Optimal Parameters for Doubling Algorithms (Journal of Mathematical Study, 2021)
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-dependentlinear fractional transformation $z→(z−\gamma)/(z+\gamma)$ is first performed in orderto bring all the eigenvalues of the associated Hamiltonian matrix in the open lefthalf-plane into the open unit disk. The closer the eigenvalues are brought to the originby the transformation via judiciously selected parameter $\gamma,$ the faster the convergenceof the doubling iteration will be later on. As the first goal of this paper, we considerseveral 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 oursecond goal, we investigate the same problem arising in solving an $M$-matrix algebraicRiccati 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 fractionaltransformations $(z_1,z_2)→[(z_2−\gamma_2)/(z_2+\gamma_1)][(z_1−\gamma_1)/$$(z_1+\gamma_2)]$ that involvestwo parameters. Illustrative examples are presented to demonstrate the efficiency ofour parameter selection strategies.