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

摘要

Let A∈Crm×n be partitioned asA=A11A12A21A22,whereA11∈Crr×r.Write B=A21A11-1 and C=A11-1A12. Suppose that B ≠ 0. For finding the minimum norm least squares solution A+b of the linear systemsAx=b,many authors studied the SOR, AOR, and SSOR methods for solving the augmented systems(1.1)A^z=bˆ,and obtained many results. In this paper we deeply study the SSOR method, whose iteration matrix is written as Jω, and prove the following new conclusions:(1)If ∥B∥ < 1, then Jω is semiconvergent ⇔ ω∈(0,2). If ∥B∥ ⩾ 1, then Jω is semiconvergent ⇔ω∈(0,ωˆ2)∪(ωˆ1,2), whereωˆ2=1-‖B‖-1‖B‖+1andωˆ1=1+‖B‖-1‖B‖+1.(2)The optimal parameters of Jω areω˜2=1-‖B‖1+1+‖B‖2andω˜1=1+‖B‖1+1+‖B‖2,andminωδ(Jω)=minωmax{|λ|:λ∈σ(Jω),λ≠1}=(1-ω˜1)2=(1-ω˜2)2=‖B‖21+1+‖B‖22.In addition, we obtain other results concerning the SOR, AOR and SSOR methods.