`ISRN Computational MathematicsVolume 2012, Article ID 126908, 6 pageshttp://dx.doi.org/10.5402/2012/126908`
Research Article

## A New Iterative Algorithm for Solving a Class of Matrix Nearness Problem

1College of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin 541004, China
2Department of Mathematics, Shanghai University, Shanghai 200444, China

Received 14 September 2011; Accepted 3 October 2011

Academic Editors: T. Allahviranloo and K. Eom

Copyright © 2012 Xuefeng Duan and Chunmei Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. H. H. Bauschke and J. M. Borwein, “Dykstra′s alternating projection algorithm for two sets,” Journal of Approximation Theory, vol. 79, no. 3, pp. 418–443, 1994.
2. F. Deutsch, Best Approximation in Inner Product Spaces, Springer, New York, NY, USA, 2001.
3. N. Higham, Matrix Nearness Problems and Applications, Oxford University Press, London, UK, 1989.
4. H. Dai and P. Lancaster, “Linear matrix equations from an inverse problem of vibration theory,” Linear Algebra and Its Applications, vol. 246, pp. 31–47, 1996.
5. M. Friswell and J. Mottorshead, Finite Element Model Updating in Structure Dynamics, Kluwer Academic publisher, Dodrecht, The Netherlands, 1995.
6. A. P. Liao, Y. Lei, and S. F. Yuan, “The matrix nearness problem for symmetric matrices associated with the matrix equation [${A}^{T}XA$, ${B}^{T}XB$] = [$C,D$],” Linear Algebra and Its Applications, vol. 418, no. 2-3, pp. 939–954, 2006.
7. Y. Chen, Z. Peng, and T. Zhou, “LSQR iterative common symmetric solutions to matrix equations $AXB=E$ and $CXD=F$,” Applied Mathematics and Computation, vol. 217, no. 1, pp. 230–236, 2010.
8. J. Cai and G. Chen, “An iterative algorithm for the least squares bisymmetric solutions of the matrix equations ${A}_{1}X{B}_{1}={C}_{1}$, ${A}_{2}X{B}_{2}={C}_{2}$,” Mathematical and Computer Modelling, vol. 50, no. 7-8, pp. 1237–1244, 2009.
9. M. Dehghan and M. Hajarian, “An iterative algorithm for solving a pair of matrix equations $AXB=E$ and $CXD=F$ over generalized centro-symmetric matrices,” Computers and Mathematics with Applications, vol. 56, no. 12, pp. 3246–3260, 2008.
10. L. Fanliang, H. Xiyan, and Z. Lei, “The generalized reflexive solution for a class of matrix equations ($AX=B$, $XC=D$),” Acta Mathematica Scientia, vol. 28, no. 1, pp. 185–193, 2008.
11. Y. X. Peng, X. Y. Hu, and L. Zhang, “An iterative method for symmetric solutions and optimal approximation solution of the system of matrix equations ${A}_{1}X{B}_{1}={C}_{1}$, ${A}_{2}X{B}_{2}={C}_{2}$,” Applied Mathematics and Computation, vol. 183, no. 2, pp. 1127–1137, 2006.
12. Z. H. Peng, X. Y. Hu, and L. Zhang, “An efficient algorithm for the least-squares reflexive solution of the matrix equation ${A}_{1}X{B}_{1}={C}_{1}$, ${A}_{2}X{B}_{2}={C}_{2}$,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 988–999, 2006.
13. X. Sheng and G. Chen, “A finite iterative method for solving a pair of linear matrix equations ($AXB$, $CXD$) = ($CXD$),” Applied Mathematics and Computation, vol. 189, no. 2, pp. 1350–1358, 2007.
14. S. Yuan, A. Liao, and G. Yao, “The matrix nearness problem associated with the quaternion matrix equation ${A}^{T}XA+{B}^{T}XB=C$,” Journal of Applied Mathematics and Computing, vol. 37, no. 1-2, pp. 133–144, 2011.
15. Q. W. Wang, “A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity,” Linear Algebra and Its Applications, vol. 384, no. 1-3, pp. 43–54, 2004.
16. A. Navarra, P. L. Odell, and D. M. Young, “Representation of the general common solution to the matrix equations ${A}_{1}X{B}_{1}={C}_{1}$, ${A}_{2}X{B}_{2}={C}_{2}$ with applications,” Computers and Mathematics with Applications, vol. 41, no. 7-8, pp. 929–935, 2001.
17. J. Ding, Y. Liu, and F. Ding, “Iterative solutions to matrix equations of the form ${A}_{i}X{B}_{i}={F}_{i}$,” Computers and Mathematics with Applications, vol. 59, no. 11, pp. 3500–3507, 2010.
18. J. von Numann, Functional Operators, vol. 11, Princeton University Press, Princeton, NJ, USA, 1950.
19. I. Halperin, “The product of projection operators,” Acta Scientiarum Mathematicarum, vol. 23, pp. 96–99, 1962.
20. R.L. Dykstra, “An algorithm for restricted least-squares regression,” Journal of the American Statistical Association, vol. 78, pp. 837–842, 1983.
21. R. Escalante and M. Raydan, “Dykstra's algorithm for a constrained least-squares matrix problem,” Numerical Linear Algebra with Applications, vol. 3, no. 6, pp. 459–471, 1996.
22. H. Dai, Matrix Theory, Science Press, Beijing, China, 2001.
23. T. Meng, “Experimental design and decision support,” in Expert system the Technology of Knowledge Management and Decision Making for 21st Centry, C. Leondes, Ed., vol. 1, Academic Press, New York, NY, USA, 2001.