The delay Lyapunov equation is an important matrix boundary-value problem which arises as an analogue of the Lyapunov equation in the study of time-delay systems $\dot{x}(t) = A_0x(t)+A_1x(t-τ)+B_0u(t)$. We propose a new algorithm for the solution of the delay Lyapunov equation. Our method is based on the fact that the delay Lyapunov equation can be expressed as a linear system of equations, whose unknown is the value $U(τ/2)\in\mathbb{R}^{n\times n}$, i.e., the delay Lyapunov matrix at time $τ/2$. This linear matrix equation with $n^2$ unknowns is solved by adapting a preconditioned iterative method such as GMRES. The action of the $n^2\times n^2$ matrix associated to this linear system can be computed by solving a coupled matrix initial-value problem. A preconditioner for the iterative method is proposed based on solving a T-Sylvester equation $MX+X^TN=C$, for which there are methods available in the literature. We prove that the preconditioner is effective under certain assumptions. The efficiency of the approach is illustrated by applying it to a time-delay system stemmingfrom the discretization of a partial differential equation with delay. Approximate solutions to this problem can be obtained for problems of size up to $n\approx 1000$, i.e., a linear system with $n^2\approx 10^6$ unknowns, a dimension which is outside of the capabilities of the other existing methods for the delay Lyapunov equation.
Efficient Solution of Generalized Sylvester Equations via Preconditioned Alternating Anderson Acceleration
Nanchang University（南昌大学）；Nanchang Hangkong University（南昌航空大学）；Fujian Agriculture and Forestry University（福建农林大学）；Dalian University of Technology（大连理工大学）
Mixed-precision algorithms for solving the Sylvester matrix equation
Chalmers University of Technology and University of Gothenburg（查尔姆斯理工大学和哥德堡大学）；University of Leeds（利兹大学）；University of Manchester（曼彻斯特大学）；Max Planck Institute for Dynamics of Complex Technical Systems（马普复杂技术系统动力学研究所）
For a linear matrix function $f$ in $X \in \R^{m\times n}$ we consider inhomogeneous linear matrix equations $f(X) = E$ for $E \neq 0$ that have or do not have solutions. For such systems we compute optimal norm constrained solutions iteratively using the Conjugate Gradient and Lanczos' methods in combination with the More-Sorensen optimizer. We build codes for ten linear matrix equations, of Sylvester, Lyapunov, Stein and structured types and their T-versions, that differ only in two five times repeated equation specific code lines. Numerical experiments with linear matrix equations are perfo…
Matrix splitting iteration methods play a vital role in solving large sparse linear systems. Their performance heavily depends on the splitting parameters, however, the approach of selecting optimal splitting parameters has not been well developed. In this paper, we present a multitask kernel-learning parameter prediction method to automatically obtain relatively optimal splitting parameters, which contains simultaneous multiple parameters prediction and a data-driven kernel learning. For solving time-dependent linear systems, including linear differential systems and linear matrix systems, we…
Sign Embedding Quantum Algorithms for Matrix Equations and Matrix Functions
Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心）；Qiuzhen College, Tsinghua University（邱氏学院）；Institute for AI Industry Research (AIR), Tsinghua University（清华大学人工智能产业研究院）；Institute for Applied Mathematics, Tsinghua University（应用数学中心）；Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）
