Lyapunov, Lanczos, and inertia

Inertia theorems for operator Lyapunov inequalities

Inertia Theorems for Operator Lyapunov Equations

We study operator Lyapunov equations in which the innnitesimal generator is not necessarily stable; but it satisses a spectrum decomposition assumption and it has at most nitely many unstable eigenvalues. Under mild conditions; these have unique self-adjoint solutions. We give conditions under which the number of negative eigenvalues of this solution equals the number of unstable eigenvalues of...

On the Kalman-Yacubovich-Popov lemma and common Lyapunov solutions for matrices with regular inertia

In this paper we extend the classical Lefschetz version of the KalmanYacubovich-Popov (KYP) lemma to the case of matrices with general regular inertia. We then use this result to derive an easily verifiable spectral condition for a pair of matrices with the same regular inertia to have a common Lyapunov solution (CLS), extending a recent result on CLS existence for pairs of Hurwitz matrices.

Nested Lanczos : Implicitly Restarting a Lanczos Algorithm Nested Lanczos : Implicitly Restarting a Lanczos Algorithm

In this text, we present a generalisation of the idea of the Implicitly Restarted Arnoldi method to the nonsymmetric Lanczos algorithm, using the two-sided Gram-Schmidt process or using a full Lanczos tridi-agonalisation. The Implicitly Restarted Lanczos method can be combined with an implicit lter. It can also be used in case of breakdown and ooers an alternative for look-ahead.

Lanczos and Linear Systems Lanczos and Linear Systems

abstract Lanczos's major contributions to the numerical solution of linear equations are contained in two papers: \An Iteration Method for the Solution of the Eigenvalue Problem of Linear Diierential and Integral Operators" and \Solutions of Linear Equations by Minimized Iterations ," the second of which contains the method of conjugate gradients. In this note we retrace Lanczos's journey from ...