ABSTRACT

The fixed point iteration (FPI) method proposed by Ke [Appl Math Lett. 2020;99:105990] for solving the absolute value equations (AVE) with the form $Ax-|x|=b$ is interesting for its simplicity and efficiency. However, its convergence is only guaranteed for the case that $0<\parallel A^{-1}\parallel <\frac{\sqrt{2}}{2}$, excluding the possible case that $\frac{\sqrt{2}}{2}\le \parallel A^{-1}\parallel <1$. To complete the gap, we develop a modified FPI (MFPI) method for solving the AVE with $0<\parallel A^{-1}\parallel <1$, which, besides keeping the simplicity of FPI, improves its efficiency by judiciously choosing the involving parameter. Under mild conditions, we prove its linear convergence. We present some preliminary numerical results for $0<\parallel A^{-1}\parallel <1$, demonstrating its convergence; and compare it with FPI when $0<\parallel A^{-1}\parallel <\frac{\sqrt{2}}{2}$, illustrating its superiority.

摘要

Ke [Appl Math Lett.] 提出的定点迭代（FPI）方法。2020;99:105990] 用于求解具有以下形式的绝对值方程 (AVE)$\mathrm{一种}X-|X|=b$它的简单性和效率很有趣。但是，它的收敛仅在以下情况下才能保证$0<\parallel {\mathrm{一种}}^{-1}\parallel <\frac{\sqrt{2}}{2}$, 排除可能的情况$\frac{\sqrt{2}}{2}\le \parallel {\mathrm{一种}}^{-1}\parallel <1$. 为了弥补这一差距，我们开发了一种改进的 FPI (MFPI) 方法来解决 AVE$0<\parallel {\mathrm{一种}}^{-1}\parallel <1$，除了保持 FPI 的简单性外，还通过明智地选择涉及参数来提高其效率。在温和条件下，我们证明了它的线性收敛性。我们提出了一些初步的数值结果$0<\parallel {\mathrm{一种}}^{-1}\parallel <1$，证明其收敛性；并将其与 FPI 进行比较$0<\parallel {\mathrm{一种}}^{-1}\parallel <\frac{\sqrt{2}}{2}$，说明了它的优越性。