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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), 682-689. 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), 1675-1689. 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 "Recent Advances in Scientific Computing and Matrix Analysis" (eds. X. Q. Jin, H. W. Sun and S. W. Vong), International Press of Boston, Boston and Higher Education Press, Beijing, (2011), 1-7. Google Scholar

[4]

D. Bini and M. Capovani, Spectral and computational properties of band symmetric Toeplitz matrices, Linear Algebra Appl., 52/53 (1983), 99-126.  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), 7-20. 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), 2450-2455. 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), 885-903. doi: 10.1137/0729054.  Google Scholar

[8]

M. T. Chu, Inverse eigenvalue problems, SIAM Rev., 40 (1998), 1-39. doi: 10.1137/S0036144596303984.  Google Scholar

[9]

M. T. Chu and G. H. Golub, Structured inverse eigenvalue problems, Acta Numer., 11 (2002), 1-71. doi: 10.1017/S0962492902000016.  Google Scholar

[10]

M. T. Chu and G. H. Golub, "Inverse Eigenvalue Problems: Theory, Algorithms and Applications," Oxford University Press, Oxford, 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), 285-294. 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), 634-667. doi: 10.1137/0724043.  Google Scholar

[13]

A. Jain, "Fundamentals of Digital Image Processing," Prentice-Hall, Englewood Cliffs, NJ, 1989. Google Scholar

[14]

E. Montaño, M. Salas and R. L. Soto, Nonnegative matrices with prescribed extremal singular values, Comput. Math. Appl., 56 (2008), 30-42.  Google Scholar

[15]

E. Montaño, M. Salas and R. L. Soto, Positive matrices with prescribed singular values, Proyecciones, 27 (2008), 289-305.  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), 356-367. 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), 2916-2921. 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), 412-429. doi: 10.1137/100815748.  Google Scholar

[19]

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

show all references

References:
[1]

Z. J. Bai, Inexact Newton methods for inverse eigenvalue problems, Appl. Math. Comput., 172 (2006), 682-689. 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), 1675-1689. 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 "Recent Advances in Scientific Computing and Matrix Analysis" (eds. X. Q. Jin, H. W. Sun and S. W. Vong), International Press of Boston, Boston and Higher Education Press, Beijing, (2011), 1-7. Google Scholar

[4]

D. Bini and M. Capovani, Spectral and computational properties of band symmetric Toeplitz matrices, Linear Algebra Appl., 52/53 (1983), 99-126.  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), 7-20. 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), 2450-2455. 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), 885-903. doi: 10.1137/0729054.  Google Scholar

[8]

M. T. Chu, Inverse eigenvalue problems, SIAM Rev., 40 (1998), 1-39. doi: 10.1137/S0036144596303984.  Google Scholar

[9]

M. T. Chu and G. H. Golub, Structured inverse eigenvalue problems, Acta Numer., 11 (2002), 1-71. doi: 10.1017/S0962492902000016.  Google Scholar

[10]

M. T. Chu and G. H. Golub, "Inverse Eigenvalue Problems: Theory, Algorithms and Applications," Oxford University Press, Oxford, 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), 285-294. 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), 634-667. doi: 10.1137/0724043.  Google Scholar

[13]

A. Jain, "Fundamentals of Digital Image Processing," Prentice-Hall, Englewood Cliffs, NJ, 1989. Google Scholar

[14]

E. Montaño, M. Salas and R. L. Soto, Nonnegative matrices with prescribed extremal singular values, Comput. Math. Appl., 56 (2008), 30-42.  Google Scholar

[15]

E. Montaño, M. Salas and R. L. Soto, Positive matrices with prescribed singular values, Proyecciones, 27 (2008), 289-305.  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), 356-367. 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), 2916-2921. 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), 412-429. doi: 10.1137/100815748.  Google Scholar

[19]

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

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