On the Solution of the Rational Matrix Equation

Peter Benner,Heike Faßbender

doi:10.1155/2007/21850

Abstract

We study numerical methods for finding the maximal symmetric positive definite solution of the nonlinear matrix equation, where is symmetric positive definite and 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 algorithm to solve the DARE. This approach is compared to several fixed-point and doubling-type iterative methods suggested in the literature.

Highlights

INTRODUCTIONTwo of the ideas stated in [2] how to solve (1) can be interpreted as the numerical computation of a deflating subspace of a matrix pencil A−λB
The nonlinear matrix equationX = f (X) with f (X) = Q + LX−1LT, (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
This symplectic pencil G − λH allows the use of a doubling algorithm to compute the solution X. These methods originate from the fixed-point iteration derived from the discrete-time algebraic Riccati equation (DARE)

Summary

INTRODUCTION

Two of the ideas stated in [2] how to solve (1) can be interpreted as the numerical computation of a deflating subspace of a matrix pencil A−λB This is usually carried out by a procedure like the QZ algorithm. This symplectic pencil G − λH allows the use of a doubling algorithm to compute the solution X These methods originate from the fixed-point iteration derived from the DARE. We propose to compute the desired solution X∗ via an approximate solution of the DARE (4) by the (butterfly) SZ algorithm applied to the corresponding symplectic pencil [24,25,26].

The fixed-point iteration

The doubling algorithm

THE BUTTERFLY SZ ALGORITHM

DEFECT CORRECTION

NUMERICAL EXPERIMENTS

CONCLUSIONS