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We consider the solution of the rational matrix equations, or generalized algebraic Riccati equations with rational terms, arising in stochastic optimal control in continuousand discrete-time. Fixed-point iteration and (modified) Newton's methods will be considered. In particular, the convergence results of a new modified Newton's method, for both continuous-and discrete-time rational Riccati equations, will be presented.I. CONTINUOUS-TIME STOCHASTIC OPTIMAL CONTROL Consider the control system with state x and control u governed by the Itô differential equation [6], [7], [11], [12]:with the stochastic perturbations or disturbances {w i (t)} t∈R+ being independent zero mean real Wiener processes, and the outputHere A, A i ∈ K n×n , B, B i ∈ K n×m , C ∈ K l×n and D ∈ K n×m for K = R or C, and i = 1, · · · , N . For the stabilization of (1), we choose u to minimize the quadratic cost functionalwhere E denotes the expectation operator and T ≡ Q L L * R ≥ 0. This gives rise to the continuous-time rational Riccati equation (CRRE)with the linear operatorandThe operator Π is said to be positive as it satisfies Π(X) ≥ 0 for X ≥ 0. Note that the positivity of Π implies that for Π 1 and Π 2 . The optimal control is given bybeing the maximal stabilizing solution X to the CRRE. A good detailed account of the topic by Damm can be found in [6]. Useful results can also be found in [7], [11], [12], [15]. Newton's method was applied in [6], [7] and more efficient modified Newton's methods in [14], [15], with the former considering only the special case with R > 0 andwhere the control u is not disturbed by stochastically. The Riccati equation (4) has a degenerate and constant rational term R −1 and is referred to as an CRRE 0 . The CRRE 0 (4) was first investigated by Wonham [22], [23]. In this paper, we are interested in the efficient numerical solution of (2) (or its discrete-time cousin (27)) by a new faster modified Newton's method. Standard continuous-time (or discrete-time) algebraic Riccati equations CAREs (or DAREs) have to be solved in each iteration, possibly applying the efficient doubling algorithms in [3], [4], [18].