IMPROVING COMPARISON RESULTS ON PRECONDITIONED GENERALIZED ACCELERATED OVERRELAXATION METHODS

Abstract

In this paper, we present preconditioned generalized accelerated overrelaxation (GAOR) methods for solving weighted linear least square problems. We compare the spectral radii of the iteration matrices of the preconditioned and the original methods. The comparison results show that the preconditioned GAOR methods converge faster than the GAOR method whenever the GAOR method is convergent. Finally, we give a numerical example to confirm our theoretical results.

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.