Condition Numbers for a Linear Function of the Solution to the Constrained and Weighted Least Squares Problem and Their Statistical Estimation

Abstract

In this paper, we consider the condition number theory for a linear function of the solution to the constrained and weighted least squares problem. We first present two explicit expressions without Kronecker product of normwise condition number using the classical method for condition numbers. Then, we derive the explicit expression of mixed and componentwise condition numbers by the dual techniques. To estimate these condition numbers with high reliability, we choose the probabilistic spectral norm estimator and the small-sample statistical condition estimation method and devise three algorithms. Numerical experiments are provided to illustrate the obtained results.

Conditioning Theory for ML-Weighted Pseudoinverse and ML-Weighted Least Squares Problem

“…In this section, first we present reliable condition estimation algorithms for normwise, mixed, and componentwise condition numbers using small sample statistical condition estimation (SCE) method then we show the accuracy of the propose condition estimation algorithms by numerical experiments. Kenny and Laub [23] provided small sample statistical condition estimation (SCE) as a reliable method to estimate condition numbers for linear least squares problems [13,28,29], indefinite least squares problems [20,30] and total least squares problems [31][32][33]. We proposed Algorithms A, B and C based on the SSCE method [23] to estimate the normwise, mixed, and componentwise condition numbers of 𝒦 † ℳℒ and for the ℳℒ -WLS solution.…”

Conditioning Theory for Generalized Inverse CA‡ and Their Estimations

“…This section proposes three algorithms for estimating the normwise, mixed and componentwise condition numbers for the generalized inverse C ‡ A . Algorithm 1 is based on a probabilistic condition estimator method [27] and utilized to examine the normwise condition number for K-weighted pseudoinverse L † K [23], ILS problem [33], constrained and weighted least squares problem [34] and Tikhonov regularization of total least squares problem [35]. Based on the SSCE method [28], we develop Algorithm 2 to estimate the normwise condition number; for details, see [23,33,[36][37][38].…”

“…To estimate the mixed and componentwise condition numbers, we need the following SSCE method, which is from [28] and has been applied to many problems (see, e.g., [23,[32][33][34][35]).…”