Iterative solution of the Lyapunov matrix equation
Applied Mathematics Letters
Alternating direction implicit iteration for systems with complex spectra
SIAM Journal on Numerical Analysis
Optimal alternating direction implicit parameters for nonsymmetric systems of linear equations
SIAM Journal on Numerical Analysis
A Cyclic Low-Rank Smith Method for Large Sparse Lyapunov Equations
SIAM Journal on Scientific Computing
Solution of the matrix equation AX + XB = C [F4]
Communications of the ACM
Low Rank Solution of Lyapunov Equations
SIAM Journal on Matrix Analysis and Applications
On the ADI method for Sylvester equations
Journal of Computational and Applied Mathematics
On the ADI method for the Sylvester equation and the optimal-H2 points
Applied Numerical Mathematics
| Hi-index | 0.09 |
The Lyapunov matrix equation AX+XA^@?=B is N-stable when all eigenvalues of the real nxn matrix A have positive real part. When the real nxn matrix B is spd the solution X is spd. It is of low rank when B=CC^@? where C is nxr with r@?n. An efficient algorithm has been found for solving the low-rank equation. This algorithm is a result of over fifty years of research starting with seemingly unrelated development of alternating direction implicit (ADI) iterative solution of elliptical systems. The low rank algorithm may be applied to a full rank equation if one can approximate the right-hand side by a sum of low rank matrices. This may be attempted with the Lanczos algorithm.