scholarly journals Condition numbers of the least squares problems with multiple right-hand sides

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1667-1676
Author(s):  
Lingsheng Meng ◽  
Bing Zheng
Keyword(s):  
Least Squares ◽  
Numerical Experiments ◽  
Upper Bounds ◽  
Condition Numbers ◽  
Least Squares Problems ◽  
Least Squares Problem ◽  
Right Hand

In this paper, we investigate the normwise, mixed and componentwise condition numbers of the least squares problem min X?Rnxd ||X - B||F, where A ? Rmxn is a rank-deficient matrix and B ? Rmxd. The closed formulas or upper bounds for these condition numbers are presented, which extend the earlier work for the least squares problem with single right-hand side (i.e. B ? b is an m-vector) of several authors. Numerical experiments are given to confirm our results.

scholarly journals Normwise condition numbers of the indefinite least squares problem with multiple right-hand sides

2021 ◽  
Vol 7 (3) ◽  
pp. 3692-3700
Author(s):  
Limin Li ◽  
Keyword(s):  
Least Squares ◽  
Numerical Experiments ◽  
Upper Bounds ◽  
Condition Numbers ◽  
Frobenius Norm ◽  
Least Squares Problem ◽  
Right Hand ◽  
Weighted Frobenius Norm

<abstract><p>In this paper, we investigate the normwise condition numbers of the indefinite least squares problem with multiple right-hand sides with respect to the weighted Frobenius norm and $ 2 $-norm. The closed formulas or upper bounds for these condition numbers are presented, which extend the earlier work for the indefinite least squares problem with single right-hand side. Numerical experiments are performed to illustrate the tightness of the upper bounds.</p></abstract>

scholarly journals On the Condition Number Theory of the Equality Constrained Indefinite Least Squares Problem

2018 ◽  
Vol 34 ◽  
pp. 619-638
Author(s):  
Shaoxin Wang ◽  
Hanyu Li ◽  
Hu Yang
Keyword(s):  
Number Theory ◽  
Least Squares ◽  
Condition Number ◽  
Condition Numbers ◽  
Least Squares Problems ◽  
Unified Framework ◽  
Least Squares Problem ◽  
Practical Applications ◽  
New Findings ◽  
Special Cases

In this paper, within a unified framework of the condition number theory, the explicit expression of the \emph{projected} condition number of the equality constrained indefinite least squares problem is presented. By setting specific norms and parameters, some widely used condition numbers, like the normwise, mixed and componentwise condition numbers follow as its special cases. Considering practical applications and computation, some new compact forms or upper bounds of the projected condition numbers are given to improve the computational efficiency. The new compact forms are of particular interest in calculating the exact value of the 2-norm projected condition numbers. When the equality constrained indefinite least squares problem degenerates into some specific least squares problems, our results give some new findings on the condition number theory of these specific least squares problems. Numerical experiments are given to illustrate our theoretical results.

An Iterative algorithm for $\eta$-(anti)-Hermitian least-squares solutions of quaternion matrix equations

2015 ◽  
Vol 30 ◽  
Author(s):  
Fatemeh Beik ◽  
Salman Ahmadi-Asl
Keyword(s):  
Least Squares ◽  
Iterative Algorithm ◽  
Numerical Experiments ◽  
Iterative Algorithms ◽  
Iterative Approach ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Hermitian Matrices ◽  
Convergence Properties ◽  
Least Squares Problems

Recently, some research has been devoted to finding the explicit forms of the η-Hermitian and η-anti-Hermitian solutions of several kinds of quaternion matrix equations and their associated least-squares problems in the literature. Although exploiting iterative algorithms is superior than utilizing the explicit forms in application, hitherto, an iterative approach has not been offered for finding η-(anti)-Hermitian solutions of quaternion matrix equations. The current paper deals with applying an efficient iterative manner for determining η-Hermitian and η-anti-Hermitian least-squares solutions corresponding to the quaternion matrix equation AXB + CY D = E. More precisely, first, this paper establishes some properties of the η-Hermitian and η-anti-Hermitian matrices. These properties allow for the demonstration of how the well-known conjugate gradient least- squares (CGLS) method can be developed for solving the mentioned problem over the η-Hermitian and η-anti-Hermitian matrices. In addition, the convergence properties of the proposed algorithm are discussed with details. In the circumstance that the coefficient matrices are ill-conditioned, it is suggested to use a preconditioner for accelerating the convergence behavior of the algorithm. Numerical experiments are reported to reveal the validity of the elaborated results and feasibility of the proposed iterative algorithm and its preconditioned version.

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