Perturbation analysis and condition numbers of scaled total least squares problems

The standard approaches to solving an overdetermined linear system Ax ≈ b find minimal corrections to the vector b and/or the matrix A such that the corrected system is consistent, such as the least squares (LS), the data least squares (DLS) and the total least squares (TLS). The scaled total least squares (STLS) method unifies the LS, DLS and TLS methods. The classical normwise condition numbers for the LS problem have been widely studied. However, there are no such similar results for the TLS and the STLS problems. In this paper, we first present a perturbation analysis of the STLS problem, which is a generalization of the TLS problem, and give a normwise condition number for the STLS problem. Different from normwise condition numbers, which measure the sizes of both input perturbations and output errors using some norms, componentwise condition numbers take into account the relation of each data component, and possible data sparsity. Then in this paper we give explicit expressions for the estimates of the mixed and componentwise condition numbers for the STLS problem. Since the TLS problem is a special case of the STLS problem, the condition numbers for the TLS problem follow immediately from our STLS results. All the discussions in this paper are under the Golub-Van Loan condition for the existence and uniqueness of the STLS solution. Yimin Wei is supported by the National Natural Science Foundation of China under grant 10871051, Shanghai Science & Technology Committee under grant 08DZ2271900 and Shanghai Education Committee under grant 08SG01. Sanzheng Qiao is partially supported by Shanghai Key Laboratory of Contemporary Applied Mathematics of Fudan University during his visiting.     

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.     

On condition numbers for Moore–Penrose inverse and linear least squares problem involving Kronecker products

In this paper, we investigate the normwise, mixed, and componentwise condition numbers and their upper bounds for the Moore–Penrose inverse of the Kronecker product and more general matrix function compositions involving Kronecker products. We also present the condition numbers and their upper bounds for the associated Kronecker product linear least squares solution with full column rank. In practice, the derived upper bounds for the mixed and componentwise condition numbers for Kronecker product linear least squares solution can be efficiently estimated using the Hager–Higham Algorithm. Copyright © 2012 John Wiley & Sons, Ltd.     

On mixed and componentwise condition numbers for Moore-Penrose inverse and linear least squares problems

Classical condition numbers are normwise: they measure the size of both input perturbations and output errors using some norms. To take into account the relative of each data component, and, in particular, a possible data sparseness, componentwise condition numbers have been increasingly considered. These are mostly of two kinds: mixed and componentwise. In this paper, we give explicit expressions, computable from the data, for the mixed and componentwise condition numbers for the computation of the Moore-Penrose inverse as well as for the computation of solutions and residues of linear least squares problems. In both cases the data matrices have full column (row) rank.

     

Condition numbers and perturbation analysis for the Tikhonov regularization of discrete ill‐posed problems

One of the most successful methods for solving the least‐squares problem min_x∥Ax?b∥₂ with a highly ill‐conditioned or rank deficient coefficient matrix A is the method of Tikhonov regularization. In this paper, we derive the normwise, mixed and componentwise condition numbers and componentwise perturbation bounds for the Tikhonov regularization. Our results are sharper than the known results. Some numerical examples are given to illustrate our results. Copyright © 2010 John Wiley & Sons, Ltd.     

Perturbation theory for generalized and constrained linear least squares

The perturbation analysis of weighted and constrained rank‐deficient linear least squares is difficult without the use of the augmented system of equations. In this paper a general form of the augmented system is used to get simple perturbation identities and perturbation bounds for the general linear least squares problem both for the full‐rank and rank‐deficient problem. Perturbation identities for the rank‐deficient weighted and constrained case are found as a special case. Interesting perturbation bounds and condition numbers are derived that may be useful when considering the stability of a solution of the rank‐deficient general least squares problem. Copyright © 2000 John Wiley & Sons, Ltd.     

Indefinite Least Squares Problem with Quadratic Constraint and Its Condition Numbers

In this paper,we consider the indefinite least squares problem with quadratic constraint and its condition numbers.The conditions under which the problem has the unique solution are first presented.Then,the normwise,mixed,and componentwise condition numbers for solution and residual of this problem are derived.Numerical example is also provided to illustrate these results.     

带多重右边的不定最小二乘问题的条件数

本文研究了带多重右边的不定最小二乘问题的条件数,给出了范数型、混合型及分量型条件数的表达式,同时,也给出了相应的结构条件数的表达式.所考虑的结构矩阵包含Toeplitz 矩阵、Hankel矩阵、对称矩阵、三对角矩阵等线性结构矩阵与Vandermonde矩阵、Cauchy矩阵等非线性结构矩阵.数值例子显示结构条件数总是紧于非结构条件数.     

Using dual techniques to derive componentwise and mixed condition numbers for a linear function of a linear least squares solution

We prove duality results for adjoint operators and product norms in the framework of Euclidean spaces. We show how these results can be used to derive condition numbers especially when perturbations on data are measured componentwise relatively to the original data. We apply this technique to obtain formulas for componentwise and mixed condition numbers for a linear function of a linear least squares solution. These expressions are closed when perturbations of the solution are measured using a componentwise norm or the infinity norm and we get an upper bound for the Euclidean norm.      

On the weighting method for mixed least squares–total least squares problems

It is well known that the standard algorithm for the mixed least squares–total least squares (MTLS) problem uses the QR factorization to reduce the original problem into a standard total least squares problem with smaller size, which can be solved based on the singular value decomposition (SVD). In this paper, the MTLS problem is proven to be closely related to a weighted total least squares problem with its error‐free columns multiplied by a large weighting factor. A criterion for choosing the weighting factor is given; and for the sake of stability in solving the MTLS problem, the Cholesky factorization‐based inverse (Cho‐INV) iteration and Rayleigh quotient iteration are also considered. For large‐scale MTLS problems, numerical tests show that Cho‐INV is superior to the standard QR‐SVD method, especially for the case with big gap between the desired and undesired singular values and the case when the coefficient matrix has much more error‐contaminated columns. Rayleigh quotient iteration behaves more efficient than QR‐SVD for most cases and fails occasionally, and in some cases, it converges much faster than Cho‐INV but still less efficient due to its higher computation cost.     

关于TLS和LS解的扰动分析

1．引言本文采用卜］的记号．最小二乘（LS）和总体最小二乘（TLS）是科学计算中的两种重要方法．尤是TLS，近来已有多篇论文讨论［1－6，8－16］．奇异值分解（SVD）和CS分解是研究TLS和LS的重要工具．令ACm，BCm,C=(A,B)，A和C的SVD分别为（1．1）（1．2）其中P51为某个正整数，U，U，V，V均为西矩阵，UI，UI，VI，VI为上述矩阵的前P列，z1一山。g（。1，…，内），】2＝di。g（内十l，…，。小】1＝dl。g（61；…，站，】2二diag（4＋1；…，dk），。l三··2。120和dl三…三d。20分别为C和A的奇异值，Z＝mhfm．n十以…     

On Frobenius normwise condition numbers for Moore–Penrose inverse and linear least‐squares problems

Condition numbers play an important role in numerical analysis. Classical condition numbers are normwise: they measure the size of both input perturbations and output errors using norms. In this paper, we give explicit, computable expressions depending on the data, for the normwise condition numbers for the computation of the Moore–Penrose inverse as well as for the solutions of linear least‐squares problems with full‐column rank. Copyright © 2007 John Wiley & Sons, Ltd.     

Ill-Conditionedness Need not be Componentwise Near to Ill-Posedness for Least Squares Problems

The condition number of a problem measures the sensitivity of the answer to small changes in the input, where small refers to some distance measure. A problem is called ill-conditioned if the condition number is large, and it is called ill-posed if the condition number is infinity. It is known that for many problems the (normwise) distance to the nearest ill-posed problem is proportional to the reciprocal of the condition number. Recently it has been shown that for linear systems and matrix inversion this is also true for componentwise distances. In this note we show that this is no longer true for least squares problems and other problems involving rectangular matrices. Problems are identified which are arbitrarily ill-conditioned (in a componentwise sense) whereas any componentwise relative perturbation less than 1 cannot produce an ill-posed problem. Bounds are given using additional information on the matrix.     

ON THE ACCURACY OF THE LEAST SQUARES AND THE TOTAL LEAST SQUARES METHODS

Consider solving an overdetermined system of linear algebraic equations by both the least squares method (LS) and the total least squares method (TLS). Extensive published computational evidence shows that when the original system is consistent. one often obtains more accurate solutions by using the TLS method rather than the LS method. These numerical observations contrast with existing analytic perturbation theories for the LS and TLS methods which show that the upper bounds for the LS solution are always smaller than the corresponding upper bounds for the TLS solutions. In this paper we derive a new upper bound for the TLS solution and indicate when the TLS method can be more accurate than the LS method.Many applied problems in signal processing lead to overdetermined systems of linear equations where the matrix and right hand side are determined by the experimental observations (usually in the form of a lime series). It often happens that as the number of columns of the matrix becomes larger, the ra     

Mixed,componentwise condition numbers and small sample statistical condition estimation of Sylvester equations

We present a componentwise perturbation analysis for the continuous‐time Sylvester equations. Componentwise, mixed condition numbers and new perturbation bounds are derived for the matrix equations. The small sample statistical method can also be applied for the condition estimation. These condition numbers and perturbation bounds are tested on numerical examples and compared with the normwise condition number. The numerical examples illustrate that the mixed condition number gives sharper bounds than the normwise one. Copyright © 2011 John Wiley & Sons, Ltd.     

Backward perturbation analysis for scaled total least‐squares problems

The scaled total least‐squares (STLS) method unifies the ordinary least‐squares (OLS), the total least‐squares (TLS), and the data least‐squares (DLS) methods. In this paper we perform a backward perturbation analysis of the STLS problem. This also unifies the backward perturbation analyses of the OLS, TLS and DLS problems. We derive an expression for an extended minimal backward error of the STLS problem. This is an asymptotically tight lower bound on the true minimal backward error. If the given approximate solution is close enough to the true STLS solution (as is the goal in practice), then the extended minimal backward error is in fact the minimal backward error. Since the extended minimal backward error is expensive to compute directly, we present a lower bound on it as well as an asymptotic estimate for it, both of which can be computed or estimated more efficiently. Our numerical examples suggest that the lower bound gives good order of magnitude approximations, while the asymptotic estimate is an excellent estimate. We show how to use our results to easily obtain the corresponding results for the OLS and DLS problems in the literature. Copyright © 2009 John Wiley & Sons, Ltd.     

COMPONENTWISE CONDITION NUMBERS FOR GENERALIZED MATRIX INVERSION AND LINEAR LEAST SQUARES

We present componentwise condition numbers for the problems of Moore-Penrose generalized matrix inversion and linear least squares. Also, the condition numbers for these condition numbers are given.     

HR分解的精化扰动界（英文）

给出了HR分解的分量型和范数型的一阶扰动界.对于范数型,新的精化扰动界至少优于已有结果,特别的,新的关于R因子的扰动界远远优于已有的扰动界.     

Normwise, mixed and componentwise condition numbers of nonsymmetric algebraic Riccati equations

This paper is devoted to the perturbation analysis for nonsymmetric algebraic Riccati equations. The upper bounds for the normwise, mixed and componentwise condition numbers are presented. The results are illustrated by numerical examples.     

Small sample statistical condition estimation for the total least squares problem

In this paper, under the genericity condition, we study the condition estimation of the total least squares (TLS) problem based on small sample condition estimation (SCE), which can be incorporated into the direct solver for the TLS problem via the singular value decomposition (SVD) of the augmented matrix [A, b]. Our proposed condition estimation algorithms are efficient for the small and medium size TLS problem because they utilize the computed SVD of [A, b] during the numerical solution to the TLS problem. Numerical examples illustrate the reliability of the algorithms. Both normwise and componentwise perturbations are considered. Moreover, structured condition estimations are investigated for the structured TLS problem.