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.