Abstract
The TTS iterative method is proposed to solve non-Hermitian positive definite linear systems and some convergence conditions are presented. Subsequently, these convergence conditions are applied to the ALUS method proposed by Xiang et al. in 2012 and comparison of some convergence theorems is made. Furthermore, an example is given to demonstrate the results obtained in this paper.
1. Introduction
Many problems in scientific computing give rise to a system of linear equations in unknowns, where is a large and sparse non-Hermitian matrix. In this paper we consider the important case where is positive definite; that is, the Hermitian part is Hermitian positive definite, where denotes the conjugate transpose of the matrix . Large and sparse systems of this type arise in many applications, including discretizations of convection-diffusion problems [1], regularized weighted least-squares problems [2], real-valued formulations of certain complex symmetric systems [3].
There have been several studies on the convergence of splitting iterative methods for non-Hermitian positive definite linear systems. In [4, pages 190–193], some convergence conditions for the splitting of non-Hermitian positive definite matrices have been established. More recently, [5, 6] give some conditions for the convergence of splittings for this class of linear systems.
Recently, there has been considerable interest in the HSS (Hermitian and skew-Hermitian splitting) method introduced by Bai et al. in order to solve non-Hermitian positive definite linear systems; see [7]; we further note the generalizations and extensions of this basic method proposed in [8–13]. Furthermore, these methods and their convergence theories have been shown to apply to (generalized) saddle point problems, either directly or indirectly (as a preconditioner); see [5, 6, 8, 9, 11–16].
Continuing in this direction, in this paper we establish new results on splitting methods for solving system (1) iteratively, focusing on a particular class of splittings—the Triangular and Triangular splitting (TTS). According to the idea of HSS method, we construct another alternative iterative method—the TTS method to solve non-Hermitian positive definite linear systems; furthermore, we will prove the convergence of this alternative method.
2. The TTS Method
Let be non-Hermitian positive definite with the Hermitian part . We split as , where , is strictly lower triangular matrix, and is strictly upper triangular matrix. Let and let . Then the splitting is called the Triangular and Triangular splitting (TTS). Subsequently, we give the TTS method.
The TTS Method. Given an initial guess , for , until converges, compute where is a given positive constant.
Eliminating from (3), we obtain the iterative process where now is the iteration matrix of the iterative scheme (4) and . It is easy to see that the iterative scheme (3) is convergent if and only if the iterative scheme (4) is convergent. Thus, we only consider the convergence of the iterative scheme (4) and consequently investigate the spectral radius, , of the iterative matrix .
3. Convergence Condition for TTS Method
In this section, a convergence condition for TTS method is presented to solve non-Hermitian positive definite linear systems. The following lemma will be used in this section.
Lemma 1. Let be non-Hermitian. Then the iteration matrix of the iterative scheme (4) is
Proof. Since , which completes the proof.
Theorem 2. Let be non-Hermitian positive definite with the TTS as in (2). Then the TTS method converges to the unique solution of (1) for any choice of the initial guess if and only if , where , , , , and .
Proof. Since the iterative scheme (3) is convergent if and only if the iterative scheme (4) is convergent, it follows from [17] that (4) converges for any given if and only if , where denotes the spectral radius of the matrix . Assume . Then . It follows from Lemma 1 that Multiplying (7) on the left side by , We assert . Otherwise, , and consequently . As a result, Equations (9) yield that Equation (10) shows that is not Hermitian positive definite; that is, is not non-Hermitian positive definite. Therefore, a contradiction appears to indicate . Thus, which indicates that if and only if where , , , , and . Since and , (12) holds if and only if . This completes the proof.
Theorem 2 presents a convergence condition for the TTS method. But, in fact, this condition is difficult to be applied. It is necessary to give a practical condition. It follows from Theorem 2 that we can get the following conclusion.
Theorem 3. Let be non-Hermitian positive definite with the TTS as in (2), and let where and denote the minimal and maximal eigenvalues of the matrix , respectively, , , and . If , then the TTS method converges to the unique solution of (1) for any choice of the initial guess .
Although the TTS method does not always converge for non-Hermitian positive definite linear systems, it always converges for Hermitian positive definite linear systems.
Theorem 4. Let be Hermitian positive definite with the TTS as in (2). Then, for all , the TTS method converges to the unique solution of (1) for any choice of the initial guess . Furthermore, it holds that where and and are the minimal and maximal singular values of , respectively.
Proof. Since is Hermitian positive definite, , and is Hermitian positive definite. Following (11), which shows the convergence of the TTS method. Following (15) where . As a result, Since is gradually increasing if , is gradually decreasing if , and, consequently, when , gets its maximum . Therefore, when which shows that we completed the proof.
4. Remark on and Comparison of Convergence Theorems
In fact, the TTS method is a special case of the generalized assymmetric SOR iteration method with parameter matrices when specifically choosing the parameter matrices. This scheme is called the ALUS method in [18, 19].
The ALUS Method. Given an initial guess , for , until converges, compute where is a given positive constant, , , , and , , and are defined in (3).
We easily generalize the convergence theorems on TTS method to ALUS method.
Theorem 5. Let be non-Hermitian positive definite with the ALUS as in (20). Then the ALUS method converges to the unique solution of (1) for any choice of the initial guess if and only if , where , , , , and .
Theorem 6. Let be non-Hermitian positive definite with the ALUS as in (20), and let where and denote the minimal and maximal eigenvalues of the matrix , respectively, , , and . If , then the ALUS method converges to the unique solution of (1) for any choice of the initial guess .
The proofs of Theorems 5 and 6 directly result from the proofs of Theorems 2 and 3.
Like Theorem 4 in [18] or Theorem 6 in [19], Theorem 5 has only theoretical significance since it is difficult to be applied. However, Theorem 6, along with Theorem 3 in [18] or Theorem 4 in [19], proposes a practical condition on convergence of ALUS method. But the condition in Theorem 6 is wider than Theorem 3 in [18] or Theorem 4 in [19]. The following will give an example to demonstrate this fact.
Example 7. The coefficient matrix of linear system (1) is given as
Now, we consider solving this system by ALUS method. Let . Then , and . In order that the ALUS method converges to the unique solution of (1) for any choice of the initial guess Theorem 3 in [18] or Theorem 4 in [19] shows that and are both positive definite for all and . As a result, the set is not empty. In fact, the set (23) is empty since the inequality fails to hold when and . Therefore, Theorem 3 in [18] or Theorem 4 in [19] does not give the convergence of the ALUS method if we solve the linear system (1) with the coefficient matrix in (22).
Using Theorem 6, set and compute and . Then when the ALUS method converges to the unique solution of (1) for any choice of the initial guess .
Example 8. Linear system (1) is shifted skew-Hermitian linear system (see [7–14]) whose coefficient matrix is given as where is a positive constant, is the identity matrix, and is a skew-Hermitian matrix with a lower triangular matrix.
The shifted skew-Hermitian linear system, arising in the HSS iterative method, can be much more problematic; in some cases this solution is as difficult as that of the original linear system [10]. Since HSS method fails to solve this linear system, we consider the ALUS method and TTS method.
We assert that if with and and being the maximum and minimum eigenvalues of the matrix , respectively, Theorem 3 in [18] or Theorem 4 in [19] fails to give the convergence of the ALUS method when solving the skew-Hermitian linear system (1).
Let . Then , , and . In order that the ALUS method converges to the unique solution of (1) for any choice of the initial guess Theorem 3 in [18] or Theorem 4 in [19] shows that and are both positive definite for all . Thus, it is easy to get It follows from (26) that which shows that when Theorem 3 in [18] or Theorem 4 in [19] fails to give the convergence of the ALUS method.
We consider using Theorem 6. Compute . Theorem 6 shows that if the ALUS method converges to the unique solution of (1) for any choice of the initial guess . Furthermore, when . Thus, if , Theorem 6 shows that the ALUS method converges to the unique solution of (1) for any choice of the initial guess and for all ; otherwise, it follows from Theorem 6 that the ALUS method converges to the unique solution of (1) for any choice of the initial guess and for all .
In particular, if , Theorem 3 yields that the TTS method converges to the unique solution of (1) for any choice of the initial guess and for all ; otherwise, it follows from Theorem 3 that the TTS method converges to the unique solution of (1) for any choice of the initial guess and for all .
5. Numerical Examples
In this section we describe the results of a numerical simple example with the TTS method on a set of linear systems arising from a finite element discretization of a convection-diffusion equation in two dimensions.
Example 1. The coefficient matrix of linear system (1) is given as
Now, we investigate convergence of TTS method for linear system (1) by Theorem 3. It is known that is non-Hermitian positive definite. By Matlab computations, we get . As a result, when , the TTS method theoretically converges to the unique solution of (1) for any choice of the initial guess . But, when , this method either converges very slowly or fails to converge since .
By numerical experiments on Matlab program, one has Table 1.
Table 1 shows that, for given matrix , the TTS method converges for . Further, is gradually decreasing when and while it is gradually increasing when and . Thus, has two minimal values: and . However, is the minimal value. As a result, . It follows from Table 1 that Theorem 3 is true. Furthermore, the interval obtained in Theorem 3 includes the optimal point .
6. Conclusions
This paper studies convergence of TTS and ALUS iterative methods for non-Hermitian positive definite linear systems. Some sufficient and necessary conditions for convergence are proposed. But these conditions are only theoretically significant and difficult to apply to practical computations. In what follows, several conditions are presented such that the TTS method and ALUS method converge for non-Hermitian positive definite linear systems.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
Cheng-Yi Zhang was partly supported by the Science Foundation of the Education Department of Shaanxi Province of China (2013JK0593), the Scientific Research Foundation (BS1014) and the Education Reform Foundation (2012JG40) of Xi’an Polytechnic University, and the National Natural Science Foundations of China (11201362 and 11271297); Yu-Qian Yang was partly supported by the National Natural Science Foundation of China (no. 61201297); Qiang Sun was partly supported by the National Natural Science Foundation of China (no. 61001140) and the Scientific Research Program Funded by Shaanxi Provincial Education Department (no. 12JK0544).