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1.
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. 相似文献
2.
Huaian Diao Weiguo Wang Yimin Wei Sanzheng Qiao 《Numerical Linear Algebra with Applications》2013,20(1):44-59
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. 相似文献
3.
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. 相似文献
4.
Delin Chu Lijing Lin Roger C. E. Tan Yimin Wei 《Numerical Linear Algebra with Applications》2011,18(1):87-103
One of the most successful methods for solving the least‐squares problem minx∥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. 相似文献
5.
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. 相似文献
6.
E. A. Nikolaevskaya A. N. Khimich 《Computational Mathematics and Mathematical Physics》2009,49(3):409-417
The weighted least squares problem is considered. Given a generally inconsistent system of linear algebraic equations, error estimates are obtained for its weighted minimum-norm least squares solution under perturbations of the matrix and the right-hand side, including the case of rank modifications of the perturbed matrix. 相似文献
7.
关于TLS的可解性及扰动分析 总被引：2，自引：0，他引：2
尽管有关总体最小二乘问题的研究工作是大量的，然而ＴＬＳ可解的充分必要条件一直没有得到。本文首先给出完整的可解性分析，然后建立了ＴＬＳ的扰动上界。 相似文献
8.
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. 相似文献
9.
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. 相似文献
10.
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… 相似文献
11.
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. 相似文献
12.
Marc Baboulin Jack Dongarra Serge Gratton Julien Langou 《Numerical Linear Algebra with Applications》2009,16(7):517-533
In this paper, we address the accuracy of the results for the overdetermined full rank linear least‐squares problem. We recall theoretical results obtained in (SIAM J. Matrix Anal. Appl. 2007; 29 (2):413–433) on conditioning of the least‐squares solution and the components of the solution when the matrix perturbations are measured in Frobenius or spectral norms. Then we define computable estimates for these condition numbers and we interpret them in terms of statistical quantities when the regression matrix and the right‐hand side are perturbed. In particular, we show that in the classical linear statistical model, the ratio of the variance of one component of the solution by the variance of the right‐hand side is exactly the condition number of this solution component when only perturbations on the right‐hand side are considered. We explain how to compute the variance–covariance matrix and the least‐squares conditioning using the libraries LAPACK (LAPACK Users' Guide (3rd edn). SIAM: Philadelphia, 1999) and ScaLAPACK (ScaLAPACK Users' Guide. SIAM: Philadelphia, 1997) and we give the corresponding computational cost. Finally we present a small historical numerical example that was used by Laplace (Théorie Analytique des Probabilités. Mme Ve Courcier, 1820; 497–530) for computing the mass of Jupiter and a physical application if the area of space geodesy. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
13.
14.
关于特征值的Hoffman-Wielandt型相对扰动界 总被引：4，自引：0，他引：4
本文主要研究了关于特征值的Hoffman-wielandt型相对扰动界，改进了LiRC和Ipsen I等人关于这方面的相应结果． 相似文献
15.
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. 相似文献
16.
This paper deals with the normwise perturbation theory for linear (Hermitian) matrix equations. The definition of condition number for the linear (Hermitian) matrix equations is presented. The lower and upper bounds for the condition number are derived. The estimation for the optimal backward perturbation bound for the Hermitian matrix equations is obtained. Copyright © 2010 John Wiley & Sons, Ltd. 相似文献
17.
考虑非线性矩阵方程X-A~*X~(-1)A=Q,其中A是n阶复矩阵,Q是n阶Hermite正定解,A~*是矩阵A的共轭转置.本文证明了此方程存在唯一的正定解,并推导出此正定解的扰动边界和条件数的显式表达式.以上结果用数值例子加以说明. 相似文献
18.
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. 相似文献
19.
For a complex matrix $Ain mathbb{C}^{mtimes n}$, the relationship between the weighted Moore-Penrose inverse $A^dag_{M_1N_1}$ and $A^dag_{M_2N_2}$ is studied, and an important formula is derived,where $M_1in mathbb{C}^{mtimes m}, N_1inmathbb{C}^{ntimes n}$ and $M_2in mathbb{C}^{mtimes m}, N_2inmathbb{C}^{ntimes n}$ are different pair of positive definite hermitian matrices. Based on this formula, this paper initiates the study of the perturbationestimations for $A^dag_{MN}$ in the case that $A$ is fixed, whereas both $M$ and $N$ are variable. The obtained norm upper bounds are then applied to the perturbation estimations for the solutions to the weighted linear least squares problems. 相似文献
20.
利用Pena距离对加权线性最小二乘估计的影响问题进行讨论,得到加权最小二乘估计的Pena距离的表达式,对其性质进行讨论,从而得到高杠异常点的判别方法.文中对Pena距离与Cook距离的性能进行了对比,得到在一定条件下Pena距离优于Cook距离的结论.并通过数值实验对此方法的有效性进行验证. 相似文献