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SOR-like iteration and FPI are consistent when they are equipped with certain optimal iterative parameters

Jiayu Liu, Tingting Luo, Cairong Chen, Deren Han

TL;DR

This work addresses solving the absolute value equation $A x - |x| = b$ by two iterative schemes: the SOR-like iteration and fixed-point iteration (FPI). It delivers new convergence analyses that broaden the permissible parameter ranges and derives analytical optimal parameters $\omega^*=1$ for the SOR-like method and $\tau^*=1$ for FPI, showing the two methods become identical under these optima. The results establish extended convergence domains: $0<\nu=\|A^{-1}\|_2<1$ with $0<\omega<\frac{2-2\sqrt{\nu}}{1-\nu}$ for SOR-like and $0<\tau<\frac{2}{\nu+1}$ for FPI, without modifying the original methods. Numerical experiments on SPD matrices corroborate faster convergence and confirm the analytic equivalence of the two schemes at the optimal parameters. These findings provide robust, analytically tractable guidelines for efficiently solving AVEs in practice.

Abstract

Two common methods for solving absolute value equations (AVE) are SOR-like iteration method and fixed point iteration (FPI) method. In this paper, novel convergence analysis, which result wider convergence range, of the SOR-like iteration and the FPI are given. Based on the new analysis, a new optimal iterative parameter with a analytical form is obtained for the SOR-like iteration. In addition, an optimal iterative parameter with a analytical form is also obtained for FPI. Surprisingly, the SOR-like iteration and the FPI are the same whenever they are equipped with our optimal iterative parameters. As a by product, we give two new constructive proof for a well known sufficient condition such that AVE has a unique solution for any right hand side. Numerical results demonstrate our claims.

SOR-like iteration and FPI are consistent when they are equipped with certain optimal iterative parameters

TL;DR

This work addresses solving the absolute value equation by two iterative schemes: the SOR-like iteration and fixed-point iteration (FPI). It delivers new convergence analyses that broaden the permissible parameter ranges and derives analytical optimal parameters for the SOR-like method and for FPI, showing the two methods become identical under these optima. The results establish extended convergence domains: with for SOR-like and for FPI, without modifying the original methods. Numerical experiments on SPD matrices corroborate faster convergence and confirm the analytic equivalence of the two schemes at the optimal parameters. These findings provide robust, analytically tractable guidelines for efficiently solving AVEs in practice.

Abstract

Two common methods for solving absolute value equations (AVE) are SOR-like iteration method and fixed point iteration (FPI) method. In this paper, novel convergence analysis, which result wider convergence range, of the SOR-like iteration and the FPI are given. Based on the new analysis, a new optimal iterative parameter with a analytical form is obtained for the SOR-like iteration. In addition, an optimal iterative parameter with a analytical form is also obtained for FPI. Surprisingly, the SOR-like iteration and the FPI are the same whenever they are equipped with our optimal iterative parameters. As a by product, we give two new constructive proof for a well known sufficient condition such that AVE has a unique solution for any right hand side. Numerical results demonstrate our claims.

Paper Structure

This paper contains 10 sections, 10 theorems, 28 equations, 2 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. New convergence and new optimal iterative parameter of SOR-like iteration
  4. New convergence analysis
  5. Optimal iterative parameter of SOR-like iteration
  6. New convergence and optimal iterative parameter of FPI method
  7. New convergence result of FPI
  8. Optimal iterative parameter of FPI method
  9. Numerical experiments
  10. Conclusions

Key Result

Theorem 1.1

Assume that $A \in \mathbb{R}^{n\times n}$ is a nonsingular matrix and $b\in \mathbb{R}^{n}$. Denote For the sequence $\{(x^{(k)},y^{(k)})\}$ generated by eq:sor, if the following inequality holds for $k=0,1,2,\cdots$. Here the norm $\| |\cdot|\|_{\omega}$ is defined by

Figures (2)

  • Figure 1: Comparison of convergence domains for the SOR-like method. The light blue area represents the range of $\omega$ obtained from \ref{['eq:con-sor']}, and the red striped area represents the range of $\omega$ obtained from chyh2024.
  • Figure 2: Comparison of convergence domains for the FPI method. The light blue area represents the range of $\tau$ obtained from \ref{['eq:con-fpi']}, and the red striped area represents the range of $\tau$ obtained from \ref{['eq:cfpi']}.

Theorems & Definitions (18)

  • Theorem 1.1: kema2017
  • Theorem 1.2: ke2020
  • Proposition 1.3: mame2006
  • Lemma 2.1: youn1971
  • Lemma 2.2: e.g., saad2003
  • Lemma 2.3: e.g., saad2003
  • Theorem 3.1
  • Remark 1
  • Remark 2
  • Theorem 3.2
  • ...and 8 more