Structured condition numbers of structured Tikhonov regularization problem and their estimations

Diao, Huaian; Wei, Yimin; Qiao, Sanzheng

doi:10.1016/j.cam.2016.05.023

Cited by 19 publications

(12 citation statements)

References 48 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, when the TLS problem is structured, it is suitable to study the structured perturbation analysis because this will help us to understand the structured preserved algorithms; see [26]. Structured perturbation analysis for linear system, linear least squares and Tikhonov regularization problem had been investigated in [27,28,29,30,31,32], respectively.…”

Section: Introductionmentioning

confidence: 99%

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

Diao

Sun

2018

Linear Algebra and its Applications

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

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

Diao

Sun

2018

Linear Algebra and its Applications

Self Cite

View full text Add to dashboard Cite

show abstract

“…Therefore, compared with the normwise condition number estimation algorithm proposed in Reference [6], our proposed condition number estimations algorithms in this paper are more efficient in terms of the computational complexity, which are also applicable for estimating the componentwise and structured perturbations for (3). For recent SCE's developments for (structured) linear systems, linear least squares, Tikhonov regularization and TLS problem, we refer to References [22,[24][25][26][27] and references therein.…”

Section: Introductionmentioning

confidence: 96%

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

Diao

Bai

2021

Numerical Linear Algebra App

View full text Add to dashboard Cite

In this paper, we present explicit expressions for the mixed and componentwise condition numbers of the truncated total least squares (TTLS) solution of Ax ≈ b under the genericity condition, where A is a m × n real data matrix and 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 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. We devise reliable condition estimation algorithms for the TTLS problem by utilizing small-sample statistical condition estimation techniques. The proposed condition estimation algorithms employ the singular value decomposition (SVD) of the augmented matrix [A b] to reduce the computational complexity, where both unstructured and structured normwise, mixed, and componentwise condition estimations are considered. The 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 condition estimation algorithms.

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

Structured condition numbers of structured Tikhonov regularization problem and their estimations

Cited by 19 publications

References 48 publications

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

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

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

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

Contact Info

Product

Resources

About