Optimal Parameters for Doubling Algorithms

Abstract

In using the structure-preserving algorithm (SDA) [Linear Algebra Appl., 2005, vol. 396, pp. 55–80] to solve a continuous-time algebraic Riccati equation, a parameter-dependent linear fractional transformation $z→(z−\gamma)/(z+\gamma)$ is first performed in order to bring all the eigenvalues of the associated Hamiltonian matrix in the open left half-plane into the open unit disk. The closer the eigenvalues are brought to the origin by the transformation via judiciously selected parameter $\gamma,$ the faster the convergence of the doubling iteration will be later on. As the first goal of this paper, we consider several common regions that contain the eigenvalues of interest and derive the best $\gamma$ so that the images of the regions under the transform are closest to the origin. For our second goal, we investigate the same problem arising in solving an $M$-matrix algebraic Riccati equation by the alternating-directional doubling algorithm (ADDA) [SIAM J. Matrix Anal. Appl., 2012, vol. 33, pp. 170–194] which uses the product of two linear fractional transformations $(z_1,z_2)→[(z_2−\gamma_2)/(z_2+\gamma_1)][(z_1−\gamma_1)/$$(z_1+\gamma_2)]$ that involves two parameters. Illustrative examples are presented to demonstrate the efficiency of our parameter selection strategies.