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Abstract and Applied Analysis
VolumeΒ 2012Β (2012), Article IDΒ 406232, 9 pages
http://dx.doi.org/10.1155/2012/406232
Research Article

Residual Iterative Method for Solving Absolute Value Equations

1Mathematics Department, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan
2Mathematics Department, College of Science, King Saud University, Riyadh, Saudi Arabia

Received 30 November 2011; Accepted 13 December 2011

Academic Editor: Khalida InayatΒ Noor

Copyright Β© 2012 Muhammad Aslam Noor et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We suggest and analyze a residual iterative method for solving absolute value equations 𝐴π‘₯βˆ’|π‘₯|=𝑏 where π΄βˆˆπ‘…π‘›Γ—π‘›, π‘βˆˆπ‘…π‘› are given and π‘₯βˆˆπ‘…π‘› is unknown, using the projection technique. We also discuss the convergence of the proposed method. Several examples are given to illustrate the implementation and efficiency of the method. Comparison with other methods is also given. Results proved in this paper may stimulate further research in this fascinating field.

1. Introduction

The residual methods were proposed for solving large sparse systems of linear equations 𝐴π‘₯=𝑏,(1.1) where π΄βˆˆπ‘…π‘›Γ—π‘› is a positive definite matrix and π‘₯,π‘βˆˆπ‘…π‘›. Paige and Saunders [1] minimized the residual norm over the Krylov subspace and proposed an algorithm for solving indefinite systems. Saad and Schultz [2] used Arnoldi process and suggested generalized minimal residual method which minimize norm of the residual at each step. The residual methods have been studied extensively [3–5].

We show that the Petrov-Galerkin process can be extended for solving absolute value equations of the form: 𝐴π‘₯βˆ’|π‘₯|=𝑏,(1.2) where π΄βˆˆπ‘…π‘›Γ—π‘›, π‘βˆˆπ‘…π‘›. Here |π‘₯| is the vector in 𝑅𝑛 with absolute values of the components of π‘₯ and π‘₯βˆˆπ‘…π‘›, is unknown. The absolute value equations (1.1) were investigated extensively in [6]. It was Managasarian [7, 8], who proved that the absolute value equations (1.2) are equivalent to the linear complementarity problems. This equivalent formulation was used by Managasarian [7, 8] to solve the absolute value equations. We would like to remark that the complementarity problems are also equivalent to the variational inequalities. Thus, we conclude that the absolute value equations are equivalent to the variational inequalities. There are several methods for solving the variational inequalities; see Noor [9–11], Noor et al. [12, 13] and the references therein. To the best our knowledge, this alternative equivalent formulation has not exploited up to now. This is another direction for future direction. We hope that this interlink among these fields may lead to discover novel and innovative techniques for solving the absolute value equations and related optimization problems. Noor et al. [14, 15] have suggested some iterative methods for solving absolute value equation (1.2) using minimization technique with symmetric positive definite matrix. For more details, see [3, 4, 6–12, 14–19].

In this paper, we suggest and analyse residual iterative method for solving absolute value equations (1.2) using projection technique. Our method is easy to implement. We discuss the convergence of the residual method for nonsymmetric positive definite matrices.

We denote by 𝐾 and 𝐿 the search subspace and the constraints subspace, respectively, and let π‘š be their dimension and π‘₯0βˆˆπ‘…π‘› an initial guess. A projection method onto the subspace 𝐾 and orthogonal to 𝐿 is a process to find an approximate solution π‘₯βˆˆπ‘…π‘› to (1.2) by imposing the Petrov-Galerkin conditions that π‘₯ belong to affine space π‘₯0+𝐾 such that the new residual vector orthogonal to 𝐿, that is, findπ‘₯∈π‘₯0+𝐾suchthatπ‘βˆ’(π΄βˆ’π·(π‘₯))π‘₯βŸ‚πΏ,(1.3) where 𝐷(π‘₯) is diagonal matrix corresponding to sign(π‘₯). For different choices of the subspace 𝐿, we have different iterative methods. Here we use the constraint space 𝐿=(π΄βˆ’π·(π‘₯))𝐾. The residual method approximates the solution of (1.2) by the vector π‘₯∈π‘₯0+𝐾that minimizes the norm of residual.

The inner product is denoted by βŸ¨β‹…,β‹…βŸ© in the 𝑛-dimensional Euclidean space 𝑅𝑛. For π‘₯βˆˆπ‘…π‘›, sign(π‘₯) will denote a vector with components equal to 1,0,βˆ’1 depending on whether the corresponding component of π‘₯ is positive, zero, or negative. The diagonal matrix 𝐷(π‘₯) corresponding to sign(π‘₯) is defined as𝐷(π‘₯)=πœ•|π‘₯|=diag(sign(π‘₯)),(1.4) where πœ•|π‘₯| represent the generalized Jacobean of |π‘₯| based on a subgradient [20, 21].

We denote the following by π‘Ž=βŸ¨πΆπ‘£1,𝐢𝑣1⟩,𝑐=βŸ¨πΆπ‘£1,𝐢𝑣2⟩,𝑑=βŸ¨πΆπ‘£2,𝐢𝑣2π‘βŸ©,1=ξ«π‘βˆ’π΄π‘₯π‘˜+||π‘₯π‘˜||,𝐢𝑣1=βŸ¨π‘βˆ’πΆπ‘₯π‘˜,𝐢𝑣1π‘βŸ©,2=ξ«π‘βˆ’π΄π‘₯π‘˜+||π‘₯π‘˜||,𝐢𝑣2=βŸ¨π‘βˆ’πΆπ‘₯π‘˜,𝐢𝑣2⟩,(1.5) where 0≠𝑣1, 𝑣2βˆˆπ‘…π‘›, and 𝐢=π΄βˆ’π·(π‘₯π‘˜). We consider 𝐴 such that 𝐢 is a positive definite matrix. We remark that 𝐷(π‘₯π‘˜)π‘₯π‘˜=|π‘₯π‘˜|.

2. Residual Iterative Method

Consider the iterative scheme of the type: π‘₯π‘˜+1=π‘₯π‘˜+𝛼𝑣1+𝛽𝑣2,0≠𝑣1,𝑣2βˆˆπ‘…π‘›,π‘˜=0,1,2,….(2.1) These vectors can be chosen by different ways. To derive residual method for solving absolute value equations in the first step, we choose the subspace𝐾1𝑣=span1ξ€Ύ,𝐿1ξ€½=span𝐢𝑣1ξ€Ύ,π‘₯0=π‘₯π‘˜.(2.2) For 𝐷(Μƒπ‘₯π‘˜+1)=𝐷(π‘₯π‘˜), we write the residual in the following form: π‘βˆ’π΄Μƒπ‘₯π‘˜+1+||Μƒπ‘₯π‘˜+1||ξ€·ξ€·=π‘βˆ’π΄βˆ’π·Μƒπ‘₯π‘˜+1ξ€Έξ€ΈΜƒπ‘₯π‘˜+1ξ€·ξ€·π‘₯=π‘βˆ’π΄βˆ’π·π‘˜ξ€Έξ€ΈΜƒπ‘₯π‘˜+1=π‘βˆ’πΆΜƒπ‘₯π‘˜+1.(2.3) From (1.3) and (2.3), we calculate Μƒπ‘₯π‘˜+1∈π‘₯π‘˜+𝐾1suchthatπ‘βˆ’πΆΜƒπ‘₯π‘˜+1βŸ‚πΏ1;(2.4) that is, we find the approximate solution by the iterative scheme Μƒπ‘₯π‘˜+1=π‘₯π‘˜+𝛼𝑣1.(2.5) Now, we rewrite (2.4) in the inner product as ξ«π‘βˆ’πΆΜƒπ‘₯π‘˜+1,𝐢𝑣1=0;(2.6) from the above discussion, we have βŸ¨π‘βˆ’πΆπ‘₯π‘˜βˆ’π›ΌπΆπ‘£1,𝐢𝑣1⟩=βŸ¨π‘βˆ’πΆπ‘₯π‘˜,𝐢𝑣1βŸ©βˆ’π›ΌβŸ¨πΆπ‘£1,𝐢𝑣1⟩=𝑝1βˆ’π‘Žπ›Ό=0,(2.7) from which we have 𝑝𝛼=1π‘Ž.(2.8) The next step is to choose the subspace 𝐾2𝑣=span2ξ€Ύ,𝐿2ξ€½=span𝐢𝑣2ξ€Ύ,π‘₯0=Μƒπ‘₯π‘˜+1,(2.9) and to find the approximate solution π‘₯π‘˜+1 such that π‘₯π‘˜+1βˆˆΜƒπ‘₯π‘˜+1+𝐾2suchthatπ‘βˆ’πΆπ‘₯π‘˜+1βŸ‚πΏ2,(2.10) where π‘₯π‘˜+1=Μƒπ‘₯π‘˜+1+𝛽𝑣2,π‘βˆ’π΄π‘₯π‘˜+1+||π‘₯π‘˜+1||=π‘βˆ’πΆπ‘₯π‘˜+1ξ€·π‘₯,π·π‘˜+1ξ€Έξ€·π‘₯=π·π‘˜ξ€Έ.(2.11) Rewriting (2.10) in terms of the inner product, we haveξ«π‘βˆ’πΆπ‘₯π‘˜+1,𝐢𝑣2=0.(2.12) Thus, we haveξ«π‘βˆ’πΆπ‘₯π‘˜+1,𝐢𝑣2=βŸ¨π‘βˆ’πΆπ‘₯π‘˜βˆ’π›ΌπΆπ‘£1βˆ’π›½πΆπ‘£2,𝐢𝑣2⟩=βŸ¨π‘βˆ’πΆπ‘₯π‘˜,𝐢𝑣2βŸ©βˆ’π›ΌβŸ¨πΆπ‘£1,𝐢𝑣2βŸ©βˆ’π›½βŸ¨πΆπ‘£2,𝐢𝑣2⟩=𝑝2βˆ’π‘π›Όβˆ’π‘‘π›½=0.(2.13) From (2.8) and (2.13), we obtain𝛽=π‘Žπ‘2βˆ’π‘π‘1π‘Žπ‘‘.(2.14)

We remark that one can choose 𝑣1=π‘Ÿπ‘˜ and 𝑣2 in different ways. However, we consider the case 𝑣2=π‘ π‘˜ (π‘ π‘˜ is given in Algorithm 2.1).

Based upon the above discussion, we suggest and analyze the following iterative method for solving the absolute value equations (1.2) and this is the main motivation of this paper.

Algorithm 2.1. Choose an initial guess π‘₯0βˆˆπ‘…π‘›,For π‘˜=0,1,2,… until convergence doπ‘Ÿπ‘˜=π‘βˆ’π΄π‘₯π‘˜+|π‘₯π‘˜|π‘”π‘˜=(π΄βˆ’π·(π‘₯π‘˜))𝑇(𝐴π‘₯π‘˜βˆ’|π‘₯π‘˜|βˆ’π‘)π»π‘˜=((π΄βˆ’π·(π‘₯π‘˜))βˆ’1(π΄βˆ’π·(π‘₯π‘˜)))π‘‡π‘ π‘˜=βˆ’π»π‘˜π‘”π‘˜If β€–π‘Ÿπ‘˜β€–=0, then stop; elseπ›Όπ‘˜=𝑝1/π‘Ž,π›½π‘˜=(π‘Žπ‘2βˆ’π‘π‘1)/π‘Žπ‘‘Set π‘₯π‘˜+1=π‘₯π‘˜+π›Όπ‘˜π‘Ÿπ‘˜+π›½π‘˜π‘ π‘˜ If β€–π‘₯π‘˜+1βˆ’π‘₯π‘˜β€–<10βˆ’6then stopEnd ifEnd for π‘˜.

If 𝛽=0, then Algorithm 2.1 reduces to minimal residual method; see [2, 5, 21, 22]. For the convergence analysis of Algorithm 2.1, we need the following result.

Theorem 2.2. Let {π‘₯π‘˜} and {π‘Ÿπ‘˜} be generated by Algorithm 2.1; if 𝐷(π‘₯π‘˜+1)=𝐷(π‘₯π‘˜), then β€–β€–π‘Ÿπ‘˜β€–β€–2βˆ’β€–β€–π‘Ÿπ‘˜+1β€–β€–2=𝑝21π‘Ž+ξ€·π‘Žπ‘2βˆ’π‘π‘1ξ€Έ2π‘Ž2𝑑,(2.15) where π‘Ÿπ‘˜+1=π‘βˆ’π΄π‘₯π‘˜+1+|π‘₯π‘˜+1| and 𝐷(π‘₯π‘˜+1)=diag(sign(π‘₯π‘˜+1)).

Proof. Using (2.1), we obtain π‘Ÿπ‘˜+1=π‘βˆ’π΄π‘₯π‘˜+1+||π‘₯π‘˜+1||ξ€·ξ€·π‘₯=π‘βˆ’π΄βˆ’π·π‘˜+1π‘₯ξ€Έξ€Έπ‘˜+1ξ€·ξ€·π‘₯=π‘βˆ’π΄βˆ’π·π‘˜π‘₯ξ€Έξ€Έπ‘˜+1ξ€·ξ€·π‘₯=π‘βˆ’π΄βˆ’π·π‘˜π‘₯ξ€Έξ€Έπ‘˜ξ€·ξ€·π‘₯βˆ’π›Όπ΄βˆ’π·π‘˜π‘£ξ€Έξ€Έ1ξ€·ξ€·π‘₯βˆ’π›½π΄βˆ’π·π‘˜π‘£ξ€Έξ€Έ2=π‘βˆ’π΄π‘₯π‘˜+||π‘₯π‘˜||βˆ’π›ΌπΆπ‘£1βˆ’π›½πΆπ‘£2=π‘Ÿπ‘˜βˆ’π›ΌπΆπ‘£1βˆ’π›½πΆπ‘£2.(2.16) Now consider β€–β€–π‘Ÿπ‘˜+1β€–β€–2=ξ«π‘Ÿπ‘˜+1,π‘Ÿπ‘˜+1=βŸ¨π‘Ÿπ‘˜βˆ’π›ΌπΆπ‘£1βˆ’π›½πΆπ‘£2,π‘Ÿπ‘˜βˆ’π›ΌπΆπ‘£1βˆ’π›½πΆπ‘£2⟩=βŸ¨π‘Ÿπ‘˜,π‘Ÿπ‘˜βŸ©βˆ’2π›ΌβŸ¨π‘Ÿπ‘˜,𝐢𝑣1βŸ©βˆ’2π›Όπ›½βŸ¨πΆπ‘£1,𝐢𝑣2βŸ©βˆ’2π›½βŸ¨π‘Ÿπ‘˜,𝐢𝑣2⟩+𝛼2βŸ¨πΆπ‘£1,𝐢𝑣1⟩+𝛽2βŸ¨πΆπ‘£2,𝐢𝑣2⟩=β€–β€–π‘Ÿπ‘˜β€–β€–2βˆ’2𝛼𝑝1+2π‘π›Όπ›½βˆ’2𝛽𝑝2+π‘Žπ›Ό2+𝛽2𝑑.(2.17) From (2.8), (2.14), and (2.17), we have β€–β€–π‘Ÿπ‘˜β€–β€–2βˆ’β€–β€–π‘Ÿπ‘˜+1β€–β€–2=𝑝21π‘Ž+ξ€·π‘Žπ‘2βˆ’π‘π‘1ξ€Έ2π‘Ž2𝑑,(2.18) the required result (2.15).

Since 𝑝21/π‘Ž+(π‘Žπ‘2βˆ’π‘π‘1)2/π‘Ž2𝑑β‰₯0, so from (2.18) we have β€–β€–π‘Ÿπ‘˜β€–β€–2βˆ’β€–β€–π‘Ÿπ‘˜+1β€–β€–2=𝑝21π‘Ž+ξ€·π‘Žπ‘2βˆ’π‘π‘1ξ€Έ2π‘Ž2𝑑β‰₯0.(2.19) From (2.19) we have β€–π‘Ÿπ‘˜+1β€–2β‰€β€–π‘Ÿπ‘˜β€–2. For any arbitrary vectors 0≠𝑣1,𝑣2βˆˆπ‘…π‘›, 𝛼,𝛽 are defined by (2.8), and (2.14) minimizes norm of the residual.

We now consider the convergence criteria of Algorithm 2.1, and it is the motivation of our next result.

Theorem 2.3. If 𝐢 is a positive definite matrix, then the approximate solution obtained from Algorithm 2.1 converges to the exact solution of the absolute value equations (1.2).

Proof. From (2.15), we have β€–β€–π‘Ÿπ‘˜β€–β€–2βˆ’β€–β€–π‘Ÿπ‘˜+1β€–β€–2β‰₯𝑝21π‘Ž=βŸ¨π‘Ÿπ‘˜,πΆπ‘Ÿπ‘˜βŸ©2βŸ¨πΆπ‘Ÿπ‘˜,πΆπ‘Ÿπ‘˜βŸ©β‰₯πœ†2minβ€–β€–π‘Ÿπ‘˜β€–β€–4πœ†2maxβ€–β€–π‘Ÿπ‘˜β€–β€–2=πœ†2minπœ†2maxβ€–β€–π‘Ÿπ‘˜β€–β€–2.(2.20) This means that the sequence β€–π‘Ÿπ‘˜β€–2 is decreasing and bounded. Thus the above sequence is convergent which implies that the left-hand side tends to zero. Hence β€–π‘Ÿπ‘˜β€–2 tends to zero, and the proof is complete.

3. Numerical Results

To illustrate the implementation and efficiency of the proposed method, we consider the following examples. All the experiments are performed with Intel(R) Core(TM) 2 Γ— 2.1 GHz, 1 GB RAM, and the codes are written in Mat lab 7.

Example 3.1. Consider the ordinary differential equation: 𝑑2π‘₯𝑑𝑑2ξ€·βˆ’|π‘₯|=1βˆ’π‘‘2ξ€Έ,0≀𝑑≀1,π‘₯(0)=βˆ’1π‘₯(1)=0.(3.1) We discredited the above equation using finite difference method to obtain the system of absolute value equations of the type: 𝐴π‘₯βˆ’|π‘₯|=𝑏,(3.2) where the system matrix 𝐴 of size 𝑛=10 is given by π‘Žπ‘–,𝑗=⎧βŽͺ⎨βŽͺβŽ©ξ‚»βˆ’242,for𝑗=𝑖,121,for𝑗=𝑖+1,𝑖=1,2,…,π‘›βˆ’1,𝑗=π‘–βˆ’1,𝑖=2,3,…,𝑛,0,otherwise.(3.3) The exact solution is ξ‚»π‘₯=.1915802528sinπ‘‘βˆ’4cos𝑑+3βˆ’π‘‘2,π‘₯<0,βˆ’1.462117157π‘’βˆ’π‘‘βˆ’0.5378828428𝑒𝑑+1+𝑑2,π‘₯>0.(3.4) In Figure 1, we compare residual method with Noor et al. [14, 15]. The residual iterative method, minimization method [14], and the iterative method [10] solve (3.1) in 51, 142, and 431 iterations, respectively. For the next two examples, we interchange 𝑣1,𝑣2 with each other as Algorithm 2.1 converges for nonzero vectors 𝑣1,𝑣2βˆˆπ‘…π‘›.

406232.fig.001
Figure 1

Example 3.2 (see [17]). We first chose a random 𝐴 from a uniform distribution on [βˆ’10,10], then we chose a random π‘₯ from a uniform distribution on [βˆ’1, 1]. Finally we computed 𝑏=𝐴π‘₯βˆ’|π‘₯|. We ensured that the singular values of each 𝐴 exceeded 1 by actually computing the minimum singular value and rescaling 𝐴 by dividing it by the minimum singular value multiplied by a random number in the interval [0, 1]. The computational results are given in Table 1.

tab1
Table 1

In Table 1, GNM and RIM denote generalized Newton method [17] and residual iterative method. From Table 1 we conclude that residual method for solving absolute value equations (1.2) is more effective.

Example 3.3 (see [23]). Consider random matrix 𝐴 and 𝑏 in Mat lab code as 𝑛=input("dimensionofmatrix𝐴=");rand("state",0);𝑅=rand(𝑛,𝑛);𝑏=rand(𝑛,1);𝐴=π‘…ξ…žβˆ—Runβˆ—eye(𝑛),(3.5) with random initial guess. The comparison between the residual iterative method and the Yong method [23] is presented in Table 2.

tab2
Table 2

In Table 2 TOC denotes time taken by CPU. Note that for large problem sizes the residual iterative method converges faster than the Yong method [23].

4. Conclusions

In this paper, we have used the projection technique to suggest an iterative method for solving the absolute value equations. The convergence analysis of the proposed method is also discussed. Some examples are given to illustrate the efficiency and implementation of the new iterative method. The extension of the proposed iterative method for solving the general absolute value equation of the form 𝐴π‘₯+𝐡|π‘₯|=𝑏 for suitable matrices is an open problem. We have remarked that the variational inequalities are also equivalent to the absolute value equations. This equivalent formulation can be used to suggest and analyze some iterative methods for solving the absolute value equations. It is an interesting and challenging problem to consider the variational inequalities for solving the absolute value equations.

Acknowledgments

This research is supported by the Visiting Professor Program of the King Saud University, Riyadh, Saudi Arabia, and Research Grant no. KSU.VPP.108. The authors are also grateful to Dr. S. M. Junaid Zaidi, Rector, COMSATS Institute of Information Technology, Pakistan, for providing the excellent research facilities.

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