1. Introduction
Consider the linear system
(1)
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
(2)
Assume that
without loss of generality we can write
(3)
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
(4)
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,
(5)
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.
2. Main Result
In this section we replace Sl 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:
(6)
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 AS is a diagonally dominant Z-matrix.
Proof. The elements 
of 
are given by
(7)
Since A is a diagonally dominant Z-matrix, so we have
(8)
Therefore, the following inequalities hold:
We denote that 
Then the following inequality holds:
Furthermore, if
, and
, for some
, then we have
(9)
Let
, 
and 
be the sums of the elements in row i of
, 
, and
, respectively. The following equations hold:
(10)
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
(11)
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
(12)
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.
3. Numerical Results
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.