Accurate solutions of M-matrix algebraic Riccati equations

AbstractThis paper is concerned with the relative perturbation theory and its entrywise relatively accurate numerical solutions of an M-matrix Algebraic Riccati Equations (MARE) $$XDX - AX - XB + C = 0 $$by which we mean the following conformally partitioned matrix $$ \left( \begin{array}{ll}B\, \, -D\\-C\, \, A\end{array}\right)$$is a nonsingular or an irreducible singular M-matrix. It is known that such an MARE has a unique minimal nonnegative solution $${\Phi}$$ . It is proved that small relative perturbations to the entries of A, B, C, and D introduce small relative changes to the entries of the nonnegative solution $${\Phi}$$ . Thus the smaller entries $${\Phi}$$ do not suffer bigger relative errors than its larger entries, unlike the existing perturbation theory for (general) Algebraic Riccati Equations. We then discuss some minor but crucial implementation changes to three existing numerical methods so that they can be used to compute $${\Phi}$$ as accurately as the input data deserve. Current study is based on a previous paper of the authors’ on M-matrix Sylvester equation for which D = 0.

[1]  Chun-Hua Guo,et al.  On the Doubling Algorithm for a (Shifted) Nonsymmetric Algebraic Riccati Equation , 2007, SIAM J. Matrix Anal. Appl..

[2]  Qiang Ye,et al.  Accurate computation of the smallest eigenvalue of a diagonally dominant M-matrix , 2002, Math. Comput..

[3]  Chun-Hua Guo,et al.  Convergence Analysis of the Doubling Algorithm for Several Nonlinear Matrix Equations in the Critical Case , 2009, SIAM J. Matrix Anal. Appl..

[4]  Shufang Xu,et al.  Accurate solutions of M-matrix Sylvester equations , 2012, Numerische Mathematik.

[5]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[6]  Chun-Hua Guo,et al.  Nonsymmetric Algebraic Riccati Equations and Wiener-Hopf Factorization for M-Matrices , 2001, SIAM J. Matrix Anal. Appl..

[7]  Peter Benner,et al.  On the ADI method for Sylvester equations , 2009, J. Comput. Appl. Math..

[8]  Nicholas J. Higham,et al.  Iterative Solution of a Nonsymmetric Algebraic Riccati Equation , 2007, SIAM J. Matrix Anal. Appl..

[9]  G. Golub,et al.  A Hessenberg-Schur method for the problem AX + XB= C , 1979 .

[10]  Yimin Wei,et al.  Normwise, mixed and componentwise condition numbers of nonsymmetric algebraic Riccati equations , 2008 .

[11]  L. Rogers.  Fluid Models in Queueing Theory and Wiener-Hopf Factorization of Markov Chains , 1994 .

[12]  Wen-Wei Lin,et al.  A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation , 2006, Numerische Mathematik.

[13]  Wen-Wei Lin,et al.  Nonsymmetric Algebraic Riccati Equations and Hamiltonian-like Matrices , 1998, SIAM J. Matrix Anal. Appl..

[14]  Israel Koltracht,et al.  Mixed componentwise and structured condition numbers , 1993 .

[15]  Richard H. Bartels,et al.  Algorithm 432 [C2]: Solution of the matrix equation AX + XB = C [F4] , 1972, Commun. ACM.

[16]  James Demmel,et al.  Applied Numerical Linear Algebra , 1997 .

[17]  R. A. Smith.  Matrix Equation $XA + BX = C$ , 1968 .

[18]  B. O. Anderson.  Second-order convergent algorithms for the steady-state Riccati equation , 1977, 1977 IEEE Conference on Decision and Control including the 16th Symposium on Adaptive Processes and A Special Symposium on Fuzzy Set Theory and Applications.

[19]  C. O'Cinneide.  Entrywise perturbation theory and error analysis for Markov chains , 1993 .

[20]  Alan J. Laub,et al.  On the Iterative Solution of a Class of Nonsymmetric Algebraic Riccati Equations , 2000, SIAM J. Matrix Anal. Appl..

[21]  Jonq Juang,et al.  Existence of algebraic matrix Riccati equations arising in transport theory , 1995 .