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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article On the Solution of the Rational Matrix Equation X = Q + LX −1LT | Hindawi Publishing Corporation EURASIP Journal on Advances in Signal Processing Volume 2007 Article ID 21850 10 pages doi 10.1155 2007 21850 Research Article On the Solution of the Rational Matrix Equation X Q LX-1 Lt Peter Benner1 and Heike FaBbender2 1 Fakultat fur Mathematik Technische Universitat Chemnitz 09107 Chemnitz Germany 2 Institut Computational Mathematics Technische Universitat 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-1LT 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. Fafibender. 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 The nonlinear matrix equation X f X with f X Q LX-1Lt 1 where Q QT G R x is positive definite and L G R x 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 .
