## An iterative procedure for solving the Riccati equation $A_2R-RA_1 = A_3+RA_4R$

### Volume 147 / 2001

#### Abstract

Let $X_1$ and $X_2$ be complex Banach spaces, and let
$A_1\in {\rm BL}(X_1)$, $A_2\in {\rm BL}(X_2)$,
$A_3\in {\rm BL}(X_1,X_2)$ and $A_4\in {\rm
BL}(X_2,X_1)$. We propose an iterative procedure which is a
modified form of Newton's iterations for obtaining
approximations for the solution $R\in {\rm
BL}(X_1,X_2)$ of the *Riccati equation*
$A_2R-RA_1 = A_3+RA_4R$, and show that the convergence of the
method is quadratic. The advantage of the present procedure is
that the conditions imposed on the operators $A_1, A_2, A_3,
A_4$ 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.