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

Diao, Huaian; Li, Dongmei; Qiao, Sanzheng

doi:10.1016/j.amc.2017.06.028

Cited by 3 publications

(2 citation statements)

References 35 publications

(61 reference statements)

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“…Based on the adjoint method and SCE, Cao and Petzold [23] proposed an efficient method for estimating the error in the solution of the Sylvester matrix equations. Diao et al applied the SCE to the Sylvester equations [22,14], algebraic Riccati equations [24] and the structured Tikhonov regularization problem [25].…”

Section: Small-sample Condition Estimationsmentioning

confidence: 99%

Backward errors and small-sample condition estimation for ⋆-Sylveter equations

Diao

Hong

Chu

2017

International Journal of Computer Mathematics

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In this paper, we adopt a componentwise perturbation analysis for ⋆-Sylvester equations. Based on the small condition estimation (SCE), we devise the algorithms to estimate normwise, mixed and componentwise condition numbers for ⋆-Sylvester equations. We also define a componentwise backward error with a sharp and easily computable bound. Numerical examples illustrate that our algorithm under componentwise perturbations produces reliable estimates, and the new derived computable bound for the componentwise backward error is sharp and reliable for well conditioned and moderate ill-conditioned ⋆-Sylvester equations under large or small perturbations.

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Section: Small-sample Condition Estimationsmentioning

confidence: 99%

Backward errors and small-sample condition estimation for ⋆-Sylveter equations

Diao

Hong

Chu

2017

International Journal of Computer Mathematics

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“…Especially, the multiple right-hand side linear system (1.2) arises naturally in many applications such as Quantum Chromo Dynamics [40], dynamics of structures [8], quasi-Newton methods for solving nonlinear equations with multiple secant equations [28], computing the lengths of nucleon-nucleon scattering [42], wave propagation phenomena [45]. For perturbation analysis for (generalized) Sylvester equation, *-Sylvester equation and algebraic Riccati equation, we refer to papers [12][13][14][15]30]. The low-rank structured matrix has been studied extensively in numerical linear algebra and has many applications; see the recent books [20,21] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Structured condition number for multiple right-hand side linear systems with parameterized quasiseparable coefficient matrix

Diao

2020

Journal of Computational and Applied Mathematics

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In this paper, we consider the structured perturbation analysis for multiple right-hand side linear systems with parameterized coefficient matrix. Especially, we present the explicit expressions for structured condition numbers for multiple right-hand sides linear systems with {1;1}-quasiseparable coefficient matrix in the quasiseparable and the Givens-vector representations. In addition, the comparisons of these two condition numbers between themselves, and with respect to unstructured condition number are investigated. Moreover, the effective structured condition number for multiple righthand sides linear systems with {1;1}-quasiseparable coefficient matrix is proposed. The relationships between the effective structured condition number and structured condition numbers with respect to the quasiseparable and the Givens-vector representations are also studied. Numerical experiments show that there are situations in which the effective structured condition number can be much smaller than the unstructured ones.

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