• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.4 no.1
• /
• pp.79-92
• /
• 2000

We investigate numerical properties of Newton's algorithm for improving an eigenpair of a real matrix A using only fixed precision arithmetic. We show that under natural assumptions it produces an eigenpair of a componentwise small relative perturbation of the data matrix A.

• Transactions of the Korean Society of Mechanical Engineers A
• /
• v.21 no.5
• /
• pp.707-718
• /
• 1997

This paper presents an efficient numerical method for the computation of eigenpair derivatives for a real symmetric eigenvalue problem with distinct and multiple eigenvalues. The method has a very simple algorithm and gives an exact solution. Furthermore, it saves computer sotrage and CPU time. The algorithm preserves not only the symmetricity but also the band width of the matrices, allowing efficient computer storage and solution techniques. Results from the proposed method for calculating the eigenpair derivatives are compared with those from Rudisill and Chu's method and Nelson's method which is known efficient one in the case of distinct natural frequencies. As an example to demonstrate the efficiency of the proposed method in the case of distinct eigenvalues, a cantilever plate is considered. The design parameter of the cantilever plate is its thickness. For the eigenvalue problem with multiple natural frequencies, the adjacent eigenvectors are used in the algebraic equation as side conditions, lying adjacent to the multiplicity of multiple natural frequency distinct eigenvalues, which appear when design parameter varies. A cantilever beam is used to demonstrate the efficiency of the proposed method in the case of multiple natural frequencies. Results form the proposed method for calculating the eigenpair derivatives are compared with those from Dailey's method(an amendation of Ojalvo's work) which finds the exact eigenvector derivatives. The design parameter of the cantilever beam is its height. Data is presented showing the amount of CPU time used to compute the first ten eigenpair derivatives by each method. It is important to note that the numerical stability of the proposed method is proved.

• Proceedings of the Computational Structural Engineering Institute Conference
• /
• 1995.10a
• /
• pp.225-233
• /
• 1995

This paper presents an efficient numerical method for computation of eigenpair derivatives for the real symmetric eigenvalue problem with distinct and multiple eigenvalues. The method has very simple algorithm and gives an exact solution. Furthermore, it saves computer storage and CPU time. The algorithm preserves the symmetry and band of the matrices, allowing efficient computer storage and solution techniques. Thus, the algorithm of the proposed method will be inserted easily in the commercial FEM codes. Results of the proposed method for calculating the eigenpair derivatives are compared with those of Rudisill and Chu's method and Nelson's method which is efficient one in the case of distinct natural frequencies. As an example to demonstrate the efficiency of the proposed method in the case of distinct eigenvalues, a cantilever plate is considered. The design parameter of the cantilever plate is its thickness. For the eigenvalue problem with multiple natural frequencies, the adjacent eigenvectors are used in the algebraic equation as side conditions, they lie adjacent to the m (multiplicity of multiple natural frequency) distinct eigenvalues, which appear when design parameter varies. As an example to demonstrate the efficiency of the proposed method in the case of multiple natural frequencies, a cantilever beam is considered. Results of the proposed method fDr calculating the eigenpair derivatives are compared with those of Bailey's method (an amendation of Ojalvo's work) which finds the exact eigenvector derivatives. The design parameter of the cantilever beam is its height. Data is persented showing the amount of CPU time used to compute the first ten eigenpair derivatives by each method. It is important to note that the numerical stability of the proposed method is proved.

• Journal of Industrial Technology
• /
• v.17
• /
• pp.255-260
• /
• 1997

The analysis of eigenvalue and eigenvector is a crucial procedure for many electromagnetic computation problems. Although it is always the case in practice that only selected eigenpairs are needed, computation of eigenpair still seems to be a time-consuming task. In order to compute the eigenpair more quickly, there are two resorts: one is to select a good algorithm with care and another is to use parallelization technique to improve the speed of the computing. In this paper, one of the best eigensolver, the Davidson method, is parallelized on a cluster of workstations. We apply this scheme to a ridged waveguide design problem and obtain promising linear speedup and scalability.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
• /
• 2004.11a
• /
• pp.721-726
• /
• 2004

A simplified method for the computation of first second and higher order derivatives of eigenvalues and eigenvectors derivatives associated with repeated eigenvalues is presented. Adjacent eigenvectors and orthonormal conditions are used to compose an algebraic equation whose order is (n+m)x(n+m), where n is the number of coordinates and m is the number of multiplicity of the repeated eigenvalues. The algebraic equation developed can be used to compute derivatives of both eigenvalues and eigenvectors simultaneously. Since the coefficient matrix in the proposed algebraic equation is non-singular, symmetric and based on N-space it is numerically stable and very efficient compared to previous methods. This method can be consistently applied to structural systems with structural design parameters and mechanical systems with lumped design parameters. To verify the effectiveness of the proposed method, the finite element model of the cantilever beam is considered.

• Proceedings of the KIEE Conference
• /
• 1997.07a
• /
• pp.12-14
• /
• 1997

The analysis of eigenvalue and eigenvector is a crucial procedure for many electromagnetic computation problems. However, eigenpair computation is timing-consuming task. Thus, its parallelization is required for designing large-scale and precision three-dimensional electromagnetic machines. In this paper, the Davidson method is parallelized on a cluster of workstations. Performance of the parallelization scheme is reported. This scheme is applied to a ridged waveguide design problem.

• Journal of KIISE:Computing Practices and Letters
• /
• v.16 no.4
• /
• pp.427-431
• /
• 2010

Multidimensional scaling (MDS) is a widely used method for dimensionality reduction, of which purpose is to represent high-dimensional data in a low-dimensional space while preserving distances among objects as much as possible. MDS has mainly been applied to data visualization and feature selection. Among various MDS methods, the classical MDS is not readily applicable to data which has large numbers of objects, on normal desktop computers due to its computational complexity. More precisely, it needs to solve eigenpair problems on dissimilarity matrices based on Euclidean distance. Thus, running time and required memory of the classical MDS highly increase as n (the number of objects) grows up, restricting its use in large-scale domains. In this paper, we propose an efficient approximation algorithm for the classical MDS based on divide-and-conquer and CUDA. Through a set of experiments, we show that our approach is highly efficient and effective for analysis and visualization of data consisting of several thousands of objects.

• Proceedings of the Earthquake Engineering Society of Korea Conference
• /
• 2000.10a
• /
• pp.176-183
• /
• 2000

A simplified for the eigenpair sensitivities of damped systems is presented. This approach employs a reduced equation to determine the sensitivities of eigenpairs of the damped vibratory systems with distinct eigenvalues. The derivatives of eigenpairs are obtained by solving an algebraic equation with a symmetric coefficient matrix of (n+1) b (n+1) dimension where n is the number of degree of freedom. This is an improved method of the previous work of Lee and Jung. Two equations are used to find eigenvalues derivatives and eigenvector derivatives in their paper. A significant advantage of this approach over Lee and Jung is that one algebraic equation newly developed is enough to compute such eigenvalue derivatives and eigenvector derivatives. Simulation results indicate that the new method is highly efficient in determining the sensitivities of engenpairs of the damped vibratory systems with distrinct eigenvalues.

• Proceedings of the Computational Structural Engineering Institute Conference
• /
• 2000.04b
• /
• pp.277-285
• /
• 2000

This paper presents a very simple procedure for determining the sensitivities of the eigenpairs of damped vibratory system with distinct eigenvalues. The eigenpairs derivatives can be obtained by solving algebraic equation with a symmetric coefficient matrix whose order is (n＋1)×(n＋1), where n is the number of degree of freedom the method is an improvement of recent work by I. W. Lee, D. O. Kim and G. H. Junng; the key idea is that the eigenvalue derivatives and the eigenvector derivatives are obtained at once via only one algebraic equation, instead of using two equations separately as like in Lee and Jung's method Of course, the method preserves the advantages of Lee and Jung's method.

• Journal of the Korean Society for Aviation and Aeronautics
• /
• v.14 no.1
• /
• pp.28-35
• /
• 2006

Substructure coupling or component mode synthesis may be employed in the solution of dynamic problems for large, flexible structures. The model is partitioned into several subdomains, and a generalized Craig-Bampton representation is derived. In this paper the mode sets (normal modes, constraint modes) is employed for model reduction. A generalized model reduction procedure is described. Vaious reduction methods that use constraint modes is described in detail. As examples, a flexible structure and a 10 DOF damped system are analyzed. Comparison with a conventional reduction method based on a complete model is made via eigenpair and dynamic responses.