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

Volume 147 / 2001

M. Thamban Nair Studia Mathematica 147 (2001), 15-26 MSC: 45B05, 47A10, 65F35, 65R20. DOI: 10.4064/sm147-1-2

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.

Authors

  • M. Thamban NairDepartment of Mathematics
    Indian Institute of Technology Madras
    Chennai 600 036, India
    e-mail

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