ICCAD '05 Proceedings of the 2005 IEEE/ACM International conference on Computer-aided design
A more reliable reduction algorithm for behavioral model extraction
ICCAD '05 Proceedings of the 2005 IEEE/ACM International conference on Computer-aided design
Proceedings of the 43rd annual Design Automation Conference
Block variants of Hammarling's method for solving Lyapunov equations
ACM Transactions on Mathematical Software (TOMS)
On the ADI method for Sylvester equations
Journal of Computational and Applied Mathematics
A Riemannian Optimization Approach for Computing Low-Rank Solutions of Lyapunov Equations
SIAM Journal on Matrix Analysis and Applications
Bi-parameter incremental unknowns ADI iterative methods for elliptic problems
Numerical Algorithms
Solving large-scale continuous-time algebraic Riccati equations by doubling
Journal of Computational and Applied Mathematics
On the ADI method for the Sylvester equation and the optimal-H2 points
Applied Numerical Mathematics
Large-scale Stein and Lyapunov equations, Smith method, and applications
Numerical Algorithms
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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+XAT = -BBT. The coefficient matrix A is assumed to be large, and the rank of the right-hand side -BBT 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.