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Low-rank generalized alternating direction implicit iteration method for solving matrix equations

Juan Zhang, Wenlu Xun

TL;DR

This work addresses solving large-scale Lyapunov and continuous-time algebraic Riccati equations by introducing a low-rank generalized alternating direction implicit iteration (R-GADI) that represents the solution as $X\approx VW^{T}$. The method combines GADI with low-rank Cholesky-like factorization and, for Riccati problems, pairs with Kleinman-Newton iterations to reduce the outer iterations to solving Lyapunov equations via RGADI. The authors prove convergence of the RGADI scheme and its consistency with GADI, and provide practical parameter-selection guidelines. Numerical experiments demonstrate that RGADI and Kleinman-Newton-RGADI outperform existing low-rank ADI-based methods in accuracy and efficiency on large-scale problems, highlighting their practical impact for control and model reduction tasks.

Abstract

This paper presents an effective low-rank generalized alternating direction implicit iteration (R-GADI) method for solving large-scale sparse and stable Lyapunov matrix equations and continuous-time algebraic Riccati matrix equations. The method is based on generalized alternating direction implicit iteration (GADI), which exploits the low-rank property of matrices and utilizes the Cholesky factorization approach for solving. The advantage of the new algorithm lies in its direct and efficient low-rank formulation, which is a variant of the Cholesky decomposition in the Lyapunov GADI method, saving storage space and making it computationally effective. When solving the continuous-time algebraic Riccati matrix equation, the Riccati equation is first simplified to a Lyapunov equation using the Newton method, and then the R-GADI method is employed for computation. Additionally, we analyze the convergence of the R-GADI method and prove its consistency with the convergence of the GADI method. Finally, the effectiveness of the new algorithm is demonstrated through corresponding numerical experiments.

Low-rank generalized alternating direction implicit iteration method for solving matrix equations

TL;DR

This work addresses solving large-scale Lyapunov and continuous-time algebraic Riccati equations by introducing a low-rank generalized alternating direction implicit iteration (R-GADI) that represents the solution as . The method combines GADI with low-rank Cholesky-like factorization and, for Riccati problems, pairs with Kleinman-Newton iterations to reduce the outer iterations to solving Lyapunov equations via RGADI. The authors prove convergence of the RGADI scheme and its consistency with GADI, and provide practical parameter-selection guidelines. Numerical experiments demonstrate that RGADI and Kleinman-Newton-RGADI outperform existing low-rank ADI-based methods in accuracy and efficiency on large-scale problems, highlighting their practical impact for control and model reduction tasks.

Abstract

This paper presents an effective low-rank generalized alternating direction implicit iteration (R-GADI) method for solving large-scale sparse and stable Lyapunov matrix equations and continuous-time algebraic Riccati matrix equations. The method is based on generalized alternating direction implicit iteration (GADI), which exploits the low-rank property of matrices and utilizes the Cholesky factorization approach for solving. The advantage of the new algorithm lies in its direct and efficient low-rank formulation, which is a variant of the Cholesky decomposition in the Lyapunov GADI method, saving storage space and making it computationally effective. When solving the continuous-time algebraic Riccati matrix equation, the Riccati equation is first simplified to a Lyapunov equation using the Newton method, and then the R-GADI method is employed for computation. Additionally, we analyze the convergence of the R-GADI method and prove its consistency with the convergence of the GADI method. Finally, the effectiveness of the new algorithm is demonstrated through corresponding numerical experiments.
Paper Structure (13 sections, 12 theorems, 73 equations, 7 figures, 7 tables, 3 algorithms)

This paper contains 13 sections, 12 theorems, 73 equations, 7 figures, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Low rank GADI for solving Lyapunov equation
  3. Derivation of iterative format
  4. Convergence analysis
  5. Selection of parameters
  6. Complexity analysis
  7. Numerical experiments
  8. Low rank GADI for solving Riccati equation
  9. Derivation of iterative format
  10. Convergence analysis
  11. Complexity analysis
  12. Numerical experiments
  13. Conclusions

Key Result

Lemma 1

ref24 Let $\alpha\in\mathbb C$, $A_{1}\in\mathbb C^{m\times n},\ \ B_{1}\in\mathbb C^{p\times q},\ \ X_{1}\in\mathbb C^{n\times p},\ \ C_{1}\in\mathbb C^{m\times n},\ \ D_{1}\in\mathbb C^{p\times q}$, then

Figures (7)

  • Figure 1: The residual curve of Example 2.5.1 $n=1024$
  • Figure 2: The time curve of Example 2.5.1
  • Figure 3: The residual curve of Example 2.5.2 $n=1024$
  • Figure 4: The time curve of Example 2.5.2
  • Figure 5: The residual curve of Example 3.4.1 $n=1024$
  • ...and 2 more figures

Theorems & Definitions (19)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 9 more