The simplified modulus-based matrix splitting iteration method for the nonlinear complementarity problem

1.   Introduction

We are concerned with the following nonlinear complementarity problem (NCP($A, \varphi$)),

here, $z\in {R}^{n}$ is to be found, and $A = (a_{ij})\in{R}^{n \times n}$ is given with $\varphi(z) = (\varphi_{1}(z_{1}), \varphi_{2}(z_{2}), \ldots, \varphi_{n}(z_{n}))^{\mathrm{T}}$ satisfying $|\frac{d\varphi_{i}}{dz_{i}}|\leq \bar{\psi}_{i}$, where $\bar{\psi}_{i}$ is the upper boundary of $|\frac{d\varphi_{i}}{dz_{i}}|$, for $i = 1, 2, ..., n$. For such a problem and its general cases, some researchers have studied this (see ^{[1,2,3,4,5,6,7,8]} and the cited references). Other types of complementary problems (CP) have also been explored, such as, the linear complementarity problem (LCP), the implicit complementarity problem (ICP), the horizontal linear complementarity problem (HLCP), the vertical linear complementarity problems (VLCP), and the horizontal nonlinear complementarity problem (HNCP), etc. (see ^{[9,10,11,12,13,14,15,16]} for details).

To calculate the numerical solution of the NCP, many researchers have proposed many practical methods, such as, the projection-filter method ^[17], the interior proximal point algorithm ^[18], and the smoothing least square method ^[19]. If $\varphi(z) = q\in R^n$ in (1.1), the NCP($A, \varphi$) is transformed into the simplest form, i.e., the linear complementarity problem LCP($A, q$), which has attracted many researchers to study this numerically, and various kinds of popular numerical solving approaches have been proposed in recent decades (^[20,21,22]). Among these numerical approaches, the modulus-based type matrix splitting iteration methods are very effective for solving the LCP with some particular system matrices (see ^{[9,23,24,25,26]}). For the high effectiveness of such modulus-based type iteration methods, some researchers have generalized these methods to deal with other complementarity problems, such as, the nonlinear complementarity problems ^[3,4,6,7,8], the implicit complementarity problems ^[5,10,27], the horizontal complementarity problems ^[11,28], and the vertical linear complementarity problems ^[12,14,15].

Motivated by the recent works in the LCPs and the developments of the NCPs, we continue to study the modulus-based type iteration methods for the NCP in this paper. Wu and Li proposed a new modulus-based matrix splitting iteration method to solve the LCP in ^[22], which differs to the existing modulus-based type matrix splitting iteration methods, appears simple in form, and shows high efficiency in the given experiments. We extend such methods to solve a kind of NCP and further discuss the convergence. In our discussion, the general convergence conditions of this method are first presented. Then, in order to facilitate practical applications, for the special matrix splitting of $A$ and the special parameter matrix $\Phi$, we propose some concrete convergence regions under certain conditions. Moreover, the quasi-optimal parameter matrix of the method is discussed and provided with respect to the spectral radius. The major results are shown in Section 3. Besides, some numerical examples are given to verify the effectiveness and the convergence of the method.

2.   The SMMS iteration method

Initially, we propose the simplified modulus-based matrix splitting (SMMS) iteration method for solving (1.1), and then give some preliminaries for later discussion. Some related definitions and notations, such as, $M$-matrix, $H_{+}$-matrix, the comparison matrix $\langle A\rangle$, and $H$-splitting, readers refer to ^[9,23] and the cited references.

For a positive diagonal matrix $\Phi$, the NCP (1.1) is equivalent to

thus, based on Lemma 3.1 in ^[22], the NCP (1.1) can be reformulated as an equivalent fixed-point equation

It follows that if $A = F-G$ is a splitting of $A$, Eq (2.2) becomes

So, if $\Phi+F$ is invertible, we obtain the SMMS iteration method below for solving the NCP (1.1).

The simplified modulus-based matrix splitting (SMMS) iteration method

$\bullet$ Given any initial victor $z^{(0)}\in R^{n}$, for k = 0, 1, 2, ..., compute $z^{(k+1)}$ by solving

$\bullet$ If $\|\mathrm{min}(z^{(k)}, Az^{(k)}+\varphi(z^{(k)}))\|_2 < \varepsilon$, the iteration stops, where $\|\cdot\|_2$ denotes the 2-norm of vectors and $\varepsilon$ is a given positive constant.

The SMMS iteration method differs from the modulus-based matrix splitting (MMS) iteration method ^[3,4], and one difference is that the former does not involve new vector $x$. Obviously, the iterative form of the SMMS iteration method is relatively simple. For the SMMS iteration method dealing with the linear complementarity problem (LCP), as well as the numerical experiments compared with other methods, readers refer to ^[22] for details.

Suppose that $z^{(*)}$ is the solution of the NCP (1.1), then from (2.3) and (2.4), there is a relationship if $\varphi_{i}(z_{i})$ is differentiable for $i = 1, 2, ..., n$, that is,

where $\Psi^{(k)} = \mathrm{diag}((\frac{d\varphi_1}{d_{z_1}}|_{\xi_1^{(k)}}, \frac{d\varphi_2}{d_{z_2}}|_{\xi_2^{(k)}}, \ldots, \frac{d\varphi_n}{d_{z_n}}|_{\xi_n^{(k)}}))$ is a diagonal matrix, and $\xi^{(k)} = (\xi_1^{(k)}, \xi_2^{(k)}, \ldots, \xi_n^{(k)})^{\mathrm{T}}$ with $\xi_i^{(k)}\in [z^{(k)}_i, z^{(*)}_i]$ or $\xi_i^{(k)}\in[z^{(*)}_i, z^{(k)}_i]$ for $i = 1, 2, \ldots, n$. In our later discussion, without special statements, we always assume that

3.   Convergence analysis

We discuss the convergence of the SMMS iteration method (2.4) for solving the NCP (1.1) with $\varphi(z)$ satisfying (2.6). In our discussion, we assume that the NCP (1.1) has a unique solution. We first give the convergence conclusions based on (2.5) from the spectral radius and the matrix norm, and then study some concrete convergence conditions in terms of the special matrix splittings.

Theorem 3.1. Let $A = F-G$ be a splitting of $A$ with $\Phi+F$ being nonsingular. If either of the conditions below holds:

then $\{z^{(k)}\}_{k = 0}^{+\infty}$ produced by the SMMS iteration method (2.4) converges for any $z^{(0)}\in R^{n}$.

Proof. Let $z^{(*)}$ be the unique solution of (1.1), for $\Phi+F$ is invertible, from Eq (2.5),

We take the absolute value function on both sides of Eq (3.2), then

Since $\Psi^{(k)}$ is a diagonal matrix and satisfies $|\Psi^{(k)}|\leq\bar{\Psi}$, we know that both $\sup_{|\Psi^{(k)}|\leq\bar{\Psi}}\big\{|(\Phi+ F)^{-1}|\; (|G-\Psi^{(k)}|+|A-\Phi+\Psi^{(k)}|)\big\}$ and $\sup_{|\Psi^{(k)}|\leq\bar{\Psi}}\big\{|(\Phi+ F)^{-1}|\; (|F-\Phi|+2|G-\Psi^{(k)}|)\big\}$ exist. Thus, if either of the inequalities

holds, from (3.3), it is easy to know that $\{z^{(k)}\}_{k = 0}^{+\infty}$ produced by the SMMS iteration method (2.4) converges for any $z^{(0)}\in R^{n}$. □

Similarly, if we take the 2-norm $\|\cdot\|_2$ on both sides of Eq (3.2) in Theorem 3.1, the relationship below holds,

Based on (3.4), we obtain the convergence conclusion below from the 2-norm $\|\cdot\|_2$.

Theorem 3.2. Let $A = F-G$ be a splitting of $A$ with $\Phi+F$ being nonsingular. If either of the conditions below holds:

then $\{z^{(k)}\}_{k = 0}^{+\infty}$ produced by the SMMS iteration method (2.4) converges for any $z^{(0)}\in R^{n}$.

We remark here that for $\delta_1, \delta_2$ in Theorem 3.1, and $\sigma_1, \sigma_2$ in Theorem 3.2, the symbol '$\mathrm{sup}_{|\Psi^{(k)}|\leq\bar{\Psi}}$' can be put inside, that is

Obviously, these formulas are more convenient in practice. In addition, if the detailed range of $\Psi^{(k)}$ is known, these convergence conditions in these two theorems can be more refined. For example, if we know that $\bar{\Psi}_{1}\leq \Psi^{(k)}\leq \bar{\Psi}_{2}$ for $k = 1, 2, ..., n$, and both bounds $\bar{\Psi}_1$ and $\bar{\Psi}_2$ can be reached, then the symbol '$\mathrm{sup}_{|\Psi^{(k)}|\leq\bar{\Psi}}$' can be replaced by '$\mathrm{sup}_{\bar{\Psi}_{1}\leq \Psi^{(k)}\leq \bar{\Psi}_{2}}$' in the formulas. Moreover, for this case, $\delta_1$ can be refined as

where '$\max$' is a function in matlab software. In addition, we can see that all the conditions are only sufficient conditions to ensure that the SMMS iteration method converges, not necessary conditions. When these conditions are not satisfied, the iteration method may also converge. It is apparent that convergence conditions $\delta_1 < 1$ and $\sigma_1 < 1$ are more precise than the other two convergence conditions.

Next, we consider some special convergence conditions. We assume that $F$ is symmetric positive definite in the matrix splitting $A = F-G$, and $\Phi = \phi I$ is a given positive scalar matrix. Then, we obtain the convergence conclusion.

Theorem 3.3. Let $A = F-G$ be a matrix splitting of $A$ with $F$ being symmetric positive definite. Denote the smallest and the largest eigenvalues of $F$ by $\lambda_{1}$ and $\lambda_{n}$, respectively, and $\tau = \sup_{|\Psi^{(k)}|\leq\bar{\Psi}}\{\|G-\Psi^{(k)}\|_{2}\}$. Set $\Phi = \phi I (\phi > 0)$, if $\tau < \lambda_{1}$, then when

$\{z^{(k)}\}_{k = 0}^{+\infty}$ produced by the SMMS iteration method (2.4) converges for any $z^{(0)}\in R^{n}$. In addition, $\phi = \dfrac{\lambda_{1}+\lambda_{n}}{2}$ is a quasi-optimal parameter.

Proof. Since $F$ is symmetric positive definite, the expression of $\sigma_{2}$ in Theorem 3.2 can be refined as

where $\lambda$ represents the eigenvalue of $F$. Therefor, based on (ⅱ) in Theorem 3.2, solving

in turn, we obtain the convergence region of parameter $\phi$ as follows.

From (Ⅰ), if $\tau < \lambda_{1}$, the parameter $\phi$ satisfies

From (Ⅱ), if $\tau < \lambda_{1}$, the parameter $\phi$ satisfies

Combining (Ⅰ) with (Ⅱ), the convergence region of $\phi$ is

Then, the first part of this theorem is proved.

For the second part of Theorem 3.3, we consider the functions appeared in (3.8), that is

respectively. It is easy to know that if $\tau < \lambda_{1}$, $f_{1}(\phi)$ is decreasing when

and $f_{2}(\phi)$ is increasing when

So, $\phi = \dfrac{\lambda_{1}+\lambda_{n}}{2}$ is the minimum value point of the above two functions. According to that $\sigma_{2} < 1$ is a convergence condition of the SMMS iteration method (2.4), and the smaller the value of $\sigma_2$, the better the convergence in general, we know that $\phi = \dfrac{\lambda_{1}+\lambda_{n}}{2}$ is a quasi-optimal parameter when $\Phi = \phi I\; (\phi > 0)$. Then the second part of this theorem is verified. □

Corollary 3.1. Let $A = F-G$ be a matrix splitting with $F$ satisfying $F = tI\; (t > 0)$. Denote $\tau = \sup_{|\Psi^{(k)}|\leq\bar{\Psi}}\{\|N-\Psi^{(k)}\|_{2}\}$. Set $\Phi = \phi I (\phi > 0)$, if $\tau < t$, then when

$\{z^{(k)}\}_{k = 0}^{+\infty}$ produced by the SMMS iteration method (2.4) converges for any $z^{(0)}\in R^{n}$. In addition, $\phi = t$ is a quasi-optimal parameter.

Proof. Based on (ⅱ) in Theorem 3.2, by the similar proof way of Theorem 3.3, we can have

Then, solving

in turn, for (Ⅰ), if $\tau < t$, $\phi$ satisfies $\phi\in(\tau, t]$, and for (Ⅱ), if $\tau < t$, $\phi$ satisfies $\phi\in(t, +\infty)$. So, combining these two cases, the convergence region of $\phi$ is $(\tau, +\infty)$ if $\tau < t$. Then, the first part of Corollary 3.1 is verified.

Similar to the proof of Theorem 3.3, for the function $f_{1}(\phi) = \dfrac{t-\phi+2\tau}{\phi+t}$ in (3.11) is decreasing when $\phi\in(\tau, t]$, and the function $f_{2}(\phi) = \dfrac{2\tau+\phi-t}{\phi+t}$ in (3.11) is increasing when $\phi\in[t, +\infty)$, we know that $\phi = t$ is a quasi-optimal parameter when $\Phi = \phi I$ with $\phi\in(\tau, +\infty)$. Therefore, the second part of Corollary 3.1 is proved. □

Next, we assume that the system matrix $A$ is an $H_{+}$-matrix, and consider the convergence region of $\Phi$ in the SMMS iteration method when the matrix splitting $A = F-G$ satisfies certain conditions.

Theorem 3.4. Assume that $A$ is an $H_{+}$-matrix and $A = F-G$ is a splitting of $A$ with

being an $M$-matrix for any $|\Psi^{(k)}|\leq\bar{\Psi}$. If $\Phi$ satisfies

then $\{z^{(k)}\}_{k = 0}^{+\infty}$ produced by the SMMS iteration method (2.4) converges for any $z^{(0)}\in R^{n}$, where $D$ represents the diagonal part of $A$.

Proof. From the first relationship of (3.3) in the proof of Theorem 3.1, that is

we will verify that

for any $|\Psi^{(k)}|\leq\bar{\Psi}$ under the conditions given in this theorem.

Since $\langle F \rangle-|G|+\Psi^{(k)}-|\Psi^{(k)}|$ is an $F$-matrix for any $|\Psi^{(k)}|\leq\bar{\Psi}$, according to $\langle F \rangle-|G|\geq \langle F \rangle-|G|+\Psi^{(k)}-|\Psi^{(k)}|$ and $\langle F \rangle\geq \langle F \rangle-|G|$, we know that both $\langle F \rangle-|G|$ and $\langle F \rangle$ are $M$-matrices. Thus $\Phi+\langle F\rangle$ is an $M$-matrix. Then, these two inequalities

hold. For the nonnegative matrix $(\Phi+\langle F\rangle)^{-1}\; \big(|G-\Psi^{(k)}|+|\Phi-A-\Psi^{(k)}|\big)$ in the second inequality of (3.14), we consider the matrix splitting

here, we use the inequality relationship $D-|B|\geq\langle F\rangle-|G|$ appeared in ^[8]. Therefor, according to the condition that $\langle F\rangle-|G|-|\Psi^{(k)}|+\Psi^{(k)}$ is an $M$-matrix for any $|\Psi^{(k)}|\leq\bar{\Psi}$, we know that $(\Phi+\langle F\rangle)-\big(|G-\Psi^{(k)}|+|\Phi-A-\Psi^{(k)}|\big)$ is an $M$-matrix too. Thus, the matrix splitting $(\Phi+\langle F\rangle)-\big(|G-\Psi^{(k)}|+|\Phi-A-\Psi^{(k)}|\big)$ in (3.15) is an $M$-splitting, so

(see ^[9,29]). Thus, based on the spectral theories of nonnegative matrices (see ^[30]) and (3.14), we obtain that

for any $|\Psi^{(k)}|\leq\bar{\Psi}$. So, the inequality

holds. Therefor, from (ⅰ) of Theorem 3.1, this theorem is established. □

Next, a special matrix splitting of $A$ is considered for the SMMS iteration method (2.4), that is, the accelerated overrelaxation (AOR) splitting, which is defined as

and has been studied by many researchers in the complementarity literatures ^[3,9,21], where $D$ is the the diagonal part of $A$, $-L$ and $-U$ are the strictly lower triangular and strictly upper triangular parts of $A$, respectively. For such matrix splitting, the iteration method (2.4) is accordingly called the simplified modulus-based accelerated overrelaxation (SMAOR) iteration method. When we let $\nu, \omega$ be some special values in (3.16), the SMAOR iteration method turns to be some special cases, i.e., the simplified modulus-based successive overrelaxation (SMSOR) iteration method ($\nu = \omega$), the simplified modulus-based Gauss Seidel (SMGS) iteration method ($\nu = \omega = 1$), and the simplified modulus-based Jacobian (SMJ) iteration method ($\nu = 1, \omega = 0$). We have the following conclusion for the SMAOR iteration method.

Theorem 3.5. Assume that $A$ is an $H_{+}$-matrix and $A = F-G$ is the AOR splitting with $\Phi$ satisfying $\Phi\geq D+\bar{\Psi}$. If any of the conditions below holds:

then $\{z^{(k)}\}_{k = 0}^{+\infty}$ produced by the SMAOR iteration method converges for any $z^{(0)}\in R^{n}$.

Proof. Since $A = F-G$ is the AOR splitting and $A$ is an $H_{+}$ matrix, we know that $\langle F\rangle$ is an $M$-matrix and $\langle F\rangle+\Phi$ is an $M$-matrix for any nonnegative diagonal matrix $\Phi$. For $\Phi\geq D+\bar{\Psi}$, just as the proof of Theorem 3.4, for the expression (3.15), we have

Under the conditions (3.17), we know that each of four matrices in the last inequality of (3.18) is an $M$-matrix. Then, the splitting $(\Phi+\langle F\rangle)-\big(|G-\Psi^{(k)}|+|\Phi-A-\Psi^{(k)}|\big)$ appeared in (3.18) is an $M$-splitting. Then

for any $|\Psi^{(k)}|\leq\bar{\Psi}$. So the convergence condition

holds. Therefor, from Theorem 3.1, the theorem is established. □

According to the proofs of Theorems 3.4 and 3.5, we can find that if $\Psi^{(k)}\geq O$ for any nonnegative integer $k$, and then $-|\Psi^{(k)}|+\Psi^{(k)} = O$ holds. Thus, $-|\Psi^{(k)}|+\Psi^{(k)}$ appeared in (3.15) and (3.18) can be deleted. Thus, we can obtain the following two corollaries derived from these two Theorems, respectively, and the proofs are omitted.

Corollary 3.2. Assume that $A$ is an $H_{+}$-matrix and $A = F-G$ is an $H$-splitting with $O\leq \Psi^{(k)}\leq\bar{\Psi}$ for any $k = 1, 2, \cdots$. If $\Phi$ satisfies

then $\{z^{(k)}\}_{k = 0}^{+\infty}$ produced by the SMMS iteration method (2.4) converges for any $z^{(0)}\in R^{n}$.

Corollary 3.3. Assume that $A$ is an $H_{+}$-matrix and $A = F-G$ is the AOR splitting with $O\leq \Psi^{(k)}\leq\bar{\Psi}$ for any $k = 1, 2, \cdots$. If $\Phi$ satisfies $\Phi\geq D+\bar{\Psi}$ and any of the following conditions holds:

then $\{z^{(k)}\}_{k = 0}^{+\infty}$ produced by the SMAOR iteration method converges for any $z^{(0)}\in R^{n}$.

We remark here that for an $H_{+}$-matrix $A$, the inequality $\rho\big(D^{-1}(|L|+|U|)\big) < 1$ holds, and it follows that the SMAOR iteration method always converges for any $z^{(0)}\in R^n$ when $0 < \nu\leq 1\; \mathrm{and}\; \omega\leq\nu$ according to (i) of Corollary 3.3. In addition, although some proposed convergence conditions are similar to that of ^[8], the SMMS iteration method discussed here differs to the MMS iteration method involved in ^[3,4,8].

4.   Numerical examples

We illustrate some numerical examples in this section. We denote the number of iteration steps and the elapsed time by IT and CPU, respectively. The norm of residual vector of the NCP is denoted by RES$(z)$, which is defined as follows

$z^{(k)}$ represents the $k$th numerical solution, and we set the iteration to cease if IT reaches $1000$ or RES$(z^{(k)}) < 1.0e-5$. In the first two experiments, we use $A(\eta, \mu) = M+\eta N+\mu S$ to generate the matrix $A$ in the NCP (1.1), where $\eta$ and $\mu$ are two constants, $M, N$ and $S$ are three given matrices. $M = \mathrm{Tridiag}(-I, T, -I) \in R^{n\times n}$ is a block-tridiagonal matrix, where $S$ is a diagonal matrix, $N = \mathrm{tridiag}(0, 0, 1)\in R^{n\times n}$ and $T = \mathrm{tridiag}(-1, 4, -1)\in R^{m\times m}$ are two tridiagonal matrix with $n = m^2$. We set $z^{(*)} = (0, 1, 0, 1, \ldots)^{\mathrm{T}}\in R^{n}$ and $z^{(0)} = (0, 0, 0, 0, \ldots)^{\mathrm{T}}\in R^{n}$ to be the solution of (1.1) and the initial vector, respectively.

Example 4.1. We test the convergence conditions given in Theorem 3.1 and compare the simplified modulus-based Gauss-Seidel(SMGS) iteration method with the modulus-based Gauss-Seidel(MGS) iteration method ^[3]. We set $\varphi(z)$ in the NCP (1.1) to be

with $q = -(Az^{(*)}+\varphi(z^{(*)}))$, then

We set $S = \mathrm{diag}((1, 2, 3, 1, 2, 3, \ldots))\in R ^{n\times n}$, and consider cases $A(0, 2)$ and $A(1, 1)$, which are a symmetric positive definite matrix and a nonsymmetric $P$-matrix, respectively. Let $\Phi$ be the diagonal part of $A$, i.e., $\Phi = D$ in both the SMGS iteration method (2.4) and the MGS iteration method ^[3]. Then for $A(0, 2)$, $\delta_1 = 0.9080$ when $n = 2500$ and $\delta_1 = 0.9609$ when $n = 3600$, and for $A(1, 1)$, $\delta_1 = 0.7024$ when $n = 2500$ and $\delta_1 = 0.8218$ when $n = 3600$. So the the SMGS is convergent for any $z^{(0)}\in R ^{n}$ based on Theorem 3.1. Table 1 below shows the numerical comparison.

From Table 1, for cases $A(0, 2)$ and $A(1, 1)$, although the ITs are similar for the two iteration methods, the elapsed time is significantly different, i.e., the former SMGS iteration method costs less time than the latter MGS iteration method. This example shows that the SMMS iteration method (2.4) is usually more efficient than the MMS iteration method.

Example 4.2. We test the convergence conditions given in Theorem 3.3. We set $\varphi(z)$ in (1.1) the NCP (1.1) to be

with $q = -(Az^{(*)}+\varphi(z^{(*)}))$, then

Four cases are considered for $A$ in the NCP (1.1), that is $A(0, 4)$, $A(0, 6)$, $A(2, 3)$ and $A(1, 3)$ with

respectively. Set $F = \mathrm{triu}(A, -1)-\mathrm{triu}(A, 2)$ in $A = F-G$ for the first two cases, and set $F = \mathrm{triu}(A)-\mathrm{triu}(A, 1)+\mathrm{triu}(A, -1)-\mathrm{triu}(A)+(\mathrm{triu}(A, -1)-\mathrm{triu}(A))^{\mathrm{T}}$ for the other two cases. We set $n = 1600$, then all cases satisfy the condition $\tau < \lambda_{1}$ given Theorem 3.3, and the SMMS iteration method converges for any $z^{(0)}\in R^{n}$ if we set $\Phi = \phi I$ with $\phi \in (\dfrac{\lambda_{n}-\lambda_{1}}{2}+\tau, +\infty)$. In order to see the numerical results clearly, we set

where $\delta = \dfrac{\lambda_{1}-\tau}{5}$. Then $\dfrac{\lambda_{n}+\lambda_{1}}{2}$ is the fourth point in the 11 points. We test the convergence condition $\sigma_{2} < 1$ given in Theorem 3.2. Table 2, Figures 1 and 2 below show the numerical results.

From Table 2, Figures 1 and 2, when $\Phi = \dfrac{\lambda_{n}+\lambda_{1}}{2}$, that is, the quasi-optimal parameter given in Theorem 3.3, IT is not very large, and the best parameter is sometimes near the quasi-optimal parameter in this example. In addition, we also can see that though the convergence condition is obtained by inequality reduction, the size of the boundary value is not exactly consistent with IT.

Example 4.3. We test the SMAOR iteration method for solving (1.1). We let $A$ be the block tridialgonal matrix in ^[4], i.e., $A = \mathrm{Tridiag}(T, K, T) \in R^{n\times n}$ with $K = \mathrm{tridiag}(-4, 20, -4)\in R^{n_{s}\times n_{s}}$ and $T = \mathrm{tridiag}(-1, 4, -1)\in R^{n_{s}\times n_{s}}$ being two tridiagonal matrices with $n = n_{t}\times n_{s}$, $n_{t} = 3\cdot2^{t}-1$ and $n_{s} = 2\cdot2^{t}-1$, where $t$ is a positive integer. We let $\varphi(z)\in R^{n}$ in (1.1) be

Then,

We let $t = 4$ and $\Phi$ be $\Phi = D+\bar{\Psi}$ in the SMAOR iteration method. Then, $A$ is an $H_{+}$-matrix and $0\leq \Psi^{(k)}\leq\bar{\Psi}$ for any $k = 1, 2, \ldots$. According to (i) in Corollary 3.3, we know that when $0 < \nu\leq 1 \; \mathrm{and}\; \omega\leq\nu$, the SMAOR iteration method is convergent. In our experiment, we set the discrete values of $\nu$ and $\omega$ to be

Table 3 and Figure 3 below show the numerical results.

From Table 3 and Figure 2, we can see that this example mainly verifies the first conclusion (i) given in Corollary 3.3, i.e., when $0 < \nu\leq 1 \; \mathrm{and}\; \omega\leq\nu$. For other conclusions given in Corollary 3.3, such as, $\omega > \nu$, $\nu > 1$, $\omega > 1$, the numerical results are not obvious, and only when $\omega_7 > \nu_6$, $\delta_{1} = \rho\Big(\sup_{|\Psi^{(k)}|\leq\bar{\Psi}}\big\{|(\Phi+ F)^{-1}|\; (|G-\Psi^{(k)}|+|A-\Phi+\Psi^{(k)}|)\big\}\Big) = 0.9984 < 1$, which can ensure the convergence of the SMAOR iteration method (see Theorem 3.1). However, if we decrease the order of $A$, for instance, let $t = 2$, then the cases related to (ⅱ)–(ⅳ) given in Corollary 3.3 will be more, and the corresponding numerical results are omitted here. In addition, we can also see that the convergence conditions given in this paper are only sufficient, not necessary, and when $\nu = \omega$, i.e., the SMSOR iteration method is relatively better. Specially, the case $\nu = \omega = 1$, i.e., the SMGS iteration method is good although IT is not the smallest in this example.

5.   Concluding remarks

The SMMS iteration method was extended to solve a kind of nonlinear complementarity problem in this paper. Both the general convergence conditions and the concrete convergence conditions were proposed. By comparing the SMGS iteration method with the MGS iteration method, the high efficiency of the SMMS iteration method was shown. The quasi-optimal parameter conclusion for the SMMS iteration method was also illustrated by the numerical experiments.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research is supported by The Natural Science Foundation of Guangdong Province (No. 2022A1515011081 and No. 2023A1515011911), The Technology Innovation Guidance Project of Zhaoqing (No. 2022040315016) and The Innovative Research Team Project of Zhaoqing University.

Conflict of interest

The author has no conflict of interest.