Lyapunov and Sylvester equations: A quadrature framework
For researchers in numerical linear algebra, this work offers a new perspective on solving Lyapunov and Sylvester equations by connecting them to numerical integration, but the equivalence to ADI suggests incremental novelty.
This paper presents a quadrature framework for solving large-scale Lyapunov and Sylvester equations, using Runge-Kutta methods to produce low-rank approximations. The framework is shown to be equivalent to the ADI iteration in one instance, and a realification approach is provided for real problems.
This paper introduces a novel framework for the solution of (large-scale) Lyapunov and Sylvester equations derived from numerical integration methods. Suitable systems of ordinary differential equations are introduced. Low-rank approximations of their solutions are produced by Runge-Kutta methods. Appropriate Runge-Kutta methods are identified following the idea of geometric numerical integration to preserve a geometric property, namely a low rank residual. For both types of equations we prove the equivalence of one particular instance of the resulting algorithm to the well known ADI iteration. As the general approach suggested here leads to complex valued computation even for real problems, we present a general realification approach based on similarity transformation.