Solution of Lur'e equations via deflating subspaces

2011 ◽  
Author(s):  
Timo Reis
Keyword(s):  
Lur’E Equations ◽  
Deflating Subspaces

scholarly journals Rayleigh-Ritz Majorization Error Bounds for the Linear Response Eigenvalue Problem

2019 ◽  
Vol 17 (1) ◽  
pp. 653-667
Author(s):  
Zhongming Teng ◽  
Hong-Xiu Zhong
Keyword(s):  
Eigenvalue Problem ◽  
Quantum Chemistry ◽  
Error Bounds ◽  
Linear Response ◽  
Computational Quantum Chemistry ◽  
Matrix Anal ◽  
Deflating Subspaces ◽  
Important Notion ◽  
Theoretical Foundations ◽  
Computational Quantum

Abstract In the linear response eigenvalue problem arising from computational quantum chemistry and physics, one needs to compute a few of smallest positive eigenvalues together with the corresponding eigenvectors. For such a task, most of efficient algorithms are based on an important notion that is the so-called pair of deflating subspaces. If a pair of deflating subspaces is at hand, the computed approximated eigenvalues are partial eigenvalues of the linear response eigenvalue problem. In the case the pair of deflating subspaces is not available, only approximate one, in a recent paper [SIAM J. Matrix Anal. Appl., 35(2), pp.765-782, 2014], Zhang, Xue and Li obtained the relationships between the accuracy in eigenvalue approximations and the distances from the exact deflating subspaces to their approximate ones. In this paper, we establish majorization type results for these relationships. From our majorization results, various bounds are readily available to estimate how accurate the approximate eigenvalues based on information on the approximate accuracy of a pair of approximate deflating subspaces. These results will provide theoretical foundations for assessing the relative performance of certain iterative methods in the linear response eigenvalue problem.

scholarly journals The ADI method for bounded real and positive real Lur’e equations

2016 ◽  
Vol 135 (2) ◽  
pp. 431-458 ◽  
Author(s):  
Arash Massoudi ◽  
Mark R. Opmeer ◽  
Timo Reis
Keyword(s):  
Positive Real ◽  
Adi Method ◽  
Lur’E Equations

A Newton-like method for computing deflating subspaces

2015 ◽  
Vol 23 (4) ◽  
Author(s):  
Kirill V. Demyanko ◽  
Yuri M. Nechepurenko ◽  
Miloud Sadkane
Keyword(s):  
Stability Problem ◽  
Initial Guess ◽  
Sylvester Equation ◽  
Gmres Method ◽  
Regular Matrix ◽  
Subspace Iteration ◽  
Generalized Sylvester Equation ◽  
Matrix Pencils ◽  
Deflating Subspaces ◽  
Preconditioned Gmres

AbstractThis work is devoted to computations of deflating subspaces associated with separated groups of finite eigenvalues near specified shifts of large regular matrix pencils. The proposed method is a combination of inexact inverse subspace iteration and Newton’s method. The first one is slow but reliably convergent starting with almost an arbitrary initial subspace and it is used as a preprocessing to obtain a good initial guess for the second method which is fast but only locally convergent. The Newton method necessitates at each iteration the solution of a generalized Sylvester equation and for this task an iterative algorithm based on the preconditioned GMRES method is devised. Numerical properties of the proposed combination are illustrated with a typical hydrodynamic stability problem.

scholarly journals Computing periodic deflating subspaces associated with a specified set of eigenvalues

2007 ◽  
Vol 47 (4) ◽  
pp. 763-791 ◽  
Author(s):  
Robert Granat ◽  
Bo Kågström ◽  
Daniel Kressner
Keyword(s):  
Deflating Subspaces
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