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Solutions of symmetry‐constrained least‐squares problems
Author(s) -
Peng ZhenYun
Publication year - 2008
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.578
Subject(s) - mathematics , matrix (chemical analysis) , iterative method , combinatorics , least squares function approximation , linear algebra , rate of convergence , linear least squares , algebra over a field , algorithm , pure mathematics , geometry , channel (broadcasting) , statistics , materials science , estimator , singular value decomposition , electrical engineering , composite material , engineering
In this paper, two new matrix‐form iterative methods are presented to solve the least‐squares problem:\documentclass{article}\usepackage[mathscr]{euscript}\usepackage{amssymb,amsbsy,rotating}\footskip=0pc\pagestyle{empty}\begin{document}\[ \min\limits_{X\in {\mathscr{S}}_1,Y\in {\mathscr{S}}_2}\|AXB+CYD-E\| \]\end{document}and matrix nearness problem:\documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}\[ \min\limits_{[X: Y]\in S_{XY}}\|[X: Y]-[\widetilde{X}:\widetilde{Y}]\| \] \end{document}where matrices $A\in R^{p\times n_1},B\in R^{n_2\times q},C\in R^{p\times m_1},D\in R^{m_2\times q},E\in R^{p\times q},\widetilde{X}\in R^{n_1\times n_2}$ and $\widetilde{Y}\in R^{m_1\times m_2}$ are given; 1 and 2 are the set of constraint matrices, such as symmetric, skew symmetric, bisymmetric and centrosymmetric matrices sets and S XY is the solution pair set of the minimum residual problem. These new matrix‐form iterative methods have also faster convergence rate and higher accuracy than the matrix‐form iterative methods proposed by Peng and Peng ( Numer. Linear Algebra Appl . 2006; 13 : 473–485) for solving the linear matrix equation AXB + CYD = E . Paige's algorithms, which are based on the bidiagonalization procedure of Golub and Kahan, are used as the framework for deriving these new matrix‐form iterative methods. Some numerical examples illustrate the efficiency of the new matrix‐form iterative methods. Copyright © 2008 John Wiley & Sons, Ltd.