On semi-convergence of a class of relaxation methods for singular saddle point problems

A note on the optimal parameters of USSOR method for solving linear least squares problems

“…Notice B T B is singular, it is more reasonable to choose a singular matrix as its approximation, so, in this note Q is chosen by a symmetric positive semi-definite (and potentially singular) matrix. M(ω 1 , ω 2 ) will act as the preconditioner for (6). The coefficient matrix A of (6) is singular, so, a singular matrix M(ω 1 , ω 2 ) may also be more reasonable to approximate A.…”

“…Fan et al [6] studied these conditions and obtained the semi-convergence results for USSOR by follows. Theorem 3.1.…”

“…Theorem 3.1. ( [6]) Let Q be symmetric positive semi-definite with N(Q) ⊆ N(B), and ρ be the spectral radius of Q † B T B. Then USSOR is semi-convergent if ω 1 and ω 2 satisfy the following conditions…”

“…For rank deficient linear least squares problems, the symmetric SOR(SSOR) method is also studied, see, e.g, [3][4][5]15]. Recently, Yun et al [6,13] proposed the unsymmetric SOR(USSOR) method to solve saddle point problems. And Song et al [10] constructed the USSOR method to solve rank deficient linear least squares problems, which was based on the block consistent linear system (5).…”