Modular method for the computation of the defining polynomial of the algebraic Riccati equation

In this paper, we compute the defining polynomial of the solution of an algebraic Riccati equation (ARE) with an unknown parameter k. Letting each entry of the solution matrix of ARE be unknown variables, ARE can be viewed as m simultaneous algebraic equations with m variables and a parameter k, where m is the number of entries of the unknown matrix. Hence, computing Groebner basis of the algebraic equations with lexicographic ordering, we obtain the polynomial whose roots are the solution of ARE, which is the defining polynomial of ARE. Although this method of Groebner basis theoretically computes the defining polynomial of the solution of any ARE, it is not practical and easily collapses when the size of a given system is large, because of its heavy computational complexities. Thus, we applied modular techniques to the method and present an algorithm that is practical and is easily parallelizable (it is advantageous under multi-CPU environments).