AN ITERATIVE ALGORITHM FOR THE LEAST SQUARES SOLUTIONS OF MATRIX EQUATIONS OVER SYMMETRIC ARROWHEAD MATRICES

- Journal title : Journal of the Korean Mathematical Society
- Volume 52, Issue 2, 2015, pp.349-372
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2015.52.2.349

Title & Authors

AN ITERATIVE ALGORITHM FOR THE LEAST SQUARES SOLUTIONS OF MATRIX EQUATIONS OVER SYMMETRIC ARROWHEAD MATRICES

Ali Beik, Fatemeh Panjeh; Salkuyeh, Davod Khojasteh;

Ali Beik, Fatemeh Panjeh; Salkuyeh, Davod Khojasteh;

Abstract

This paper concerns with exploiting an oblique projection technique to solve a general class of large and sparse least squares problem over symmetric arrowhead matrices. As a matter of fact, we develop the conjugate gradient least squares (CGLS) algorithm to obtain the minimum norm symmetric arrowhead least squares solution of the general coupled matrix equations. Furthermore, an approach is offered for computing the optimal approximate symmetric arrowhead solution of the mentioned least squares problem corresponding to a given arbitrary matrix group. In addition, the minimization property of the proposed algorithm is established by utilizing the feature of approximate solutions derived by the projection method. Finally, some numerical experiments are examined which reveal the applicability and feasibility of the handled algorithm.

Keywords

matrix equation;projection technique;iterative algorithm;least squares problem;arrowhead matrix;

Language

English

Cited by

1.

2.

References

1.

F. P. A. Beik and D. K. Salkuyeh, On the global Krylov subspace methods for solving general coupled matrix equation, Comput. Math. Appl. 62 (2011), no. 12, 4605-4613.

2.

F. P. A. Beik and D. K. Salkuyeh, The coupled Sylvester-transpose matrix equations over generalized centro- symmetric matrices, Int. J. Comput. Math. 90 (2013), no. 7, 1546-1566.

3.

D. S. Bernstein, Matrix Mathematics: theory, facts, and formulas, Second edition, Princeton University Press, 2009.

4.

A. Bjorck, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, 1996.

5.

C. F. Borges, R. Frezza, and W. B. Gragg, Some inverse eigenproblems for Jacobi and arrow matrices, Numer. Linear Algebra Appl. 2 (1995), no. 3, 195-203.

6.

A. Bouhamidi and K. Jbilou, A note on the numerical approximate solutions for gener- alized Sylvester matrix equations with applications, Appl. Math. Comput. 206 (2008), no. 2, 687-694.

7.

M. Dehghan and M. Hajarian, An iterative algorithm for solving a pair of matrix equa- tion AY B = E, CY D = F over generalized centro-symmetric matrices, Comput. Math. Appl. 56 (2008), no. 12, 3246-3260.

8.

M. Dehghan and M. Hajarian, The general coupled matrix equations over generalized bisymmetric matrices, Linear Algebra Appl. 432 (2010), no. 6, 1531-1552.

9.

M. Dehghan and M. Hajarian, Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations, Appl. Math. Model. 35 (2011), no. 7, 3285-3300.

10.

F. Ding and T. Chen, Gradient based iterative algorithms for solving a class of matrix equations, IEEE Trans. Automat. Control 50 (2005), no. 8, 1216-1221.

11.

F. Ding and T. Chen, On iterative solutions of general coupled matrix equations, SIAM J. Control Optim. 44 (2006), no. 6, 2269-2284.

12.

F. Ding, P. X. Liu, and J. Ding, Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle, Appl. Math. Comput. 197 (2008), no. 1, 41-50.

13.

A. El Guennouni, K. Jbilou, and A. J. Riquet, Block Krylov subspace methods for solving large Sylvester equations, Numer. Algorithms 29 (2002), no. 1-3, 75-96.

14.

D. Fisher, G. Golub, O. Hald, C. Leiva, and O. Widlund, On Fourier-Toeplitz methods for separable elliptic problems, Math. Comp. 28 (1974), 349-368.

15.

M. Hajarian, Developing the CGLS algorithm for the least squares solutions of the general coupled matrix equations, Math. Methods Appl. Sci. 37 (2014), no. 17, 2782- 2798.

16.

M. Hajarian and M. Dehghan, The generalized centro-symmetric and least squares gen- eralized centro-symmetric solutions of the matrix equation AY B + C$Y^T$ D = E, Math. Methods Appl. Sci. 34 (2011), no. 13, 1562-1579.

17.

D. Y. Hu and L. Reichel, Krylov-subspace methods for the Sylvester equation, Linear Algebra Appl. 172 (1992), 283-313.

18.

G. X. Huang, F. Ying, and K. Gua, An iterative method for skew-symmetric solution and the optimal approximate solution of the matrix equation AXB = C, J. Comput. Appl. Math. 212 (2008), no. 2, 231-244.

19.

K. Jbilou and A. J. Riquet, Projection methods for large Lyapunov matrix equations, Linear Algebra Appl. 415 (2006), no. 2, 344-358.

20.

T. Jiang and M. Wei, On solutions of the matrix equations X − AXB = C and X − A X B = C, Linear Algebra Appl. 367 (2003), 225-233.

21.

H. Li, Z. Gao, and D. Zhao, Least squares solutions of the matrix equation AXB + CY D = E with the least norm for symmetric arrowhead matrices, Appl. Math. Comput. 226 (2014), 719-724.

22.

J. F. Li, X. Y. Hu, X.-F. Duan, and L. Zhang, Iterative method for mirror-symmetric solution of matrix equation AXB +CY D = E, Bull. Iranian Math. Soc. 36 (2010), no. 2, 35-55.

23.

D. P. O'leary and G. Stewart, Computing the eigenvalues and eigenvectors of symmetric arrowhead matrices, J. Comput. Phys. 90 (1990), no. 2, 497-405.

24.

B. Parlett, The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cilffs, 1980.

25.

B. Parlett and B. Nour-Omid, The use of a refined error bound when updating eigen- values of tridiagonals, Linear Algebra Appl. 68 (1985), 179-219.