Let X1 and X2 be complex Banach spaces, and let A1 ∈ BL(X1), A2 ∈ BL(X2), A3 ∈ BL(X1, X2) and A4 ∈ BL(X2, X1). We propose an iterative procedure which is a modified form of Newton's iterations for obtaining approximations for the solution R ∈ BL(X1, X2) of the Riccati equation A2R - RA1 = A3 + RA4R, and show that the convergence of the method is quadratic. The advantage of the present procedure is that the conditions imposed on the operators A1, A2, A3, A4 are weaker than the corresponding conditions for Newton's iterations, considered earlier by Demmel (1987), Nair (1989) and Nair (1990) in the context of obtaining error bounds for approximate spectral elements. Also, we discuss an application of the procedure to spectral approximation under perturbations of the operator.