Two new fixed point iterative schemes for absolute value equations

Abstract

Absolute value equations (AVEs) can be used to solve many engineering, management science, and operations research problems. This paper proposes two new iterative schemes for solving $Ax-{|x|}= b$, where A is an M-matrix. These methods depend on the splitting of the coefficient matrix. The convergence conditions for these two methods are given. Some numerical examples are given to demonstrate that the iterative schemes are valid and efficient. The results are inspiring and may animate more study in this direction.

Appendix

This Appendix demonstrates how to execute the proposed iterative methods. Method I for the AVE:

$$\begin{aligned} x^{m+1}=x^{m}-\lambda E[-M_{A} x^{m+1}+N_{A}x^{m}-(| x^{m}|+b)]. \end{aligned}$$

Method II for the AVE:

$$\begin{aligned} x^{m+1}=x^{m}+D_{A}^{-1}M_{A}x^{m+1}-\lambda E\left[ Ax^{m}-| x^{m}|-b \right] -D_{A}^{-1}M_{A}x^{m}. \end{aligned}$$

Both iterative schemes on the right-hand side include $x^{m+1}$, which defines the unknown vector. Based on $Ax-|x|=b,$ we obtain

$$\begin{aligned} x=A^{-1}(| x | +b). \end{aligned}$$

Thus, $x^{m+1}$ can be approximated as follows:

$$\begin{aligned} x^{m+1}\approx A^{-1}(| x^{m} | +b). \end{aligned}$$

This procedure is named the Picard scheme [27]. The subsequent step is to describe the Method I algorithm. Algorithm for Method I:

$$\begin{aligned} y^{m}= & {} A^{-1}(| x^{m} | +b), \\ x^{m+1}= & {} x^{m}-\lambda E[-M_{A} y^{m}+N_{A}x^{m}-(| x^{m}|+b)]. \end{aligned}$$

Method II follows the similar procedure.