To the Beginning of the Third Millennium SOLUTIONS OF MATRIX EQUATIONS IN PROBLEMS OF MECHANICS AND CONTROL V. B. Larin The paper considers algorithms for solving linear matrix equations related to problems of mechanics and control, namely, the Lyapunov and Sylvester matrix equations and Riccati-type nonlinear matrix equations. These algorithms are capable of solving both linear equations and linear matrix inequalities. Algorithms based on the Bass relations are used to solve Riccati-type nonlinear matrix equations in so-called special cases where some eigenvalues of the matrix pencil are on a unit circle. These algorithms are compared with those of other authors by way of examples. It is shown that the algorithms can be implemented in symbolic computing routings, which allows solving these equations with high accuracy Keywords: Lyapunov matrix equation, Sylvester matrix equation, Riccati-type matrix equation, Bass relation, linear matrix inequalities, solution algorithms 1. Introduction. Many problems of mechanics and control are reduced to setting up algorithms and solving elementary linear and nonlinear matrix equations [8–12], such as the Lyapunov equations AX XA C + = T (1.1) (hereafter, the superscript “T” denotes transposition); the Sylvester equation X AXB C - = , (1.2) the generalized Sylvester equation EX AXB C - = , (1.3) nonlinear Riccati-type equations [45] X A X A Q + = - T 1 , (1.4) X A X A Q - = - T 1 . (1.5) The fact that the term “Riccati type” as applied in [45] to (1.4) and (1.5) is due possibly to the close relationship between these equations and the algebraic Riccati equation [16]. According to [46], Eq. (1.4) is a special case of the discrete algebraic Riccati equation (DARE): X F XF B XF N R B XB B XF N D = - + + + + - T T T T T T T ( ) ( ) ( ) 1 (1.6) International Applied Mechanics, Vol. 45, No. 8, 2009 1063-7095/09/4508-0847 ©2009 Springer Science+Business Media, Inc. 847 S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, 3 Nesterov St., Kyiv, Ukraine 03057, e-mail: model@inmech.kiev.ua. Translated from Prikladnaya Mekhanika, Vol. 45, No. 8, pp. 54–85, August 2009. Original article submitted February 27, 2009.