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This work presents a new two-step iterative technique for solving absolute value equations. The developed technique is valuable and effective for solving the absolute value equation. Various examples are given to demonstrate the accuracy and efficacy of the suggested technique.

In this paper, we present a new two-step iterative technique for solving absolute value equations. This technique is very straightforward, and due to the simplicity of this approach, it can be used to solve large systems with great effectiveness. Moreover, under certain assumptions, we examine the convergence of the proposed method using various theorems. Numerical outcomes are conducted to present the feasibility of the proposed technique.

This paper gives numerical experiments on how to solve a system of absolute value equations.

Nowadays, two-step approaches are very popular for solving equations (1). For solving AVEs, Liu in Shams (2021), Ning and Zhou (2015) demonstrated two-step iterative approaches. Moosaei et al. (2015) introduced a novel approach that utilizes a generalized Newton’s approach and Simpson’s rule to solve AVEs. Zainali and Lotfi (2018) presented a two-step Newton technique for AVEs that converges linearly. Feng and Liu (2016) have proposed minimization approaches for AVEs and presented their convergence under specific circumstances. Khan et al. (2023), suggested a nonlinear CSCS-like technique and a Picard-CSCS approach. Based on the benefits and drawbacks of the previously mentioned methods, we will provide a two-step iterative approach to efficiently solve equation (1). The numerical results show that our proposed technique converges rapidly and provides a more accurate solution.

This study explores whether targeted microcredit can effectively alleviate households’ multidimensional relative poverty (MdRP) in rural China in the new era following the poverty elimination campaign and discusses it from a gendered perspective.

This study applies a fixed-effects model, propensity score matching (PSM) and two-stage instrumental variable method to two-period panel data collected from 611 households in rural western China in 2018 and 2021 to explore the effects, mechanisms and heterogenous performance of targeted microcredit on households’ MdRP in the new era.

(i) Targeted microcredit can alleviate MdRP among rural households in the new era, mainly by reducing income and opportunity inequality. (ii) Targeted microcredit can promote women’s empowerment, mainly by enhancing their social participation, thereby helping alleviate households’ MdRP. The effect of the targeted microcredit on MdRP is more significant in medium-educated women households and non-left-behind women households. (iii) The MdRP alleviation effect is stronger in villages with a high degree of digitalization.

Learn from the experience of targeted microcredit. Accurately identify poor groups and integrate loan design into financial health and women empowerment. Particularly, pay attention to less-educated and left-behind women households and strengthen coordination between targeted microcredit and digital village strategies.

This study clarifies the effect of targeted microcredit on women’s empowerment and households’ MdRP alleviation in the new era. It also explores its various effects on households with different female characteristics and regional digitalization levels, providing ideas for optimizing microcredit.