Exact solution to Lyapunov´s equation using algebraic methods

Let A´P + PA = -Q be a Lyapunov equation with A being a stability matrix and both A and Q matrices with rational entries. Multiplying A and Q by a suitable positive Integer an equivalent Lyapunov equation A´1P + PA1 = -Q1 is obtained, with A1 and Q1 having integer entries, Let I(x,y) be the ring of polynomials in x and y over the integers I, and E be the set of all square matrices with integer entries. The solution P to this equation is given by: Pu = (emn) = fA 1 (qu(x,y), Q1) P = 1u2 ?? Pu where: qu(x,y) ?? I(x,y) and u ?? I fA 1 : I(x,y) ?? E ?? E defined as fA 1 (h(x,y), M) = ??j,khjk(A´1)j ?? M??A1 k which is a finite sum. The calculation of u and qu(x,y) requires finding the characterstic polynomial of A1, as well as using the Euclidean Algorithm, computations which lead to polynomial coefficient growth. In order to eliminate the space consuming manipulation of large integers in intermediate steps, modular arithimetic is used to obtain the matrix p iPu= (emnmodpi) and p iu=mu modpi with pi a prime, for a sufficient number of primes. The Chinese Remainder Theorem is then applied to obtain the solution P. The algorithm has been programmed on MACSYMA which is a very suitable computer programming system for all the numerical computations involved. Numerical results as well as extensions to solving the Algebraic Riccati Equation are presented.