American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A filter successive linear programming method for nonlinear semidefinite programming problems
  • NACO Home
  • This Issue
  • Next Article
    A DC programming approach for a class of bilevel programming problems and its application in Portfolio Selection
2012, 2(1): 187-192. doi: 10.3934/naco.2012.2.187

On some inverse singular value problems with Toeplitz-related structure

Zheng-Jian Bai 1, , Xiao-Qing Jin 2, and  Seak-Weng Vong 2,

1. 

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
2. 

Department of Mathematics, University of Macau, Macau, China, China

Received  May 2011 Revised  September 2011 Published  March 2012

In this paper, we consider some inverse singular value problems for Toeplitz-related matrices. We construct a Toeplitz-plus-Hankel matrix from prescribed singular values including a zero singular value. Then we find a solution to the inverse singular value problem for Toeplitz matrices which have double singular values including a double zero singular value.
Keywords: Toeplitz-plus-Hankel matrix., Inverse singular value problem, Toeplitz matrix.
Mathematics Subject Classification: Primary: 65F15; Secondary: 65F1.
Citation: Zheng-Jian Bai, Xiao-Qing Jin, Seak-Weng Vong. On some inverse singular value problems with Toeplitz-related structure. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 187-192. doi: 10.3934/naco.2012.2.187
References:
[1]

Z. J. Bai, Inexact Newton methods for inverse eigenvalue problems,, Appl. Math. Comput., 172 (2006), 682.  doi: 10.1016/j.amc.2004.11.023.  Google Scholar

[2]

Z. J. Bai, R. H. Chan and B. Morini, An inexact Cayley transform method for inverse eigenvalue problems,, Inverse Problems, 20 (2004), 1675.  doi: 10.1088/0266-5611/20/5/022.  Google Scholar

[3]

Z. J. Bai and X. Q. Jin, A note on the Ulm-like method for inverse eigenvalue problems,, In, (2011), 1.   Google Scholar

[4]

D. Bini and M. Capovani, Spectral and computational properties of band symmetric Toeplitz matrices,, Linear Algebra Appl., 52/53 (1983), 99.   Google Scholar

[5]

R. H. Chan, H. L. Chung and S. F. Xu, The inexact Newton-like method for inverse eigenvalue problem,, BIT, 43 (2003), 7.  doi: 10.1023/A:1023611931016.  Google Scholar

[6]

X. S. Chen, A backward error for the inverse singular value problem,, J. Comput. Appl. Math., 234 (2010), 2450.  doi: 10.1016/j.cam.2010.03.003.  Google Scholar

[7]

M. T. Chu, Numerical methods for inverse singular value problems,, SIAM J. Numer. Anal., 29 (1992), 885.  doi: 10.1137/0729054.  Google Scholar

[8]

M. T. Chu, Inverse eigenvalue problems,, SIAM Rev., 40 (1998), 1.  doi: 10.1137/S0036144596303984.  Google Scholar

[9]

M. T. Chu and G. H. Golub, Structured inverse eigenvalue problems,, Acta Numer., 11 (2002), 1.  doi: 10.1017/S0962492902000016.  Google Scholar

[10]

M. T. Chu and G. H. Golub, "Inverse Eigenvalue Problems: Theory, Algorithms and Applications,", Oxford University Press, (2005).  doi: 10.1093/acprof:oso/9780198566649.001.0001.  Google Scholar

[11]

F. Diele, T. Laudadio and N. Mastronardi, On some inverse eigenvalue problems with Toeplitz-related structure,, SIAM J. Matrix Anal. Appl., 26 (2004), 285.  doi: 10.1137/S0895479803430680.  Google Scholar

[12]

S. Friedland, J. Nocedal and M. L. Overton, The formulation and analysis of numerical methods for inverse eigenvalue problems,, SIAM J. Numer. Anal., 24 (1987), 634.  doi: 10.1137/0724043.  Google Scholar

[13]

A. Jain, "Fundamentals of Digital Image Processing,", Prentice-Hall, (1989).   Google Scholar

[14]

E. Montaño, M. Salas and R. L. Soto, Nonnegative matrices with prescribed extremal singular values,, Comput. Math. Appl., 56 (2008), 30.   Google Scholar

[15]

E. Montaño, M. Salas and R. L. Soto, Positive matrices with prescribed singular values,, Proyecciones, 27 (2008), 289.   Google Scholar

[16]

W. P. Shen, C. Li and X. Q. Jin, A Ulm-like method for inverse eigenvalue problems,, Appl. Numer. Math., 61 (2011), 356.  doi: 10.1016/j.apnum.2010.11.001.  Google Scholar

[17]

J. A. Tropp, I. S. Dhillon and R. W. Heath Jr., Finite-step algorithms for constructing optimal CDMA signature sequences,, IEEE Trans. Inform. Theory, 50 (2004), 2916.  doi: 10.1109/TIT.2004.836698.  Google Scholar

[18]

S. W. Vong, Z. J. Bai and X. Q. Jin, An Ulm-like method for inverse singular value problems,, SIAM J. Matrix Anal. Appl., 32 (2011), 412.  doi: 10.1137/100815748.  Google Scholar

[19]

S. F. Xu, "An Introduction to Inverse Eigenvalue Problems,", Peking University Press and Vieweg Publishing, (1998).   Google Scholar

show all references

References:
[1]

Z. J. Bai, Inexact Newton methods for inverse eigenvalue problems,, Appl. Math. Comput., 172 (2006), 682.  doi: 10.1016/j.amc.2004.11.023.  Google Scholar

[2]

Z. J. Bai, R. H. Chan and B. Morini, An inexact Cayley transform method for inverse eigenvalue problems,, Inverse Problems, 20 (2004), 1675.  doi: 10.1088/0266-5611/20/5/022.  Google Scholar

[3]

Z. J. Bai and X. Q. Jin, A note on the Ulm-like method for inverse eigenvalue problems,, In, (2011), 1.   Google Scholar

[4]

D. Bini and M. Capovani, Spectral and computational properties of band symmetric Toeplitz matrices,, Linear Algebra Appl., 52/53 (1983), 99.   Google Scholar

[5]

R. H. Chan, H. L. Chung and S. F. Xu, The inexact Newton-like method for inverse eigenvalue problem,, BIT, 43 (2003), 7.  doi: 10.1023/A:1023611931016.  Google Scholar

[6]

X. S. Chen, A backward error for the inverse singular value problem,, J. Comput. Appl. Math., 234 (2010), 2450.  doi: 10.1016/j.cam.2010.03.003.  Google Scholar

[7]

M. T. Chu, Numerical methods for inverse singular value problems,, SIAM J. Numer. Anal., 29 (1992), 885.  doi: 10.1137/0729054.  Google Scholar

[8]

M. T. Chu, Inverse eigenvalue problems,, SIAM Rev., 40 (1998), 1.  doi: 10.1137/S0036144596303984.  Google Scholar

[9]

M. T. Chu and G. H. Golub, Structured inverse eigenvalue problems,, Acta Numer., 11 (2002), 1.  doi: 10.1017/S0962492902000016.  Google Scholar

[10]

M. T. Chu and G. H. Golub, "Inverse Eigenvalue Problems: Theory, Algorithms and Applications,", Oxford University Press, (2005).  doi: 10.1093/acprof:oso/9780198566649.001.0001.  Google Scholar

[11]

F. Diele, T. Laudadio and N. Mastronardi, On some inverse eigenvalue problems with Toeplitz-related structure,, SIAM J. Matrix Anal. Appl., 26 (2004), 285.  doi: 10.1137/S0895479803430680.  Google Scholar

[12]

S. Friedland, J. Nocedal and M. L. Overton, The formulation and analysis of numerical methods for inverse eigenvalue problems,, SIAM J. Numer. Anal., 24 (1987), 634.  doi: 10.1137/0724043.  Google Scholar

[13]

A. Jain, "Fundamentals of Digital Image Processing,", Prentice-Hall, (1989).   Google Scholar

[14]

E. Montaño, M. Salas and R. L. Soto, Nonnegative matrices with prescribed extremal singular values,, Comput. Math. Appl., 56 (2008), 30.   Google Scholar

[15]

E. Montaño, M. Salas and R. L. Soto, Positive matrices with prescribed singular values,, Proyecciones, 27 (2008), 289.   Google Scholar

[16]

W. P. Shen, C. Li and X. Q. Jin, A Ulm-like method for inverse eigenvalue problems,, Appl. Numer. Math., 61 (2011), 356.  doi: 10.1016/j.apnum.2010.11.001.  Google Scholar

[17]

J. A. Tropp, I. S. Dhillon and R. W. Heath Jr., Finite-step algorithms for constructing optimal CDMA signature sequences,, IEEE Trans. Inform. Theory, 50 (2004), 2916.  doi: 10.1109/TIT.2004.836698.  Google Scholar

[18]

S. W. Vong, Z. J. Bai and X. Q. Jin, An Ulm-like method for inverse singular value problems,, SIAM J. Matrix Anal. Appl., 32 (2011), 412.  doi: 10.1137/100815748.  Google Scholar

[19]

S. F. Xu, "An Introduction to Inverse Eigenvalue Problems,", Peking University Press and Vieweg Publishing, (1998).   Google Scholar

[1]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012
[2]

Shengxin Zhu, Tongxiang Gu, Xingping Liu. AIMS: Average information matrix splitting. Mathematical Foundations of Computing, 2020, 3 (4) : 301-308. doi: 10.3934/mfc.2020012
[3]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018
[4]

Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124
[5]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435
[6]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375
[7]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016
[8]

Nalin Fonseka, Jerome Goddard II, Ratnasingham Shivaji, Byungjae Son. A diffusive weak Allee effect model with U-shaped emigration and matrix hostility. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020356
[9]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, 2021, 15 (1) : 159-183. doi: 10.3934/ipi.2020076
[10]

Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108
[11]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442
[12]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354
[13]

Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088
[14]

Nguyen Huu Can, Nguyen Huy Tuan, Donal O'Regan, Vo Van Au. On a final value problem for a class of nonlinear hyperbolic equations with damping term. Evolution Equations & Control Theory, 2021, 10 (1) : 103-127. doi: 10.3934/eect.2020053
[15]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367
[16]

Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021004
[17]

Shahede Omidi, Jafar Fathali. Inverse single facility location problem on a tree with balancing on the distance of server to clients. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021017
[18]

Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021005
[19]

Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021006
[20]

Yueh-Cheng Kuo, Huan-Chang Cheng, Jhih-You Syu, Shih-Feng Shieh. On the nearest stable $ 2\times 2 $ matrix, dedicated to Prof. Sze-Bi Hsu in appreciation of his inspiring ideas. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020358

 Impact Factor: 

Tools

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences