The Picard-HSS-SOR iteration method for absolute value equations

Abstract

In this paper, we present the Picard-HSS-SOR iteration method for finding the solution of the absolute value equation (AVE), which is more efficient than the Picard-HSS iteration method for AVE. The convergence results of the Picard-HSS-SOR iteration method are proved under certain assumptions imposed on the involved parameter. Numerical experiments demonstrate that the Picard-HSS-SOR iteration method for solving absolute value equations is feasible and effective.

Modified Picard-like Method for Solving Absolute Value Equations

“…), γ is a positive constant, and ω = 1, then this method will reduce to the Picard-HSS Method in [25]; if M 1 = γI + H, and M 2 = γI + S, where H = 1 2 (A + A H ), S = 1 2 (A − A H ), γ is a positive constant, and ω ∈ (0, 2), then this method will reduce to the Picard-HSS-SOR Method in [26]. Obviously, by choosing other splittings for the matrix A, more methods can be also extracted.…”

“…Guo, Wu, and Li, in [22], presented some new convergence conditions obtained from the involved iteration matrix of the SOR-like iteration method in [21]; also based on (3), Li and Wu, in [23], improved the SOR-like iteration method proposed by Ke and Ma in [21] and obtained a modified SOR-like iteration method; Ali et al proposed two modified generalized GaussSeidel (MGGS) iteration techniques to determine the AVE (1) in [24]. To accelerate the convergence, in [25], Salkuyeh proposed the Picard-HSS iteration method for the AVE; Zheng extended this method to the Picard-HSS-SOR Method in [26]; in [27], Ma proposed Picard methods for solving the AVE by combining matrix splitting iteration algorithms, such as the Jacobi, SSOR, or SAOR; Dehghan et al, in [28], proposed the following matrix multisplitting Picard-iterative methods (PIM), see Algorithm 1, under the condition σ min (A) > nσ max (B). In addition, for A being an M-matrix case, in [29], Ali et al presented two new generalized Gauss-Seidel iteration methods; finally, in [30], Yu et al proposed an inverse-free dynamical system to solve AVE (1).…”

“…(1) Based on a splitting of the two-by-two block coeffificient matrix, the Modified Picard-Like Iteration Method (MPIM) is proposed for solving AVE (1). It unifies some existing matrix splitting algorithms, such as the algorithms in [21,[23][24][25][26]28];…”

An iterative method based on Simpson's 3/8 rule to solve absolute value equations

“…So far, some numerical algorithms have been provided to obtain the approximate solution of AVE [7,8,9,20]. The most important of these algorithms are the non-smooth Newton method [13], the mixed-type splitting technique [6], the Levenberg-Marquardt technique [15], the SOR-like method [24], the Picard iteration method [26], the Gauss-Seidel iteration method [5], a method based on interval matrix [19], the complementarity and smoothing functions method [1], the smoothing techniques for non-Lipschitz absolute value equations [25], the alternating projections method [2], and the combination of Newton method and Simpson rule [11]. In this paper, we present a two-step iterative method for solving the absolute value equation (1.2).…”