Aider Ya. BULGAKOV
Mathematical Depart.
Selcuk University
KONYA, TURKEY
e-mail: bulgk@trselcuk.bitnet
  		        
Algorithm for Solving Riccati Matrix Equation with Guaranteed Accuracy

			Abstract
     
        Speaking about the algorithms with guaranteed accuracy we mean
such algorithms which result either in the solution of the problem with
all true digits or in detecting its ill-condition. This scientific direction
is presented by S.K.Godunov, U.Kulisch, A.J.Laub, G.W.Stewart, L.N.Trefethen
and others.
        The paper presents the algorithm for solving Riccati matrix equation
based on the solution of the two point boundary problem. The algorithm
includes the series of orthogonal transformations of the matrix made up of
the Riccati equation coefficients. As a result we obtain the system of
equations whose condition is not worse than the condition of the Riccati
matrix equation. As a mesure of condition we use four numerical characteristics
which are the norms of the solution of the four matrix equations: the initial
and dual equations and two Lyapunov matrix equations.
        This characteristics is the development of the notion of the quantity
of stability parametre suggested by author (1980). The parametre is equivalent
to the notion of the distance from unstable matrices ( C. Van Loan) but is
more effective in computer realization. The similar parametre was suggested
independently by Eising (1984). At present such parametres found application in
the Stability Theory and in the Automatical Control Theory.
        A complete analysis of the influence of round-off errors and
uncertainties in the data has been made. The analysis has shown that the
algorithm is as fast as the known algorithm of the sign-function type but
is more stable.