Real Cholesky Factor-ADI Method for Low-Rank Solution of Projected Generalized Lyapunov Equations

Yuichi TANJI  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E99-A   No.3   pp.702-709
Publication Date: 2016/03/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E99.A.702
Type of Manuscript: PAPER
Category: Nonlinear Problems
Keyword: 
Lyapunov equations,  low-rank solution,  balanced truncation,  RLC networks,  signal/power integrity,  Krylov subspace method,  

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Summary: 
The alternating direction implicit (ADI) method is proposed for low-rank solution of projected generalized continuous-time algebraic Lyapunov equations. The low-rank solution is expressed by Cholesky factor that is similar to that of Cholesky factorization for linear system of equations. The Cholesky factor is represented in a real form so that it is useful for balanced truncation of sparsely connected RLC networks. Moreover, we show how to determine the shift parameters which are required for the ADI iterations, where Krylov subspace method is used for finding the shift parameters that reduce the residual error quickly. In the illustrative examples, we confirm that the real Cholesky factor certainly provides low-rank solution of projected generalized continuous-time algebraic Lyapunov equations. Effectiveness of the shift parameters determined by Krylov subspace method is also demonstrated.