Iterative solution of the Lyapunov matrix equation
Applied Mathematics Letters
Direct methods for sparse matrices
Direct methods for sparse matrices
SIAM Journal on Control and Optimization
Matrix computations (3rd ed.)
Algorithm 782: codes for rank-revealing QR factorizations of dense matrices
ACM Transactions on Mathematical Software (TOMS)
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
The Quadratic Eigenvalue Problem
SIAM Review
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Direct Methods for Sparse Linear Systems (Fundamentals of Algorithms 2)
Direct Methods for Sparse Linear Systems (Fundamentals of Algorithms 2)
A New Iterative Method for Solving Large-Scale Lyapunov Matrix Equations
SIAM Journal on Scientific Computing
On the ADI method for Sylvester equations
Journal of Computational and Applied Mathematics
SIAM Journal on Matrix Analysis and Applications
On the numerical solution of large-scale sparse discrete-time Riccati equations
Advances in Computational Mathematics
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The solution of large-scale Lyapunov equations is a crucial problem for several fields of modern applied mathematics. The low-rank Cholesky factor version of the alternating directions implicit method (LRCF-ADI) is one iterative algorithm that computes approximate low-rank factors of the solution. In order to achieve fast convergence it requires adequate shift parameters, which can be complex if the matrices defining the Lyapunov equation are unsymmetric. This will require complex arithmetic computations as well as storage of complex data and thus, increase the overall complexity and memory requirements of the method. In this article we propose a novel reformulation of LRCF-ADI which generates real low-rank factors by carefully exploiting the dependencies of the iterates with respect to pairs of complex conjugate shift parameters. It significantly reduces the amount of complex arithmetic calculations and requirements for complex storage. It is hence often superior in terms of efficiency compared to other real formulations.