Parallel methods for computating the matrix sign function with applications to the algebraic Riccati equation

Several methods of solving the algebraic Riccati equation (ARE) are presented. Functional iterations with acceleration techniques am introduced. Also variations of the Newton method, the subspace iteration, and Krylov sequence are proposed for solving the ARE. The relation between the matrix sign function and the solution of the algebraic Riccati equation is stated, and several iterative schemes for the matrix sign function are described. Specifically, higher order rational functions for computing the matrix sign function of complex matrices has been developed, where parallel implementation of the matrix sign function is developed through partial fractions expansion. A QR inverse free method for computing the matrix sign function for symmetric matrices is derived. the matrix sign function is then used to solve the algebraic Riccati equation. The performance of these methods is demonstrated by several examples.