The sensitivity of the algebraic and differential riccati equations
SIAM Journal on Control and Optimization
Krylov subspace methods for solving large Lyapunov equations
SIAM Journal on Numerical Analysis
Low-Rank Solution of Lyapunov Equations
SIAM Review
Approximation of Large-Scale Dynamical Systems (Advances in Design and Control) (Advances in Design and Control)
Low rank approximate solutions to large Sylvester matrix equations
Applied Mathematics and Computation
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
On the numerical solution of large-scale sparse discrete-time Riccati equations
Advances in Computational Mathematics
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We consider the solution of large-scale Lyapunov and Stein equations. For Stein equations, the well-known Smith method will be adapted, with $A_k = A^{2^k}$ not explicitly computed but in the recursive form $A_k = A_{k-1}^{2}$ , and the fast growing but diminishing components in the approximate solutions truncated. Lyapunov equations will be first treated with the Cayley transform before the Smith method is applied. For algebraic equations with numerically low-ranked solutions of dimension n, the resulting algorithms are of an efficient O(n) computational complexity and memory requirement per iteration and converge essentially quadratically. An application in the estimation of a lower bound of the condition number for continuous-time algebraic Riccati equations is presented, as well as some numerical results.