1. Introduction
Consider the weighted linear least squares problem
where W is the variance-covariance matrix. The problem has many scientific applications. A typical source is parameter estimation in mathematical modeling. This problem has been discussed in many books and articles. In order to solve it, one has to solve a nonsingular linear system as
where
is an invertible matrix with
In order to solve the linear system using the GAOR method, we split H as
Then, for ω ≠ 0, one GAOR method can be defined by
where
is the iteration matrix and
In order to decrease the spectral radius of the iteration matrix, an effective method is to precondition the linear system (1.1), namely,
then the preconditioned GAOR (PGAOR) method can be defined by
where
This paper is organized as follows. In Section 2, we propose three preconditioners and give the comparison theorems between the preconditioned and original methods. These results show that the preconditioned GAOR methods converge faster than the GAOR method whenever the GAOR method is convergent. In Section 3, we give one example to confirm our theoretical results.
2. Comparison results
In paper [5], the preconditioners introduced by Zhou et al. are of the form
In paper [3], the following preconditioned linear system was considered
where with
S is a p × p matrix with 1 < p < n. And S was taken as follows:
The preconditioned GAOR methods for solving (2.1) are
where
are iteration matrices for i = 1, 2, 3.
In paper [4], the preconditioners introduced by Yun are of the form
In this paper, we will consider new preconditioners
where Si are defined as above and
Then
The preconditioned GAOR methods for solving are defined as follows
where for i = 1, 2, 3,
Lemma 2.1 ([1,2]). Let A ∈ Rn×n be nonnegative and irreducible.Then
Theorem 2.1. Let be the iteration matrices associated of the GAOR and preconditioned GAOR methods, respectively. If the matrix H is irreducible with D ≤ 0, E ≤ 0, B ≥ 0, C ≥ 0, bi,i+1 > 0, bi+1,i > 0, ci,i+1 > 0, ci+1,i > 0 for some i ∈ {1, 2, · · · , p − 1}, 0 < ω ≤ 1, 0 ≤ r < 1, then either
or
Proof. Since 0 < ω ≤ 1, 0 ≤ r < 1, D ≤ 0, E ≤ 0, B ≥ 0, C ≥ 0, it is easy to prove that both and Lr,ω are irreducible and non-negative. By Lemma 2.1, there is a positive vector x such that Lr,ωx = λx, where λ = ρ(Lr,ω). Then
Since bi,i+1 > 0, bi+1,i > 0, ci,i+1 > 0, ci+1,i > 0 then S1 > 0, V1 > 0 and
If λ < 1, then By Lemma 2.1, we get
If λ > 1, then By Lemma 2.1, we get □
By the analogous proof of Theorem 2.1, we can prove the following two theorems.
Theorem 2.2. Let be the iteration matrices associated of the GAOR and preconditioned GAOR methods, respectively. If the matrix H is irreducible with D ≤ 0, E ≤ 0, B ≥ 0, C ≥ 0, bi,i+1 > 0, bi+1,i > 0, ci,i+1 > 0, ci+1,i > 0 for some i ∈ {1, 2, · · · , p − 1}, 0 < ω ≤ 1, 0 ≤ r < 1, then either
Theorem 2.3. Let be the iteration matrices associated of the GAOR and preconditioned GAOR methods, respectively. If the matrix H is irreducible with D ≤ 0, E ≤ 0, B ≥ 0, C ≥ 0, bi,i+1 > 0, bi+1,i > 0, ci,i+1 > 0, ci+1,i > 0 for some i ∈ {1, 2, · · · , p − 1}, 0 < ω ≤ 1, 0 ≤ r < 1, then either
Theorem 2.4. Under the assumptions of Theorem 2.1, then either
or
Proof. By Lemma 2.1, there is a positive vector x ,such that
where λ = ρ(Lr,ω). Then
Under the conditions of Theorem 2.1, we know that
Thus
Then
□
By the analogous proof of Theorem 2.4, we can prove the following one theorem.
Theorem 2.5. Under the assumptions of Theorem 2.1, then either
or
Theorem 2.6. Under the assumptions of Theorem 2.1, then either
or
Proof. By Lemma 2.1, there is a positive vector x ,such that
where Then
By assumptions, V1 > 0. Hence we obtain the following results.
If λ < 1, then By Lemma 2.1, we get
If λ > 1, then By Lemma 2.1, we get □
By the analogous proof of Theorem 2.6, we can prove the following two theorems.
Theorem 2.7. Let be the iteration matrices associated of the GAOR and preconditioned GAOR methods, respectively. If the matrix H is irreducible with D ≤ 0, E ≤ 0, B ≥ 0, C ≥ 0, bi,i+1 > 0, bi+1,i > 0, ci,i+1 > 0, ci+1,i > 0 for some i ∈ {1, 2, · · · , p − 1}, 0 < ω ≤ 1, 0 ≤ r < 1, then either
Theorem 2.8. Let be the iteration matrices associated of the GAOR and preconditioned GAOR methods, respectively. If the matrix H is irreducible with D ≤ 0, E ≤ 0, B ≥ 0, C ≥ 0, bi,i+1 > 0, bi+1,i > 0, ci,i+1 > 0, ci+1,i > 0 for some i ∈ {1, 2, · · · , p − 1}, 0 < ω ≤ 1, 0 ≤ r < 1, then either
3. Numerical exampleg
Now, we present an example to illustrate our theoretical results.
Example 3.1. The coefficient matrix H in (1.1) is given by
Table 1 displays the spectral radii of the corresponding iteration matrices with some randomly chosen parameters r, ω, p. From Table 1, we see that these results accord with Theorems 2.1-2.8.
Table 1.Here
Remark: In this paper, we propose three preconditioners and give the comparison theorems between the preconditioned and original methods. These results show that the preconditioned GAOR methods converge faster than the GAOR method whenever the GAOR method is convergent.
참고문헌
- A. Berman, R.J. Plemmons. Nonnegative Matrices in the Mathematical Sciences, SIAM Press, Philadelphia, 1994.
- R.S. Varga. Matrix Iterative Analysis, in: Springer Series in Computational Mathematics, vol. 27, Springer-Verlag, Berlin, 2000.
- G.B. Wang, T. Wang, F.P. Tan, Some results on preconditioned GAOR methods, Appl. Math. Comput., 219 (2013), 5811-5816. https://doi.org/10.1016/j.amc.2012.12.021
- J.H. Yun, Comparison results on the preconditioned GAOR method for generalized least squares problems, Int. J. Comput. Math., 89 (2012), 2094-2105. https://doi.org/10.1080/00207160.2012.702898
- X.X. Zhou, Y.Z. Song, L. Wang and Q.S. Liu, Preconditioned GAOR methods for solving weighted linear least squares problems, J. Comput. Appl. Math., 224 (2009), 242-249. https://doi.org/10.1016/j.cam.2008.04.034