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Let X1 and X2 be complex Banach spaces, and let A12 BL(X1), A22 BL(X2), A3 2 BL(X1;X2) and A4 2 BL(X2;X1). We propose an iterative procedure which is a modied form of Newton's iterations for obtaining approximations for the so- lution R2 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 corre- sponding conditions for Newton's iterations, considered earlier by Demmel (1987), Nair (1989) and Nair (1990) in the context of obtaining error bounds for approximate spec- tral elements. Also, we discuss an application of the procedure to spectral approximation under perturbations of the operator.

An iterative procedure for solving the Riccati equation A2R-RA1= A3+RA4R

An iterative procedure for solving the Riccati equation A₂R-RA₁= A₃+RA₄R