Residual bounds of approximate solutions of the algebraic Riccati equation

Abstract

LetX ≥ 0 approximate the unique Hermitian positive semi-definite solution X to the algebraic Riccati equation (ARE)be the residual of the ARE with respect toX , and define the linear operator L byBy applying a new forward perturbation bound to the optimal backward perturbation corresponding to the approximate solutionX , we obtained the following result: If A − FX is stable, and if 4 L −1 L −1R F < 1 for any unitarily invariant norm , then

A POD projection method for large-scale algebraic Riccati equations

“…• Sun sharpened the original residual error bound of Kenney, Laub, and Wette, and also relaxed the assumptions [66]. It may be possible to extend Sun's approach to the situation here; we leave this to be considered elsewhere.…”

“…To test the accuracy of the approximate solution obtained in the projection method for AREs, many researchers have computed the standard or Frobenius norm of the matrix residual for the ARE. The actual approximation error can be bounded using the matrix residual norm [44,66]. Many existing approaches to solving the ARE (1) are linked with specialized algorithms that rapidly compute (or estimate) the residual.…”

Backward Perturbation Analysis of the Periodic Discrete-Time Algebraic Riccati Equation

“…After that, Kågström [9] evaluates the normwise backward error of an approximate solution to the generalized Sylvester equation, and determines the sensitivity of the equation; Ghavimi and Laub [4] present a new backward error criterion, together with a sensitivity measure, for assessing solution accuracy of nonsymmetric and symmetric continuous-time algebraic Riccati equations. Normwise backward errors and residual bounds for continuous-time and discrete-time algebraic Riccati equations are obtained by the author [14], [15], [16]. This work, as a generalization of the results given by [15] and [16], derives normwise backward errors and residual bounds for an approximate Hermitian p.s.d.…”

Residual bounds of approximate solutions of the discrete-time algebraic Riccati equation

“…Perturbation theory for the continuous-time algebraic Riccati equation (CARE) has been investigated by a number of authors (see [14] and the references contained therein). However, perturbation theory for the DARE (1.1) (or equivalently, for the DARE (1.2)) has not been studied in such depth up to now [5,7].…”

“…In this paper we shall derive residual bounds for the approximate solutioñ X by using a technique different from that of [14]. In [14], we derive residual bounds of an approximate solutionX to the CARE by using the method of backward perturbation analysis, i.e., by applying a forward perturbation theorem to the optimal backward perturbation corresponding to the approximate solutionX . But in this paper, we obtain residual bounds of an approximate solutionX to the DARE directly from a perturbation equation.…”