# Preprint No. MPIMD/15-06

#### Author(s): Peter Benner, Matthias Heinkenschloss, Jens Saak, Heiko K. Weichelt

#### Email: weichelt@mpi-magdeburg.mpg.de

#### Date: 2015-05-05

#### Revised: 2015-09-14

#### Abstract:

This paper improves the inexact Kleinman-Newton method by incorporating a line search and by systematically integrating the low-rank structure resulting from ADI methods for the approximate solution of the Lyapunov equation that needs to be solved to compute the Kleinman-Newton step. A convergence result is presented that tailors the convergence proof for general inexact Newton methods to the structure of Riccati equations and avoids positive semi-deniteness ssumptions on the Lyapunov equation residual, which in general do not hold for ow-rank approaches. On a test example, the improved inexact Kleinman-Newton ethod is seven to twelve times faster than the exact Kleinman-Newton method ithout line search; the addition of the line search to the inexact Kleinman-Newton method alone can reduce computation time by up to a factor of two

#### BibTeX:

@TECHREPORT{MPIMD15-06,

author = {Peter Benner and Matthias Heinkenschloss and Jens Saak and Heiko K. Weichelt},

title = {Inexact low-rank Newton-ADI method for large-scale algebraic Riccati equations},

number = {MPIMD/15-06},

month = may,

year = 2015,

institution = {Max Planck Institute Magdeburg},

type = {Preprint},

note = {Available from \url{http://www.mpi-magdeburg.mpg.de/preprints/}},

}