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Keywords:
inverse singular value problem; two-step; Ulm-Chebyshev-like method; cubically convergent; multiple singular values
Summary:
We study the convergence of two-step Ulm-Chebyshev-like method for solving the inverse singular value problems. We focus on the case when the given singular values are positive and multiple. This work extends the result of W. Ma (2022). We show that the new method is cubically convergent. Moreover, numerical experiments are given in the last section, which show that the proposed method is practical and efficient.
References:
[1] Alonso, P., Flores-Becerra, G., Vidal, A. M.: Sequential and parallel algorithms for the inverse Toeplitz singular value problem. Proceedings of the 2006 International Conference on Scientific Computing, CSC 2006 CSREA Press, Las Vegas (2006), 91-96.
[2] Bai, Z.-J., Chu, D., Sun, D.: A dual optimization approach to inverse quadratic eigenvalue problems with partial eigenstructure. SIAM J. Sci. Comput. 29 (2007), 2531-2561. DOI 10.1137/060656346 | MR 2357626 | Zbl 1154.65312
[3] Bai, Z.-J., Jin, X.-Q., Vong, S.-W.: On some inverse singular value problems with Toeplitz-related structure. Numer. Algebra Control Optim. 2 (2012), 187-192. DOI 10.3934/naco.2012.2.187 | MR 2904125 | Zbl 1246.65060
[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 (2007), 344-360. DOI 10.1016/j.cam.2005.06.050 | MR 2260673 | Zbl 1110.65030
[5] Bai, Z.-J., Xu, S.: An inexact Newton-type method for inverse singular value problems. Linear Algebra Appl. 429 (2008), 527-547. DOI 10.1016/j.laa.2008.03.008 | MR 2419944 | Zbl 1154.65021
[6] Chen, X. S., Sun, H.-W.: On the unsolvability for inverse singular value problems almost everywhere. Linear Multilinear Algebra 67 (2019), 987-994. DOI 10.1080/03081087.2018.1440521 | MR 3923041 | Zbl 1411.65062
[7] Chu, M. T.: Numerical methods for inverse singular value problems. SIAM J. Numer. Anal. 29 (1992), 885-903. DOI 10.1137/0729054 | MR 1163362 | Zbl 0757.65041
[8] Chu, M. T., Golub, G. H.: Structured inverse eigenvalue problems. Acta Numerica 11 (2002), 1-71. DOI 10.1017/S0962492902000016 | MR 2008966 | Zbl 1105.65326
[9] Chu, M. T., Golub, G. H.: Inverse Eigenvalue Problems: Theory, Algorithms, and Applications. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2005). DOI 10.1093/acprof:oso/9780198566649.001.0001 | MR 2263317 | Zbl 1075.65058
[10] Ezquerro, J. A., Hernández, M. A.: An Ulm-type method with $R$-order of convergence three. Nonlinear Anal., Real World Appl. 13 (2012), 14-26. DOI 10.1016/j.nonrwa.2011.07.039 | MR 2846814 | Zbl 1238.65048
[11] Flores-Becerra, G., Garcia, V. M., Vidal, A. M.: Parallelization and comparison of local convergent algorithms for solving the inverse additive singular value problem. WSEAS Trans. Math. 5 (2006), 81-88. MR 2194662
[12] Freund, R. W., Nachtigal, N. M.: QMR: A quasi-minimal residual method for non-Hermitian linear systems. Numer. Math. 60 (1991), 315-339. DOI 10.1007/BF01385726 | MR 1137197 | Zbl 0754.65034
[13] Friedland, S., Nocedal, J., Overton, M. L.: The formulation and analysis of numerical methods for inverse eigenvalue problems. SIAM. J. Numer. Anal. 24 (1987), 634-667. DOI 10.1137/0724043 | MR 0888754 | Zbl 0622.65030
[14] Ma, W.: Two-step Ulm-Chebyshev-like method for inverse singular value problems. Numer. Linear Algebra Appl. 29 (2022), Article ID e2440, 20 pages. DOI 10.1002/nla.2440 | MR 4541746 | Zbl 07584148
[15] Ma, W., Bai, Z.-J.: A regularized directional derivative-based Newton method for inverse singular value problems. Inverse Prob. 28 (2012), Article ID 125001, 24 pages. DOI 10.1088/0266-5611/28/12/125001 | MR 2997010 | Zbl 1268.65053
[16] Ma, W., Chen, X.-S.: Two-step inexact Newton-type method for inverse singular value problems. Numer. Algorithms 84 (2020), 847-870. DOI 10.1007/s11075-019-00783-x | MR 4110687 | Zbl 1442.65097
[17] Montaño, E., Salas, M., Soto, R. L.: Nonnegative matrices with prescribed extremal singular values. Comput. Math. Appl. 56 (2008), 30-42. DOI 10.1016/j.camwa.2007.11.030 | MR 2427682 | Zbl 1145.15304
[18] Montaño, E., Salas, M., Soto, R. L.: Positive matrices with prescribed singular values. Proyecciones 27 (2008), 289-305. DOI 10.4067/S0716-09172008000300005 | MR 2470405 | Zbl 1178.15006
[19] Politi, T.: A discrete approach for the inverse singular value problem in some quadratic group. Computational science -- ICCS 2003 Lecture Notes in Computer Science 2658. Springer, Berlin (2003), 121-130. DOI 10.1007/3-540-44862-4_14 | MR 2088389 | Zbl 1147.65306
[20] Potra, F. A., Qi, L., Sun, D.: Secant methods for semismooth equations. Numer. Math. 80 (1998), 305-324. DOI 10.1007/s002110050369 | MR 1645041 | Zbl 0914.65051
[21] Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18 (1993), 227-244. DOI 10.1287/moor.18.1.227 | MR 1250115 | Zbl 0776.65037
[22] Queiró, J. F.: An inverse problem for singular values and the Jacobian of the elementary symmetric functions. Linear Algebra Appl. 197-198 (1994), 277-282. DOI 10.1016/0024-3795(94)90491-X | MR 1275618 | Zbl 0793.15007
[23] Rockafellar, R. T.: Convex Analysis. Princeton Mathematical Series 28. Princeton University Press, Princeton (1970). DOI 10.1515/9781400873173 | MR 0274683 | Zbl 0193.18401
[24] Saunders, C. S., Hu, J., Christoffersen, C. E., Steer, M. B.: Inverse singular value method for enforcing passivity in reduced-order models of distributed structures for transient and steady-state simulation. IEEE Trans. Microwave Theory Tech. 59 (2011), 837-847. DOI 10.1109/TMTT.2011.2108311
[25] Shen, W.-P., Li, C., Jin, X.-Q., Yao, J.-C.: Newton-type methods for inverse singular value problems with multiple singular values. Appl. Numer. Math. 109 (2016), 138-156. DOI 10.1016/j.apnum.2016.06.008 | MR 3541948 | Zbl 1348.65073
[26] Sun, D., Sun, J.: Strong semismoothness of eigenvalues of symmetric matrices and its application to inverse eigenvalue problems. SIAM J. Numer. Anal. 40 (2002), 2352-2367. DOI 10.1137/s0036142901393814 | MR 1974190 | Zbl 1041.65037
[27] J. A. Tropp, I. S. Dhillon, R. W. Heath, Jr.: Finite-step algorithms for constructing optimal CDMA signature sequences. IEEE Trans. Inf. Theory 50 (2004), 2916-2921. DOI 10.1109/TIT.2004.836698 | MR 2097014 | Zbl 1288.94006
[28] Vong, S.-W., Bai, Z.-J., Jin, X.-Q.: An Ulm-like method for inverse singular value problems. SIAM J. Matrix Anal. Appl. 32 (2011), 412-429. DOI 10.1137/100815748 | MR 2817496 | Zbl 1232.65063
[29] Xu, S.-F.: An Introduction to Inverse Algebraic Eigenvalue Problems. Peking University Press, Peking (1998). MR 1682124 | Zbl 0927.65057
[30] Yuan, S., Liao, A., Yao, G.: Parameterized inverse singular value problem for anti-bisymmetric matrices. Numer. Algorithms 60 (2012), 501-522. DOI 10.1007/s11075-011-9526-x | MR 2927669 | Zbl 1247.65053
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