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Lopsided HSS Iterative Method and Preconditioner for a class of Complex Symmetric Linear System

Yusong Zhang, Zeng-Qi Wang

TL;DR

This work tackles solving complex symmetric linear systems $A=W+iT$ with $W\succ0$ and $T$ symmetric indefinite. It introduces the LHSS Iteration, derives rigorous convergence bounds, and identifies an optimal parameter $\alpha^*=\mu_{\min}^{2}/\lambda_{\max}$; it then augments LHSS with preconditioning to form PLHSS, and derives two real, efficient preconditioners $\mathcal P_{PLW}$ and $\mathcal P_{PLT}$ that ensure favorable spectral properties. The authors analyze the eigenstructure of the preconditioned matrices and demonstrate that eigenvalues cluster and eigenvectors remain well-conditioned, enabling fast convergence for GMRES and COCG. Numerical experiments on surface acoustic wave simulations show mesh-size independence and strong robustness to indefiniteness, with PLHSS-based methods outperforming several competing schemes in both iteration count and runtime, highlighting practical impact for large-scale complex symmetric indefinite systems.

Abstract

In this study, we propose the lopsided HSS (LHSS) iteration method for solving a class of complex symmetric indefinite systems of linear equations. This method employs an alternating iterative scheme, where each iteration entails solving two systems of equations with symmetric real coefficient matrices. This design is intended to reduce the high computational costs associated with complex arithmetic. Theoretical analysis shows that the upper bound of the convergence rate depends only on the maximum and minimum eigenvalues of the real symmetric matrices, as well as the iteration parameters. When the eigenvalues satisfy certain conditions, the method guarantees convergence for any positive iteration parameter. Building on this insight, we developed the preconditioned lopsided HSS iteration method (PLHSS). Theoretical results demonstrate that PLHSS exhibits superior convergence properties compared to the original method. Additionally, we derived the optimal parameters for the new approaches and corresponding optimal convergence rate. Furthermore, we derive the PLHSS preconditioner on the basis of the iterative method. The eigenvalues of the preconditioned matrix are well-clustered, and the eigenvectors are orthogonal with a specific inner product. Numerical experiments demonstrate the efficiency of the preconditioned GMRES and COCG methods. LHSS iteration methods and the relevant preconditioners show mesh size independent and parameter-insensitive convergence behavior for the test numerical examples.

Lopsided HSS Iterative Method and Preconditioner for a class of Complex Symmetric Linear System

TL;DR

This work tackles solving complex symmetric linear systems with and symmetric indefinite. It introduces the LHSS Iteration, derives rigorous convergence bounds, and identifies an optimal parameter ; it then augments LHSS with preconditioning to form PLHSS, and derives two real, efficient preconditioners and that ensure favorable spectral properties. The authors analyze the eigenstructure of the preconditioned matrices and demonstrate that eigenvalues cluster and eigenvectors remain well-conditioned, enabling fast convergence for GMRES and COCG. Numerical experiments on surface acoustic wave simulations show mesh-size independence and strong robustness to indefiniteness, with PLHSS-based methods outperforming several competing schemes in both iteration count and runtime, highlighting practical impact for large-scale complex symmetric indefinite systems.

Abstract

In this study, we propose the lopsided HSS (LHSS) iteration method for solving a class of complex symmetric indefinite systems of linear equations. This method employs an alternating iterative scheme, where each iteration entails solving two systems of equations with symmetric real coefficient matrices. This design is intended to reduce the high computational costs associated with complex arithmetic. Theoretical analysis shows that the upper bound of the convergence rate depends only on the maximum and minimum eigenvalues of the real symmetric matrices, as well as the iteration parameters. When the eigenvalues satisfy certain conditions, the method guarantees convergence for any positive iteration parameter. Building on this insight, we developed the preconditioned lopsided HSS iteration method (PLHSS). Theoretical results demonstrate that PLHSS exhibits superior convergence properties compared to the original method. Additionally, we derived the optimal parameters for the new approaches and corresponding optimal convergence rate. Furthermore, we derive the PLHSS preconditioner on the basis of the iterative method. The eigenvalues of the preconditioned matrix are well-clustered, and the eigenvectors are orthogonal with a specific inner product. Numerical experiments demonstrate the efficiency of the preconditioned GMRES and COCG methods. LHSS iteration methods and the relevant preconditioners show mesh size independent and parameter-insensitive convergence behavior for the test numerical examples.

Paper Structure

This paper contains 8 sections, 9 theorems, 93 equations, 3 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Lopsided HSS iteration method
  3. The Preconditioned LHSS iteration method
  4. PLHSS Preconditioners and Preconditioned Krylov Subspace Methods
  5. Numerical examples
  6. Conclusion
  7. Proof of Theorem 4.1
  8. Proof of Theorem 4.2

Key Result

Theorem 2.1

Let $A=W+\mathrm{i}T$, with symmetric positive definite matrix $W\in\mathbb{R}^{n\times n}$ and symmetric indefinite matrix $T\in\mathbb{R}^{n\times n}$. $\lambda_{\max}$ and $\mu_{\min}$ are largest and smallest eigenvalues of $W$ and $T$ in magnitude, respectively. Then, the spectral radius of ite Furthermore, it holds that (i) If $\lambda_{\max}\le\mu_{\min}$, then $\sigma(\alpha)<1$ for any $\

Figures (3)

  • Figure 1: Numbers of iterations versus variable values of $\alpha$ for PLHSS iteration method when $\omega=8\times 10^{9}$.
  • Figure 2: Relative residual versus number of iterations for different iteration methods when $\omega=8\times 10^{9}$.
  • Figure 3: Relative residual versus number of iterations for different iterative solvers when $N=208170$.

Theorems & Definitions (22)

  • Theorem 2.1
  • proof : Proof
  • Theorem 2.2
  • proof : proof
  • Theorem 3.1
  • proof : Proof
  • Theorem 3.2
  • proof : proof
  • Theorem 3.3
  • Remark 3.1
  • ...and 12 more