Applications of the matrix sign function to solving Lyapunov and Riccati equations and to other systems theory calculations are reviewed and the spectral implications of its definition discussed, Tho disadvantages of reduced order formulations are presented. Several theorems related to the mapping of eigenvalues under accelerated sign function algorithms are developed together with a consistent metric for measuring the distance of eigenvalues from their ultimate destination. After listing the desirable properties of an accelerated map, several previous attempts at acceleration are analysed and the fundamental reasons for their degree of success or otherwise determined. An optimally accelerated method and another rapidly convergent one are then developed. All of the methods are compared on two 28 x 28 examples arising from a Lyapunov and a Riccati equation. Finally, global and higher order convergence of the three successfully accelorated algorithms are proved.