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Further study on two fixed point iterative schemes for absolute value equations

Jiayu Liu, Tingting Luo, Cairong Chen

TL;DR

This work addresses solving the absolute value equation $Ax-|x|=b$ via two predictor–corrector fixed-point schemes (s3 and s4) derived from a matrix split $A=N_A-M_A$. It establishes convergence conditions for both schemes: for s3, a bound $ lambda ext{(something)}$ involving a diagonal $E>0$ and the split components $N_A,M_A$ guarantees a unique solution and convergence; for s4, an analogous bound involving $D_A^{-1}M_A$, $E$, and $A$ ensures convergence, pending nonsingularity requirements. The analysis further identifies optimal iteration parameters in specific setups, showing the optimal $ lambda$ is $1$ under common choices $E=N_A^{-1}$ (for s3) and $E=D_A^{-1}$ with $N_A=D_A$ (for s4), supported by numerical tests that compare the four variants and confirm convergence behavior. These results provide practical guidance for selecting parameters when applying these AVE solvers to matrices with M-matrix structure, and point to future work on general optimal-parameter selection beyond the special cases examined.

Abstract

In this paper, we reconsider two new iterative methods for solving absolute value equations (AVE), which is proposed by Ali and Pan (Jpn. J. Ind. Appl. Math. 40: 303--314, 2023). Convergence results of the two iterative schemes and new sufficient conditions for the unique solvability of AVE are presented. In addition, for a special case, the optimal iteration parameters of the two algorithms are analyzed, respectively. Numerical results demonstrate our claims.

Further study on two fixed point iterative schemes for absolute value equations

TL;DR

This work addresses solving the absolute value equation via two predictor–corrector fixed-point schemes (s3 and s4) derived from a matrix split . It establishes convergence conditions for both schemes: for s3, a bound involving a diagonal and the split components guarantees a unique solution and convergence; for s4, an analogous bound involving , , and ensures convergence, pending nonsingularity requirements. The analysis further identifies optimal iteration parameters in specific setups, showing the optimal is under common choices (for s3) and with (for s4), supported by numerical tests that compare the four variants and confirm convergence behavior. These results provide practical guidance for selecting parameters when applying these AVE solvers to matrices with M-matrix structure, and point to future work on general optimal-parameter selection beyond the special cases examined.

Abstract

In this paper, we reconsider two new iterative methods for solving absolute value equations (AVE), which is proposed by Ali and Pan (Jpn. J. Ind. Appl. Math. 40: 303--314, 2023). Convergence results of the two iterative schemes and new sufficient conditions for the unique solvability of AVE are presented. In addition, for a special case, the optimal iteration parameters of the two algorithms are analyzed, respectively. Numerical results demonstrate our claims.
Paper Structure (8 sections, 9 theorems, 20 equations, 3 figures, 1 table)

This paper contains 8 sections, 9 theorems, 20 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convergence analysises of Algorithm \ref{['alg:s3']} and Algorithm \ref{['alg:s4']}
  4. The optimal iteration parameters for Algorithm \ref{['alg:s3']} and Algorithm \ref{['alg:s4']}
  5. The optimal iteration parameter for Algorithm \ref{['alg:s3']}
  6. The optimal iteration parameter for Algorithm \ref{['alg:s4']}
  7. Numerical results
  8. Conclusions

Key Result

Theorem 1.1

Let $A = N_A - M_A$ be non-singular, $\lambda > 0$ and $E\in \mathbb{R}^{n\times n}$ be a diagonal matrix with positive diagonal elements. If with $R = I - \lambda E |M_A|$ and $S = \lambda E + |I - \lambda EN_A|$, then for any initial point $x^{(0)}\in \mathbb{R}^n$, the sequence $\{x^{(k)}\}_{k=0}^\infty$ generated by Algorithm alg:s1 converges to the unique solution of AVE eq:ave.

Figures (3)

  • Figure 1: Iterative curves for Example \ref{['exam:1']} with $\lambda=0.5$ and $\mu=4$.
  • Figure 2: The trend diagram of IT versus $\lambda$ for Algorithm \ref{['alg:s3']} in Example \ref{['exam:1']}.
  • Figure 3: The trend diagrams of IT versus $\lambda$ for Algorithms \ref{['alg:s3']} and \ref{['alg:s4']} in Example \ref{['exam:2']}.

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 3.1
  • Remark 1
  • Theorem 3.2
  • Remark 2
  • Theorem 4.1
  • ...and 3 more