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Relevant bibliographies by topics / Generalized eigenvalue algorithm

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Academic literature on the topic 'Generalized eigenvalue algorithm'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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Journal articles on the topic "Generalized eigenvalue algorithm"

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Kangal, Fatih, and Emre Mengi. "Nonsmooth algorithms for minimizing the largest eigenvalue with applications to inner numerical radius." IMA Journal of Numerical Analysis 40, no. 4 (November 13, 2019): 2342–76. http://dx.doi.org/10.1093/imanum/drz041.

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Abstract Nonsmoothness at optimal points is a common phenomenon in many eigenvalue optimization problems. We consider two recent algorithms to minimize the largest eigenvalue of a Hermitian matrix dependent on one parameter, both proven to be globally convergent unaffected by nonsmoothness. One of these algorithms models the eigenvalue function with a piece-wise quadratic function and is effective in dealing with nonconvex problems. The other algorithm projects the Hermitian matrix into subspaces formed of eigenvectors and is effective in dealing with large-scale problems. We generalize the latter slightly to cope with nonsmoothness. For both algorithms we analyze the rate of convergence in the nonsmooth setting, when the largest eigenvalue is multiple at the minimizer and zero is strictly in the interior of the generalized Clarke derivative, and prove that both algorithms converge rapidly. The algorithms are applied to, and the deduced results are illustrated on the computation of the inner numerical radius, the modulus of the point on the boundary of the field of values closest to the origin, which carries significance for instance for the numerical solution of a symmetric definite generalized eigenvalue problem and the iterative solution of a saddle point linear system.

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Nedamani, F. Abbasi, A. H. Refahi Sheikhani, and H. Saberi Najafi. "A New Algorithm for Solving Large-Scale Generalized Eigenvalue Problem Based on Projection Methods." Mathematical Problems in Engineering 2020 (December 14, 2020): 1–10. http://dx.doi.org/10.1155/2020/8895856.

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In this paper, we consider four methods for determining certain eigenvalues and corresponding eigenvectors of large-scale generalized eigenvalue problems which are located in a certain region. In these methods, a small pencil that contains only the desired eigenvalue is derived using moments that have obtained via numerical integration. Our purpose is to improve the numerical stability of the moment-based method and compare its stability with three other methods. Numerical examples show that the block version of the moment-based (SS) method with the Rayleigh–Ritz procedure has higher numerical stability than respect to other methods.

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Zheng, Wenming. "Class-Incremental Generalized Discriminant Analysis." Neural Computation 18, no. 4 (April 1, 2006): 979–1006. http://dx.doi.org/10.1162/neco.2006.18.4.979.

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Generalized discriminant analysis (GDA) is the nonlinear extension of the classical linear discriminant analysis (LDA) via the kernel trick. Mathematically, GDA aims to solve a generalized eigenequation problem, which is always implemented by the use of singular value decomposition (SVD) in the previously proposed GDA algorithms. A major drawback of SVD, however, is the difficulty of designing an incremental solution for the eigenvalue problem. Moreover, there are still numerical problems of computing the eigenvalue problem of large matrices. In this article, we propose another algorithm for solving GDA as for the case of small sample size problem, which applies QR decomposition rather than SVD. A major contribution of the proposed algorithm is that it can incrementally update the discriminant vectors when new classes are inserted into the training set. The other major contribution of this article is the presentation of the modified kernel Gram-Schmidt (MKGS) orthogonalization algorithm for implementing the QR decomposition in the feature space, which is more numerically stable than the kernel Gram-Schmidt (KGS) algorithm. We conduct experiments on both simulated and real data to demonstrate the better performance of the proposed methods.

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Oustry, François. "A Superlinear Algorithm to Solve Generalized Eigenvalue Problems." IFAC Proceedings Volumes 30, no. 27 (October 1997): 291–95. http://dx.doi.org/10.1016/s1474-6670(17)41197-9.

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Dai, Hua. "An algorithm for symmetric generalized inverse eigenvalue problems." Linear Algebra and its Applications 296, no. 1-3 (July 1999): 79–98. http://dx.doi.org/10.1016/s0024-3795(99)00109-3.

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Zheng, Wenming, Li Zhao, and Cairong Zou. "A Modified Algorithm for Generalized Discriminant Analysis." Neural Computation 16, no. 6 (June 1, 2004): 1283–97. http://dx.doi.org/10.1162/089976604773717612.

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Generalized discriminant analysis (GDA) is an extension of the classical linear discriminant analysis (LDA) from linear domain to a nonlinear domain via the kernel trick. However, in the previous algorithm of GDA, the solutions may suffer from the degenerate eigenvalue problem (i.e., several eigenvectors with the same eigenvalue), which makes them not optimal in terms of the discriminant ability. In this letter, we propose a modified algorithm for GDA (MGDA) to solve this problem. The MGDA method aims to remove the degeneracy of GDA and find the optimal discriminant solutions, which maximize the between-class scatter in the subspace spanned by the degenerate eigenvectors of GDA. Theoretical analysis and experimental results on the ORL face database show that the MGDA method achieves better performance than the GDA method.

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Rajakumar, C., and C. R. Rogers. "The Lanczos algorithm applied to unsymmetric generalized eigenvalue problem." International Journal for Numerical Methods in Engineering 32, no. 5 (October 5, 1991): 1009–26. http://dx.doi.org/10.1002/nme.1620320506.

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Aishima, Kensuke. "A quadratically convergent algorithm for inverse generalized eigenvalue problems." Journal of Computational and Applied Mathematics 367 (March 2020): 112485. http://dx.doi.org/10.1016/j.cam.2019.112485.

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Li, Kuiyuan, Tien-Yien Li, and Zhonggang Zeng. "An algorithm for the generalized symmetric tridiagonal eigenvalue problem." Numerical Algorithms 8, no. 2 (September 1994): 269–91. http://dx.doi.org/10.1007/bf02142694.

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Spiridonov, Alexander O., Anna I. Repina, Ilya V. Ketov, Sergey I. Solov’ev, and Evgenii M. Karchevskii. "Exponentially Convergent Galerkin Method for Numerical Modeling of Lasing in Microcavities with Piercing Holes." Axioms 10, no. 3 (August 11, 2021): 184. http://dx.doi.org/10.3390/axioms10030184.

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The paper investigates an algorithm for the numerical solution of a parametric eigenvalue problem for the Helmholtz equation on the plane specially tailored for the accurate mathematical modeling of lasing modes of microring lasers. The original problem is reduced to a nonlinear eigenvalue problem for a system of Muller boundary integral equations. For the numerical solution of the obtained problem, we use a trigonometric Galerkin method, prove its convergence, and derive error estimates in the eigenvalue and eigenfunction approximation. Previous numerical experiments have shown that the method converges exponentially. In the current paper, we prove that if the generalized eigenfunctions are analytic, then the approximate eigenvalues and eigenfunctions exponentially converge to the exact ones as the number of basis functions increases. To demonstrate the practical effectiveness of the algorithm, we find geometrical characteristics of microring lasers that provide a significant increase in the directivity of lasing emission, while maintaining low lasing thresholds.

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Dissertations / Theses on the topic "Generalized eigenvalue algorithm"

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Maeda, Kazuki. "Theory of Discrete and Ultradiscrete Integrable Finite Lattices Associated with Orthogonal Polynomials and Its Applications." 京都大学 (Kyoto University), 2014. http://hdl.handle.net/2433/188859.

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Adlerborn, Björn. "Parallel Algorithms and Library Software for the Generalized Eigenvalue Problem on Distributed Memory Computer Systems." Licentiate thesis, Umeå universitet, Institutionen för datavetenskap, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-119439.

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We present and discuss algorithms and library software for solving the generalized non-symmetric eigenvalue problem (GNEP) on high performance computing (HPC) platforms with distributed memory. Such problems occur frequently in computational science and engineering, and our contributions make it possible to solve GNEPs fast and accurate in parallel using state-of-the-art HPC systems. A generalized eigenvalue problem corresponds to finding scalars y and vectors x such that Ax = yBx, where A and B are real square matrices. A nonzero x that satisfies the GNEP equation is called an eigenvector of the ordered pair (A,B), and the scalar y is the associated eigenvalue. Our contributions include parallel algorithms for transforming a matrix pair (A,B) to a generalized Schur form (S,T), where S is quasi upper triangular and T is upper triangular. The eigenvalues are revealed from the diagonals of S and T. Moreover, for a specified set of eigenvalues an associated pair of deflating subspaces can be computed, which typically is requested in various applications. In the first stage the matrix pair (A,B) is reduced to a Hessenberg-triangular form (H,T), where H is upper triangular with one nonzero subdiagonal and T is upper triangular, in a finite number of steps. The second stage reduces the matrix pair further to generalized Schur form (S,T) using an iterative QZ-based method. Outgoing from a one-stage method for the reduction from (A,B) to (H,T), a novel parallel algorithm is developed. In brief, a delayed update technique is applied to several partial steps, involving low level operations, before associated accumulated transformations are applied in a blocked fashion which together with a wave-front task scheduler makes the algorithm scale when running in a parallel setting. The potential presence of infinite eigenvalues makes a generalized eigenvalue problem ill-conditioned. Therefore the parallel algorithm for the second stage, reduction to (S,T) form, continuously scan for and robustly deflate infinite eigenvalues. This will reduce the impact so that they do not interfere with other real eigenvalues or are misinterpreted as real eigenvalues. In addition, our parallel iterative QZ-based algorithm makes use of multiple implicit shifts and an aggressive early deflation (AED) technique, which radically speeds up the convergence. The multi-shift strategy is based on independent chains of so called coupled bulges and computational windows which is an important source of making the algorithm scalable. The parallel algorithms have been implemented in state-of-the-art library software. The performance is demonstrated and evaluated using up to 1600 CPU cores for problems with matrices as large as 100000 x 100000. Our library software is described in a User Guide. The software is, optionally, tunable via a set of parameters for various thresholds and buffer sizes etc. These parameters are discussed, and recommended values are specified which should result in reasonable performance on HPC systems similar to the ones we have been running on.

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Rippl, Michael [Verfasser], Thomas [Akademischer Betreuer] Huckle, Bruno [Gutachter] Lang, and Thomas [Gutachter] Huckle. "Parallel Algorithms for the Solution of Banded Symmetric Generalized Eigenvalue Problems / Michael Rippl ; Gutachter: Bruno Lang, Thomas Huckle ; Betreuer: Thomas Huckle." München : Universitätsbibliothek der TU München, 2020. http://d-nb.info/1230985379/34.

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Lee, Ling, and 李. 凌. "Design and Implementation of Modified Generalized Eigenvalue Decomposition Processor based on Square-Root Algorithm for Leakage-based Precoding in MU-MIMO Systems." Thesis, 2016. http://ndltd.ncl.edu.tw/handle/fe23hm.

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碩士
國立清華大學
通訊工程研究所
104
In order to improve the speed and reliability in current wireless communication system, multiuser MIMO (MU-MIMO) has become a popular research topic. For suppressing co-channel interference, it is necessary to design a precoding scheme for MU-MIMO downlink communication system. Leakage-based precoding scheme is a popular scheme for MU-MIMO communication system because of good performance. By adopting leakage-based precoding scheme, generalized eigenvalue decomposition (GEVD) is not only an inevitable process but also a complicated operation to calculate the precoder. Therefore, a GEVD hardware algorithm is proposed for decreasing computational time in this thesis. Compared to the conventional GEVD algorithm, the proposed algorithm has the less number of multiplications and shorter latency according to the theoretical analysis and practical implementation. The architecture of the proposed algorithm is presented in the following content. The proposed algorithm is implemented and verified by FPGA. The synthesis results in terms of FPGA and TSMC 90nm/40nm are shown. In the end of thesis, the architecture of the proposed algorithm is implemented as chip with TSMC 40nm and the specifications of the chip are presented.

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Men, Quei Sung, and 門桂松. "Some Davidson - Type Algorithms for the Generalized Eigenvalue Problems." Thesis, 1993. http://ndltd.ncl.edu.tw/handle/26205912869350159059.

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ZENG, RONG-CHUAN, and 曾榮川. "HMDR AND FMDR ALGORITHMS AND ITS APPLICATIONS TO THE GENERALIZED EIGENVALUE PROBLEMS AND OPTIMAL CONTROL PROBLEMS." Thesis, 1988. http://ndltd.ncl.edu.tw/handle/21916418430457789331.

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Books on the topic "Generalized eigenvalue algorithm"

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L, Patrick Merrell, and Langley Research Center, eds. The use of Lanczos's method to solve the large generalized symmetric definite eigenvalue problem. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1989.

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Amin, Nabil Ahmed Fouad. Adaptive antenna algorithms for maximizing the signal to jamming-plus-noise ratio as solving the generalized Eigenvalue problem. 1988.

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Book chapters on the topic "Generalized eigenvalue algorithm"

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Liu, Yifan, and Zheng Su. "Generalized Rayleigh Quotient Shift Strategy in QR Algorithm for Eigenvalue Problems." In Lecture Notes in Computer Science, 391–98. Berlin, Heidelberg: Springer Berlin Heidelberg, 2005. http://dx.doi.org/10.1007/978-3-540-31852-1_47.

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Chen, G., H. B. Keller, S. H. Lui, and B. Roux. "Parallel homotopy algorithm for large sparse generalized eigenvalue problems: Application to hydrodynamic stability analysis." In Parallel Processing: CONPAR 92—VAPP V, 331–42. Berlin, Heidelberg: Springer Berlin Heidelberg, 1992. http://dx.doi.org/10.1007/3-540-55895-0_427.

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Cullum, Jane K., and Ralph A. Willoughby. "Real Symmetric Generalized Problems." In Lanczos Algorithms for Large Symmetric Eigenvalue Computations Vol. II Programs, 228–72. Boston, MA: Birkhäuser Boston, 1985. http://dx.doi.org/10.1007/978-1-4684-9178-4_5.

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Hara, Satoshi, Yoshinobu Kawahara, Takashi Washio, and Paul von Bünau. "Stationary Subspace Analysis as a Generalized Eigenvalue Problem." In Neural Information Processing. Theory and Algorithms, 422–29. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-17537-4_52.

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Beattie, C., A. Beex, and M. Fargues. "Rank-One Extensions of the Generalized Hermitian Eigenvalue Problem for Adaptive High Resolution Array Processing." In Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms, 417–18. Berlin, Heidelberg: Springer Berlin Heidelberg, 1991. http://dx.doi.org/10.1007/978-3-642-75536-1_22.

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"Chapter 5: Generalized and Matrix Polynomial Eigenvalue Problems." In Core-Chasing Algorithms for the Eigenvalue Problem, 89–104. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2018. http://dx.doi.org/10.1137/1.9781611975345.ch5.

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Ericsson, Thomas. "A Generalised Eigenvalue Problem and The Lanczos Algorithm." In Large Scale Eigenvalue Problems, Proceedings of the IBM Europe Institute Workshop on Large Scale Eigenvalue Problems, 95–119. Elsevier, 1986. http://dx.doi.org/10.1016/s0304-0208(08)72642-2.

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Conference papers on the topic "Generalized eigenvalue algorithm"

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Hang, Tana, Guoren Yang, Bo Yu, Xuesong Liang, and Ying Tang. "Neural Network Based Algorithm for Generalized Eigenvalue Problem." In 2013 International Conference on Information Science and Cloud Computing Companion (ISCC-C). IEEE, 2013. http://dx.doi.org/10.1109/iscc-c.2013.93.

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Song, Junxiao, Prabhu Babu, and Daniel P. Palomar. "A fast algorithm for sparse generalized eigenvalue problem." In 2014 48th Asilomar Conference on Signals, Systems and Computers. IEEE, 2014. http://dx.doi.org/10.1109/acssc.2014.7094747.

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Rong Wang, Feifei Gao, Minli Yao, and Hongxing Zou. "Low complexity adaptive algorithm for generalized eigenvalue decomposition." In 2013 8th International Conference on Communications and Networking in China (CHINACOM). IEEE, 2013. http://dx.doi.org/10.1109/chinacom.2013.6694681.

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Beex, A. A., D. M. Wilkes, and M. P. Fargues. "The C-RISE algorithm and the generalized eigenvalue problem." In [Proceedings] ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing. IEEE, 1991. http://dx.doi.org/10.1109/icassp.1991.150101.

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Yuan, Ganzhao, Li Shen, and Wei-Shi Zheng. "A Decomposition Algorithm for the Sparse Generalized Eigenvalue Problem." In 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). IEEE, 2019. http://dx.doi.org/10.1109/cvpr.2019.00627.

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Zhou, Wenliang, and David Chelidze. "Generalized Eigenvalue Decomposition in Time Domain Modal Parameter Identification." In ASME 2006 International Mechanical Engineering Congress and Exposition. ASMEDC, 2006. http://dx.doi.org/10.1115/imece2006-14702.

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This paper is intended to point out the relationship among current time domain modal analysis methods by employing the generalized eigenvalue decomposition. Various well-known time domain modal analysis algorithms are reviewed. Ibrahim Time Domain (ITD), Least Square Complex Exponent (LSCE) and Eigensystem Realization Algorithm (ERA) methods are chosen to do the comparison. Reformulation to these original forms show these three methods can all be attributed to a generalized eigenvalue problem with different matrix pairs. With this general format, we can see that Single-Input Multi-Output (SIMO) methods can easily be extended to Multi-Input Multi-Output (MIMO) case by taking advantage of the generalized Hankel matrix or generalized Toeplitz matrix.

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Abed-Meraim, Karim, and Samir Attallah. "A new adaptive algorithm for the generalized symmetric eigenvalue problem." In 2007 9th International Symposium on Signal Processing and Its Applications (ISSPA). IEEE, 2007. http://dx.doi.org/10.1109/isspa.2007.4555621.

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Zhou, Youling, Angli Liu, Wencai Du, and Jiping Jiao. "EF-ESPRIT: A simplified ESPRIT algorithm without generalized eigenvalue calculation." In 2013 3rd International Conference on Computer Science and Network Technology (ICCSNT). IEEE, 2013. http://dx.doi.org/10.1109/iccsnt.2013.6967222.

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Lin, R. M., Z. Wang, and M. K. Lim. "A Practical Algorithm for the Efficient Computation of Eigenvector Derivatives." In ASME 1995 Design Engineering Technical Conferences collocated with the ASME 1995 15th International Computers in Engineering Conference and the ASME 1995 9th Annual Engineering Database Symposium. American Society of Mechanical Engineers, 1995. http://dx.doi.org/10.1115/detc1995-0486.

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Abstract Derivatives of eigenvalues and eigenvectors have become increasingly important in the development of modern numerical methods for areas such as structural design optimization, dynamic system identification and dynamic control, and the development of effective and efficient methods for the calculation of such derivatives has remained to be an active research area for several decades. In this paper, a practical algorithm has been developed for efficiently computing eigenvector derivatives of generalized symmetric eigenvalue problems. For eigenvector derivative of a separate mode, the computation only requires the knowledge of eigenvalue and eigenvector of the mode itself and an inverse of system matrix accounts for most computation cost involved. In the case of two close modes, the modal information of both modes is required and the eigenvector derivatives can be accurately determined simultaneously at minor additional computational cost. Further, the method has been extended to the case of practical structural design where structural modifications are made locally and the eigenvalues and eigenvectors and their derivatives are of interest. By combining the proposed algorithm together with the proposed inverse iteration technique and singular value decomposition theory, eigenproperties and their derivatives can be very efficiently computed. Numerical results from a practical finite element model have demonstrated the practicality of the proposed method. The proposed method can be easily incorporated into commercial finite element packages to improve the computational efficiency of eigenderivatives needed for practical applications.

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Rezaei, Shahram. "Generalizing of Numerically Solving Methods of Eigenvalue Problems to Asymmetrical, Damping Included Case." In ASME 2001 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2001. http://dx.doi.org/10.1115/detc2001/cie-21270.

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Abstract In this paper, “Subspace” method is generalized to asymmetrical case. In the new algorithm described here, “Lanczos” method is used to find the first subspace and to solve the eigenvalue problem resulted in generalized subspace method. To solve the standard eigenvalue problem developed by “Lanczos” method “Jacoby” method is used. If eigenvalue problem includes damping matrix, that will be imported in new defined mass and stiffness matrices.

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