A Two-Step Newton-Type Method for Solving System of Absolute Value Equations

where A ∈ R and b ∈ R are known vectors and |x| denotes the absolute values of the components of x ∈ R. )e system of absolute value equations arises in optimization, the economies with institutional restrictions upon prices, the free boundary problems for journal bearing lubrication, and the network equilibrium problems, for example, see [1–12]. Mansoori and Erfanian [13] suggested a dynamic model to obtain the exact solution of equation (1).)e development of multistep methods has gained popularity in the field of computational mathematics. Recently, several multistep methods are proposed to solve equation (1), for example, see [3–6]. In this paper, we present a Newton-type method for solving the system of absolute value equations. )e new method is a two-step method where the well-known numerical quadrature technique is used in the corrector step. )e Newton-type method is very simple and easy to implement in practice. )e existence and uniqueness of solution shows the importance of the suggested method. )e convergence of the new method is discussed in Section 3. Numerical results prove that the Newton-type method is very effective for solving large systems. For x ∈ R, consider


Introduction
We consider the system of absolute value equations of the form: where A ∈ R n×n and b ∈ R n are known vectors and |x| denotes the absolute values of the components of x ∈ R n . e system of absolute value equations arises in optimization, the economies with institutional restrictions upon prices, the free boundary problems for journal bearing lubrication, and the network equilibrium problems, for example, see [1][2][3][4][5][6][7][8][9][10][11][12]. Mansoori and Erfanian [13] suggested a dynamic model to obtain the exact solution of equation (1). e development of multistep methods has gained popularity in the field of computational mathematics. Recently, several multistep methods are proposed to solve equation (1), for example, see [3][4][5][6].
In this paper, we present a Newton-type method for solving the system of absolute value equations.
e new method is a two-step method where the well-known numerical quadrature technique is used in the corrector step.
e Newton-type method is very simple and easy to implement in practice. e existence and uniqueness of solution shows the importance of the suggested method. e convergence of the new method is discussed in Section 3.
Numerical results prove that the Newton-type method is very effective for solving large systems.
For x ∈ R n , consider e diagonal matrix S corresponding to sign(x) is defined as where z|x| represents the generalized Jacobian of |x|. Note that S(x)x � |x|.

Proposed Method
Consider e generalized Jacobian of H at x is where S(x) is a diagonal matrix and S(x)x � |x|. We suggest the Newton-type method as e predictor step of equation (6) is the generalized Newton method [8], and the corrector step is the Trapezoidal method. e proposed method can be described as Algorithm 1.

Convergence Analysis
e predictor step is well defined (see Lemma 2 [8]): Now, we want to prove that which is nonsingular.

Lemma 1. If the singular values of
Since singular values of A exceed 1, therefore which is a contradiction; hence, 4A − S(x k ) − 2S(x k + y k ) − S(y k ) is nonsingular. Subsequently, the sequence in (6) is well defined.

□
In the next result, we establish boundedness of equation (6). Rewrite the corrector step of equation (6) as (10) For simplicity, we let en, equation (10) converts into

Lemma 2. e sequence in equation (6) is bounded and well defined if the singular values of A exceed one. Hence, there exists an accumulation point _
x such that for diagonal matrix _ S.
Proof. Proof of this Lemma is similar to the proof of Proposition 3 of [8].
□ Now, we prove the main result about the convergence of the Newton-type method.
for any diagonal matrix S with diagonal entries of 0 or ±1, then the Newton-type method converges to a solution x * of equation (1).
Since x * is the solution of equation (1), therefore From equations 15 and (16), we have or In equation (19), we have used Lipschitz continuity of the absolute value (see Lemma 5 [8]); that is, Since S(x k ), S(y k ), and 2S(x k + y k ) are diagonal matrices with elements equal to ±1 or 0, hence From equations (20) and (22), we have In equation (20), we have used the condition that ‖(E) − 1 ‖ < 1/12. Hence, the sequence x k converges linearly to x * . □ Mathematical Problems in Engineering Corollary 1. Let ‖A − 1 ‖ < (1/13) and ‖S(x k )‖, ‖S(y k )‖, and ‖2S(x k + y k )‖ are nonzero. en, equation (1) is uniquely solvable for any b and the Newton-type method is well defined and converges to the unique solution of equation (1) for any initial guess x 0 .

Numerical Results
Now, we consider some numerical examples to show the performance of the new method. Let IT denote the number of iterations, RES is the residual, and CPU denotes the time in seconds. All the experiments are done with Intel(R) Core (TM)-5Y10c CPU @ 0.80 GHz 1.00 GHz, 4 GB RAM.
We compare the Newton-type method (NTM) with the generalized Traub method (GTM) [5] and improved generalized Newton method (INM) [2]. Numerical results are given in Table 1.
From the last column of Table 1, we conclude that the Newton-type method converges to the approximate solution of (1.1) with high accuracy.
We take m � 1000 (problem size) and compare the Newton-type method (NTM) with the improved generalized Newton method (INM) [2] and the two-step iterative (TSI) method [3] graphically. Numerical results are illustrated in Figure 1.
Convergence curves show the efficiency of the proposed method. Figure 1 illustrates that the convergence of the NTM is better than other two methods.

Example 3. Consider the heat equation as follows:
We use the finite differences to approximate the solution of heat equation. Define the distance step-size h � (1/m) for m > 0 and the time step-size k such that μ i � ih, i �, 1, 2, . . . , m − 1, and t j � jk, j � 1, 2, . . . ,. e partial derivative at the grid point (μ i , t j ) can be written as Comparison of equation (6) with Maple solutions is given in Figure 2.
In this example, we compare the Newton-type method with the SOR-like method [6] and NINA [15]. Comparison Step 1: select an initial guess x (0) ∈ R n Step 2: start for k compute y k � (A − S(x k )) − 1 b Step 3: using y k compute x k+1 � x k − 4(H′(x k ) + 2H′(x k + y k /2) + H′(y k )) − 1 H(x k ) Step 4: if x k+1 � x k , then stop. If not, then put k � k + 1 and turn back to step 2     is given in Table 2, where ERR � ‖x k − x * ‖ and x * is the exact solution. Table 2 shows that NTM converges to the solution in just 2 iterations with high accuracy.
We take n � 1000 (problem size) and compare the Newton-type method (NTM) with the SOR-like method [6] and NINA [11] graphically in Figure 3.

Conclusion
In this paper, we developed the Newton-type method for solving the system of absolute value equations.
is new iterative method is very simple and easy to implement. e future work is to extend this idea by taking more than three points in the corrector step. e numerical results show the efficiency and accuracy of the proposed method.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.