Convergence of TTS Iterative Method for Non-Hermitian Positive Definite Linear Systems

where A is a large and sparse non-Hermitian matrix. In this paper we consider the important case whereA is positive definite; that is, the Hermitian partH = (A+A)/2 is Hermitian positive definite, where A denotes the conjugate transpose of the matrixA. Large and sparse systems of this type arise in many applications, including discretizations of convectiondiffusion 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.


Introduction
Many problems in scientific computing give rise to a system of  linear equations in  unknowns,  = ,  = (  ) ∈ C × nonsingular, , ∈ C  , (1) 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  = (+ * )/2 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 convectiondiffusion 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.
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.
Eliminating  (+1/2) 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   .

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  ∈ C × be non-Hermitian.Then the iteration matrix of the iterative scheme (4) is Proof.Since  = L + U, which completes the proof.
Theorem 2. Let  ∈ C × 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  (0) if and only if  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  (0) if and only if (  ) < 1, where (  ) denotes the spectral radius of the matrix   .
Equation (10) shows that  = ( * + )/2 is not Hermitian positive definite; that is,  is not non-Hermitian positive definite.Therefore, a contradiction appears to indicate  =  * ( 2  +  + LU) ̸ = 0. Thus, which indicates that (  ) < 1 if and only if where  =  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  ∈ C × be Hermitian positive definite with the TTS as in (2).Then, for all  > 0, the TTS method converges to the unique solution of (1) for any choice of the initial guess  (0) .Furthermore, it holds that where  ∈ { ∈ C  :    = ,  *  = 1 and || = (  )} and   and  1 are the minimal and maximal singular values of L, respectively.

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].

Theorem 5. Let 𝐴 ∈ C 𝑛×𝑛 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 𝑥
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.
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  ≤  max () −  min () with  =  +  * and  max () and  min () 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).

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 convectiondiffusion 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  min (H) +  = −3.0471.As a result, when  ∈ (1.7456, +∞), the TTS method theoretically converges to the unique solution of (1) for any choice of the initial guess  (0) .But, when  → +∞, this method either converges very slowly or fails to converge since (  ) → 1.
By numerical experiments on Matlab program, one has Table 1.

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.

Table 1 :
The comparison results of (  ) with different parameter pairs .