FINITE ITERATIVE ALGORITHM FOR THE COMPLEX GENERALIZED SYLVESTER TENSOR EQUATIONS

Yifen Ke

doi:10.11948/20190178

2020 Volume 10 Issue 3

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2020 > 10(3): 972-985

Next Article Previous Article

Yifen Ke. FINITE ITERATIVE ALGORITHM FOR THE COMPLEX GENERALIZED SYLVESTER TENSOR EQUATIONS[J]. Journal of Applied Analysis & Computation, 2020, 10(3): 972-985. doi: 10.11948/20190178

Citation:

Yifen Ke. FINITE ITERATIVE ALGORITHM FOR THE COMPLEX GENERALIZED SYLVESTER TENSOR EQUATIONS[J]. Journal of Applied Analysis & Computation, 2020, 10(3): 972-985. doi: 10.11948/20190178

FINITE ITERATIVE ALGORITHM FOR THE COMPLEX GENERALIZED SYLVESTER TENSOR EQUATIONS

Yifen Ke

College of Mathematics and Informatics & FJKLMAA, Fujian Normal University, Fuzhou 350117, China

Author Bio: Email address: keyifen@fjnu.edu.cn
Fund Project: This work was supported by National Key Research and Development Program of China (Nos. 2018YFC1504200 and 2018YFC0603500) and National Natural Science Foundation of China (Nos. 11901098 and U1839207)

Abstract

Abstract

A finite iterative algorithm is proposed to solve a class of complex generalized Sylvester tensor equations. The properties of this proposed algorithm are discussed based on a real inner product of two complex tensors and the finite convergence of this algorithm is obtained. Two numerical examples are offered to illustrate the effectiveness of the proposed algorithm.
- Sylvester tensor equation /
- finite iterative algorithm /
- convergence
MSC: 65H10, 12A24

FullText(HTML)

References

[1]	J. Ballani and L. Grasedyck, A Projection method to solve linear systems in tensor format, Numer. Linear. Algebra. Appl., 2013, 20, 27-43. doi: 10.1002/nla.1818 CrossRef Google Scholar
[2]	F. P. A. Beik, F. S. Movahed and S. Ahmadi-Asl, On the Krylov subspace methods based on tensor format for positive definite Sylvester tensor equations, Numer. Linear. Algebra. Appl., 2016, 23, 444-466. doi: 10.1002/nla.2033 CrossRef Google Scholar
[3]	Z. Chen and L. Lu, A projection method and Kronecker product preconditioner for solving Sylvester tensor equations, Sci. China Math., 2012, 55(6), 1281-1292. doi: 10.1007/s11425-012-4363-5 CrossRef Google Scholar
[4]	Z. Chen and L. Lu, A gradient based iterative solutions for Sylvester tensor equations, Math. Probl. Eng., Volume 2013, Article ID 819479, 7 pages. Google Scholar
[5]	F. Ding and T. Chen, On iterative solutions of general coupled matrix equations, SIAM J. Control Optim., 2006, 44(6), 2269-2284. doi: 10.1137/S0363012904441350 CrossRef Google Scholar
[6]	G. H. Golub, S. Nash and C. F. Van Loan, A Hessenberg-Schur method for the problem $AX + XB = C$, IEEE Trans. Automat. Contr., 1979, 24, 909-913. doi: 10.1109/TAC.1979.1102170 CrossRef $AX + XB = C$" target="_blank">Google Scholar
[7]	L. Grasedyck, Existence and computation of low Kronecker-rank approximations for large linear systems of tensor product structure, Computing, 2004, 72(3-4), 247-265. Google Scholar
[8]	Y. Ke and C. Ma, A preconditioned nested splitting conjugate gradient iterative method for the large sparse generalized Sylvester equation, Comput. Math. Appl., 2014, 68, 1409-1420. doi: 10.1016/j.camwa.2014.09.009 CrossRef Google Scholar
[9]	Y. Ke and C. Ma, Alternating direction method for generalized Sylvester matrix equation $AXB + CYD=E$, Appl. Math. Comput., 2015, 260, 106-125. $AXB + CYD=E$" target="_blank">Google Scholar
[10]	Y. Ke and C. Ma, The unified frame of alternating direction method of multipliers for three classes of matrix equations arising in control theory, Asian J. Control, 2017, 20(3), 1-18. Google Scholar
[11]	T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev., 2009, 51(3), 455-500. doi: 10.1137/07070111X CrossRef Google Scholar
[12]	T. Kolda, B. Bader et al., MATLAB Tensor Toolbox Version 2.6 (released Feb. 6, 2015). (Available from: http://www.sandia.gov/tgkolda/TensorToolbox). Google Scholar
[13]	D. Kressner and C. Tobler, Krylov subspace methods for linear systems with tensor product structure, SIAM J. Matrix Anal. Appl., 2010, 31, 1688-1714. doi: 10.1137/090756843 CrossRef Google Scholar
[14]	B. Li, S. Tian, Y. Sun and Z. Hu, Schur-decomposition for 3D matrix equations and its application in solving radiative discrete ordinates equations discretized by Chebyshev collocation spectral method, J. Comput. Phys., 2010, 229(4), 1198-1212. doi: 10.1016/j.jcp.2009.10.025 CrossRef Google Scholar
[15]	L. Liang and B. Zheng, Sensitivity analysis of the Lyapunov tensor equation, Linear Multilinear A., 2018, 1-18. Google Scholar
[16]	A. Malek and S. H. M. Masuleh, Mixed collocation-finite difference method for 3D microscopic heat transport problems, J. Comput. Appl. Math., 2008, 217, 137-147. doi: 10.1016/j.cam.2007.06.023 CrossRef Google Scholar
[17]	A. Malek, Z. K. Bojdi and P. N. N. Golbarg, Solving fully three-dimensional microscale dual phase lag problem using mixed-collocation finite difference discretization, J. Heat Transfer, 2012, 134, 0945041-0945046. Google Scholar
[18]	S. H. M. Masuleh and T. N. Phillips, Viscoelastic flow in an undulating tube using spectral methods, Comput. Fluids, 2004, 33, 1075-1095. doi: 10.1016/j.compfluid.2003.09.002 CrossRef Google Scholar
[19]	A. Wu, L. Lv and M. Hou, Finite iterative algorithms for extended Sylvester-conjugate matrix equations, Math. Comput. Model., 2011, 54, 2363-2384. Google Scholar
[20]	X. Zhang, Matrix Analysis and Applications, Tsinghua University Press, Beijing, 2004. Google Scholar
[21]	H. Zhang, A finite iterative algorithm for solving the complex generalized coupled Sylvester matrix equations by using the linear operators, J. Franklin Inst., 2017, 354, 1856-1874. doi: 10.1016/j.jfranklin.2016.12.011 CrossRef Google Scholar