A homotopy method for solving optimal projection equations for the reduced order model problem

An algorithm using probability-1 homotopy theory is proposed for solving the optimal projection equations for the reduced order model problem. There is a family of systems (the homotopy) which make a continuous transformation from some initial system to the final system. The process of solving each system involves computing the derivative of the Drazin inverse of a matrix and solving a number of Sylvester equations. The central theorem of the present works shows the validity of the whole process, i.e., determines the class of initial systems which certainly lead to the final system along a homotopy path. Another theorem shows that the differentiation of the Drazin inverse is justified, i.e., that the derivative of the Drazin inverse exists. Finally, it is shown how the optimal solution to the reduced order model problem can be easily computed from a solution to the modified Lyapunov equations