A general alternating direction implicit iteration method for solving complex symmetric linear systems

TL;DR

This work develops the generalized alternating direction implicit (GADI) iteration for large sparse complex symmetric systems of the form $A x=b$ with $A=W+iT$, where $W\succeq 0$ and $T\succeq 0$. It extends the real GADI framework to the complex case, deriving a two-phase update that requires solving two SPD subsystems per iteration and proving convergence under standard PSD conditions, with an optimal parameter $\tilde{\alpha}=\sqrt{\gamma_{\min}\gamma_{\max}}$. The paper further adapts GADI to Lyapunov equations with complex coefficients via vectorization and Kronecker structure, and to Riccati equations using a Newton-GADI scheme that reduces CARE to Lyapunov steps; convergence is analyzed under stability and detectability assumptions. Numerical experiments show GADI often outperforms established methods (MHSS, PMHSS, CRI, TSCSP) in iteration count and CPU time, and demonstrate its effectiveness for complex-coefficient Lyapunov and Riccati problems. Overall, GADI offers a scalable, efficient framework for solving large-scale complex symmetric systems with practical impact in scientific computing and engineering applications.

Abstract

We have introduced the generalized alternating direction implicit iteration (GADI) method for solving large sparse complex symmetric linear systems and proved its convergence properties. Additionally, some numerical results have demonstrated the effectiveness of this algorithm. Furthermore, as an application of the GADI method in solving complex symmetric linear systems, we utilized the flattening operator and Kronecker product properties to solve Lyapunov and Riccati equations with complex coefficients using the GADI method. In solving the Riccati equation, we combined inner and outer iterations, first simplifying the Riccati equation into a Lyapunov equation using the Newton method, and then applying the GADI method for solution. Finally, we provided convergence analysis of the method and corresponding numerical results.

Figures (7)

• Figure 1: $n=1024$
• Figure 2: $n=1024$
• Figure 3: $t=0.01$
• Figure 4: $t=0.1$
• Figure 5: $t=0.01$

Theorems & Definitions (14)

• Lemma 1
• Corollary 1
• proof
• Lemma 2
• Theorem 1
• proof
• Corollary 2
• proof
• Theorem 2
• proof