Numerical solution and perturbation theory for generalized Lyapunov equations

By Tatjana Stykel. 2002

53 citations recorded by the source corpus.

We discuss the numerical solution and perturbation theory for the generalized continuoustime Lyapunov equation E ⁄ XA+A ⁄ XE =iG with a singular matrix E. If this equation has a solution, it is not unique. We generalize a Bartels-Stewart method and a Hammarling method to compute a partial solution of the generalized Lyapunov equation with a special right-hand side. A spectral condition number is introduced and perturbation bounds for such an equation are presented. Numerical examples are given.

Subfields: Computational Theory and Mathematics, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/74a6c0b449612c73e44a84c64ac96922dc06deea

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In Computational Theory and Mathematics

In Numerical Analysis

In Statistical and Nonlinear Physics

Citations in the corpus

Balanced Truncation Model Reduction for Large-Scale Systems in Descriptor Form (2005, methodology): “These factors can be determined using the generalized Schur-Hammarling method [69, 70 ] without computing the solutions of Lyapunov equations explicitly.”
On Iterative Solutions of General Coupled Matrix Equations (2006, methodology): “In this area, Chu gave an algorithm for solving generalized/coupled Sylvester equations [7]; Syrmos, Misra, and Aripirala [29], Stykel [28], and Takaba, Morihhira, and Katayama [30] discussed numerical solutions to generalized coupled Lyapunov equations; many authors studied least squares solutions ”
Accelerating a Recurrent Neural Network to Finite-Time Convergence for Solving Time-Varying Sylvester Equation by Using a Sign-Bi-power Activation Function (2012, methodology): “For static Sylvester equation without time-varying parameters, a classical algorithm for the numerical solution is the Bartels–Stewart algorithm [6] and its extensions [7,8].”
Analysis and Numerical Solution of Generalized Lyapunov Equations (2002, background): “Due to the practical importance the numerical solution of Lyapunov equations has received a lot of attention, see [9, 17, 55, 64, 72, 80, 100, 109, 126, 127, 136, 146] and the references therein.”
Computational Methods for Linear Matrix Equations (2016, background): “, [239], [178] and their references for further details on projected Sylvester equations.”
Gradient based iterative algorithms for solving a class of matrix equations (2005, methodology): “In this note, the problem will be tackled in a new way—by applying the so-called hierarchical identification principle [6], [ 7 ].”
Low rank iterative methods for projected generalized Lyapunov equations (2005, methodology): “Until now, only direct methods have been extended to projected Lyapunov equations [33].”
Krylov subspace methods for projected Lyapunov equations (2012, methodology): “All these methods except for Krylov subspace methods have been extended to projected Lyapunov equations [35, 39, 40].”