In the numerical treatment of large-scale Sylvester and Lyapunov equations, projection methods require solving a reduced problem to check convergence. As the approximation space expands, this solution takes an increasing portion of the overall computational effort. When data are symmetric, we show that the Frobenius norm of the residual matrix can be computed at significantly lower cost than with available methods, without explicitly solving the reduced problem. For certain classes of problems, the new residual norm expression combined with a memory-reducing device make classical Krylov strategies competitive with respect to more recent projection methods. Numerical experiments illustrate the effectiveness of the new implementation for standard and extended Krylov subspace methods.
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…
The nonsymmetric T-Riccati equation is a quadratic matrix equation where the linear part corresponds to the so-called T-Sylvester or T-Lyapunov operator that has previously been studied in the literature. It has applications in macroeconomics and policy dynamics. So far, it presents an unexplored problem in numerical analysis, and both, theoretical results and computational methods, are lacking in the literature. In this paper we provide some sufficient conditions for the existence and uniqueness of a nonnegative minimal solution and its efficient computation is deeply analyzed. Both the small…
One of the most computationally expensive steps of the low-rank ADI method for large-scale Lyapunov equations is the solution of a shifted linear system at each iteration. We propose the use of the extended Krylov subspace method for this task. In particular, we illustrate how a single approximation space can be constructed to solve all the shifted linear systems needed to achieve a prescribed accuracy in terms of Lyapunov residual norm. Moreover, we show how to fully merge the two iterative procedures in order to obtain a novel, efficient implementation of the low-rank ADI method, for an impo…
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）
