Mixed and componentwise condition numbers of nonsymmetric algebraic Riccati equation

Liu, Landong

doi:10.1016/j.amc.2012.01.026

Cited by 9 publications

(4 citation statements)

References 16 publications

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“…Consider the NDRE (1), constructed according to the rules given in [15]. This scheme is used in [10] to analyze the effectiveness of mixed and componentwise condition numbers, and in [11], to illustrate the validity of a condition number and backward errors of nonsymmetric algebraic Riccati equation.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…To estimate the actual error in the computed solution, it is important to find a bound of the error in the computed solution in terms of the perturbations in the data. Perturbation analysis of the nonsymmetric algebraic Riccati equation is given in [8], normwise, mixed and componentwise condition numbers, as well as residual bounds are proposed in [9][10][11]. To the best of our knowledge, the sensitivity and the conditioning of the NDRE are not yet analyzed.This work has two goals.…”

Section: Introduction and Notationsmentioning

confidence: 99%

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Sensitivity of the Solution to Nonsymmetric Differential Matrix Riccati Equation

2021

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Nonsymmetric differential matrix Riccati equations arise in many problems related to science and engineering. This work is focusing on the sensitivity of the solution to perturbations in the matrix coefficients and the initial condition. Two approaches of nonlocal perturbation analysis of the symmetric differential Riccati equation are extended to the nonsymmetric case. Applying the techniques of Fréchet derivatives, Lyapunov majorants and fixed-point principle, two perturbation bounds are derived: the first one is based on the integral form of the solution and the second one considers the equivalent solution to the initial value problem of the associated differential system. The first bound is derived for the nonsymmetric differential Riccati equation in its general form. The perturbation bound based on the sensitivity analysis of the associated linear differential system is formulated for the low-dimensional approximate solution to the large-scale nonsymmetric differential Riccati equation. The two bounds exploit the existing sensitivity estimates for the matrix exponential and are alternative.

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Section: Numerical Examplesmentioning

confidence: 99%

Section: Introduction and Notationsmentioning

confidence: 99%

Sensitivity of the Solution to Nonsymmetric Differential Matrix Riccati Equation

2021

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“…A condition number is a measurement of the sensitivity. Liu studied mixed and componentwise condition numbers of nonsymmetric algebraic Riccati equation in [36]. For the perturbation analysis of the CARE (1.3) or DARE (1.4), we refer papers [6,17,22,29,31] and their references therein.…”

Section: Introductionmentioning

confidence: 99%

Structured condition numbers and small sample condition estimation of symmetric algebraic Riccati equations

Diao

Qiao

2017

Applied Mathematics and Computation

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This paper is devoted to a structured perturbation analysis of the symmetric algebraic Riccati equations by exploiting the symmetry structure. Based on the analysis, the upper bounds for the structured normwise, mixed and componentwise condition numbers are derived. Due to the exploitation of the symmetry structure, our results are improvements of the previous work on the perturbation analysis and condition numbers of the symmetric algebraic Riccati equations. Our preliminary numerical experiments demonstrate that our condition numbers provide accurate estimates for the change in the solution caused by the perturbations on the data. Moreover, by applying the small sample condition estimation method, we propose a statistical algorithm for practically estimating the condition numbers of the symmetric algebraic Riccati equations.

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“…In this case, componentwise analysis can be one alternative approach by which much tighter and revealing bounds can be obtained. There are two kinds of alternative condition numbers called mixed and componentwise condition numbers, respectively, which are developed by Gohberg and Koltracht [17], and we refer to [16,22,34,35,[39][40][41][42][43] for more details of these two kinds of condition numbers.…”

Section: Introductionmentioning

confidence: 99%

Condition Numbers and Backward Error of a Matrix Polynomial Equation Arising in Stochastic Models

2018

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We consider a matrix polynomial equation (MPE) A n X n + A n−1 X n−1 + · · · + A 0 = 0, where A n , A n−1 , . . . , A 0 ∈ R m×m are the coefficient matrices, and X ∈ R m×m is the unknown matrix. A sufficient condition for the existence of the minimal nonnegative solution is derived, where minimal means that any other solution is componentwise no less than the minimal one. The explicit expressions of normwise, mixed and componentwise condition numbers of the matrix polynomial equation are obtained. A backward error of the approximated minimal nonnegative solution is defined and evaluated. Some numerical examples are given to show the sharpness of the three kinds of condition numbers.

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Mixed and componentwise condition numbers of nonsymmetric algebraic Riccati equation

Cited by 9 publications

References 16 publications

Sensitivity of the Solution to Nonsymmetric Differential Matrix Riccati Equation

Sensitivity of the Solution to Nonsymmetric Differential Matrix Riccati Equation

Structured condition numbers and small sample condition estimation of symmetric algebraic Riccati equations

Condition Numbers and Backward Error of a Matrix Polynomial Equation Arising in Stochastic Models

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