**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30184

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

**Authors:**
Minghui Wang

**Abstract:**

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.

**Keywords:**
Matrix equation,
bisymmetric matrix,
least squares problem,
like-minimum norm,
iterative algorithm.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1070047

**References:**

[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.