Large-scale Stein and Lyapunov equations, Smith method, and applications

DOI10.1007/s11075-012-9650-2zbMath1271.65079OpenAlexW2094517106MaRDI QIDQ2391820

Peter Chang-Yi Weng, Tiexiang Li, Wen-Wei Lin, Eric King-Wah Chu

Publication date: 5 August 2013

Published in: Numerical Algorithms (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s11075-012-9650-2

zbMATH Keywords

algorithmcomputational complexitynumerical resultsLyapunov equationcondition numberKrylov subspace methodStein equationCayley transformalgebraic Riccati equationsSmith method

Mathematics Subject Classification ID

Matrix equations and identities (15A24) Iterative numerical methods for linear systems (65F10) Complexity and performance of numerical algorithms (65Y20)

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Uses Software

• Matlab
• MESS

Cites Work

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• A low-rank Krylov squared Smith method for large-scale discrete-time Lyapunov equations
• A mixed-precision algorithm for the solution of Lyapunov equations on hybrid CPU-GPU platforms
• On the numerical solution of large-scale sparse discrete-time Riccati equations
• ADI preconditioned Krylov methods for large Lyapunov matrix equations
• On the ADI method for Sylvester equations
• Block Krylov subspace methods for large algebraic Riccati equations
• A structure-preserving doubling algorithm for continuous-time algebraic Riccati equations
• A modified low-rank Smith method for large-scale Lyapunov equations
• Dimension reduction of large-scale systems. Proceedings of a workshop, Oberwolfach, Germany, October 19--25, 2003.
• An Arnoldi based algorithm for large algebraic Riccati equations
• Projection methods for large Lyapunov matrix equations
• Low rank approximate solutions to large Sylvester matrix equations
• The Sensitivity of the Algebraic and Differential Riccati Equations
• A New Iterative Method for Solving Large-Scale Lyapunov Matrix Equations
• Numerical solution of large‐scale Lyapunov equations, Riccati equations, and linear‐quadratic optimal control problems
• Direct methods and ADI‐preconditioned Krylov subspace methods for generalized Lyapunov equations
• Krylov Subspace Methods for Solving Large Lyapunov Equations
• Low-Rank Solution of Lyapunov Equations
• Matrix Equation $XA + BX = C$
• Approximation of Large-Scale Dynamical Systems

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