An Iterative Method for the Symmetric Arrowhead Solution of Matrix Equation

Authors: Minghui Wang, Luping Xu, Juntao Zhang

Abstract:

In this paper, according to the classical algorithm LSQR for solving the least-squares problem, an iterative method is proposed for least-squares solution of constrained matrix equation. By using the Kronecker product, the matrix-form LSQR is presented to obtain the like-minimum norm and minimum norm solutions in a constrained matrix set for the symmetric arrowhead matrices. Finally, numerical examples are also given to investigate the performance.

Keywords: Symmetric arrowhead matrix, iterative method, like-minimum norm, minimum norm, Algorithm LSQR.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1338568

References:

[1] Y. F. Xu, An inverse eigenvalue problem for a special kind of matrices. Math. Appl., vol. 1, 1996, pp. 68-75.
[2] G. P. Xu, M. S. Wei, D. S. Zhang, On solutions of matrix equations AXB+CYD=F. Linear Algebra Appl., vol. 279, 1998, pp. 93-109.
[3] C. C. Paige, A. Saunders, LSQR: An algorithm for sparse linear equations and sparse least squares. Appl. Math. Comput., vol. 8, 1982, pp. 43-71.
[4] F. K. Toutounian, S. Karimi, Global least squares method (Gl-LSQR) for solving general linear systems with several right-hand sides. Appl. Math. Comput., vol. 178, 2006, pp. 452-460.
[5] A. P. Liao, Z. Z. Bai, Y. Lei, Best approximate solution of matrix equation AXB+CYD=E. SIAM J. Matrix Anal. Appl., vol. 27, 2006, pp. 675-688.
[6] Z. Y. Peng, A matrix LSQR iterative method to solve matrix equation AXB=C. International Journal of Computer Mathematics, vol. 87, 2010, pp. 1820-1830.
[7] Hongyi Li, Zongsheng Gao, Di Zhao, Least squares solutions of the matrix equation with the least norm for symmetric arrowhead matrices. Appl. Math. Comput., vol. 226, 2014, pp. 719-724.
[8] M. H. Wang, An iterative method for the least-squares symmetric solution of AXB+CYD=E and its application. International Journal of Math. Comput. Sciences, vol. 6, 2010, pp. 196-199.