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Two-step inexact Newton-type method for inverse singular value problems

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Abstract

In this paper, based on the two-step Newton iterative procedure, we propose a two-step inexact Newton-type method for solving inverse singular value problems. Under some mild assumptions, our results show that the two-step inexact Newton-type method is super quadratically convergent. Numerical implementations demonstrate the effectiveness of the new method.

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References

  1. Alonso, P., Flores-Becerra, G., Vidal, A.M.: Sequential and parallel algorithms for the inverse Toeplitz singular value problem. Proc CSC, pp. 91–96 (2006)

  2. Bai, Z.J., Chu, Delin, Sun, D.F.: A dual optimization approach to inverse quadratic eigenvalue problems with partial eigenstructure. SIAM J. Sci. Comput. 29 (6), 2531–2561 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bai, Z.J., Jin, X.Q., Vong, S.W.: On some inverse singular value problems with Toeplitz-related structure. Numerical Algebra, Control and Optimization 2, 187–192 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bai, Z.J., Morini, B., Xu, S.F.: On the local convergence of an iterative approach for inverse singular value problems. J. Comput. Appl. Math. 198, 344–360 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bai, Z.J., Xu, S.F.: An inexact Newton-type method for inverse singular value problems. Linear Algebra Appl. 429, 527–547 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, X.S., Sun, H.W.: On the unsolvability for inverse singular vlaue problems almost everywhere. Linear and Multilinear Algebra 67, 987–994 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, X.S., Wen, C.T., Sun, H.W.: Two-step Newton type method for solving inverse eigenvalue problems. Numer. Linear Algebra Appl. https://doi.org/10.1002/nla.e2185

  8. Chu, M.T.: Numerical methods for inverse singular value problems. SIAM J. Numer. Anal. 29, 885–903 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chu, M.T., Golub, G.H.: Structured inverse eigenvalue problems. Acta Numer. 11, 1–71 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chu, M.T., Golub, G.H.: Inverse Eigenvalue Problems: Theory, Algorithms, and Applications. Oxford University Press, Oxford (2005)

    Book  MATH  Google Scholar 

  11. Flores-Becerra, G., Garcia, V.M., Vidal, A.M.: Parallelization of a method for the solution of the inverse additive singular value problem. WSEAS Transactions on Mathematics 5, 81–88 (2006)

    MathSciNet  Google Scholar 

  12. Friedland, S., Nocedal, J., Overton, M.L.: The formulation and analysis of numerical methods for inverse eigenvalue problems. SIAM J. Numer. Anal. 24, 634–667 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  13. Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  14. Ma, W., Bai, Z.J.: A regularized directional derivative-based Newton method for inverse singular value problems. Inverse Prob. 28, 125001(24pp) (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Montaño, E., Salas, M., Soto, R.L.: Nonnegative matrices with prescribed extremal singular values. Comput. Math. Appl. 56, 30–42 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Montaño, E., Salas, M., Soto, R.L.: Positive matrices with prescribed singular values. Proyecciones 27, 289–305 (2008)

    MATH  MathSciNet  Google Scholar 

  17. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)

    MATH  Google Scholar 

  18. Politi, T.: A discrete approach for the inverse singular value problem in some quadratic group. Lecture Notes in Comput. Sci. 2658, 121–130 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Queiró, J.F.: An inverse problem for singular values and the Jacobian of the elementary symmetric functions. Linear Algebra Appl. 197–198, 277–282 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. Saad, Y.: Iterative Methods for Sparse Linear Systems. PWS Publishing Company, Boston (1996)

    MATH  Google Scholar 

  21. Saunders, C., Hu, J., Christoffersen, C., Steer, M.: Inverse singular value method for enforcing passivity in reduced-order models of distributed structures for transient and steady-state simulation. IEEE Trans. Microwave Theory and Techniques 59, 837–847 (2011)

    Article  Google Scholar 

  22. Stewart, G.W., Sun, J.G.: Matrix Perturbation Theory. Academic, San Diego (1990)

    MATH  Google Scholar 

  23. Tropp, J.A., Dhillon, I.S., Heath, R.W.: Finite-step algorithms for constructing optimal CDMA signature sequences. IEEE Trans. Inform. Theory 50, 2916–2921 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Vong, S.W., Bai, Z.J., Jin, X.Q.: A Ulm-like method for inverse singular value problems. SIAM J. Matrix. Anal. Appl. 32, 412–429 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  26. Yuan, S.F., Liao, A.P., Yao, G.Z.: Parameterized inverse singular value problem for anti-bisymmetric matrices. Numer. Algorithms 60, 501–522 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

We would like to thank anonymous referees for their valuable comments.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Nanyang Normal University, Nanyang, 473061, Henan, People’s Republic of China

    Wei Ma

  2. School of Mathematical Sciences, South China Normal University, Guangzhou, 530631, People’s Republic of China

    Xiao-shan Chen

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  1. Wei Ma
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  2. Xiao-shan Chen
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Correspondence to Xiao-shan Chen.

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The research of Wei Ma was partially supported by the Special Project Grant of Nanyang Normal University (ZX2014078).

The research of Xiao-shan Chen was partially supported by National Natural Science Foundations of China (11771159, U1811464).

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Ma, W., Chen, Xs. Two-step inexact Newton-type method for inverse singular value problems. Numer Algor 84, 847–870 (2020). https://doi.org/10.1007/s11075-019-00783-x

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