Small-sample statistical condition estimation of rational Riccati equations

Abstract We study the small-sample statistical condition estimation of the rational Riccati equation, which may be incorporated into a direct solver applying the homotopy method. Our proposed condition estimation algorithms are efficient for small and medium size rational Riccati equations. Illustrative numerical examples are presented.

[1]  A. Laub,et al.  Statistical Condition Estimation for Linear Least Squares , 1998, SIAM J. Matrix Anal. Appl..

[2]  Gene H. Golub,et al.  Matrix computations , 1983 .

[3]  Matthew M. Lin,et al.  Perturbation analysis of the stochastic algebraic Riccati equation , 2013 .

[4]  Yimin Wei,et al.  Effective condition numbers and small sample statistical condition estimation for the generalized Sylvester equation , 2013 .

[5]  Jianlin Xia,et al.  Applications of statistical condition estimation to the solution of linear systems , 2008, Numer. Linear Algebra Appl..

[6]  Hua Xiang,et al.  Mixed, componentwise condition numbers and small sample statistical condition estimation of Sylvester equations , 2012, Numer. Linear Algebra Appl..

[7]  Hung-Yuan Fan,et al.  Numerical solution to generalized Lyapunov/Stein and rational Riccati equations in stochastic control , 2015, Numerical Algorithms.

[8]  A. Laub,et al.  Benchmarks for the numerical solution of algebraic Riccati equations , 1997 .

[9]  Yimin Wei,et al.  Perturbation analysis and condition numbers of symmetric algebraic Riccati equations , 2009, Autom..

[10]  Dongmei Liu,et al.  Structured condition numbers and small sample condition estimation of symmetric algebraic Riccati equations , 2016, Appl. Math. Comput..

[11]  Yimin Wei,et al.  Homotopy for Rational Riccati Equations Arising in Stochastic Optimal Control , 2015, SIAM J. Sci. Comput..

[12]  V. Mehrmann.  The Autonomous Linear Quadratic Control Problem: Theory and Numerical Solution , 1991 .

[13]  Alan J. Laub,et al.  Small-Sample Statistical Condition Estimates for General Matrix Functions , 1994, SIAM J. Sci. Comput..

[14]  Gerhard Freiling,et al.  On a class of rational matrix differential equations arising in stochastic control , 2004 .

[15]  Huai-An Diao,et al.  Backward errors and small-sample condition estimation for ⋆-Sylveter equations , 2017, Int. J. Comput. Math..

[16]  Alan J. Laub,et al.  A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case , 1998 .