On the Solution of the Rational Matrix Equation
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Research Article On the Solution of the Rational Matrix Equation X = Q + LX −1LT Peter Benner1 and Heike Faßbender2 1 Fakult¨ at 2 Institut
f¨ur Mathematik, Technische Universit¨at Chemnitz, 09107 Chemnitz, Germany Computational Mathematics, Technische Universit¨at Braunschweig, 38106 Braunschweig, Germany
Received 30 September 2006; Revised 9 February 2007; Accepted 22 February 2007 Recommended by Paul Van Dooren We study numerical methods for finding the maximal symmetric positive definite solution of the nonlinear matrix equation X = Q + LX −1 LT , where Q is symmetric positive definite and L is nonsingular. Such equations arise for instance in the analysis of stationary Gaussian reciprocal processes over a finite interval. Its unique largest positive definite solution coincides with the unique positive definite solution of a related discrete-time algebraic Riccati equation (DARE). We discuss how to use the butterfly SZ algorithm to solve the DARE. This approach is compared to several fixed-point and doubling-type iterative methods suggested in the literature. Copyright © 2007 P. Benner and H. Faßbender. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1.
INTRODUCTION
Lagrangian deflating subspace of the matrix pencil
The nonlinear matrix equation −1 T
X = f (X) with f (X) = Q + LX L ,
(1)
where Q = QT ∈ Rn×n is positive definite and L ∈ Rn×n is nonsingular, arises in the analysis of stationary Gaussian reciprocal processes over a finite interval. The solutions of certain 1D stochastic boundary value problems are reciprocal processes. For instance, the steady state distribution of the temperature along a heated ring or beam subjected to random loads along its length can be modeled in terms of such reciprocal processes. A different example is a ship surveillance problem: given a Gauss-Markov state-space model of the ship’s trajectory, it is desired to assign a probability distribution not only to the initial state, but also to the final state, corresponding to some predictive information about the ship’s destination. This has the effect of modeling the trajectory as a reciprocal process. For references to these examples see, for example, [1]. The problem considered here is to find the (unique) largest positive definite symmetric solution X+ of (1). This equation has been considered, for example, in [2–7]. In [2], the set of Hermitian solutions of (1) is characterized in terms of the spectral factors of the matrix Laurent polynomial L(z) = Q + Lz − LT z−1 . These factors are related to the
LT 0 0 I . −λ G − λH = L 0 −Q I
(2)
In particular, one can conclude from the results in [2, Section 2] that this matrix pencil does not have any eigenvalues on the unit circle and that the spectral radius ρ(X+−1 LT ) is less than 1 as [ XI+ ] spans the stable Lagrangian deflating subspace of G − λH. Alternatively, one could rewrite (1) as the di
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