Solutions of symmetry-constrained least-squares problems

In this paper, two new matrix-form iterative methods are presented to solve the least-squares problem: min(X epsilon y1,X epsilon y1) parallel to AX B + CY D - E parallel to and matrix nearness problem: min([X : Y]epsilon SXY) parallel to[X : Y] - [(X) over tilde : (Y) over tilde]parallel to where matrices A epsilon R-pxn1, B epsilon R-n2xq, C epsilon R-pxm1, D epsilon R-m2xq, E epsilon R-pxq, (X) over tilde epsilon R-n1xn2 and (Y) over tilde epsilon R-m1xm2 are given; y(1) and y(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 AX B + CY D = 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 (C) 2008 John Wiley & Sons, Ltd.