Comparison Results on Preconditioned GAOR Methods for Weighted Linear Least Squares Problems

Definition 2.1. Let $$ and $$. We say A ≥ B if a_ij ≥ b_ij for all i, j = 1,2, …, n.

This definition can be carried over to vectors by identifying them with n × 1 matrices.

In this paper, ρ(·) denotes the spectral radius of a matrix.

Lemma 2.2 (see [6].)Let A ∈ R^n×n be nonnegative and irreducible. Then

• (i)

  A  has a positive real eigenvalue equal to its spectral radius ρ(A);

• (ii)

  for ρ(A), there corresponds an eigenvector x > 0.

Lemma 2.3 (see [7].)Let A ∈ R^n×n be nonnegative and irreducible. If

for some nonnegative vector x, then α < ρ(A) < β and x is a positive vector.

Theorem 3.1. Let $$ be the iteration matrices associated with the GAOR and preconditioned GAOR methods, respectively. If the matrix H in (1.2) is irreducible with C ≤ 0, U ≤ 0, B₁ ≥ 0, B₂ ≥ 0, 0 < ω ≤ 1, 0 ≤ r < 1, b_i,i+1 > 0, b_i+1,i > 0 for some i ∈ {2, …, p}, when 0 ≤ b_ii < 1  (i ∈ {2, …, p}),

or when b_ii ≥ 1, α_i > 0, β_i > 0  (i ∈ {2, …, p}), then either or

Proof. By direct operation, we have

Since 0 < ω ≤ 1, 0 ≤ r < 1, C ≤ 0, U ≤ 0, B₁ ≥ 0, B₂ ≥ 0, then

and L_r,ω is nonnegative. Since H is irreducible, from (3.13), it is easy to see that the matrix L_r,ω is nonnegative and irreducible.

Similarly, we can prove that the matrix $$ is a nonnegative and irreducible matrix.

By Lemma 2.2, there is a positive vector x such that

where λ = ρ(L_r,ω). Since the matrix H is nonsingular, λ ≠ 1. Hence, we get either λ > 1 or λ < 1.

Now, from (3.15) and by the definitions of L_r,ω and $$, we have

Since b_i,i+1 > 0, b_i+1,i > 0 for some i ∈ {2, …, p}, when 0 ≤ b_ii < 1 (i ∈ {2, …, p}),

or when b_ii ≥ 1, α_i > 0, β_i > 0 (i ∈ {2, …, p}), then S₁ ≥ 0 and S₁ ≠ 0. So we have $$, $$.

If λ < 1, then $$, $$.

By Lemma 2.3, the inequality (3.11) is proved.

If λ > 1, then $$, $$.

By Lemma 2.3, the inequality (3.12) is proved.

Theorem 3.2. Let $$ be the iteration matrices associated with the GAOR and preconditioned GAOR methods, respectively. If the matrix H in (1.2) is irreducible with C ≤ 0, U ≤ 0, B₁ ≥ 0, B₂ ≥ 0, 0 < ω ≤ 1, 0 ≤ r < 1, b_i,1 > 0, b_1,i > 0 for some i ∈ {2,3, …, p}, when 0 ≤ b₁₁ < 1, 0 < β_i < 1/(1 − b₁₁),

or when b₁₁ ≥ 1, α_i > 0, β_i > 0, i ∈ {2,3, …, p}, then either or

By the analogous proof of Theorem 3.1, we can prove Theorem 3.2.

Example 4.1. The coefficient matrix H in (1.2) is given by

where $$, $$, $$, and $$ with

Table 1 displays the spectral radii of the corresponding iteration matrices with some randomly chosen parameters r, ω, p. The randomly chosen parameters α_i and β_i satisfy the conditions of two theorems.

From Table 1, we see that these results accord with Theorems 3.1-3.2.