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

On some inverse singular value problems with Toeplitz-related structure

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.
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.

[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.

[3]

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

[4]

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

[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.

[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.

[7]

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

[8]

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

[9]

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

[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.

[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.

[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.

[13]

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

[14]

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

[15]

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

[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.

[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.

[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.

[19]

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

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.

[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.

[3]

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

[4]

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

[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.

[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.

[7]

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

[8]

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

[9]

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

[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.

[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.

[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.

[13]

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

[14]

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

[15]

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

[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.

[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.

[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.

[19]

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

[1]

María Isabel Cortez. $Z^d$ Toeplitz arrays. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 859-881. doi: 10.3934/dcds.2006.15.859

[2]

Lori Alvin. Toeplitz kneading sequences and adding machines. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3277-3287. doi: 10.3934/dcds.2013.33.3277

[3]

Sonia Martínez, Jorge Cortés, Francesco Bullo. A catalog of inverse-kinematics planners for underactuated systems on matrix groups. Journal of Geometric Mechanics, 2009, 1 (4) : 445-460. doi: 10.3934/jgm.2009.1.445

[4]

Wei-guo Wang, Wei-chao Wang, Ren-cang Li. Deflating irreducible singular M-matrix algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 491-518. doi: 10.3934/naco.2013.3.491

[5]

Lanzhe Liu. Mean oscillation and boundedness of Toeplitz Type operators associated to pseudo-differential operators. Communications on Pure & Applied Analysis, 2015, 14 (2) : 627-636. doi: 10.3934/cpaa.2015.14.627

[6]

Adel Alahmadi, Hamed Alsulami, S.K. Jain, Efim Zelmanov. On matrix wreath products of algebras. Electronic Research Announcements, 2017, 24: 78-86. doi: 10.3934/era.2017.24.009

[7]

Paul Skerritt, Cornelia Vizman. Dual pairs for matrix groups. Journal of Geometric Mechanics, 2019, 11 (2) : 255-275. doi: 10.3934/jgm.2019014

[8]

Meijuan Shang, Yanan Liu, Lingchen Kong, Xianchao Xiu, Ying Yang. Nonconvex mixed matrix minimization. Mathematical Foundations of Computing, 2019, 0 (0) : 0-0. doi: 10.3934/mfc.2019009

[9]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121

[10]

Zhengshan Dong, Jianli Chen, Wenxing Zhu. Homotopy method for matrix rank minimization based on the matrix hard thresholding method. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 211-224. doi: 10.3934/naco.2019015

[11]

K. T. Arasu, Manil T. Mohan. Optimization problems with orthogonal matrix constraints. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 413-440. doi: 10.3934/naco.2018026

[12]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355

[13]

Weishi Liu. Geometric approach to a singular boundary value problem with turning points. Conference Publications, 2005, 2005 (Special) : 624-633. doi: 10.3934/proc.2005.2005.624

[14]

Lei Zhang, Anfu Zhu, Aiguo Wu, Lingling Lv. Parametric solutions to the regulator-conjugate matrix equations. Journal of Industrial & Management Optimization, 2017, 13 (2) : 623-631. doi: 10.3934/jimo.2016036

[15]

Heide Gluesing-Luerssen, Fai-Lung Tsang. A matrix ring description for cyclic convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 55-81. doi: 10.3934/amc.2008.2.55

[16]

Houduo Qi, ZHonghang Xia, Guangming Xing. An application of the nearest correlation matrix on web document classification. Journal of Industrial & Management Optimization, 2007, 3 (4) : 701-713. doi: 10.3934/jimo.2007.3.701

[17]

Angelo B. Mingarelli. Nonlinear functionals in oscillation theory of matrix differential systems. Communications on Pure & Applied Analysis, 2004, 3 (1) : 75-84. doi: 10.3934/cpaa.2004.3.75

[18]

A. Cibotarica, Jiu Ding, J. Kolibal, Noah H. Rhee. Solutions of the Yang-Baxter matrix equation for an idempotent. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 347-352. doi: 10.3934/naco.2013.3.347

[19]

Haixia Liu, Jian-Feng Cai, Yang Wang. Subspace clustering by (k,k)-sparse matrix factorization. Inverse Problems & Imaging, 2017, 11 (3) : 539-551. doi: 10.3934/ipi.2017025

[20]

Leda Bucciantini, Angiolo Farina, Antonio Fasano. Flows in porous media with erosion of the solid matrix. Networks & Heterogeneous Media, 2010, 5 (1) : 63-95. doi: 10.3934/nhm.2010.5.63

 Impact Factor: 

Metrics

  • PDF downloads (3)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]