ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

20th Conference of the International Linear Algebra Society (ILAS)

20th ILAS Conference

In minisymposium: Matrix Equations

Mon 17:30–18:00, Room AV 02.17
Interval arithmetic methods to verify the stabilizing solution of an algebraic Riccati equation
Federico Poloni (Università di Pisa)
Joint work with Tayyebe Haqiri (Shahid Bahonar University of Kerman)

We present a method which uses interval arithmetic to compute a certified solution enclosure for the stabilizing solution $X_s$ of a dense continuous-time algebraic Riccati equation $A^*X+XA+Q=XGX$. The algorithm is based on the modified Krawczyk's method used e.g. in [1], and includes a few improvements such as a preprocessing of the equation using permuted bases methods [2]. The algorithm has been tested on a suite of standard benchmark examples, and achieves results comparable to the state-of-the-art method in [3], surpassing it in some examples. An alternative algorithm which does not require the approximate diagonalization of the closed-loop matrix $A-GX_s$, based on a different fixed-point formulation, is also presented.

  1. Andreas Frommer and Behnam Hashemi. Verified computation of square roots of a matrix. SIAM J. Matrix Anal. Appl., 31(3):1279–1302, 2009.
  2. Volker Mehrmann and Federico Poloni. Doubling algorithms with permuted Lagrangian graph bases. SIAM J. Matrix Anal. Appl., 33(3):780–805, 2012.
  3. Shinya Miyajima. Fast verified computation for solutions of continuous-time algebraic Riccati equations. Jpn. J. Ind. Appl. Math., 32(2):529–544, 2015.