arXiv subjects

Huai-An Diao

Publications and source records attributed to Huai-An Diao.

9 recordsLinked to original sources

arXiv2020

Condition numbers for the truncated total least squares problem and their estimations

In this paper, we present explicit expressions for the mixed and componentwise condition numbers of the truncated total least squares (TTLS) solution of $A\boldsymbol{x} \approx \boldsymbol{b} $ under the genericity condition, where $A$ is a $m\times n$ real data matrix and $\boldsymbol{b}$ is a real $m$-vector. Moreover, we reveal that normwise, componentwise and mixed condition numbers for the TTLS problem can recover the previous corresponding counterparts for the total least squares (TLS) problem when the truncated level of for the TTLS problem is $n$. When $A$ is a structured matrix, the structured perturbations for the structured truncated TLS (STTLS) problem are investigated and the corresponding explicit expressions for the structured normwise, componentwise and mixed condition numbers for the STTLS problem are obtained. Furthermore, the relationships between the structured and unstructured normwise, componentwise and mixed condition numbers for the STTLS problem are studied. Based on small sample statistical condition estimation (SCE), reliable condition estimation algorithms for both unstructured and structured normwise, mixed and componentwise are devised, which utilize the SVD of the augmented matrix $[A~\boldsymbol{b} ]$. The efficient proposed condition estimation algorithms can be integrated into the SVD-based direct solver for the small and medium size TTLS problem to give the error estimation for the numerical TTLS solution. Numerical experiments are reported to illustrate the reliability of the proposed estimation algorithms, which coincide with our theoretical results.

math.NA↗

arXiv2018

Linearized estimate of the backward error for the equality constrained indefinite least squares problem

In this note, we concentrate on the backward error of the equality constrained indefinite least squares problem. For the normwise backward error of the equality constrained indefinite least square problem, we adopt the linearization method to derive the tight estimate for the exact backward normwise error. The numerical examples show that the linearization estimate is effective for the normwise backward errors.

math.NA↗

arXiv2017

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

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.

math.NA↗

arXiv2017

A Condition Analysis of the Weighted Linear Least Squares Problem Using Dual Norms

In this paper, based on the theory of adjoint operators and dual norms, we define condition numbers for a linear solution function of the weighted linear least squares problem. The explicit expressions of the normwise and componentwise condition numbers derived in this paper can be computed at low cost when the dimension of the linear function is low due to dual operator theory. Moreover, we use the augmented system to perform a componentwise perturbation analysis of the solution and residual of the weighted linear least squares problems. We also propose two efficient condition number estimators. Our numerical experiments demonstrate that our condition numbers give accurate perturbation bounds and can reveal the conditioning of individual components of the solution. Our condition number estimators are accurate as well as efficient.

math.NA↗

arXiv2016

Mixed and componentwise condition numbers for a linear function of the solution of the total least squares problem

In this paper, we consider the mixed and componentwise condition numbers for a linear function of the solution to the total least squares (TLS) problem. We derive the explicit expressions of the mixed and componentwise condition numbers through the dual techniques. The sharp upper bounds for the derived mixed and componentwise condition numbers are obtained. For the structured TLS problem, we consider the structured perturbation analysis and obtain the corresponding expressions of the mixed and componentwise condition numbers. We prove that the structured ones are smaller than their corresponding unstructured ones based on the derived expressions. Moreover, we point out that the new derived expressions can recover the previous results on the condition analysis for the TLS problem. The numerical examples show that the derived condition numbers can give sharp perturbation bounds, on the other hand normwise condition numbers can severely overestimate the relative errors because normwise condition numbers ignore the data sparsity and scaling. Meanwhile, from the observations of numerical examples, it is more suitable to adopt structured condition numbers to measure the conditioning for the structured TLS problem.

math.NA↗

arXiv2016

Backward error and condition number analysis for the indefinite linear least squares problem

In this paper, we concentrate on the backward error and condition number of the indefinite least squares problem. For the normwise backward error of the indefinite least square problem, we adopt the linearization method to derive the tight estimations for the exact normwise backward errors. Using the dual techniques of condition number theory \cite{22.0}, we derive the explicit expressions of the mixed and componentwise condition numbers for the linear function of the solution for the indefinite least squares problem. The tight upper bounds for the derived mixed and componentwise condition numbers are obtained, which can be estimated efficiently by means of the classical power method for estimating matrix 1-norm \cite[Chapter 15]{Higham2002Book} during using the QR-Cholesky method \cite{1.0} for solving the indefinite least squares problem. The numerical examples show that the derived condition numbers can give sharp perturbation bound with respect to the interested component of the solution. And the linearization estimations are effective for the normwise backward errors.

math.NA↗

arXiv2016

Mixed and componentwise condition numbers for a linear function of the solution of the linear least squares problem with equality constrains

In this paper, we consider the mixed and componentwise condition numbers for a linear function of the solution to the linear least squares problem with equality constrains (LSE). We derive the explicit expressions of the mixed and componentwise condition numbers through the dual techniques. The sharp upper bounds for the derived mixed and componentwise condition numbers are obtained, which can be estimated efficiently by means of the classical Hager-Higham algorithm for estimating matrix one-norm during using the generalized QR factorization method for solving LSE. The numerical examples show that the derived condition numbers can give sharp perturbation bounds, on the other hand normwise condition numbers can severely overestimate the relative errors because normwise condition numbers ignore the data sparsity and scaling.

math.NA↗

arXiv2016

Backward Errors and Small Sample Condition Estimation for $\star$-Sylveter Equations

In this paper, we adopt a componentwise perturbation analysis for $\star$-Sylvester equations. Based on the small condition estimation (SCE), we devise the algorithms to estimate normwise, mixed and componentwise condition numbers for $\star$-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 $\star$-Sylvester equations under large or small perturbations.

math.NA↗

arXiv2016

Structured Condition Numbers of Structured Tikhonov Regularization Problem and their Estimations

Both structured componentwise and structured normwise perturbation analysis of the Tikhonov regularization are presented. The structured matrices under consideration include: Toeplitz, Hankel, Vandermonde, and Cauchy matrices. Structured normwise, mixed and componentwise condition numbers for the Tikhonov regularization are introduced and their explicit expressions are derived. For the general linear structure, we prove the structured condition numbers are smaller than their corresponding unstructured counterparts based on the derived expressions. By means of the power method and small sample condition estimation, the fast condition estimation algorithms are proposed. Our estimation methods can be integrated into Tikhonov regularization algorithms that use the generalized singular value decomposition (GSVD). The structured condition numbers and perturbation bounds are tested on some numerical examples and compared with their unstructured counterparts. Our numerical examples demonstrate that the structured mixed condition numbers give sharper perturbation bounds than existing ones, and the proposed condition estimation algorithms are reliable.

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