Eigenfactor solution of the matrix Riccati equation - A continuous square root algorithm

This paper introduces a new algorithm for solving the matrix Riccati equation. Differential equations for the eigenvalues and eigenvectors of the solution matrix are developed in which their derivatives are expressed in terms of the eigenvalues and eigenvectors themselves and not as functions of the solution matrix. The solution of these equations yields, then, the time behavior of the eigenvalues and eigenvectors of the solution matrix. A reconstruction of the matrix itself at any desired time is immediately obtained through a trivial similarity transformation. This algorithm serves two purposes. First, being a square-root solution, it entails all the advantages of square root algorithms such as nonnegativedefiniteness and accuracy. Secondly, it furnishes the eigenvalues and eigenvectors of the solution matrix continuously without resorting to the complicated route of solving the equation directly and then decomposing the solution matrix into its eigenvalues and eigenvectors. The accuracy and the stability of the new algorithm are demonstrated through an example. It is shown that the algorithm works in a case where the ordinary algorithm fails and the closed-form solution cannot be computed as a result of numerical difficulties.