ISRN Computational Mathematics

Volume 2012 (2012), Article ID 126908, 6 pages

http://dx.doi.org/10.5402/2012/126908

Research Article

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

^{1}College of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin 541004, China^{2}Department 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.

#### Linked References

- 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. View at Publisher · View at Google Scholar · View at Scopus - F. Deutsch,
*Best Approximation in Inner Product Spaces*, Springer, New York, NY, USA, 2001. - N. Higham,
*Matrix Nearness Problems and Applications*, Oxford University Press, London, UK, 1989. - 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. View at Publisher · View at Google Scholar · View at Scopus - M. Friswell and J. Mottorshead,
*Finite Element Model Updating in Structure Dynamics*, Kluwer Academic publisher, Dodrecht, The Netherlands, 1995. - 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. View at Publisher · View at Google Scholar · View at Scopus - 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. View at Publisher · View at Google Scholar · View at Scopus - 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. View at Publisher · View at Google Scholar · View at Scopus - 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. View at Publisher · View at Google Scholar · View at Scopus - 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. View at Publisher · View at Google Scholar · View at Scopus - 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. View at Publisher · View at Google Scholar · View at Scopus - 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. View at Publisher · View at Google Scholar · View at Scopus - 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. View at Publisher · View at Google Scholar · View at Scopus - 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. View at Publisher · View at Google Scholar - 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. View at Publisher · View at Google Scholar · View at Scopus - 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. View at Publisher · View at Google Scholar · View at Scopus - 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. View at Publisher · View at Google Scholar · View at Scopus - J. von Numann,
*Functional Operators*, vol. 11, Princeton University Press, Princeton, NJ, USA, 1950. - I. Halperin, “The product of projection operators,”
*Acta Scientiarum Mathematicarum*, vol. 23, pp. 96–99, 1962. View at Google Scholar - R.L. Dykstra, “An algorithm for restricted least-squares regression,”
*Journal of the American Statistical Association*, vol. 78, pp. 837–842, 1983. View at Google Scholar - 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. View at Google Scholar · View at Scopus - H. Dai,
*Matrix Theory*, Science Press, Beijing, China, 2001. - 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. View at Google Scholar