ESTONIAN ACADEMY
PUBLISHERS
eesti teaduste
akadeemia kirjastus
PUBLISHED
SINCE 1952
 
Proceeding cover
proceedings
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
Impact Factor (2020): 1.045

A dynamically parameterized inversion-free iteration for a system of nonlinear matrix equation; pp. 311–322

Full article in PDF format | 10.3176/proc.2020.4.04

Authors
Ning Dong, Bo Yu, Zhaoyun Meng

Abstract

Computation of the stabilizing solution pair of a system of nonlinear matrix equations is of great interest in calculating the Green’s function of nanoparticles. By noting that each solution of the pair might have various sizes, an inversion-free iteration with dynamical parameters is proposed in this paper. Under proper assumptions the convergence of the algorithm is established, as well as the bound of the iteration sequence. Preliminary numerical experiments indicate that the dynamically parameterized inversion-free iteration is very efficient to compute the stabilizing solution pair.


References

1. Datta, S. Nanoscale device modeling: the Green’s function method. Superlattices Microstructures, 2000, 28(4), 253–278.
https://doi.org/10.1006/spmi.2000.0920

2. John, D. L. and Pulfrey, D. L. Green’s function calculations for semi-infinite carbon nanotubes. Phys. Status Solidi B, 2006, 243(2), 442–448.
https://doi.org/10.1002/pssb.200541227

3. Tomfohr, J. and Sankey, O. F. Theoretical analysis of electron transport through organic molecules. J. Chem. Phys., 2004, 120(3), 1542–1554.
https://doi.org/10.1063/1.1625911

4. Guo, C.-H., Kuo, Y.-C., and Lin, W.-W. Complex symmetric stabilizing solution of the matrix equation ATX−1QLinear Algebra Appl., 2011, 435(6), 1187–1192.

5. Guo, C.-H., Kuo, Y.-C., and Lin, W.-W. Numerical solution of nonlinear matrix equations arising from Green’s function calculations in nano research. J. Comput. Appl. Math., 2012, 236(17), 4166–4180.
https://doi.org/10.1016/j.cam.2012.05.012

6. Dai, Z. F., Dong, X. D., Kang, J., and Hong, L. Y. Forecasting stock market returns: new technical indicators and two-step economic constraint method. North Am. J. Econ. Finance, 2020, 53, 101216.
https://doi.org/10.1016/j.najef.2020.101216

7. Engwerda, J. C., Ran, A. C. M., and Rijkeboer, A. L. Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation +A*X−1QLinear Algebra Appl., 1993, 186, 255–275.
https://doi.org/10.1016/0024-3795(93)90295-Y

8. Meini, B. Efficient computation of the extreme solutions of +A*X−1and A*X−1QMath. Comput., 2002, 71(239), 1189–1204.
https://doi.org/10.1090/S0025-5718-01-01368-0

9. Weng, P. C.-Y., Chu, E. K.-W., Kuo, Y.-C., and Lin, W.-W. Solving large-scale nonlinear matrix equations by doubling. Linear Algebra. Appl., 2013, 439, 914–932.
https://doi.org/10.1016/j.laa.2012.08.008

10. Huang, N., Ma, C.-F., and Tang, J. The inversion-free iterative methods for a system of nonlinear matrix equations. Int. J. Comput. Math., 2016, 93(9), 1470–1483.
https://doi.org/10.1080/00207160.2015.1059934

11. Peng, Z.-Y., El-Sayed, S. M., and Zhang, X.-L. Iterative methods for the extremal positive definite solution of the matrix equation +ATXαQJ. Comput. Appl. Math., 2007, 200(2), 520–527.
https://doi.org/10.1002/nla.510

12. Al-Dubiban, A. M. Iterative algorithm for solving a system of nonlinear matrix equations. J. Appl. Math., 2012, ID 461407.
https://doi.org/10.1155/2012/461407

13. Al-Dubiban, A. M. On the iterative method for the system of nonlinear matrix equations. Abstr. Appl. Anal., 2013, ID 685753.
https://doi.org/10.1155/2013/685753

14. Zhan, X. Computing the extremal positive definite solutions of a matrix equation. SIAM J. Sci. Comput., 1996, 17(5), 1167– 1174.
https://doi.org/10.1137/S1064827594277041

15. Guo, C.-H. and Lancaster, P. Iterative solution of two matrix equations. Math. Comput., 1999, 68(228), 1589–1603.
https://doi.org/10.1090/S0025-5718-99-01122-9

16. El-Sayed, S. M. and Al-Dbiban, A. M. A new inversion free iteration for solving the equation +A*X−1QJ. Comput. Appl. Math., 2005, 181(1), 148–156.
https://doi.org/10.1016/j.cam.2004.11.025

17. Zhang, L. An improved inversion-free method for solving the matrix equation +A*XαQJ. Comput. Appl. Math., 2013, 253, 200–203.
https://doi.org/10.1016/j.cam.2013.04.007

18. Dong, N. and Yu, B. On the tripling algorithm for large-scale nonlinear matrix equations with low rank structure. J. Comput. Appl. Math., 2015, 288, 18–32.
https://doi.org/10.1016/j.cam.2015.03.036

19. Yu, B., Li, D.-H., and Dong, N. Low memory and low complexity iterative schemes for a nonsymmetric algebraic Riccati equation arising from transport theory. J. Comput. Appl. Math., 2013, 250, 175–189.
https://doi.org/10.1016/j.cam.2013.03.017

20. Golub, G. H. and Van Loan, C. F. Matrix computations, 3rd Edition. The Johns Hopkins University Press, Baltimore, MD, 1996.

21. Lin, W.-W. and Xu, S.-F. Convergence analysis of structure-preserving doubling algorithms for Riccati-type matrix equations. SIAM J. Matrix Anal. Appl., 2006, 28(1), 26–39.
https://doi.org/10.1137/040617650

22. Monsalve, M. and Raydan, M. A new inversion-free method for a rational matrix equation. Linear Algebra Appl., 2010, 433(1), 64–71.
https://doi.org/10.1016/j.laa.2010.02.006

23. Parodi, M. La localisation des valeurs caracterisiques des matrices etses applications. Gauthier Villars, Paris, 1959.

24. Bhatia, R. Matrix analysis, Graduate Texts in Mathematics. Springer, Berlin, 1997.
https://doi.org/10.1007/978-1-4612-0653-8


Back to Issue