The sensitivity of the algebraic and differential riccati equations
SIAM Journal on Control and Optimization
Perturbation Theory for Algebraic Riccati Equations
SIAM Journal on Matrix Analysis and Applications
Low-Rank Solution of Lyapunov Equations
SIAM Review
Approximation of Large-Scale Dynamical Systems (Advances in Design and Control) (Advances in Design and Control)
Convergence Analysis of Structure-Preserving Doubling Algorithms for Riccati-Type Matrix Equations
SIAM Journal on Matrix Analysis and Applications
Low rank approximate solutions to large Sylvester matrix equations
Applied Mathematics and Computation
A First Course in the Numerical Analysis of Differential Equations
A First Course in the Numerical Analysis of Differential Equations
On the numerical solution of large-scale sparse discrete-time Riccati equations
Advances in Computational Mathematics
| Hi-index | 7.29 |
We consider the solution of large-scale algebraic Riccati equations with numerically low-ranked solutions. For the discrete-time case, the structure-preserving doubling algorithm has been adapted, with the iterates for A not explicitly computed but in the recursive form A"k=A"k"-"1^2-D"k^(^1^)S"k^-^1[D"k^(^2^)]^@?, with D"k^(^1^) and D"k^(^2^) being low-ranked and S"k^-^1 being small in dimension. For the continuous-time case, the algebraic Riccati equation will be first treated with the Cayley transform before doubling is applied. With n being the dimension of the algebraic equations, the resulting algorithms are of an efficient O(n) computational complexity per iteration, without the need for any inner iterations, and essentially converge quadratically. Some numerical results will be presented. For instance in Section 5.2, Example 3, of dimension n=20209 with 204 million variables in the solution X, was solved using MATLAB on a MacBook Pro within 45 s to a machine accuracy of O(10^-^1^6).