A necessary and sufficient condition for semiconvergence and optimal parameter of the SSOR method for solving the rank deficient linear least squares problem

Abstract

Let A ∈ C r m × n be partitioned as A = A 11 A 12 A 21 A 22 , where A 11 ∈ C r r × r . Write B = A 21 A 11 - 1 and C = A 11 - 1 A 12 . Suppose that B ≠ 0. For finding the minimum norm least squares solution A + b of the linear systems Ax = b , many authors studied the SOR, AOR, and SSOR methods for solving the augmented systems (1.1) A ^ z = b ˆ , and obtained many results. In this paper we deeply study the SSOR method, whose iteration matrix is written as J ω , and prove the following new conclusions: (1) If ∥ B∥ < 1, then J ω is semiconvergent ⇔ ω ∈ ( 0 , 2 ) . If ∥ B∥ ⩾ 1, then J ω is semiconvergent ⇔ ω ∈ ( 0 , ω ˆ 2 ) ∪ ( ω ˆ 1 , 2 ) , where ω ˆ 2 = 1 - ‖ B ‖ - 1 ‖ B ‖ + 1 and ω ˆ 1 = 1 + ‖ B ‖ - 1 ‖ B ‖ + 1 . (2) The optimal parameters of J ω are ω ˜ 2 = 1 - ‖ B ‖ 1 + 1 + ‖ B ‖ 2 and ω ˜ 1 = 1 + ‖ B ‖ 1 + 1 + ‖ B ‖ 2 , and min ω δ ( J ω ) = min ω max { | λ | : λ ∈ σ ( J ω ) , λ ≠ 1 } = ( 1 - ω ˜ 1 ) 2 = ( 1 - ω ˜ 2 ) 2 = ‖ B ‖ 2 1 + 1 + ‖ B ‖ 2 2 . In addition, we obtain other results concerning the SOR, AOR and SSOR methods.