Low-Rank Solution of Lyapunov Equations

• Published in: SIAM review Vol. 46; no. 4; pp. 693 - 713
• Main Authors: Li, Jing-Rebecca, White, Jacob
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.12.2004
• Subjects: Algebra; Algorithms; Alternating direction implicit method; Applied sciences; Approximation; Area of dominant influence; Coefficients; Computer Science; Computer science; control theory; systems; Control system analysis; Control theory. Systems; Eigenvalues; Exact sciences and technology; Linear and multilinear algebra, matrix theory; Mathematical models; Mathematical vectors; Mathematics; Matrices; Matrix equations; Minimax problems; Modeling and Simulation; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Rational choice theory; Sciences and techniques of general use; Sigest; Studies; Alternating direction method; Invariant subspace; Implicit method; Krylov subspace method; Matrix product; Iterative method; Lyapunov equation; Scientific computation; Applied mathematics; Cholesky method; Minimax problem; Best approximation; Low rank approximation
• ISSN: 0036-1445; 1095-7200
• DOI: 10.1137/S0036144504443389

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$AX+XA^{T}=-BB^{T}$. The coefficient matrix A is assumed to be large, and the rank of the right-hand side$-BB^{T}$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.