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Low-Rank Solution of Lyapunov Equations.

Authors :
Li, Jing-Rebecca
White, Jacob
Source :
SIAM Review; 2004, Vol. 46 Issue 4, p693-713, 21p, 1 Chart, 4 Graphs
Publication Year :
2004

Abstract

This paper presents the Cholesky factor-alternating direction implicit (CF-ADI) algorithm, which generates a low-rank approximation to the solution X of the Lyapunov equation AX + XA<superscript>T</superscript> = -BB<superscript>T</superscript>. The coefficient matrix A is assumed to be large, and the rank of the right-hand side -BB<superscript>T</superscript> is assumed to be much smaller than the size of A. The CF-ADI algorithm requires only matrix-vector products and matrix-vector solves by shifts of A. Hence, it enables one to take advantage of any sparsity or structure in A. This paper also discusses the approximation of the dominant invariant subspace of the solution X. We characterize a group of spanning sets for the range of X. A connection is made between the approximation of the dominant invariant subspace of X and the generation of various low-order Krylov and rational Krylov subspaces. It is shown by numerical examples that the rational Krylov subspace generated by the CF-ADI algorithm, where the shifts are obtained as the solution of a rational minimax problem, often gives the best approximation to the dominant invariant subspace of X. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
00361445
Volume :
46
Issue :
4
Database :
Complementary Index
Journal :
SIAM Review
Publication Type :
Academic Journal
Accession number :
15155134
Full Text :
https://doi.org/10.1137/S0036144504443389