On some inverse singular value problems with Toeplitz-related structure

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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

Abstract
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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 and 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.
[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.
[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.
[4]	D. Bini and M. Capovani, Spectral and computational properties of band symmetric Toeplitz matrices, Linear Algebra Appl., 52/53 (1983), 99-126.
[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.
[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.
[7]	M. T. Chu, Numerical methods for inverse singular value problems, SIAM J. Numer. Anal., 29 (1992), 885-903. doi: 10.1137/0729054.
[8]	M. T. Chu, Inverse eigenvalue problems, SIAM Rev., 40 (1998), 1-39. doi: 10.1137/S0036144596303984.
[9]	M. T. Chu and G. H. Golub, Structured inverse eigenvalue problems, Acta Numer., 11 (2002), 1-71. doi: 10.1017/S0962492902000016.
[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.
[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.
[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.
[13]	A. Jain, "Fundamentals of Digital Image Processing," Prentice-Hall, Englewood Cliffs, NJ, 1989.
[14]	E. Montaño, M. Salas and R. L. Soto, Nonnegative matrices with prescribed extremal singular values, Comput. Math. Appl., 56 (2008), 30-42.
[15]	E. Montaño, M. Salas and R. L. Soto, Positive matrices with prescribed singular values, Proyecciones, 27 (2008), 289-305.
[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.
[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.
[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.
[19]	S. F. Xu, "An Introduction to Inverse Eigenvalue Problems," Peking University Press and Vieweg Publishing, Beijing, 1998.

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.
[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.
[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.
[4]	D. Bini and M. Capovani, Spectral and computational properties of band symmetric Toeplitz matrices, Linear Algebra Appl., 52/53 (1983), 99-126.
[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.
[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.
[7]	M. T. Chu, Numerical methods for inverse singular value problems, SIAM J. Numer. Anal., 29 (1992), 885-903. doi: 10.1137/0729054.
[8]	M. T. Chu, Inverse eigenvalue problems, SIAM Rev., 40 (1998), 1-39. doi: 10.1137/S0036144596303984.
[9]	M. T. Chu and G. H. Golub, Structured inverse eigenvalue problems, Acta Numer., 11 (2002), 1-71. doi: 10.1017/S0962492902000016.
[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.
[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.
[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.
[13]	A. Jain, "Fundamentals of Digital Image Processing," Prentice-Hall, Englewood Cliffs, NJ, 1989.
[14]	E. Montaño, M. Salas and R. L. Soto, Nonnegative matrices with prescribed extremal singular values, Comput. Math. Appl., 56 (2008), 30-42.
[15]	E. Montaño, M. Salas and R. L. Soto, Positive matrices with prescribed singular values, Proyecciones, 27 (2008), 289-305.
[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.
[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.
[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.
[19]	S. F. Xu, "An Introduction to Inverse Eigenvalue Problems," Peking University Press and Vieweg Publishing, Beijing, 1998.

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