Consider the linear system

where is a known nonsingular matrix and are vectors. For any splitting A = M – N with a nonsingular matrix M, the basic splitting iterative method can be expressed as

Assume that

without loss of generality we can write

where I is the identity matrix, and are strictly lower triangular and strictly upper triangular parts of, respectively. In order to accelerate the convergence of the iterative method for solving the linear system (1), the original linear system (1) is transformed into the following preconditioned linear system

where P, called a preconditioner, is a nonsingular matrix.

In 1991, Gunawardena et al. [2] considered the modified Gauss-Seidel method with, where

Then, the preconditioned matrix can be written as

where D and E are the diagonal and strictly lower triangular parts of SL, respectively. If , then exists. Therefore, the preconditioned Gauss-Seidel iterative matrix for becomes

which is referred to as the modified Gauss-Seidel iterative matrix. Gunawardena et al. proved the following inequality:

where denotes the spectral radius of the GaussSeidel iterative matrix T. Morimoto et al. [3] have proposed the following preconditioner,

In this preconditioner, is defined by

where and

, for

. The preconditioned matrix can then be written as

where, and are the diagonal, strictly lower and strictly upper triangular parts of respectively. Assume that the following inequalities are satisfied:

Then is nonsingular. The preconditioned Gauss-Seidel iterative matrix for is then defined by

Morimoto et al. [3] proved that. To extend the preconditioning effect to the last row, Morimoto et al. [7] proposed the preconditioner

where R is defined by

The elements of are given by

And Morimoto et al. proved that holds, where is the iterative matrix for. They also presented combined preconditioners, which are given by combinations of R with any upper preconditioner, and showed that the convergence rate of the combined methods are better than those of the Gauss-Seidel method applied with other upper preconditioners [7]. In [14], Niki et al. considered the preconditioner . Denote. In [5], Niki et al. proved that if the following inequality is satisfied,

then holds, where is the iterative matrix for. For matrices that do not satisfy Equation

(5), by putting

Equation (5) is satisfied. Therefore, Niki et al. [5] proposed a new preconditioner, where

Put, and.

Replacing by and setting the Gauss-Seidel splitting of can be written as

where is constructed by the elements. Thus, if the preconditioner is used, then all of the rows of are subject to preconditioning. Niki et al. [5] proved that under the condition, where is the upper bound of those values of for which. By setting, they obtained

Niki et al. [5] proved that the preconditioner satisfies the Equation (5) unconditionally. Moreover, they reported that the convergence rate of the GaussSeidel method using preconditioner is better than that of the SOR method using the optimum found by numerical computation. They also reported that there is an optimum in the range which produces an extremely small, where is the upper bound of the values of for which, for all.

In this paper we use different preconditions for solving (1) by Gauss-Siedel method, that assuming none of the components of the matrix to be zero. If the largest component of the column j is not then the value of will be improved.

In this section we replace S_l by of Morimoto such that and define by

where s.t, and has the same form as the proposed by Morimoto et al. [3].

The precondition Matrix can then be written as

where and are the diagonal, strictly lower and strictly upper triangular parts of, respectively. Assume that the following inequalities are satisfied:

Therefore exists and the preconditioned GaussSeidel iterative matrix for is defined by

For and, we write whenever holds for all A is nonnegative if and if and only if.

Definition 2.1 (Young, [15]). A real matrix with for all is called a Zmatrix.

Definition 2.2 (Varga, [16]). A matrix A is irreducible if the directed graph associated to A is strongly connected.

Lemma 2.3. If A is an irreducible diagonally dominant Z-matrix with unit diagonal, and if the assumption (6) holds, then the preconditioned matrix A_S is a diagonally dominant Z-matrix.

Proof. The elements of are given by

Since A is a diagonally dominant Z-matrix, so we have

Therefore, the following inequalities hold:

We denote that Then the following inequality holds:

Furthermore, if, and, for some, then we have

Let, and be the sums of the elements in row i of, , and, respectively. The following equations hold:

where and are the sums of the elements in row i of L and U for, respectively. Since A is a diagonally dominant Z-matrix, by (8) and by the condition (6) the following relations hold:

Therefore, , , and is a Zmatrix. Moreover, by (9) and by the assumption, we can easily obtain

Therefore, satisfies the condition of diagonal dominance.

Lemma 2.4 [10, Lemma 2]. An upper bound on the spectral radius for the Gauss-Seidel iteration matrix T is given by

where and are the sums of the moduli of the elements in row i of the triangular matrices L and U, respectively.

Theorem 2.5. Let A be a nonsingular diagonally dominant Z-matrix with unit diagonal elements and let the condition (6) holds, then

Proof. From (11) and we have

This implies that

Hence, by Lemma (2.4) we have

Definition 2.6. Let A be an real matrix. Then, is referred to as:

1) a regular splitting, if M is nonsingular, and

2) a weak regular splitting, if M is nonsingular, and

3) a convergent splitting, if

Lemma 2.7 (Varga, [10]). Let be a nonnegative and irreducible matrix. Then 1) A has a positive real eigenvalue equal to its spectral radius;

2) for, there corresponds an eigenvector x > 0;

3) is a simple eigenvalue of A;

4) increases whenever any entry of A increases.

Corollary 2.8 [16, Corollary 3.20]. If is a real, irreducibly diagonally dominant matrix with for all, and for all, then.

Theorem 2.9 [16, Theorem 3.29]. Let be a regular splitting of the matrix A. Then, A is nonsingular with if and only if, where

Theorem 2.10 (Gunawardena et al. [2, Theorem 2.2]). Let A be a nonnegative matrix. Then 1) If for some nonnegative vector x, then

2) If for some positive vector x, then Moreover, if A is irreducible and if for some nonnegative vector x, then and is a positive vector.

Let be a real Banach space, its dual and the space of all bounded linear operator mapping B into itself. We assume that B is generated by a normal cone K [17]. As is defined in [17], the operator has the property “d” if its dual possesses a Frobenius eigenvector in the dual cone which is defined by

As is remarked in [1,17], when and, all real matrices have the property “d”. Therefore the case are discussing fulfills the property “d”. For the space of all matrices, the theorem of Marek and Szyld can be stated as follows:

Theorem 2.11 (Marek and Szyld [17, Theorem 3.15]). Let and be weak regular splitting with. Let be such that and. If and if either or with, then

Moreover, if and then

Now in the following lemma we prove that is Gauss-Seidel convergent regular splitting.

Theorem 2.12. Let A be an irreducibly diagonally dominant Z-matrix with unit diagonal, and let the condition (6) holds, then is Gauss-Seidel convergent regular splitting. Moreover

Proof. If A is an irreducibly diagonally dominant Zmatrix, then by Lemma (2.3), is a diagonally dominant Z -matrix. So we have. By hypothesis we have. Thus the strictly lower triangular matrix has nonnegative elements. By considering Neumanns series, the following inequality holds:

Direct calculation shows that holds. Thus, by definition (2.6) is the Gauss-Seidel convergent regular splitting. Also in [3] we have and

Direct comparison of the two matrix elements

and also and

we obtain

Thus

Furthermore, since, we have . From Lemma (2.7), x is an eigenvector of , and x is also a Perron vector of. Therefore, from Theorem (2.11),

holds.

Denote

and also let, , and be the iterative matrix associated to, , and respectively. Then we can prove and, similarly. In summary, we have the following inequalities:

Remark 2.13. W. Li, in [18] used the M-matrix instead of irreducible diagonally dominant Z-matrix, therefore we can say that the Lemma 2.3 and the Theorems 2.5 and 2.12 are hold for M-matrices.

In this section, we test a simple example to compare and contrast the characteristics of the different preconditioners. Consider the matrix

Applying the Gauss-Seidel method, we have . By using preconditioner we find that and have the following forms:

and.

Using the preconditioner we obtain

and.

For, we have

and.

For, we have

and.

For we have

and

From the above results, we have . Then and have the forms:

and.

For, we have

and Since the preconditioned matrices differ only in the values of their last rows, the related matrices also differ only in these values, as is shown in the above results. Thus the elements of new and are similar to elements of and, respectively than the elements of last rows. Therefore, we hereafter show only the last row.

By putting, the matrices and have the following forms:

and,

and.

For we have:

and and

and.

For we have:

and and

and.

From the numerical results, we see that this method with the preconditioner produces a spectral radius smaller than the recent preconditioners that above was introduced.