Regularized variable <i>s</i> -step CG and CR algorithms to solve shifted linear systems

Purpose This paper aims to investigate the transformation of control systems into shifted linear systems using Krylov subspace algorithms. It focuses on integrating a novel s-step technique to enhance algorithm efficiency by minimizing communication overhead during the implementation of conjugate gradient (CG) and conjugate residual (CR) algorithms. Design/methodology/approach The study presents a stable variant of CG and CR algorithms specifically adapted for control systems, proposing an innovative approach that dynamically adjusts the number of bases utilized in each iteration. This approach aims to optimize the computational process while reducing the necessary connections. Findings The proposed algorithm demonstrates a significant reduction in device communication requirements by combining the s-step technique with the CG and CR algorithms. Furthermore, a new regularization condition is introduced to maintain the stability of shifted linear systems, addressing potential issues of instability in iterative algorithms. Originality/value This paper contributes to the field by developing a dynamic algorithm that incorporates the s-step technique, offering a comprehensive evaluation of its efficiency through various numerical examples. The stabilization of shifted linear systems marks a significant advancement in the application of Krylov subspace methods to control systems.