Abstract
The preconditioner presented by Hadjidimos et al. (2003) can improve on the convergence rate of the classical iterative methods to solve linear systems. In this paper, we extend this preconditioner to solve linear complementarity problems whose coefficient matrix is M-matrix or H-matrix and present a multisplitting and Schwarz method. The convergence theorems are given. The numerical experiments show that the methods are efficient.
1. Introduction
Many science and engineering problems are usually induced as linear complementarity problems (LCP): find an such that where is a given matrix and is a vector. It is necessary to establish an efficient algorithm to solve the complementarity problem. Numerical methods for complementarity problems fall in two major kinds, direct and iterative methods. There have been lots of works on the solution of the linear complementarity problem ([1–4], etc.), which presented feasible and essential techniques for LCP. Recently some parallel multisplitting iterative methods for solving the large sparse linear complementarity problems are presented ([5–11], etc.). These methods are based on several splittings of the system matrix and are constructed with a suitable weighting combination of the solution of the sublinear complementarity problems.
For the large sparse linear complementarity problem, some accelerated modulus-based matrix splitting iteration methods and modulus-based synchronous two-stage multisplitting iteration methods are constructed [7, 11]. Numerical results show that these methods are more efficient.
Many researchers have studied preconditioners applied to linear system so that the corresponding iterative methods, such as Jacobi or GS, converge faster than the classical ones. Hadjidimos et al. [12] considered the following preconditioner: where with constants .
Consider
In (3), let ; is a preconditioner presented by Milaszewicz [13]. It eliminates the elements of the first column of below the diagonal. Reference [12] shows that the new modifications and improvements of the original preconditioners can improve on the convergence rates of the classical iterative methods (Jacobi, GS, etc.).
In this paper, with multisplitting technique, we will extend the preconditioner to solve the linear complementarity problem (1) and present a new multisplitting and Schwarz method. The new method is parallel and has high computational efficiency.
In Section 2, some preliminaries for the new method are presented. A multisplitting and Schwarz method is given in Section 3. Convergence analysis is given in Section 4. Section 5 presents the numerical experiments results.
2. Preliminaries
At first we briefly describe the notations. In and , the relation denotes the natural components partial ordering. In addition, for , we write if . A nonsingular matrix is termed -matrix, if for and . Or the nonsingular matrix is called -matrix, if , , and . Its comparison matrix is defined by and . is said to be an -matrix if is an -matrix. To simplify the notation, we may assume that .
Lemma 1 (see [2]). Let be an -matrix and let be a solution of (1).(1)If , then and therefore .(2)If , then is the solution of (1).
If the problem (1) has a nonzero solution, there at least exists an index , . In this paper, let us assume that . By Lemma 1, we have the following conclusion.
Lemma 2 (see [14]). Let be an -matrix, , and . If , then the following linear complementarity problem is equivalent to the problem (1).
Lemma 3 (see [15]). Let and for . is an -matrix if and only if there exists a positive vector such that .
Definition 4 (see [16]). (1) splitting is termed a regular splitting of matrix if and .
(2) splitting is termed -splitting of matrix if is an -matrix and .
(3) splitting is termed -compatible splitting of matrix if .
Lemma 5 (see [16]). Let be two regular splittings of , where .(1)If , then
(2)If , then
By Lemma 5, we have the following lemma.
Lemma 6. Let be two -splittings of , and If , then .
Lemma 7 (see [14]). If is an -matrix, then is a -matrix and is also an -matrix.
Lemma 8 (see [15]). is a nonsingular -matrix if and only if all the principal minors of are positive.
By (4), we have
Define the following matrices:
Consider the following splittings [12]:
Define the following matrices with the above splittings: (i);(ii);(iii);(iv);(v);(vi).
Theorem 9 (see [12]). Under the notation so far, if is an -matrix, then, for any , there exists , , such that
3. Synchronous Multisplitting and Schwarz Method
By Theorem 9, . It means that the Gauss-Seidel iterative methods associated with the new preconditional matrix will be no worse than the ones corresponding to . Similar to [6], we present a synchronous multisplitting and Schwarz algorithm corresponding to .
Algorithm 10 (synchronous multisplitting and Schwarz method). (1) Give an initial vector .
(2) Let
where , is a nonnegative diagonal matrix, and is the solution of the following LCP:
where , .
(3) Consider ; if the iteration solution is convergent, stop; else, return to step (2).
Let , , and . Define as where denotes and denotes .
Then the following lemma is obviously true.
Lemma 11. For each splitting , let be defined by (15). Then the subproblem (14) is equivalent to the following problem: find , such that
4. Convergence Analysis
In this section, we give the convergence analysis of the algorithm.
Lemma 12 (see [6]). Let be the solution of (1), and is the solution of (14); then
Theorem 13. Let be an -matrix; the sequence generated by Algorithm 10 converges to the solution of (1).
Proof. The conclusion easily resulted from Lemma 2, Lemma 12, and Theorem 9.
In Lemma 7, if , then is an -matrix. If , , then and is not an -matrix. In the sequel we will examine that is an -matrix with positive diagonal elements, where satisfies some conditions.
Lemma 14 (see [16]). Let be either a strictly diagonally dominant or an irreducibly dominant matrix. Then is an -matrix.
Lemma 15. Let be a diagonally dominant -matrix. If and for , , then is an -matrix with positive diagonal elements.
Proof. Note that , and . We have
is well defined. By the definition of , and for , , we have that(1);(2);(3)if ,
It implies that is a diagonally dominant matrix; then it is an -matrix with positive diagonal elements by Lemma 14.
Since is an -matrix, according to [8], we can solve the problem (5) using Algorithm 10, where maybe an -compatible splitting of matrix .
Lemma 16 (see [6]). Let be the solution of (1), and is the solution of (14); then
Similar to the proof in Theorem 2.1 in [8], we have the following convergence theorem.
Theorem 17. Let be an -matrix; the sequence generated by Algorithm 10 converges to the solution of the problem (1).
5. Numerical Experiments
In this section, we give two numerical examples to show that the new methods are efficient. In the numerical experiments, the stop criterion is . In the tables, MMS denotes Algorithm 10 with preconditioner, and GSOR denotes Algorithm 10, in which .
Example 1. We consider a linear complementarity problem, whose coefficient matrix is
The results are shown as Table 1.
Example 2. Let us consider the following problem:
where , is a unit matrix, , and .
For , let us choose ; then is an -matrix. In Algorithm 10 maybe an -compatible splitting for each splitting. The corresponding results are shown in Tables 2 and 3.
An accelerated modulus-based accelerated overrelaxation (AMAOR) iteration method is presented by Zheng and Yin [11]. Same as in [11], we choose , , and . In Example 2, . In Table 4, iter denotes iterative step and cputime denotes time (seconds). Table 4 shows that our preconditioned method MMS spends less time than the AMAOR.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Nature Science Projects of China 11161014 and Guangxi Experiment Center of Information Science Foundation.