_k}} \right]} \right)}^{ - 1}}} \right]\left[ {\left| {1 - {\alpha _k}} \right|\left[ {\mathit{\boldsymbol{\bar D}}} \right] + } \right.\\ \;\;\;\;\;\left| {{\alpha _k} - {\beta _k}} \right|\left[ {{{\mathit{\boldsymbol{\bar L}}}_k}} \right] + \left| {{\alpha _k} - {\gamma _k}} \right|\left[ {{{\mathit{\boldsymbol{\bar F}}}_k}} \right] + {\alpha _k}\left[ {{{\mathit{\boldsymbol{\bar U}}}_k}} \right] + \\ \;\;\;\;\;\left. {{\alpha _k}\left[ {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} - \mathit{\boldsymbol{H}}} \right]} \right]\left[ {{x^{\left( m \right)}} - {x_ * }} \right] + \left( {1 - \omega } \right)\left[ {{x^{\left( m \right)}} - {x_ * }} \right] \le \\ \;\;\;\;\;\omega \sum\limits_{k = 1}^l {\left[ {{\mathit{\boldsymbol{E}}_k}} \right]{{\left( {{\alpha _k}\left\langle \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \right\rangle + {\mathit{\boldsymbol{D}}_{\left\langle H \right\rangle }} - \left( {{\beta _k}\left[ {{\mathit{\boldsymbol{L}}_k}} \right] + {\gamma _k}\left[ {{\mathit{\boldsymbol{F}}_k}} \right]} \right)} \right)}^{ - 1}}} \times \\ \;\;\;\;\;\left[ {\left( {\left| {1 - {\alpha _k}} \right| - {\alpha _k}} \right){\mathit{\boldsymbol{D}}_{\left\langle H \right\rangle }} + \left( {\left| {{\alpha _k} - {\gamma _k}} \right| + {\alpha _k}} \right)\left[ {{{\mathit{\boldsymbol{\bar L}}}_k}} \right] + } \right.\\ \;\;\;\;\;\left( {\left| {{\alpha _k} - {\gamma _k}} \right| + {\alpha _k}} \right)\left[ {{{\mathit{\boldsymbol{\bar F}}}_k}} \right] + 2{\alpha _k}\left[ {{{\mathit{\boldsymbol{\bar U}}}_k}} \right] + \\ \;\;\;\;\;\left. {{\alpha _k}\left\langle \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \right\rangle } \right]\left[ {{x^{\left( m \right)}} - {x_ * }} \right] + \left( {1 - \omega } \right)\left[ {{x^{\left( m \right)}} - {x_ * }} \right] = \\ \;\;\;\;\;\omega \sum\limits_{k = 1}^l {\left[ {{\mathit{\boldsymbol{E}}_k}} \right]{\mathit{\boldsymbol{H}}_{{\rm{RMSBMMTOR}}}}\left[ {{x^{\left( m \right)}} - {x_ * }} \right] + } \\ \;\;\;\;\;\left( {1 - \omega } \right)\left[ {{x^{\left( m \right)}} - {x_ * }} \right] = \left\{ {\omega \sum\limits_{k = 1}^l {\left[ {{\mathit{\boldsymbol{E}}_k}} \right]{\mathit{\boldsymbol{H}}_{{\rm{RMSBMMTOR}}}}} + } \right.\\ \;\;\;\;\;\left. {\left( {1 - \omega } \right)\mathit{\boldsymbol{I}}} \right\}\left[ {{x^{\left( m \right)}} - {x_ * }} \right] = \\ \;\;\;\;\;{{\bar H}_{{\rm{RMSBMMTOR}}}}\left[ {{x^{\left( m \right)}} - {x_ * }} \right], \end{array} $

(10)

where

$ \begin{array}{l} {{\mathit{\boldsymbol{\bar H}}}_{{\rm{RMSBMMTOR}}}} = \omega \sum\limits_{k = 1}^l {\left[ {{\mathit{\boldsymbol{E}}_k}} \right]{\mathit{\boldsymbol{H}}_{{\rm{RMSBMMTOR}}}}} + \left( {1 - \omega } \right)\mathit{\boldsymbol{I}},\\ {\mathit{\boldsymbol{H}}_{{\rm{RMSBMMTOR}}}} = {\alpha _k}\left\langle \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \right\rangle + {\mathit{\boldsymbol{D}}_{\left\langle H \right\rangle }} - \left( {{\beta _k}\left[ {{\mathit{\boldsymbol{L}}_k}} \right] + } \right.\\ \;\;\;\;\;\;\;{\left. {\left. {{\gamma _k}\left[ {{\mathit{\boldsymbol{F}}_k}} \right]} \right)} \right)^{ - 1}}\left[ {\left( {\left| {1 - {\alpha _k}} \right| - {\alpha _k}} \right){\mathit{\boldsymbol{D}}_{\left\langle H \right\rangle }} + \left( {\left| {{\alpha _k} - {\beta _k}} \right| + } \right.} \right.\\ \;\;\;\;\;\;\;\left. {\left. {{\alpha _k}} \right)\left[ {{{\mathit{\boldsymbol{\bar L}}}_k}} \right] + \left( {\left| {{\alpha _k} - {\gamma _k}} \right| + {\alpha _k}} \right)\left[ {{{\mathit{\boldsymbol{\bar F}}}_k}} \right] + 2{\alpha _k}\left[ {{{\mathit{\boldsymbol{\bar U}}}_k}} \right]{\alpha _k}\left\langle \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \right\rangle } \right]. \end{array} $

(11)

The error relationship (10) is the base for proving the convergence of RMS-BMMTOR method. By making use of lemmas 1 and 2, defining ε^(m)=x^(m)-x_* and arranging similar terms together, we can obtain

$ \begin{array}{l} {\varepsilon ^{\left( {m + 1} \right)}} = \left[ {{x^{\left( {m + 1} \right)}} - {x_ * }} \right] \le {{\mathit{\boldsymbol{\bar H}}}_{{\rm{RMSBMMTOR}}}}\left[ {{x^{\left( {m + 1} \right)}} - {x_ * }} \right] = \\ \;\;\;\;\;\;\;\;\left\{ {\omega \sum\limits_{k = 1}^l {\left[ {{\mathit{\boldsymbol{E}}_k}} \right]{\mathit{\boldsymbol{H}}_{{\rm{RMSBMMTOR}}}}} + \left( {1 - \omega } \right)\mathit{\boldsymbol{I}}} \right\}\left[ {{x^{\left( {m + 1} \right)}} - {x_ * }} \right]. \end{array} $

Case 1    Let 0 < β_k, γ_k≤α_k≤1, $0 < \omega < \frac{2}{{1 + \rho ' }}$. Define

$ {\mathit{\boldsymbol{M}}_k} = {\alpha _k}\left\langle \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \right\rangle + {\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }} - \left( {{\beta _k}\left[ {{\mathit{\boldsymbol{L}}_k}} \right] + {\gamma _k}\left[ {{\mathit{\boldsymbol{F}}_k}} \right]} \right), $

$ \begin{array}{l} \mathit{\boldsymbol{N}}_k^1 = \left[ {\left( {\left| {1 - {\alpha _k}} \right| - {\alpha _k}} \right){\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }} + \left( {\left| {{\alpha _k} - {\beta _k}} \right| + {\alpha _k}} \right)\left[ {{{\mathit{\boldsymbol{\bar L}}}_k}} \right] + } \right.\\ \;\;\;\;\;\;\;\;\left( {\left| {{\alpha _k} - {\gamma _k}} \right| + {\alpha _k}} \right)\left[ {{{\mathit{\boldsymbol{\bar F}}}_k}} \right] + 2{\alpha _k}\left[ {{{\mathit{\boldsymbol{\bar U}}}_k}} \right] + {\alpha _k}\left\langle \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \right\rangle = \\ \;\;\;\;\;\;\;\;\left( {1 - 2{\alpha _k}} \right){\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }} + \left( {2{\alpha _k} - {\beta _k}} \right)\left[ {{{\mathit{\boldsymbol{\bar L}}}_k}} \right] + \\ \;\;\;\;\;\;\;\;\left( {2{\alpha _k} - {\gamma _k}} \right)\left[ {{{\mathit{\boldsymbol{\bar F}}}_k}} \right] + 2{\alpha _k}\left[ {{{\mathit{\boldsymbol{\bar U}}}_k}} \right] + {\alpha _k}\left\langle \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \right\rangle = \\ \;\;\;\;\;\;\;\;{\mathit{\boldsymbol{M}}_k} - 2{\alpha _k}{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }} + 2{\alpha _k}{\mathit{\boldsymbol{B}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}. \end{array} $

So, we have

$ \begin{array}{l} {\mathit{\boldsymbol{H}}_{{\rm{RMS - BMMTOR}}}} = \mathit{\boldsymbol{M}}_k^{ - 1}\mathit{\boldsymbol{N}}_k^1 = \mathit{\boldsymbol{M}}_k^{ - 1}\left( {{\mathit{\boldsymbol{M}}_k} - 2{\alpha _k}{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }} + } \right.\\ \;\;\;\;\;\;\;\left. {2{\alpha _k}{\mathit{\boldsymbol{B}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}} \right) = \mathit{\boldsymbol{I}} - 2{\alpha _k}\mathit{\boldsymbol{M}}_k^{ - 1}\left( {{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }} - {5_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}} \right). \end{array} $

Through further analysis, we have

$ \begin{array}{l} \left[ {{\mathit{\boldsymbol{H}}_{{\rm{RMS - BMMTOR}}}}} \right] \le \mathit{\boldsymbol{M}}_k^{ - 1}\left[ {{\mathit{\boldsymbol{M}}_k} - 2{\alpha _k}\left( {{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }} - {\mathit{\boldsymbol{B}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}} \right)} \right] \le \\ \;\;\;\;\;\;\;\mathit{\boldsymbol{I}} - 2{\alpha _k}\mathit{\boldsymbol{M}}_k^{ - 1}{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}\left( {\mathit{\boldsymbol{I}} - {\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}^{ - 1}{\mathit{\boldsymbol{B}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}} \right). \end{array} $

Since A∈L_{n, I}(n₁, n₂, …, n_p) is a block H_B^I(P, Q)-matrix, by lemmas 2 and 3, we know μ₁(PAQ)=ρ(D_〈H〉^-1B_〈H〉)=ρ(J_〈H〉) < 1, J_ε=J_〈H〉+εee^T, where e denotes the vector e=(1, 1, …, 1)^T∈Rⁿ. Since J_ε is nonnegative, the matrix J_〈H〉+εee^T has only positive entries and irreducible for any ε>0. By the Perron-Frobenius theorem, for any ε>0, there is a vector x_ε>0 such that

$ \left( {{\mathit{\boldsymbol{J}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }} + \varepsilon \mathit{\boldsymbol{e}}{\mathit{\boldsymbol{e}}^{\rm{T}}}} \right){x_\varepsilon } = {\rho _\varepsilon }{x_\varepsilon }, $

where ρ_ε=ρ(J_〈H〉+εee^T)=ρ(J_ε). Moreover, if ε>0 is small enough, we have ρ_ε < 1 by continuity of the spectral radius. Because of 0 < α_k≤1, we also have 1-2α_k+2α_kρ < 1, and 1-2α_k+2α_kρ_ε < 1. So

$ \begin{array}{l} \left[ {{\mathit{\boldsymbol{H}}_{{\rm{RMSBMMTOR}}}}} \right] \le I - 2{\alpha _k}\mathit{\boldsymbol{M}}_k^{ - 1}{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}\left[ {\mathit{\boldsymbol{I}} - \left( {\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }^{ - 1}{\mathit{\boldsymbol{B}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }} + } \right.} \right.\\ \;\;\;\;\;\;\left. {\left. {\varepsilon \mathit{\boldsymbol{e}}{\mathit{\boldsymbol{e}}^{\rm{T}}}} \right)} \right] = I - 2{\alpha _k}\mathit{\boldsymbol{M}}_k^{ - 1}{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}\left[ {\mathit{\boldsymbol{I}} - {\mathit{\boldsymbol{J}}_\varepsilon }} \right]. \end{array} $

Multiplying x_ε in two sides of the above inequality, and M_k^-1≥D_〈H〉^-1, we can obtain

$ \begin{array}{l} \left[ {{\mathit{\boldsymbol{H}}_{{\rm{RMS - BMMTOR}}}}} \right]{\mathit{\boldsymbol{x}}_\varepsilon } \le {\mathit{\boldsymbol{x}}_\varepsilon } - 2{\alpha _k}\mathit{\boldsymbol{M}}_k^{ - 1}{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}\left[ {1 - \rho \left( {{\mathit{\boldsymbol{J}}_\varepsilon }} \right)} \right]{\mathit{\boldsymbol{x}}_\varepsilon } = \\ \;\;\;\;\;\;\;{\mathit{\boldsymbol{x}}_\varepsilon } - 2{\alpha _k}\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }^{ - 1}{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}\left[ {1 - \rho \left( {{\mathit{\boldsymbol{J}}_\varepsilon }} \right)} \right]{\mathit{\boldsymbol{x}}_\varepsilon } = \\ \;\;\;\;\;\;\;\left( {1 - 2{\alpha _k} + 2{\alpha _k}\rho \left( {{\mathit{\boldsymbol{J}}_\varepsilon }} \right)} \right){\mathit{\boldsymbol{x}}_\varepsilon }. \end{array} $

Based on E_k and the definition of [·], we know that $\sum\limits_{k = 1}^l {[{\mathit{\boldsymbol{E}}_k}]} = \mathit{\boldsymbol{I}}$. By (11), we have

$ \begin{array}{l} \left[ {{{\mathit{\boldsymbol{\bar H}}}_{{\rm{RMS - BMMTOR}}}}} \right]{\mathit{\boldsymbol{x}}_\varepsilon } = \omega \sum\limits_{k = 1}^l {\left[ {{\mathit{\boldsymbol{E}}_k}} \right]\left( {1 - 2{\alpha _k} + 2{\alpha _k}\rho \left( {{\mathit{\boldsymbol{J}}_\varepsilon }} \right)} \right){x_\varepsilon } + } \\ \;\;\;\;\;\;\;\;\left| {1 - \omega } \right|{\mathit{\boldsymbol{x}}_\varepsilon } \le \omega \left( {1 - 2{\alpha _k} + 2{\alpha _k}{\rho _\varepsilon }} \right){\mathit{\boldsymbol{x}}_\varepsilon } + \\ \;\;\;\;\;\;\;\;\left| {1 - \omega } \right|{\mathit{\boldsymbol{x}}_\varepsilon } \le \left( {\omega \rho ' + \left| {1 - \omega } \right|} \right){\mathit{\boldsymbol{x}}_\varepsilon } = {\theta _1}{\mathit{\boldsymbol{x}}_\varepsilon }\left( {\varepsilon \to {0^ + }} \right), \end{array} $

where θ₁=ωρ′+|1-ω| < 1.

Case 2 0 < β_k, ${\gamma _k} < \frac{1}{{{\mu _1}(\mathit{\boldsymbol{PAQ}})}}$,

$ 1 < {\alpha _k} < \frac{1}{{{\mu _1}\left( {\mathit{\boldsymbol{PAQ}}} \right)}},\;\;\;0 < \omega < \frac{2}{{1 + \rho '}}. $

Subcase 1    α_k≥β_k and α_k≥γ_k. Define

$ \begin{array}{l} \mathit{\boldsymbol{N}}_k^2 = \left( {\left| {1 - {\alpha _k}} \right| - {\alpha _k}} \right){\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }} + \left( {\left| {{\alpha _k} - {\beta _k}} \right| + {\alpha _k}} \right)\left[ {{{\mathit{\boldsymbol{\bar L}}}_k}} \right] + \\ \;\;\;\;\;\;\left( {\left| {{\alpha _k} - {\gamma _k}} \right| + {\alpha _k}} \right)\left[ {{{\mathit{\boldsymbol{\bar F}}}_k}} \right] + 2{\alpha _k}\left[ {{{\mathit{\boldsymbol{\bar U}}}_k}} \right] + {\alpha _k}\left\langle \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \right\rangle = \\ \;\;\;\;\;\;{\mathit{\boldsymbol{M}}_k} - 2{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }} + 2{\alpha _k}{\mathit{\boldsymbol{B}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}. \end{array} $

So

$ \begin{array}{l} \left[ {{\mathit{\boldsymbol{H}}_{{\rm{RMS - BMMTOR}}}}} \right] \le \mathit{\boldsymbol{M}}_k^{ - 1}\left[ {{\mathit{\boldsymbol{M}}_k} - 2\left( {{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }} - {\alpha _k}{\mathit{\boldsymbol{B}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}} \right)} \right] \le \\ \;\;\;\;\;\;\mathit{\boldsymbol{I}} - 2\mathit{\boldsymbol{M}}_k^{ - 1}{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}\left( {\mathit{\boldsymbol{I}} - {\alpha _k}{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}^{ - 1}{\mathit{\boldsymbol{B}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}} \right). \end{array} $

Similar to the case 1, let e denote the vector e=(1, 1, …, 1)^T∈Rⁿ, and x_ε>0 such that J_εx_ε=(J_〈H〉+εee^T)x_ε=ρ(J_ε)x_ε. Moreover, if ε>0 is small enough, we have ρ_ε < 1 by continuity of the spectral radius. Because of $1 < {\alpha _k} < \frac{1}{{{\mu _1}(\mathit{\boldsymbol{PAQ}})}}$, we also have

$ 2{\alpha _k}\rho - 1 < 1\;{\rm{and}}\;{\rm{2}}{\alpha _k}{\rho _\varepsilon } - 1 < 1. $

So

$ \begin{array}{l} \left[ {{\mathit{\boldsymbol{H}}_{{\rm{RMS - BMMTOR}}}}} \right] \le \mathit{\boldsymbol{I}} - 2\mathit{\boldsymbol{M}}_k^{ - 1}{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}\left[ {\mathit{\boldsymbol{I}} - {\alpha _k}\left( {{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}^{ - 1}{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }} + } \right.} \right.\\ \;\;\;\;\;\;\;\left. {\left. {\varepsilon \mathit{\boldsymbol{e}}{\mathit{\boldsymbol{e}}^{\rm{T}}}} \right)} \right] = \mathit{\boldsymbol{I}} - 2\mathit{\boldsymbol{M}}_k^{ - 1}{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}\left[ {\mathit{\boldsymbol{I}} - {\alpha _k}{\mathit{\boldsymbol{J}}_\varepsilon }} \right]. \end{array} $

Multiplying x_ε in two sides of the above inequality, and M_k^-1≥D_〈H〉^-1, we can obtain

$ \begin{array}{l} \left[ {{\mathit{\boldsymbol{H}}_{{\rm{RMS - BMMTOR}}}}} \right]{x_\varepsilon } \le {x_\varepsilon } - 2\mathit{\boldsymbol{M}}_k^{ - 1}{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}\left[ {1 - {\alpha _k}\rho \left( {{\mathit{\boldsymbol{J}}_\varepsilon }} \right)} \right]{\mathit{\boldsymbol{x}}_\varepsilon } = \\ \;\;\;\;\;\;\;{\mathit{\boldsymbol{x}}_\varepsilon } - 2\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }^{ - 1}{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}\left[ {1 - {\alpha _k}\rho \left( {{\mathit{\boldsymbol{J}}_\varepsilon }} \right)} \right]{\mathit{\boldsymbol{x}}_\varepsilon } = \left( {2{\alpha _k}\rho \left( {{\mathit{\boldsymbol{J}}_\varepsilon }} \right) - 1} \right){\mathit{\boldsymbol{x}}_\varepsilon }. \end{array} $

Based on E_k and the definition of [·], we know that $\sum\limits_{k = 1}^l {[{\mathit{\boldsymbol{E}}_k}]} = \mathit{\boldsymbol{I}}$. By (11), we have

$ \begin{array}{l} \left[ {{{\mathit{\boldsymbol{\bar H}}}_{{\rm{RMS - BMMTOR}}}}} \right]{\mathit{\boldsymbol{x}}_\varepsilon } = \omega \sum\limits_{k = 1}^l {\left[ {{\mathit{\boldsymbol{E}}_k}} \right]\left( {2{\alpha _k}\rho \left( {{\mathit{\boldsymbol{J}}_\varepsilon }} \right) - 1} \right){\mathit{\boldsymbol{x}}_\varepsilon }} + \\ \;\;\;\;\;\;\;\left| {1 - \omega } \right|{\mathit{\boldsymbol{x}}_\varepsilon } \le \omega \left( {2{\alpha _k}{\rho _\varepsilon } - 1} \right){\mathit{\boldsymbol{x}}_\varepsilon } + \left| {1 - \omega } \right|{\mathit{\boldsymbol{x}}_\varepsilon } \le \\ \;\;\;\;\;\;\;\left( {\omega \rho ' + \left| {1 - \omega } \right|} \right){\mathit{\boldsymbol{x}}_\varepsilon } = {\theta _2}{\mathit{\boldsymbol{x}}_\varepsilon }\left( {\varepsilon \to {0^ + }} \right),\\ {\rm{where}}\;\;\;\;\;\;\;\;\;\;{\theta _2} = \omega \rho ' + \left| {1 - \omega } \right| < 1. \end{array} $

where θ₂=ωρ′+|1-ω| < 1.

Subcase 2    α_k≤β_k and α_k≤γ_k. Define

$ \begin{array}{l} \mathit{\boldsymbol{N}}_k^3 = \left( {\left| {1 - {\alpha _k}} \right| - {\alpha _k}} \right){\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }} + \left( {\left| {{\alpha _k} - {\beta _k}} \right| + {\alpha _k}} \right)\left[ {{{\mathit{\boldsymbol{\bar L}}}_k}} \right] + \\ \;\;\;\;\;\;\left( {\left| {{\alpha _k} - {\gamma _k}} \right| + {\alpha _k}} \right)\left[ {{{\mathit{\boldsymbol{\bar F}}}_k}} \right] + 2{\alpha _k}\left[ {{{\mathit{\boldsymbol{\bar U}}}_k}} \right] + {\alpha _k}\left\langle \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \right\rangle = \\ \;\;\;\;\;\;{\mathit{\boldsymbol{M}}_k} - 2{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }} + 2{\beta _k}\left[ {{{\mathit{\boldsymbol{\bar L}}}_k}} \right] + + 2{\gamma _k}\left[ {{{\mathit{\boldsymbol{\bar F}}}_k}} \right] + + 2{\alpha _k}\left[ {{{\mathit{\boldsymbol{\bar U}}}_k}} \right] \le \\ \;\;\;\;\;\;{\mathit{\boldsymbol{M}}_k} - 2{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }} + 2{\delta _k}{\mathit{\boldsymbol{B}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}, \end{array} $

where δ_k=max{β_k, γ_k}. So

$ \begin{array}{l} \left[ {{\mathit{\boldsymbol{H}}_{{\rm{RMS - BMMTOR}}}}} \right] \le \mathit{\boldsymbol{M}}_k^{ - 1}\left[ {{\mathit{\boldsymbol{M}}_k} - 2\left( {{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }} - {\delta _k}{\mathit{\boldsymbol{B}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}} \right)} \right] \le \\ \;\;\;\;\;\;\;\mathit{\boldsymbol{I}} - 2\mathit{\boldsymbol{M}}_k^{ - 1}{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}\left( {\mathit{\boldsymbol{I}} - {\delta _k}{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}^{ - 1}{\mathit{\boldsymbol{B}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}} \right). \end{array} $

Similar to the case 1, let e denote the vector e=(1, 1, …, 1)^T∈Rⁿ, and x_ε>0 such that J_εx_ε=(J_〈H〉+εee^T)x_ε=ρ(J_ε)x_ε. Moreover, if ε>0 is small enough, we have ρ_ε < 1 by continuity of the spectral radius. Because of 0 < β_k, ${\gamma _k} < \frac{1}{{{\mu _1}(\mathit{\boldsymbol{PAQ}})}}$, we also have

$ 2{\delta _k}\rho - 1 < 1\;{\rm{and}}\;{\rm{2}}{\delta _k}{\rho _\varepsilon } - 1 < 1. $

So

$ \begin{array}{l} \left[ {{\mathit{\boldsymbol{H}}_{{\rm{RMS - BMMTOR}}}}} \right] \le \mathit{\boldsymbol{I}} - 2\mathit{\boldsymbol{M}}_k^{ - 1}{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}\left[ {\mathit{\boldsymbol{I}} - {\delta _k}\left( {\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }^{ - 1}{\mathit{\boldsymbol{D}}_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }} + } \right.} \right.\\ \;\;\;\;\;\;\;\left. {\left. {\varepsilon \mathit{\boldsymbol{e}}{\mathit{\boldsymbol{e}}^{\rm{T}}}} \right)} \right] = I - 2M_k^{ - 1}{D_{\left\langle \mathit{\boldsymbol{H}} \right\rangle }}\left[ {\mathit{\boldsymbol{I}} - {\delta _k}{\mathit{\boldsymbol{J}}_\varepsilon }} \right]. \end{array} $

Multiplying x_ε in two sides of the above inequality, and M_k^-1≥D_〈H〉^-1, we can obtain

$ \begin{array}{l} \left[ {{\mathit{\boldsymbol{H}}_{{\rm{RMS - BMMTOR}}}}} \right]{\mathit{\boldsymbol{x}}_\varepsilon } \le {\mathit{\boldsymbol{x}}_\varepsilon } - 2\left[ {1 - {\delta _k}\rho \left( {{\mathit{\boldsymbol{J}}_\varepsilon }} \right)} \right]{\mathit{\boldsymbol{x}}_\varepsilon } = \\ \;\;\;\;\;\;\;\left( {2{\delta _k}\rho \left( {{\mathit{\boldsymbol{J}}_\varepsilon }} \right) - 1} \right){\mathit{\boldsymbol{x}}_\varepsilon }. \end{array} $

Based on E_k and the definition of [·], we know that $\sum\limits_{k = 1}^l {[{\mathit{\boldsymbol{E}}_k}]} = \mathit{\boldsymbol{I}}$. By (11), we have

$ \begin{array}{l} \left[ {{{\mathit{\boldsymbol{\bar H}}}_{{\rm{RMS - BMMTOR}}}}} \right]{\mathit{\boldsymbol{x}}_\varepsilon } = \omega \sum\limits_{k = 1}^l {\left[ {{\mathit{\boldsymbol{E}}_k}} \right]\left( {2{\delta _k}\rho \left( {{\mathit{\boldsymbol{J}}_\varepsilon }} \right) - 1} \right){\mathit{\boldsymbol{x}}_\varepsilon }} + \\ \;\;\;\;\;\;\;\;\left| {1 - \omega } \right|{\mathit{\boldsymbol{x}}_\varepsilon } \le \omega \left( {2{\delta _k}{\rho _\varepsilon } - 1} \right){\mathit{\boldsymbol{x}}_\varepsilon } + \left| {1 - \omega } \right|{\mathit{\boldsymbol{x}}_\varepsilon } \le \\ \;\;\;\;\;\;\;\;\left( {\omega \rho ' + \left| {1 - \omega } \right|} \right){\mathit{\boldsymbol{x}}_\varepsilon } = {\theta _3}{\mathit{\boldsymbol{x}}_\varepsilon }\left( {\varepsilon \to {0^ + }} \right), \end{array} $

where θ₃=ωρ′+|1-ω| < 1.

Remark 2    In the fact application, the parameters can be adjusted suitably so that the convergence property of method can be substantially improved. That is to say, we have more choices for the splitting A=B-C which makes the multisplitting iteration methods converge.Therefore, our convergence theories extend the scope of multisplitting iterative methods in applications.

Through the similar proving process of theorem 4, we can obtain the following con-vergence results.

Theorem 6    Let A∈L_{n, I}(n₁, n₂, …, n_p) be a block H_B^Ⅱ(P, Q)-matrix, with H∈Ω_B^Ⅱ(PAQ), and let (M_k, N_k, E_k)(k=1, 2, …, l) and (D-L_k, U_k, E_k)(k=1, 2, …, l) be a block multisplitting and a block triangular multisplitting of block H-matrix, respectively. Assume that γ>0 and the positive matrix Ω satisfy Ω≥D(H) and diag(Ω)=diag(D(H)). If H=D-L_k-U_k(k=1, 2, …, l) satisfies 〈〈H〉〉=I-[D^-1L_k]-[D^-1F_k]-[D^-1U_k]=I-B_{〈〈H〉〉}(k=1, 2, …, l), then the iteration sequence {z_m=0^(m)∞} generated by the GRM-SBMMTOR iteration method converges to the unique solution z_* of LCP(q, A) for any initial vector z⁽⁰⁾∈R₊ⁿ, provided the relaxation parameters α_k and β_k satisfy

$ \begin{array}{c} 0 < {\beta _k},{\gamma _k} \le {\alpha _k} \le 1,0 < \omega < \frac{2}{{1 + \rho '}}{\rm{or}}\\ 0 < {\beta _k},{\gamma _k} < \frac{1}{{{\mu _2}\left( {\mathit{\boldsymbol{PAQ}}} \right)}},1 < {\alpha _k} < \frac{1}{{{\mu _2}\left( {\mathit{\boldsymbol{PAQ}}} \right)}},0 < \omega < \frac{2}{{1 + \rho '}}, \end{array} $

(7)

where μ₂(PAQ)=ρ(D_{〈〈H〉〉}^-1B_{〈〈H〉〉})=ρ(J_{〈〈H〉〉}), ρ′=$\mathop {\max }\limits_{1 \le k \le l} ${1-2α_k+2α_kρ_ε, 2δ_kρ_ε-1, 2α_kρ_ε-1}, δ_k=max {β_k, γ_k}. Moreover, β_k, γ_k should be greater than or less than α_k at once.

Remark 3    From table 1, we obviously see that the MSMMAOR method in [30] and the MSBMAOR method in [23] use the same parameters α, β in different processors, but the RMS-BMMTOR method in this paper uses different parameters α_k, β_k, γ_k(k=1, 2, …, l) in different processors. Moreover, when computing x^(m+1) in method 2, we utilize relaxation extrapolation technique and add a relaxation parameter. Therefore, we may choose proper relaxation parameters to increase convergence speed and reduce the computation time when doing numerical experiments. In RMS-BMMTOR method, we may choose proper E_k to balance the load of each processor and avoid synchronization.

4 Conclusions

The manuscript discusses relaxed modulus-based synchronous block multisplitting multi-parameters methods for linear complementarity problems. We have analyzed and got the new convergence theorem under certain conditions. However, table 1 shows some existing methods for linear complementarity problems are special cases of new methods with appropriate parameters. Moreover, new convergence theorems are more general and practical.