#### Abstract

We present a preconditioned mixed-type splitting iterative method for solving the linear system , where is a Z-matrix. And we give some comparison theorems to show that the rate of convergence of the preconditioned mixed-type splitting iterative method is faster than that of the mixed-type splitting iterative method. Finally, we give one numerical example to illustrate our results.

#### 1. Introduction

For solving linear system, where is an square matrix and and are -dimensional vectors, the basic iterative method is where and is nonsingular. Thus, (2) can be written as where and .

Assuming that has unit diagonal entries, let , where is the identity matrix and and are strictly lower and strictly upper triangular parts of , respectively.

Transform the original system (1) into the preconditioned form as follows:

Then, we can define the basic iterative scheme as follows: where and is nonsingular. Thus, the equation above can also be written as where and .

In paper [1], Cheng et al. presented the mixed-type splitting iterative method as follows: with the following iterative matrix: where is an auxiliary nonnegative diagonal matrix, is an auxiliary strictly lower triangular matrix, and .

In this paper, we will establish the preconditioned mixed-type splitting iterative method with the preconditioners , , and for solving linear systems. And we obtain some comparison results which show that the rate of convergence of the preconditioned mixed-type splitting iterative method with is faster than that of the preconditioned mixed-type splitting iterative method with or . Finally, we give one numerical example to illustrate our results.

#### 2. Preconditioned Mixed-Type Splitting Iterative Method

For the linear system (1), we consider its preconditioned form as follows: with the preconditioner ; that is,

We apply the mixed-type splitting iterative method to it and have the corresponding preconditioned mixed-type splitting iterative method as follows: that is, So, the iterative matrix is where , , and are the diagonal, strictly lower, and strictly upper triangular matrices obtained from , is an auxiliary nonnegative diagonal matrix, is an auxiliary strictly lower triangular matrix, and .

If we choose , we have the following corresponding iterative matrix: And if we choose , we have the following corresponding iterative matrix:

If we choose certain auxiliary matrices, we can get the classical iterative methods as follows.(1)The PSOR method is (2)The PAOR method is

We need the following definitions and results.

*Definition 1 (see [2]). *A matrix is a -matrix if , for all , such that . A matrix is an -matrix if , and , for all , such that .

*Definition 2 (see [2]). *A matrix is an -matrix if is a nonsingular -matrix, and .

*Definition 3 (see [2, 3]). *Let . Then, is called a regular splitting if and ; is called an -splitting if is an -matrix, .

Lemma 4 (see [2]). *Let be an irreducible matrix. Then,*(1)* has a positive real eigenvalue equal to its spectral radius;*(2)*to , there corresponds an eigenvector ;*(3)* is a simple eigenvalue of .*

Lemma 5 (see [4]). *Let be a nonnegative matrix. Then,*(1)*if for some nonnegative vector , , then ;*(2)*if for some positive vector , then . Moreover, if is irreducible and if for some nonnegative vectors , then
*

Lemma 6 (see [5]). *Let be an -splitting of . Then, if and only if is a nonsingular -matrix.*

Lemma 7 (see [6, 7]). *Let be a Z-matrix. Then, is a nonsingular -matrix if and only if there is a positive vector such that .*

Lemma 8 (see [8]). *Let be a regular splitting of . Then, the splitting is convergent if and only if .*

Lemma 9 (see [9]). *Let and be two nonsingular lower triangular -matrices. If , then .*

#### 3. Convergence Analysis and Comparison Results

Theorem 10. *Let be a nonsingular Z-matrix. Assume that , , , and and are the iterative matrices given by (14) and (8), respectively. Consider the following.*(i)*If , then .*(ii)*Let be irreducible. Assume that and ;**
then,one has*(1)*, if ,*(2)*, if .*

*Proof. *Let

Then, we have
(i) Since is a nonsingular Z-matrix and , , it is clear that is a nonsingular -matrix and the splitting
is an -splitting. Since , it follows from Lemma 6 that is a nonsingular -matrix. Then, by Lemma 7, there is a positive vector such that , so .

By Lemma 7, is also a nonsingular -matrix.

Obviously, we can get that is a positive diagonal matrix. And from is nonnegative, we know that being a -matrix. Since is a strictly lower triangular matrix, so that .

So, we have .

Then, ; hence, is a nonsingular -matrix.

For , it is obvious that . And for , we have . Thus, and .

We have proven that and are both -splittings and , two splittings are nonnegative.

On the other hand, since , we get
which implies that

Therefore, . So, we have ; that is,
(ii) Let be irreducible. Since is a nonnegative and irreducible matrix, and according to the proof of Lemma 4 in paper [9], we can obtain that and are nonnegative and irreducible matrices. Thus, from Lemma 4, we know that there exists a positive vector such that , where we denote , which is equivalent to
Let , where , , and are the diagonal, lower triangular, and upper triangular parts of , respectively. So,
where , , .

Now, we consider
Since is an -matrix, and , we have the following.(1)If , then . By Lemma 5, we get .(2)If , then . By Lemma 5, we get .

Theorem 11. *Let be a nonsingular -matrix. Assume that , , , and and are the iterative matrices given by (13) and (8), respectively. Consider the following.*(i)*If , then
*(ii)*Let be irreducible. Assume that
**Then, one has *(1)* if ,*(2)* if .*

*Proof. *Let

Then, we have

(i) By a similar proof of Theorem 10, we can prove that and are both -splitting and , two splittings , are nonnegative.

On the other hand, since , we get
which implies that

Therefore, . So, we have ; that is,

(ii) Let
where , , and , , , , , and are the diagonal, strictly lower, and strictly upper triangular matrices of and , respectively.

And denote ; then according to (35), we have

By (25), we have
If , then by the proof of Theorem 10, we have .

Therefore, one has the following.(1)If , then but not equal to 0. Therefore, . By Lemma 5, we get .(2)If , then but not equal to 0. Therefore, . By Lemma 5, we get .*Remark.* If we choose in Theorem 11, we have a similar result which is showed by the following corollary.

Corollary 12. *Let be a nonsingular Z-matrix. Assume that , , , and and are the iterative matrices given by (15) and (8), respectively. Consider the following.*(i)*If , then
*(ii)*Let be irreducible. Assume that
**then, one has*(1)* if ,*(2)* if .**Now, one will provide some results to show the relations among , , and .*

Theorem 13. *Let be a nonsingular -matrix. Let and be iterative matrices given by (13) and (14), respectively. Assume that , , . If and , then*(1)* if ;*(2)* if .*

*Proof. *Since and are two lower triangular -matrices with , by Lemma 9, we have

By the proof of Theorems 10 and 11, we consider

In view of the proof of Theorem 11, we have .

Therefore, one has the following.(1)If , the right-hand side of the above inequality is more than zero. By Lemma 8, .(2)If , the right-hand side of the above inequality is more than zero. By Lemma 8, .

Theorem 14. *Let be a nonsingular Z-matrix. Let and be iterative matrices given by (13) and (15), respectively. Assume that , , . If and ,**
then*(1)* if ,*(2)* if .*

*Proof. *Since and are two lower triangular -matrices with , by Lemma 9, we have

By the proof of Corollary 12 and Theorem 11, we consider

Since , we get the following.(1)If , the right-hand side of the above inequality is more than zero. By Lemma 8, .(2)If , the right-hand side of the above inequality is more than zero. By Lemma 8, .*Remark.* The results (theorems and corollaries) in Section 3 are in some sense the generalized Stein-Rosenberg-type theorems like those in the papers [10–13]. The results (theorems and corollaries) in Section 3 are the comparisons of spectral radius of iterative matrices between the mixed-type splitting method and the preconditioned mixed-type splitting method, while the results in the papers [10–13] are the comparisons of spectral radius of iterative matrices between the parallel decomposition-type relaxation method and its special case.

#### 4. Numerical Example

Consider the following equation: in the unit square with Dirichlet boundary conditions.

If we apply the central difference scheme on a uniform grid with interior nodes () to the discretization of the above equation, we can get a system of linear equations with the coefficient matrix where denotes the Kronecker product, are tridiagonal matrices, and the step size is .

We choose ; then .

If we choose(1), , ,(2), , ,(3), , , then we can obtain the following results by Theorems 10–14.

Table 1 shows that that the rate of convergence of the preconditioned mixed-type splitting method is faster than that of the mixed-type splitting method. And it shows that the rate of convergence of the preconditioned mixed-type splitting method with is faster than that of the preconditioned mixed-type splitting method with or .

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11001144) and the Natural Science Foundation of Shandong Province of China (ZR2012AL09).