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

The computational structural mechanics testbed procedures manual

The purpose of this manual is to document the standard high level command language procedures of the Computational Structural Mechanics (CSM) Testbed software system. A description of each procedure including its function, commands, data interface, and use is presented. This manual is designed to assist users in defining and using command procedures to perform structural analysis in the CSM Testbed User's Manual and the CSM Testbed Data Library Description

Recent Experiences in Multidisciplinary Analysis and Optimization, part 1

Papers presented at the NASA Symposium on Recent Experiences in Multidisciplinary Analysis and Optimization held at NASA Langley Research Center, Hampton, Virginia April 24 to 26, 1984 are given. The purposes of the symposium were to exchange information about the status of the application of optimization and associated analyses in industry or research laboratories to real life problems and to examine the directions of future developments. Information exchange has encompassed the following: (1) examples of successful applications; (2) attempt and failure examples; (3) identification of potential applications and benefits; (4) synergistic effects of optimized interaction and trade-offs occurring among two or more engineering disciplines and/or subsystems in a system; and (5) traditional organization of a design process as a vehicle for or an impediment to the progress in the design methodology

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

A structured approach to physically-based modeling for computer graphics

This thesis presents a framework for the design of physically-based computer graphics models. The framework includes a paradigm for the structure of physically-based models, techniques for "structured" mathematical modeling, and a specification of a computer program structure in which to implement the models. The framework is based on known principles and methodologies of structured programming and mathematical modeling. Because the framework emphasizes the structure and organization of models, we refer to it as "Structured Modeling." The Structured Modeling framework focuses on clarity and "correctness" of models, emphasizing explicit statement of assumptions, goals, and techniques. In particular, we partition physically-based models, separating them into conceptual and mathematical models, and posed problems. We control complexity of models by designing in a modular manner, piecing models together from smaller components. The framework places a particular emphasis on defining a complete formal statement of a model's mathematical equations, before attempting to simulate the model. To manage the complexity of these equations, we define a collection of mathematical constructs, notation, and terminology, that allow mathematical models to be created in a structured and modular manner. We construct a computer programming environment that directly supports the implementation of models designed using the above techniques. The environment is geared to a tool-oriented approach, in which models are built from an extensible collection of software objects, that correspond to elements and tasks of a "blackboard" design of models. A substantial portion of this thesis is devoted to developing a library of physically-based model "modules," including rigid-body kinematics, rigid-body dynamics, and dynamic constraints, all built with the Structured Modeling framework. These modules are intended to serve both as examples of the framework, and as potentially useful tools for the computer graphics community. Each module includes statements of goals and assumptions, explicit mathematical models and problem statements, and descriptions of software objects that support them. We illustrate the use of the library to build some sample models, and include discussion of various possible additions and extensions to the library. Structured Modeling is an experiment in modeling: an exploration of designing via strict adherence to a dogma of structure, modularity, and mathematical formality. It does not stress issues such as particular numerical simulation techniques or efficiency of computer execution time or memory usage, all of which are important practical considerations in modeling. However, at least so far as the work carried on in this thesis, Structured Modeling has proven to be a useful aid in the design and understanding of complex physically based models.</p

Structural Dynamics and Control Interaction of Flexible Structures

A workshop on structural dynamics and control interaction of flexible structures was held to promote technical exchange between the structural dynamics and control disciplines, foster joint technology, and provide a forum for discussing and focusing critical issues in the separate and combined areas. Issues and areas of emphasis were identified in structure-control interaction for the next generation of flexible systems

Time Localization of Abrupt Changes in Cutting Process using Hilbert Huang Transform

Cutting process is extremely dynamical process influenced by different phenomena such as chip formation, dynamical responses and condition of machining system elements. Different phenomena in cutting zone have signatures in different frequency bands in signal acquired during process monitoring. The time localization of signal’s frequency content is very important. An emerging technique for simultaneous analysis of the signal in time and frequency domain that can be used for time localization of frequency is Hilbert Huang Transform (HHT). It is based on empirical mode decomposition (EMD) of the signal into intrinsic mode functions (IMFs) as simple oscillatory modes. IMFs obtained using EMD can be processed using Hilbert Transform and instantaneous frequency of the signal can be computed. This paper gives a methodology for time localization of cutting process stop during intermittent turning. Cutting process stop leads to abrupt changes in acquired signal correlated to certain frequency band. The frequency band related to abrupt changes is localized in time using HHT. The potentials and limitations of HHT application in machining process monitoring are shown

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.2327124276Chu, 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