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

M. Thamban Nair ¹

¹ Department of Mathematics Indian Institute of Technology Madras Chennai 600 036, India

Studia Mathematica, Tome 147 (2001) no. 1, pp. 15-26

Résumé

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.

Détail

DOI : 10.4064/sm147-1-2

Keywords: complex banach spaces propose iterative procedure which modified form newtons iterations obtaining approximations solution riccati equation r ra convergence method quadratic advantage present procedure conditions imposed operators weaker corresponding conditions newtons iterations considered earlier demmel nair nair context obtaining error bounds approximate spectral elements discuss application procedure spectral approximation under perturbations operator