1 Introduction

The generalized absolute value equation (GAVE) is formulated as:

$$\begin{aligned} Ax-B|x |=b, \end{aligned}$$
(1)

where \(A, \ B\in \mathbb {R}^{n\times n}\) are given large sparse matrices, \(b\in \mathbb {R}^{n}\), and \(|x |=(|x_{1}|, \ldots , |x_{n}|)^{\text {T}}\in \mathbb {R}^{n}\) denotes the componentwise absolute value of an unknown \(x\in \mathbb {R}^{n}\). If \(B = I\), where I stands for an identity matrix of suitable dimension, the GAVE (1) can be simplified to the following absolute value equation (AVE)

$$\begin{aligned} Ax-|x |=b. \end{aligned}$$
(2)

GAVEs have arisen in various scientific and engineering fields and been presented in enormous applications since they were first introduced by Rohn [1]. Among many important applications, a well-known example is the linear complementarity problem (LCP) [2,3,4,5,6]. Besides LCPs, many other optimization problems can be transformed into the GAVEs (1), including linear programming and convex quadratic programming [1, 7].

Due to the existence of the nonlinear term \(B|x |\), the GAVE (1) can be regarded as a weakly nonlinear system

$$\begin{aligned} A x=G(x), \quad \text {with}\quad G(x)=B|x |+b. \end{aligned}$$
(3)

For solving the general weakly nonlinear systems

$$\begin{aligned} A x=G(x), \end{aligned}$$
(4)

where the nonlinear function \(G:\mathbb {R}^{n} \rightarrow \mathbb {R}^{n}\) is B-differentiable, through the two-stage splitting \(A=E-F\) and \(E=M-N\), Bai for the first time introduced and studied the following two-stage iterative method [8]:

$$\begin{aligned} Mx^{(k, \ell +1)}=N x^{(k, \ell )}+F x^{(k)}+G(x^{(k)}), \quad \text {for}\quad \ell =0,1, \ldots , l_{k}-1, \end{aligned}$$
(5)

with \(x^{(k, 0)}:=x^{(k)}\) and \(x^{(k+1)}:=x^{(k, l_{k})}\). See also [9,10,11] for related methods. It is noted that the two-stage iterative method provides a general framework of matrix splitting iteration methods for solving the weakly nonlinear systems (4). For the GAVE (1), i.e., the case when \(G(x)=B|x |+b\), the two-stage iterative method includes a series of existing matrix splitting iteration methods [12,13,14,15,16] as its special cases. For example, when \(E=A\), \(F=0\), \(M=E\), \(N=0\), and \(l_{k} \equiv 1\), the two-stage iterative method reduces to the well-known Picard iteration method [12]

$$\begin{aligned} Ax^{(k+1)}=B|x^{(k)} |+b. \end{aligned}$$
(6)

Recently, by reformulating the AVE (2) as a two-by-two block nonlinear equation, Ke et al. proposed an SOR-like iteration method [17] for solving the AVE (2). This method was also analyzed in [18]. The SOR-like iteration method received wide attentions and obtained considerable achievements in recent years. Using the similar technology, other SOR-like-based methods [19,20,21] are presented to solve the AVE (2). In order to further improve computational efficiency, Ke proposed an efficient fixed point iteration (FPI) method [22] to solve the AVE (2), which can be described as

Algorithm 1

(The FPI Method for AVE). Let \(A\in \mathbb {R}^{n\times n}\) be a nonsingular matrix and \(b\in \mathbb {R}^{n}\). Given the initial vectors \(x^{(0)}, y^{(0)}\in \mathbb {R}^{n}\), compute \((x^{(k+1)},y^{(k+1)})\) for \(k=0,1,2, \ldots\) using the following iteration scheme until \(\{(x^{(k)},y^{(k)})\}_{k=0}^{+\infty }\) satisfies the stopping criterion:

$$\begin{aligned} \left\{ \begin{array}{l} x^{(k+1)}=A^{-1}(y^{(k)}+b),\\ y^{(k+1)}=(1-\omega )y^{(k)}+\omega |x^{(k+1)} |, \end{array}\right. \end{aligned}$$
(7)

where \(\omega\) is a positive constant.

Note that the FPI method reduces to the Picard iteration method for \(\omega = 1\). Owing to the simplicity and effectiveness of FPI method for solving the AVE (2), Yu et al. developed a modified FPI (MFPI) method [23], which is a generalized version of the FPI method.

Clearly, at each step of the FPI method, a linear system \(Au = f\) needs to be solved. Since A is always large and sparse, a computationally efficient way is to use matrix splitting iteration methods to obtain the approximate solution of this linear system. For solving non-Hermitian positive definite linear systems, Bai et al. first proposed the shift-splitting (SS) iteration method [24]. Motivated by its promising performance, the SS method was extended to solve many linear systems with special structure such as the saddle point problems [25], block \(3\times 3\) saddle point problems [26], and time-harmonic eddy current problems [27]. In this paper, using the shift-splitting [24] of the coefficient matrix A, we propose a shift-splitting fixed point iteration (FPI-SS) method for solving the GAVE (1). Compared with the FPI method, the coefficient matrix of the first sub-iteration scheme of our method is more diagonally dominant. Our method is more efficient than the FPI method and the SOR-like iteration method as shown in our numerical experiments.

In what follows, some notations in this work are described. For \(x\in \mathbb {R}^{n}\), \(x_{i}\) stands for the ith entry of vector x for all \(i = 1, 2, \ldots , n\). \(\mathrm {sgn}(x)\in \mathbb {R}^{n}\) denotes a vector with components equal to 1, 0, or \(-1\) depending on whether the corresponding component of the vector x is positive, zero, or negative, respectively. Let \(\mathrm {diag}(x)\in \mathbb {R}^{n\times n}\) represent a diagonal matrix with \(x_{i}\) as its ith diagonal entry for \(i = 1, 2, \ldots , n\). For matrix \(M\in \mathbb {R}^{n\times n}\), \(\Vert M\Vert\) denotes the spectral norm defined by \(\Vert M\Vert :=\mathrm {max}\{\Vert Mx\Vert :x\in \mathbb {R}^{n}, \Vert x\Vert =1\}\), where \(\Vert x\Vert\) is the 2-norm.

The organization of the remaining parts is the following. In Section 2, we present a brief introduction of the FPI method and establish the FPI-SS method for solving the GAVE (1). In Section 3, the convergence theories for the FPI-SS method are presented in detail. In Section 4, we give two numerical examples in Section 4 to verify the effectiveness of our method. Finally, the conclusions are given in Section 5.

2 The shift-splitting fixed point iteration (FPI-SS) method

Let \(y=|x |\), then the GAVE (1) is equivalent to

$$\begin{aligned} \left\{ \begin{array}{l} Ax-By=b,\\ -|x |+y=0, \end{array}\right. \end{aligned}$$
(8)

which can be reformulated as the following two-by-two block nonlinear equation

$$\begin{aligned} \left( \begin{array}{cc} A &{} -B \\ -{H}(x) &{} I\end{array}\right) \left( \begin{array}{c} x \\ y\end{array}\right) =\left( \begin{array}{c} b \\ 0\end{array}\right) , \end{aligned}$$
(9)

where \(H(x)=\mathrm {diag}(\mathrm {sign}(x))\).

If A is a nonsingular matrix, (9) yields the following fixed point equation

$$\begin{aligned} \left\{ \begin{array}{l} x^{*}=A^{-1}(By^{*}+b),\\ y^{*}=(1-\omega )y^{*}+\omega |x^{*} |, \end{array}\right. \end{aligned}$$
(10)

where the relaxation parameter \(\omega >0\).

Then, we can obtain the following fixed point iteration (FPI) method for the GAVE (1).

Algorithm 2

(The FPI Method for GAVE). Let \(A\in \mathbb {R}^{n\times n}\) be nonsingular, \(B\in \mathbb {R}^{n\times n}\) and \(b\in \mathbb {R}^{n}\). Given the initial vectors \(x^{(0)}, y^{(0)}\in \mathbb {R}^{n}\), compute \((x^{(k+1)},y^{(k+1)})\) for \(k=0,1,2, \ldots\) using the following iteration scheme until \(\{(x^{(k)},y^{(k)})\}_{k=0}^{+\infty }\) satisfies the stopping criterion:

$$\begin{aligned} \left\{ \begin{array}{l} x^{(k+1)}=A^{-1}(By^{(k)}+b),\\ y^{(k+1)}=(1-\omega )y^{(k)}+\omega |x^{(k+1)} |, \end{array}\right. \end{aligned}$$
(11)

where \(\omega\) is a positive constant.

It is evident that Algorithm 2 reduces to Algorithm 1 when we take \(B=I\). Similarly, if we set \(\omega = 1\) in Algorithm 2, the Picard iteration method (6) for solving the GAVE (1) can be obtained. Since the convergence analyses of Algorithm 2 are analogous to those of Algorithm 1 discussed in detail in [22], we do not give them here.

Importantly, by employing the following shift-splitting of the matrix A [24]

$$\begin{aligned} A=\frac{1}{2}(\alpha I+A)-\frac{1}{2}(\alpha I-A), \end{aligned}$$

where the parameter \(\alpha\) is a positive constant and the matrix \(\alpha I+A\) is invertible, we get the following fixed point equation from (9)

$$\begin{aligned} \left\{ \begin{array}{l} x^{*}=(\alpha I+A)^{-1}(\alpha I-A)x^{*}+2(\alpha I+A)^{-1}(By^{*}+b),\\ y^{*}=(1-\omega )y^{*}+\omega |x^{*} |, \end{array}\right. \end{aligned}$$
(12)

which leads to the following FPI-SS method for the GAVE (1).

Algorithm 3

(The FPI-SS Method for GAVE). Let \(A, \ B \in \mathbb {R}^{n\times n}\) and \(b\in \mathbb {R}^{n}\). Let \(\alpha\) be a positive constant such that \(\alpha I+A\in \mathbb {R}^{n\times n}\) is nonsingular. Given the initial vectors \(x^{(0)}, y^{(0)}\in \mathbb {R}^{n}\), compute \((x^{(k+1)},y^{(k+1)})\) for \(k=0,1,2, \ldots\) using the following iteration scheme until \(\{(x^{(k)},y^{(k)})\}_{k=0}^{+\infty }\) satisfies the stopping criterion:

$$\begin{aligned} \left\{ \begin{array}{l} x^{(k+1)}=(\alpha I+A)^{-1}(\alpha I-A)x^{(k)}+2(\alpha I+A)^{-1}(By^{(k)}+b),\\ y^{(k+1)}=(1-\omega )y^{(k)}+\omega |x^{(k+1)} |, \end{array}\right. \end{aligned}$$
(13)

where \(\omega\) is a positive constant.

Remark 1

If matrix A is positive semi-definite, the condition that \(\alpha I+A\) is nonsingular naturally holds. Even if matrix A is singular, we can always find some sufficiently large parameters \(\alpha\) to ensure that \(\alpha I+A\) is nonsingular. Therefore, the FPI-SS method has a broader range of application than the FPI method. In addition, owing to the positive scalar matrix \(\alpha I\), the matrix \(\alpha I+A\) is expected to be strictly diagonally dominant and better conditioned than the matrix A. Thus, our FPI-SS method may have better computing efficiency than the FPI method.

3 Convergence of the FPI-SS method

We first give some lemmas that will be used in convergence analysis of the FPI-SS method for solving the GAVE (1).

Lemma 1

[28,29,30] For any vectors \(x\in \mathbb {R}^{n}\) and \(y\in \mathbb {R}^{n}\), the following results hold:

  1. (1)

    \(\Vert |x |-|y |\Vert \le \Vert x-y\Vert\);

  2. (2)

    if \(0 \le x \le y\), then \(\Vert x\Vert _{p}\le \Vert y\Vert _{p}\), with \(\Vert \cdot \Vert _{p}\) standing for p-norm of vector;

  3. (3)

    if \(x \le y\) and P is a nonnegative matrix, then \(P x \le P y\).

Lemma 2

[28, 29] For any matrices \(A, B\in \mathbb {R}^{n \times n}\), if \(0 \le A \le B\), then \(\Vert A\Vert _{p}\le \Vert B\Vert _{p}\), with \(\Vert \cdot \Vert _{p}\) standing for p-norm of matrix.

Lemma 3

[28, 31] Both roots of the real quadratic equation \(x^{2}-ax+b=0\) are less than one in modulus if and only if \(|b |<1\) and \(|a |<1+b\).

In the remainder of this section, we assume that the GAVE (1) has a unique solution. Let \((x^{*},y^{*})\) be the solution pair of (12) and \((x^{(k)},y^{(k)})\) be generated by the FPI-SS iteration (13). The iteration errors are denoted by

$$\begin{aligned} {e_{k}^{x}}=x^{*}-x^{(k)} \quad \text {and}\quad {e_{k}^{y}}=y^{*}-y^{(k)}. \end{aligned}$$

Then, we can get the following convergence theorem by estimating the above two iteration errors.

Theorem 1

Let \(A, B\in \mathbb {R}^{n\times n}\) and \(b\in \mathbb {R}^{n}\). Let \(\alpha\) be a positive constant such that \(\alpha I+A\in \mathbb {R}^{n\times n}\) is nonsingular. Denote

$$\begin{aligned} \delta =\Vert (\alpha I+A)^{-1}(\alpha I-A)\Vert , \beta =2\Vert (\alpha I+A)^{-1}B\Vert , \gamma =|1-\omega |, \end{aligned}$$

and

$$\begin{aligned} E^{(k+1)}=\left( \begin{array}{c} \Vert e_{k+1}^{x}\Vert \\ \Vert e_{k+1}^{y}\Vert \end{array}\right) . \end{aligned}$$

Then, we have

$$\begin{aligned} \Vert E^{(k+1)}\Vert _{\infty }\le \Vert L(\alpha ,\omega )\Vert _{\infty }\cdot \Vert E^{(k)}\Vert _{\infty }, \end{aligned}$$
(14)

where \(\Vert \cdot \Vert _{\infty }\) denotes the \(\infty\)-norm of vector or matrix and

$$\begin{aligned} L(\alpha ,\omega ):=\left( \begin{array}{cc} \delta &{} \beta \\ \omega \delta &{} \omega \beta +\gamma \end{array}\right) . \end{aligned}$$

Furthermore, \(\Vert L(\alpha ,\omega )\Vert _{\infty }<1\) if and only if parameters \(\alpha\) and \(\omega\) satisfy

$$\begin{aligned} \delta +\beta<1&\ and&0<\omega <\frac{2}{1+\delta +\beta }, \end{aligned}$$
(15)

i.e., if the conditions (15) hold, the iteration sequence \(\{x^{(k)}\}_{k=0}^{+\infty }\) generated by the FPI-SS iteration converges to the unique solution \(x^{*}\) of the GAVE (1) for any initial vector.

Proof

Subtracting (13) from (12), we get

$$\begin{aligned} e_{k+1}^{x}=(\alpha I+A)^{-1}(\alpha I-A){e_{k}^{x}}+2(\alpha I+A)^{-1}B{e_{k}^{y}}, \end{aligned}$$
(16)
$$\begin{aligned} e_{k+1}^{y}=(1-\omega ){e_{k}^{y}}+\omega (|x^{*} |-|x^{(k+1)} |). \end{aligned}$$
(17)

According to (16), we can obtain

$$\begin{aligned} \Vert e_{k+1}^{x}\Vert \le \delta \Vert {e_{k}^{x}}\Vert +\beta \Vert {e_{k}^{y}}\Vert . \end{aligned}$$
(18)

From (17) and Lemma 1, we have

$$\begin{aligned} \Vert e_{k+1}^{y}\Vert\le & {} \gamma \cdot \Vert {e_{k}^{y}}\Vert +\omega \Vert |x^{*} |-|x^{(k+1)} |\Vert \nonumber \\\le & {} \gamma \cdot \Vert {e_{k}^{y}}\Vert +\omega \Vert x^{*}-x^{(k+1)}\Vert \nonumber \\= & {} \gamma \cdot \Vert {e_{k}^{y}}\Vert +\omega \Vert e_{k+1}^{x}\Vert . \end{aligned}$$
(19)

Rearranging (18) and (19), we find

$$\begin{aligned} \left( \begin{array}{cc} 1 &{} 0 \\ -\omega &{} 1 \end{array}\right) \left( \begin{array}{c} \Vert e_{k+1}^{x}\Vert \\ \Vert e_{k+1}^{y}\Vert \end{array}\right) \le \ \left( \begin{array}{cc} \delta &{} \beta \\ 0 &{} \gamma \end{array}\right) \left( \begin{array}{c} \Vert {e_{k}^{x}}\Vert \\ \Vert {e_{k}^{y}}\Vert \end{array}\right) . \end{aligned}$$
(20)

Let

$$\begin{aligned} P=\left( \begin{array}{cc}1 &{} 0 \\ \omega &{} 1\end{array}\right) \ge 0. \end{aligned}$$

Multiplying (20) from left by the nonnegative matrix P and according to Lemma 1, we have

$$\begin{aligned} \left( \begin{array}{c} \Vert e_{k+1}^{x}\Vert \\ \Vert e_{k+1}^{y}\Vert \end{array}\right) \le \ \left( \begin{array}{cc} \delta &{} \beta \\ \omega \delta &{} \omega \beta +\gamma \end{array}\right) \left( \begin{array}{c} \Vert {e_{k}^{x}}\Vert \\ \Vert {e_{k}^{y}}\Vert \end{array}\right) , \end{aligned}$$
(21)

which can be rewritten as

$$\begin{aligned} E^{(k+1)}\le L(\alpha ,\omega )\cdot E^{(k)}. \end{aligned}$$
(22)

Taking the \(\infty\)-norm on both sides of inequality (22) and according to (2) of Lemma 1, the estimation (14) is obtained. Since

$$\begin{aligned} \Vert L(\alpha ,\omega )\Vert _{\infty } =\mathrm {max}\{\delta +\beta , (\delta +\beta )\omega +\gamma \}, \end{aligned}$$

we have

$$\begin{aligned}&\Vert L(\alpha ,\omega )\Vert _\infty<1 \Leftrightarrow \left\{ \begin{array}{l} \delta +\beta<1\\ (\delta +\beta )\omega +\gamma<1 \end{array}\right. \Leftrightarrow \left\{ \begin{array}{l} \delta +\beta<1\\ |1-\omega |<1-(\delta +\beta )\omega \end{array}\right. \\&\Leftrightarrow \left\{ \begin{array}{l} \delta +\beta<1\\ 1-(\delta +\beta )\omega >0\\ (\delta +\beta )\omega -1<1-\omega<1-(\delta +\beta )\omega \end{array}\right. \Leftrightarrow \left\{ \begin{array}{l} \delta +\beta<1\\ \omega<\frac{1}{\delta +\beta }\\ 0<\omega<\frac{2}{1+\delta +\beta } \end{array}\right. \\&\Leftrightarrow \left\{ \begin{array}{l} \delta +\beta<1\\ 0<\omega <\frac{2}{1+\delta +\beta }. \end{array}\right. \end{aligned}$$

From (14), we deduce that

$$\begin{aligned} 0 \le \Vert E^{(k)}\Vert _{\infty } \le \Vert L(\alpha ,\omega )\Vert _{\infty } \cdot \Vert E^{(k-1)}\Vert _{\infty } \le \cdots \le \Vert L(\alpha ,\omega )\Vert _{\infty }^{k} \cdot \Vert E^{(0)}\Vert _{\infty }. \end{aligned}$$

Hence if the conditions (15) are satisfied, then we have \(\lim \limits _{k \rightarrow \infty }\Vert E^{(k)}\Vert _{\infty }=0.\)

As

$$\begin{aligned} \Vert E^{(k)}\Vert _{\infty }=\max \{\Vert {e_{k}^{x}}\Vert , \Vert {e_{k}^{y}}\Vert \}, \end{aligned}$$

it follows that

$$\begin{aligned} \lim _{k \rightarrow \infty }\Vert {e_{k}^{x}}\Vert =0 \quad \text {and}\quad \lim _{k \rightarrow \infty }\Vert {e_{k}^{y}}\Vert =0, \end{aligned}$$

which mean that the iteration sequence \(\{(x^{(k)},y^{(k)})\}_{k=0}^{+\infty }\) is convergent to \((x^{*},y^{*})\) under the conditions (15). This proves the theorem.

Using a different error estimate by a new weighted norm, we can obtain another convergence theorem as follows.

Theorem 2

Let the assumptions of Theorem 1 hold, \(\delta\), \(\beta\), and \(\gamma\) be defined as in Theorem 1. Denote

$$\begin{aligned} E_{\omega }^{(k+1)}=\left( \begin{array}{c} \Vert e_{k+1}^{x}\Vert \\ \omega ^{-1}\Vert e_{k+1}^{y}\Vert \end{array}\right) . \end{aligned}$$

Then, we have

$$\begin{aligned} \Vert E_{\omega }^{(k+1)}\Vert \le \Vert T(\alpha ,\omega )\Vert \cdot \Vert E_{\omega }^{(k)}\Vert , \end{aligned}$$
(23)

where

$$\begin{aligned} T(\alpha ,\omega ):=\left( \begin{array}{cc} \delta &{} \omega \beta \\ \delta &{} \omega \beta +\gamma \end{array}\right) . \end{aligned}$$

Furthermore, \(\Vert T(\alpha ,\omega )\Vert <1\) if and only if parameters \(\alpha\) and \(\omega\) satisfy

$$\begin{aligned} {\delta \gamma <1} \end{aligned}$$
(24)

and

$$\begin{aligned} 2\delta ^{2}+\omega ^{2}\beta ^{2}+(\omega \beta +\gamma )^{2}-\delta ^{2}\gamma ^{2}-1<0, \end{aligned}$$
(25)

i.e., if the conditions (24)–(25) hold, the iteration sequence \(\{x^{(k)}\}_{k=0}^{+\infty }\) generated by the FPI-SS iteration converges to the unique solution \(x^{*}\) of the GAVE (1) for any initial vector.

Proof

Denote

$$\begin{aligned} D=\left( \begin{array}{cc} 1 &{}0\\ 0&{} \omega ^{-1}\end{array}\right) >0. \end{aligned}$$

According to Lemma 1, we multiply left (21) by matrix D to obtain

$$\begin{aligned} \left( \begin{array}{c} \Vert e_{k+1}^{x}\Vert \\ \omega ^{-1}\Vert e_{k+1}^{y}\Vert \end{array}\right) \le \ \left( \begin{array}{cc} \delta &{} \omega \beta \\ \delta &{} \omega \beta +\gamma \end{array}\right) \left( \begin{array}{c} \Vert {e_{k}^{x}}\Vert \\ \omega ^{-1}\Vert {e_{k}^{y}}\Vert \end{array}\right) , \end{aligned}$$

which can be rewritten as

$$\begin{aligned} E_{\omega }^{(k+1)}\le T(\alpha ,\omega )\cdot E_{\omega }^{(k)}. \end{aligned}$$

From the above, it follows that (23) holds.

Let \(\lambda\) be an eigenvalue of the matrix \(Q:=T(\alpha ,\omega )^{\mathrm {T}}T(\alpha ,\omega )\). Since

$$\begin{aligned} Q= \left( \begin{array}{cc} 2\delta ^{2} &{} 2\omega \beta \delta +\delta \gamma \\ 2\omega \beta \delta +\delta \gamma &{} \omega ^{2}\beta ^{2}+(\omega \beta +\gamma )^{2}\end{array}\right) , \end{aligned}$$

we get

$$\begin{aligned} \text {tr}(Q)=2\delta ^{2}+\omega ^{2}\beta ^{2}+(\omega \beta +\gamma )^{2} \end{aligned}$$

and

$$\begin{aligned} \text {det}(Q)=\delta ^{2}\gamma ^{2}. \end{aligned}$$

Thus, \(\lambda\) is the root of the following real quadratic equation

$$\begin{aligned} \lambda ^{2}-(2\delta ^{2}+\omega ^{2}\beta ^{2}+(\omega \beta +\gamma )^{2})\lambda +\delta ^{2} \gamma ^{2}=0. \end{aligned}$$
(26)

From Lemma 3, it follows that \(\Vert T(\alpha ,\omega )\Vert <1\) if and only if

$$\begin{aligned} \delta ^{2}\gamma ^{2}<1, \end{aligned}$$

and

$$\begin{aligned} 2\delta ^{2}+\omega ^{2}\beta ^{2}+(\omega \beta +\gamma )^{2}<1+\delta ^{2}\gamma ^{2}. \end{aligned}$$

From (23), we conclude that

$$\begin{aligned} 0 \le \Vert E_{\omega }^{(k)}\Vert \le \Vert T(\alpha ,\omega )\Vert \cdot \Vert E_{\omega }^{(k-1)}\Vert \le \cdots \le \Vert T(\alpha ,\omega )\Vert ^{k} \cdot \Vert E_{\omega }^{(0)}\Vert . \end{aligned}$$

Hence, we have \(\lim \limits _{k \rightarrow \infty }\Vert E_{\omega }^{(k)}\Vert =0\) when the conditons (24)–(25) are satisfied.

From the definition

$$\begin{aligned} \Vert E_{\omega }^{(k)}\Vert =\sqrt{\Vert {e_{k}^{x}}\Vert ^{2}+\omega ^{-2}\Vert {e_{k}^{y}}\Vert ^{2}}, \end{aligned}$$

we get

$$\begin{aligned} \lim _{k \rightarrow \infty }\Vert {e_{k}^{x}}\Vert =0 \quad \text {and}\quad \lim _{k \rightarrow \infty }\Vert {e_{k}^{y}}\Vert =0, \end{aligned}$$

which mean that the iteration sequence \(\{(x^{(k)},y^{(k)})\}_{k=0}^{+\infty }\) is convergent to \((x^{*},y^{*})\) under the conditions (24)–(25). This completes the proof.

Theorem 2 shows that in order to obtain the convergence of the FPI-SS method, we need to find the conditions in which \(\Vert T(\alpha ,\omega )\Vert <1\) holds. Here, we give convergence conditions that are simpler than those in Theorem 2.

Corollary 1

Let the assumptions of Theorem 1 hold, \(\delta\), \(\beta\), and \(\gamma\) be defined as in Theorem 1, \(E_{\omega }^{(k+1)}\) and \(T(\alpha ,\omega )\) be defined as in Theorem 2. If

$$\begin{aligned} \delta<\frac{3-\sqrt{5}}{2},~\beta <\frac{\sqrt{5}-1}{2}, \end{aligned}$$
(27)

and

$$\begin{aligned} \frac{\sqrt{5}-1}{2}<\omega <\mathrm {min}\{\frac{3-\sqrt{5}}{2\beta },~\frac{5-\sqrt{5}}{2}\}, \end{aligned}$$
(28)

then \(\Vert T(\alpha ,\omega )\Vert <1\), i.e., the FPI-SS method is convergent when the conditions (27)–(28) hold.

Proof

Let \(\eta =\mathrm {max}\{\delta , ~\omega \beta , ~\gamma \}\), we can get

$$\begin{aligned} 0\le T(\alpha ,\omega )=\left( \begin{array}{cc} \delta &{} \omega \beta \\ \delta &{} \omega \beta +\gamma \end{array}\right) \le \left( \begin{array}{cc} \eta &{} \eta \\ \eta &{} 2\eta \end{array}\right) =\eta \left( \begin{array}{cc} 1 &{} 1 \\ 1 &{} 2\end{array}\right) :=\eta K, \end{aligned}$$

where

$$\begin{aligned} K=\left( \begin{array}{cc} 1 &{} 1 \\ 1 &{} 2\end{array}\right) . \end{aligned}$$

From lemma 2, we obtain

$$\begin{aligned} \Vert T(\alpha ,\omega )\Vert \le \Vert \eta K\Vert =\eta \Vert K\Vert =\eta \cdot \frac{3+\sqrt{5}}{2}. \end{aligned}$$

Let \(\theta =\displaystyle \frac{3-\sqrt{5}}{2}\). Hence, we have \(\Vert T(\alpha ,\omega )\Vert <1\) if \(\eta <\theta\). Then,

$$\begin{aligned} \eta<\theta\Leftrightarrow & {} \left\{ \begin{array}{l} \delta<\theta \\ \omega \beta<\theta \\ \gamma =|1-\omega |<\theta \end{array}\right. \Leftrightarrow \left\{ \begin{array}{l} \delta<\theta \\ \omega<\frac{\theta }{\beta }\\ 1-\theta<\omega<1+\theta \end{array}\right. \\\Leftrightarrow & {} \left\{ \begin{array}{l} \delta<\theta \\ 1-\theta<\omega<\mathrm {min}\{\frac{\theta }{\beta },~1+\theta \}\\ 1-\theta< \frac{\theta }{\beta } \end{array}\right. \\\Leftrightarrow & {} {\left\{ \begin{array}{l} \delta<\theta \\ \frac{\sqrt{5}-1}{2}<\omega<\mathrm {min}\{\frac{3-\sqrt{5}}{2\beta },~\frac{5-\sqrt{5}}{2}\}.\\ \beta <\frac{\theta }{1-\theta }=\frac{\sqrt{5}-1}{2} \end{array}\right. } \end{aligned}$$

Therefore, if the conditions (27)–(28) are satisfied, the iteration sequence \(\{(x^{(k)},y^{(k)})\}_{k=0}^{+\infty }\) is convergent to \((x^{*},y^{*})\).

4 Numerical experiments

In this section, two examples from LCPs are presented to show the feasibility and effectiveness of the FPI-SS method. We compare the FPI-SS method with the FPI method [22] and the SOR-like iteration method [17, 18] from aspects of the numbers of iteration steps (denoted as “IT”), elapsed CPU time in seconds (denoted as “CPU”), and relative residual error (denoted as “RES”) which is defined by

$$\begin{aligned} \mathrm {RES}(x^{(k)}):=\frac{\Vert Ax^{(k)}-B|x^{(k)}|-b\Vert _{2}}{\Vert b\Vert _{2}}. \end{aligned}$$

In our implementation, all initial guess vectors \(x^{(0)}\) and \(y^{(0)}\) are chosen to zero vectors and all iterations are terminated if \(\mathrm {RES}\le 10^{-6}\) or the maximum number of iteration steps \(k_{\mathrm {max}}\) exceeds 500. All computations are performed in MATLAB R2018b on a personal computer with 2.40GHz central processing unit (Intel(R) Core(TM) i5-6200U) and 8 GB memory.

Consider the following LCP(q, M) [2]: to derive two real vectors \(z, \omega \in \mathbb {R}^{n}\) such that

$$\begin{aligned} z\ge 0,\quad \omega =Mz+q\ge 0,\quad {z^{\text {T}}\omega =0}, \end{aligned}$$
(29)

where \(M\in \mathbb {R}^{n\times n}\) and \(q\in \mathbb {R}^{n}\) are given. From [3,4,5,6], the LCP(q, M) (29) can be formulated as the following GAVE:

$$\begin{aligned} (M+I)x-(M-I)|x |=q, \end{aligned}$$
(30)

with

$$\begin{aligned} x=\frac{1}{2}((M-I)z+q). \end{aligned}$$

Example 1

([5, 6]) The matrix \(M\in \mathbb {R}^{n \times n}\) is defined by \(M=\widehat{M}+\mu I\in \mathbb {R}^{n \times n}\) and \(q\in \mathbb {R}^{n}\) is defined by \(q=-Mz^{*}\), where

$$\begin{aligned} \widehat{M}=\mathrm {Tridiag}(-I,S,-I)=\left[ \begin{array}{llllll} S&{}-I&{}0&{}\cdots &{}0&{}0\\ -I&{}S&{}-I&{}\cdots &{}0&{}0\\ 0&{}-I&{}S&{}\cdots &{}0&{}0\\ \vdots &{}\vdots &{} &{}\ddots &{}\vdots &{}\vdots \\ 0&{}0&{}\cdots &{}\cdots &{}S&{}-I\\ 0&{}0&{}\cdots &{}\cdots &{}-I&{}S \end{array}\right] \in \mathbb {R}^{n\times n} \end{aligned}$$

is a block-tridiagonal matrix,

$$\begin{aligned} S=\mathrm {tridiag}(-1,4,-1)=\left[ \begin{array}{llllll} 4&{}-1&{}0&{}\cdots &{}0&{}0\\ -1&{}4&{}-1&{}\cdots &{}0&{}0\\ 0&{}-1&{}4&{}\cdots &{}0&{}0\\ \vdots &{}\vdots &{} &{}\ddots &{}\vdots &{}\vdots \\ 0&{}0&{}\cdots &{}\cdots &{}4&{}-1\\ 0&{}0&{}\cdots &{}\cdots &{}-1&{}4 \end{array}\right] \in \mathbb {R}^{m\times m} \end{aligned}$$

is a tridiagonal matrix, \(n=m^{2}\), and \(z^{*}=(1,2,1,2,\ldots ,1,2,\ldots )^{\mathrm {T}}\in \mathbb {R}^{n}\) is the unique solution of the LCP(q, M) (29). It can be derived that \(x^{*}=(-0.5,-1,-0.5,-1,\ldots ,-0.5,-1,\ldots )^{\mathrm {T}}\in \mathbb {R}^{n}\) is the exact solution after formulating the LCP(q, M) (29) as the GAVE (30).

For various problem sizes n, the optimal experimental parameters, the iteration steps, CPU time, and relative residual errors of three methods in the case of \(\mu =1\) and \(\mu =4\) are listed in Tables 1 and 2, respectively.

We find that each tested method converges to the exact solution and the number of iterative steps becomes smaller with the increase of \(\mu\). Notably, among these methods, the FPI-SS method requires the least iteration steps and costs the least computing time.

Table 1 Numerical results for Example 1 with \(\mu =1\)
Full size table
Table 2 Numerical results for Example 1 with \(\mu =4\)
Full size table

Example 2

([5]) Consider the LCP(q, M) (29). The matrix \(M\in \mathbb {R}^{n \times n}\) is defined by \(M=\widehat{M}+\mu I\in \mathbb {R}^{n \times n}\) and \(q\in \mathbb {R}^{n}\) is defined by \(q=-Mz^{*}\), where

$$\begin{aligned} \widehat{M}=\mathrm {Tridiag}(-1.5I,S,-0.5I)=\left[ \begin{array}{llllll} S&{}-0.5I&{}0&{}\cdots &{}0&{}0\\ -1.5I&{}S&{}-0.5I&{}\cdots &{}0&{}0\\ 0&{}-1.5I&{}S&{}\cdots &{}0&{}0\\ \vdots &{}\vdots &{} &{}\ddots &{}\vdots &{}\vdots \\ 0&{}0&{}\cdots &{}\cdots &{}S&{}-0.5I\\ 0&{}0&{}\cdots &{}\cdots &{}-1.5I&{}S \end{array}\right] \in \mathbb {R}^{n\times n} \end{aligned}$$

is a block-tridiagonal matrix,

$$\begin{aligned} S=\mathrm {tridiag}(-1.5,4,-0.5)=\left[ \begin{array}{llllll} 4&{}-0.5&{}0&{}\cdots &{}0&{}0\\ -1.5&{}4&{}-0.5&{}\cdots &{}0&{}0\\ 0&{}-1.5&{}4&{}\cdots &{}0&{}0\\ \vdots &{}\vdots &{} &{}\ddots &{}\vdots &{}\vdots \\ 0&{}0&{}\cdots &{}\cdots &{}4&{}-0.5\\ 0&{}0&{}\cdots &{}\cdots &{}-1.5&{}4 \end{array}\right] \in \mathbb {R}^{m\times m} \end{aligned}$$

is a tridiagonal matrix, \(n=m^{2}\), and \(z^{*}=(1,2,1,2,\ldots ,1,2,\ldots )^{\mathrm {T}}\in \mathbb {R}^{n}\) is the unique solution of the LCP(q, M) (29). It can be derived that \(x^{*}=(-0.5,-1,-0.5,-1,\ldots ,-0.5,-1,\ldots )^{\mathrm {T}}\in \mathbb {R}^{n}\) is the exact solution after formulating the LCP(q, M) (29) as the GAVE (30).

In Tables 3 and 4, we list the numerical results of three methods by using experimental optimal parameters in the case of \(\mu =1\) and \(\mu =4\), respectively. From those results, we get the same conclusions as Example 1.

Table 3 Numerical results for Example 2 with \(\mu =1\)
Full size table
Table 4 Numerical results for Example 2 with \(\mu =4\)
Full size table

5 Conclusion

In this paper, by combining the shift-splitting of the coefficient matrix with the fixed point iteration (FPI) method, we proposed a shift-splitting fixed point iteration (FPI-SS) method to solve the generalized absolute value equation (GAVE). We have given several different types of convergence conditions of the FPI-SS method by introducing two different norms of the iteration error. Furthermore, using two numerical examples from linear complementarity problems, we have demonstrated that the FPI-SS method outperforms the FPI method and the SOR-like iteration method in terms of iteration steps and computing times.

Finally, we should mention that the FPI-SS method can be seen as an inexact version of the FPI method. If we replace the shift-splitting in the FPI-SS algorithm with other matrix splitting such as SOR-based splitting [28, 29, 31] and HSS-based splitting [28, 32,33,34], we can establish a series of inexact FPI methods which may have similar convergence results. In real applications of inexact FPI algorithms, how to choose the optimal (or quasi-optimal) parameters is an interesting and practical topic, which is left as our future work.