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

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Nair M.

Studia Mathematica

2001

tom 147

nr 1

15-26

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.

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1 Studia Mathematica

1 Nair M.

1 2001

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