Research ArticleOpen AccessGoogle Scholar indexed
Dykstra’s Algorithm for the Optimal Approximate Symmetric Positive Semidefinite Solution of a Class of Matrix Equations
College of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin, China
College of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin, China
College of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin, China
- 1 College of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin, China
- 2 College of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin, China
- 3 College of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin, China
Advances in Linear Algebra & Matrix Theory·Volume 06 (2016)·Pages 1–10·Published 7 March 2016·DOI10.4236/alamt.2016.61001
Copy link · social · email
Abstract
Dykstra’s alternating projection algorithm was proposed to treat the problem of finding the projection of a given point onto the intersection of some closed convex sets. In this paper, we first apply Dykstra’s alternating projection algorithm to compute the optimal approximate symmetric positive semidefinite solution of the matrix equations AXB = E , CXD = F . If we choose the initial iterative matrix X 0 = 0, the least Frobenius norm symmetric positive semidefinite solution of these matrix equations is obtained. A numerical example shows that the new algorithm is feasible and effective.
KeywordsMatrix EquationDykstra’s Alternating Projection AlgorithmOptimal Approximate SolutionLeast Norm Solution
- Bauschke, H.H. and Borwein, J.M. (1994) Dykstra’s Alternating Projection Algorithm for Two Sets. Journal of Approximation Theory, 79, 418-443. http://dx.doi.org/10.1006/jath.1994.1136
- Dai, H. and Lancaster, P. (1996) Linear Matrix Equations from an Inverse Problem of Vibration Theory. Linear Algebra and Its Applications, 246, 31-47. http://dx.doi.org/10.1016/0024-3795(94)00311-4
- Meng, T. (2001) Experimental Design and Decision Support. In: Leondes, C., Ed., Expert System the Technology of Knowledge Management and Decision Making for 21st Century, Volume 1, Academic Press, San Diego.
- Navarra, A., Odell, P.L. and Young, D.M. (2001) A Representation of the General Common Solution to the Matrix Equations A 1 XB 1 = C 1 , A 2 XB 2 = C 2 with Applications. Computers & Mathematics with Applications, 41, 929-935. http://dx.doi.org/10.1016/S0898-1221(00)00330-8
- Wang, Q.W. (2004) A System of Matrix Equations over Arbitrary Regular Rings with Identity. Linear Algebra and Its Applications, 384, 44-53. http://dx.doi.org/10.1016/j.laa.2003.12.039
- Liao, A.P., Lei, Y. and Yuan, S.F. (2006) The Matrix Nearness Problem for Symmetric Matrices Associated with the Matrix Equations [ A T XA , B T XB ]=[ C , D ]. Linear Algebra and Its Applications, 418, 939-954. http://dx.doi.org/10.1016/j.laa.2006.03.032
- Liao, A.P. and Lei, Y. (2005) Least-Squares Solution with the Minimum-Norm for the Matrix Equations [AXB,GXH]=[C,D]. Computers & Mathematics with Applications, 50, 539-549. http://dx.doi.org/10.1016/j.camwa.2005.02.011
- Sheng, X.P. and Chen, G.L. (2007) A Finite Iterative Method for Solving a Pair of Linear Matrix Equation (AXB,CXD)=(C,D). Applied Mathematics and Computation, 189, 1350-1358. http://dx.doi.org/10.1016/j.amc.2006.12.026
- Ding, J., Liu, Y.J. and Ding, F. (2010) Iterative Solutions to Matrix Equation of the Form A i XB i = f i . Computers & Mathematics with Applications, 59, 3500-3507. http://dx.doi.org/10.1016/j.camwa.2010.03.041
- Peng, Y.X., Hu, X.Y. and Zhang, L. (2006) An Iterative Method for Symmetric Solutions and Optimal Approximation Solution of the System of Matrix Equations A 1 XB 1 = C 1 , A 2 XB 2 = C 2 . Applied Mathematics and Computation, 183, 1127-1137. http://dx.doi.org/10.1016/j.amc.2006.05.124
- Chen, Y.B., Peng Z.Y. and Zhou, T.J. (2010) LSQR Iterative Method Common Symmetric Solutions to Matrix Equations AXB = E and CXD = F . Applied Mathematics and Computation, 217, 230-236. http://dx.doi.org/10.1016/j.amc.2010.05.053