An extension of the Cayley transform method for a parameterized generalized inverse eigenvalue problem

[EN] Since recent studies have shown that the Cayley transform method can be an effective iterative method for solving the inverse eigenvalue problem, in this work, we consider using an extension of it for solving a type of parameterized generalized inverse eigenvalue problem and prove its locally quadratic convergence. This type of inverse eigenvalue problem, which includes multiplicative and additive inverse eigenvalue problems, appears in many applications. Also, we consider the case where the given eigenvalues are multiple. In this case, we describe a modified problem that is not overdetermined and discuss the extension of the Cayley transform method for this modified problem. Finally, to demonstrate the effectiveness of these algorithms, we present some numerical examples to show that the proposed methods are practical and efficient.The authors would like to express their heartfelt thanks to the editor and anonymous referees for their useful comments and constructive suggestions that substantially improved the quality and presentation of this article. This research was developed during a visit of Z.D. to Universitat Politecnica de Valencia. Z.D. would like to thank the hospitality shown by D. Sistemes Informatics i Computacio, Universitat Politecnica de Valencia. J.E.R. was partially supported by the Spanish Agencia Estatal de Investigacion (AEI) under grant TIN2016-75985-P, which includes European Commission ERDF funds. The authors thank Carmen Campos for useful comments on an initial draft of the article.Dalvand, Z.; Hajarian, M.; Román Moltó, JE. (2020). An extension of the Cayley transform method for a parameterized generalized inverse eigenvalue problem. Numerical Linear Algebra with Applications. 27(6):1-24. https://doi.org/10.1002/nla.2327S124276Chu, M., & Golub, G. (2005). Inverse Eigenvalue Problems. doi:10.1093/acprof:oso/9780198566649.001.0001Hajarian, M., & Abbas, H. (2018). Least squares solutions of quadratic inverse eigenvalue problem with partially bisymmetric matrices under prescribed submatrix constraints. Computers & Mathematics with Applications, 76(6), 1458-1475. doi:10.1016/j.camwa.2018.06.038Hajarian, M. (2019). An efficient algorithm based on Lanczos type of BCR to solve constrained quadratic inverse eigenvalue problems. Journal of Computational and Applied Mathematics, 346, 418-431. doi:10.1016/j.cam.2018.07.025Hajarian, M. (2018). Solving constrained quadratic inverse eigenvalue problem via conjugate direction method. Computers & Mathematics with Applications, 76(10), 2384-2401. doi:10.1016/j.camwa.2018.08.034Chu, M. T., & Golub, G. H. (2002). Structured inverse eigenvalue problems. Acta Numerica, 11, 1-71. doi:10.1017/s0962492902000016Ghanbari, K., & Parvizpour, F. (2012). Generalized inverse eigenvalue problem with mixed eigendata. Linear Algebra and its Applications, 437(8), 2056-2063. doi:10.1016/j.laa.2012.05.020Yuan, Y.-X., & Dai, H. (2009). A generalized inverse eigenvalue problem in structural dynamic model updating. Journal of Computational and Applied Mathematics, 226(1), 42-49. doi:10.1016/j.cam.2008.05.015Yuan, S.-F., Wang, Q.-W., & Xiong, Z.-P. (2013). Linear parameterized inverse eigenvalue problem of bisymmetric matrices. Linear Algebra and its Applications, 439(7), 1990-2007. doi:10.1016/j.laa.2013.05.026Dai, H., Bai, Z.-Z., & Wei, Y. (2015). On the Solvability Condition and Numerical Algorithm for the Parameterized Generalized Inverse Eigenvalue Problem. SIAM Journal on Matrix Analysis and Applications, 36(2), 707-726. doi:10.1137/140972494Gladwell, G. M. L. (1986). Inverse Problems in Vibration. Applied Mechanics Reviews, 39(7), 1013-1018. doi:10.1115/1.3149517Friedland, S., Nocedal, J., & Overton, M. L. (1987). The Formulation and Analysis of Numerical Methods for Inverse Eigenvalue Problems. SIAM Journal on Numerical Analysis, 24(3), 634-667. doi:10.1137/0724043Chan, R. H. (2003). BIT Numerical Mathematics, 43(1), 7-20. doi:10.1023/a:1023611931016Bai, Z.-J., Chan, R. H., & Morini, B. (2004). An inexact Cayley transform method for inverse eigenvalue problems. Inverse Problems, 20(5), 1675-1689. doi:10.1088/0266-5611/20/5/022Shen, W. P., Li, C., & Jin, X. Q. (2011). A Ulm-like method for inverse eigenvalue problems. Applied Numerical Mathematics, 61(3), 356-367. doi:10.1016/j.apnum.2010.11.001Shen, W., & Li, C. (2012). AN ULM-LIKE CAYLEY TRANSFORM METHOD FOR INVERSE EIGENVALUE PROBLEMS. Taiwanese Journal of Mathematics, 16(1). doi:10.11650/twjm/1500406546Aishima, K. (2018). A quadratically convergent algorithm based on matrix equations for inverse eigenvalue problems. Linear Algebra and its Applications, 542, 310-333. doi:10.1016/j.laa.2017.05.019Shen, W. P., Li, C., & Jin, X. Q. (2015). An inexact Cayley transform method for inverse eigenvalue problems with multiple eigenvalues. Inverse Problems, 31(8), 085007. doi:10.1088/0266-5611/31/8/085007Shen, W., Li, C., & Jin, X. (2016). An Ulm-like Cayley Transform Method for Inverse Eigenvalue Problems with Multiple Eigenvalues. Numerical Mathematics: Theory, Methods and Applications, 9(4), 664-685. doi:10.4208/nmtma.2016.y15030Aishima, K. (2018). A quadratically convergent algorithm for inverse eigenvalue problems with multiple eigenvalues. Linear Algebra and its Applications, 549, 30-52. doi:10.1016/j.laa.2018.03.022Li, L. (1995). Sufficient conditions for the solvability of an algebraic inverse eigenvalue problem. Linear Algebra and its Applications, 221, 117-129. doi:10.1016/0024-3795(93)00225-oBiegler-König, F. W. (1981). Sufficient conditions for the solubility of inverse eigenvalue problems. Linear Algebra and its Applications, 40, 89-100. doi:10.1016/0024-3795(81)90142-7Alexander, J. C. (1978). The additive inverse eigenvalue problem and topological degree. Proceedings of the American Mathematical Society, 70(1), 5-5. doi:10.1090/s0002-9939-1978-0487546-3Byrnes, C. I., & Wang, X. (1993). The Additive Inverse Eigenvalue Problem for Lie Perturbations. SIAM Journal on Matrix Analysis and Applications, 14(1), 113-117. doi:10.1137/0614009Wang, Z., & Vong, S. (2013). A Guass–Newton-like method for inverse eigenvalue problems. International Journal of Computer Mathematics, 90(7), 1435-1447. doi:10.1080/00207160.2012.750721Jiang, J., Dai, H., & Yuan, Y. (2013). A symmetric generalized inverse eigenvalue problem in structural dynamics model updating. Linear Algebra and its Applications, 439(5), 1350-1363. doi:10.1016/j.laa.2013.04.021Cox, S. J., Embree, M., & Hokanson, J. M. (2012). One Can Hear the Composition of a String: Experiments with an Inverse Eigenvalue Problem. SIAM Review, 54(1), 157-178. doi:10.1137/080731037Ji, X. (1998). On matrix inverse eigenvalue problems. Inverse Problems, 14(2), 275-285. doi:10.1088/0266-5611/14/2/004Dai, H., & Lancaster, P. (1997). Newton’s Method for a Generalized Inverse Eigenvalue Problem. Numerical Linear Algebra with Applications, 4(1), 1-21. doi:10.1002/(sici)1099-1506(199701/02)4:13.0.co;2-dShu, L., Bo, W., & Ji-zhong, H. (2004). Homotopy solution of the inverse generalized eigenvalue problems in structural dynamics. Applied Mathematics and Mechanics, 25(5), 580-586. doi:10.1007/bf02437606Dai, H. (1999). An algorithm for symmetric generalized inverse eigenvalue problems. Linear Algebra and its Applications, 296(1-3), 79-98. doi:10.1016/s0024-3795(99)00109-3Lancaster, P. (1964). Algorithms for lambda-matrices. Numerische Mathematik, 6(1), 388-394. doi:10.1007/bf01386088Biegler-K�nig, F. W. (1981). A Newton iteration process for inverse eigenvalue problems. Numerische Mathematik, 37(3), 349-354. doi:10.1007/bf01400314Parlett, B. N. (1998). The Symmetric Eigenvalue Problem. doi:10.1137/1.9781611971163Higham, N. J. (2008). Functions of Matrices. doi:10.1137/1.978089871777

Algorithms for Robust Pole Assignment in Singular Systems

The solution of the pole assignment problem by feedback in singular systems is parameterized and conditions are given which guarantee the regularity and maximal degree of the closed loop pencil. A robustness measure is defined, and numerical procedures are described for selecting the free parameters in the feedback to give optimal robustness

Orbital stability of periodic waves in the class of reduced Ostrovsky equations

Periodic travelling waves are considered in the class of reduced Ostrovsky equations that describe low-frequency internal waves in the presence of rotation. The reduced Ostrovsky equations with either quadratic or cubic nonlinearities can be transformed to integrable equations of the Klein--Gordon type by means of a change of coordinates. By using the conserved momentum and energy as well as an additional conserved quantity due to integrability, we prove that small-amplitude periodic waves are orbitally stable with respect to subharmonic perturbations, with period equal to an integer multiple of the period of the wave. The proof is based on construction of a Lyapunov functional, which is convex at the periodic wave and is conserved in the time evolution. We also show numerically that convexity of the Lyapunov functional holds for periodic waves of arbitrary amplitudes.Comment: 34 page

Eigenstructure Assignment by State-derivative and Partial Output-derivative Feedback for Linear Time-invariant Control Systems

This paper introduces a parametric approach for solving the problem of eigenstructure assignment via state-derivative feedback for linear control time-invariant systems. This problem is always solvable for any controllable systems if the open-loop system matrix is nonsingular. In this work, the parametric solution to the feedback gain matrix is introduced that describes the available degrees of freedom offered by the state-derivative feedback in selecting the associated eigenvectors from an admissible class. These freedoms can be utilized to improve the robustness of the closed-loop system. Finally, the eigenstructure assignment problem via partial output-derivative feedback is introduced. Numerical examples are included to show the effectiveness of the proposed approach