An Iterative Method for the Least-squares Symmetric Solution of AXB+CYD=F and its Application

Based on the classical algorithm LSQR for solving (unconstrained) LS problem, an iterative method is proposed for the least-squares like-minimum-norm symmetric solution of AXB+CYD=E. As the application of this algorithm, an iterative method for the least-squares like-minimum-norm biymmetric solution of AXB=E is also obtained. Numerical results are reported that show the efficiency of the proposed methods.

• Minghui Wang

• Matrix equation
• bisymmetric matrix
• least squares problem
• like-minimum norm
• iterative algorithm.

[1] K. Chu, Singular value and generalized singular value decomposition
and the solution of linear matrix equations, Linear Algebra Appl.,
88/89(1987), 83-98.
[2] L. Huang and Q. Zeng, The matrix equation AXB+CY D = E over a
simple Artinian ring, Linear and Multilinear Algebra, 38(1995), 225-232.
[3] A. O¨ zgu¨ler, The equation AXB + CY D = E over a principal ideal
domain, SIAM J. Matrix Anal. Appl., 12(1991), 581-591.
[4] G. Xu, M. Wei and D. Zheng, On solutions of matrix equation AXB +
CY D = F, Linear Algebra Appl., 279(1998), 93-109.
[5] S. Shim and Y. Chen, Least squares solutions of matrix equation
AXB*+CY D* = E, Siam J. Matrix Anal. Appl., 24(3)(2003)802-808.
[6] C. Paige and A. Saunders, LSQR: An algorithm for sparse linear equations
and sparse least squares, ACM Trans. Math. Software 8(1)(1982), 43-71.
[7] G. Golub and W. Kahan, Calculating the singular values and pseudoinverse
of a matrix, SIAM J. Numer. Anal., 2(1965), 205-224.
[8] Z. Peng, X. Hu and L. Zhang, One kind of inverse problem for the
bisymmetric matrices, Mathematica Numerica Sinica, 27(1)(2005), 11-
18.
[9] A. Liao and Z. Bai, Least-squares solutions of the ma trix equation
ATXA = D in bisymmetric matrix set, Mathematica Numerica Sinica,
24(1)(2002), 9-20
[10] Y. Qiu, Z. Zhang and J. Lu, Matrix iterative solutions to the least squares
problem of BXAT = F with some linear constraints, Appl. Math.
Comput., 185(2007)284-300.
[11] M. Wang, M. Wei and Y. Feng, An iterative algorithm for a least
squares solution of a matrix equation, International Journal of Computer
Mathematics, 87(2010)1289-1298.