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Accurate solutions of M-matrix algebraic Riccati equations

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Abstract

This 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.

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Authors and Affiliations

  1. School of Mathematical Science, Fudan University, Shanghai, 200433, People’s Republic of China

    Jungong Xue

  2. School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China

    Shufang Xu

  3. Department of Mathematics, University of Texas at Arlington, Arlington, P.O. Box 19408, TX, 76019, USA

    Ren-Cang Li

Authors
  1. Jungong Xue
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  2. Shufang Xu
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  3. Ren-Cang Li
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Correspondence to Ren-Cang Li.

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Xue, J., Xu, S. & Li, RC. Accurate solutions of M-matrix algebraic Riccati equations. Numer. Math. 120, 671–700 (2012). https://doi.org/10.1007/s00211-011-0421-0

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  • DOI: https://doi.org/10.1007/s00211-011-0421-0

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