• Korean Journal of Mathematics
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• v.3 no.2
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• pp.309-314
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• 1995

• Bulletin of the Korean Mathematical Society
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• v.51 no.2
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• pp.501-510
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• 2014

We characterize hypergeneralized k-projectors (i.e., $A^k=A^{\dag}$). Also, some representation for the Moore-Penrose inverse of a linear combination of hypergeneralized k-projectors is found and the invertibility for some linear combinations of commuting hypergeneralized k-projectors is considered.

• Journal of Korean Society of Steel Construction
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• v.10 no.3 s.36
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• pp.473-486
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• 1998

A numerically stable technique to remove the limitation in choosing a shift in the subspace iteration method with shift is presented. A major difficulty of the subspace iteration method with shift is that because of singularity problem, a shift close to an eigenvalue can not be used, resulting in slower convergence. This study solves the above singularity problem using side conditions without sacrifice of convergence. The method is always nonsingular even if a shift is an eigenvalue itself. This is one of the significant characteristics of the proposed method. The nonsingularity is proved analytically. The convergence of the proposed method is at least equal to that of the subspace iteration method with shift, and the operation counts of above two methods are almost the same when a large number of eigenpairs are required. To show the effectiveness of the proposed method, two numerical examples are considered.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.12 no.3
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• pp.371-383
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• 1999

An efficient and numerically stable eigensolution method for structures with multiple natural frequencies is presented. The proposed method is developed by improving the well-known subspace iteration method with shift. A major difficulty of the subspace iteration method with shift is that because of singularity problem, a shift close to an eigenvalue can not be used, resulting in slower convergence. In this paper, the above singularity problem has been solved by introducing side conditions without sacrifice of convergence. The proposed method is always nonsingular even if a shift is on a distinct eigenvalue or multiple ones. This is one of the significant characteristics of the proposed method. The nonsingularity is proved analytically. The convergence of the proposed method is at least equal to that of the subspace iteration method with shift, and the operation counts of above two methods are almost the same when a large number of eigenpairs are required. To show the effectiveness of the proposed method, two numerical examples are considered.

• Bulletin of the Korean Mathematical Society
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• v.38 no.1
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• pp.29-53
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• 2001

This note presents a an indirect adaptive control scheme for first-order continuous-time systems. The estimated plant model is controllable and then the adaptive scheme is free from singularities. The singularities are avoided through a modification of the estimated plant parameter vector so that its associated Sylvester matrix is guaranteed to be nonsingular. That properties is achieved by ensuring that the absolute value of its determinant does not lie below a positive threshold. A modification scheme based on the achievement of a modified diagonally dominant Sylvester matrix of the parameter estimates is also given as an alternative method. This diagonal dominance is achieved through estimates modification as a way to guarantee the controllability of the modified estimated model when a controllability measure of the ‘a priori’ estimated model fails. In both schemes, the use of a hysteresis switching function for the modification of the estimates is not required to ensure the nonsingularity of the Sylvester matrix of the estimates.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 1998.10a
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• pp.431-438
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• 1998

An efficient solution method is presented to solve the eigenvalue problem arising in tile dynamic analysis of non-proportionally damped structural systems with multiple or close eigenvalues. The proposed method is obtained by applying the modified Newton-Raphson technique and the orthonormal condition of the eigenvectors to the quadratic eigenvalue problem. Even if the shift value is an eigenvalue of the system, the proposed method guarantees nonsingularity, which is analytically proved. The initial values of the proposed method can be taken as the intermediate results of iteration methods or results of approximate methods. Two numerical examples are also presented to demonstrate the effectiveness of the proposed method and the results are compared with those of the well-known subspace iteration method and the Lanczos method.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 1998.10a
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• pp.84-91
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• 1998

A numerically stable technique to remove tile limitation in choosing a shift in the subspace iteration method with shift is presented. A major difficulty of the subspace iteration method with shift is that because of singularity problem, a shift close to an eigenvalue can not be used, resulting in slower convergence. This study selves the above singularity problem using side conditions without sacrifice of convergence. The method is always nonsingular even if a shiht is an eigenvalue itself. This is one of tile significant characteristics of the proposed method. The nonsingularity is proved analytically. The convergence of the proposed method is at least equal to that of the subspace iteration method with shift, and the operation counts of above two methods are almost the same when a large number of eigenpairs are required. To show the effectiveness of the proposed method, two numerical examples are considered

• Proceedings of the Earthquake Engineering Society of Korea Conference
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• 2003.03a
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• pp.481-490
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• 2003

This paper presents an efficient eigensolution method for non-proportionally damped systems. The proposed method is obtained by applying the accelerated Newton-Raphson technique and the orthonormal condition of the eigenvectors to the linearized form of the quadratic eigenproblem. A step length and a selective scheme are introduced to increase the convergence of the solution. The step length can be evaluated by minimizing the norm of the residual vector using the least square method. While the singularity may occur during factorizing process in other iteration methods such as the inverse iteration method and the subspace iteration method if the shift value is close to an exact eigenvalue, the proposed method guarantees the nonsingularity by introducing the orthonormal condition of the eigenvectors, which can be proved analytically. A numerical example is presented to demonstrate the effectiveness of the proposed method.

• Transactions on Control, Automation and Systems Engineering
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• v.3 no.4
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• pp.217-222
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• 2001

We propose a new two-degree of freedom parallel mechanism for a haptic device and will refer to the mechanism as the SenSation. The SenSation is designed in order to improve the kinematic performanced and to achieve static balance. We use the panto graph mechanisms in order to change the location of active joints, which leads to transform a direct kinematic singularity into a nonsingularity. The direct kinematic singular configurations of the SenSation occur near the workspace boundary. Using the property that position vector of rigid body rotating about a fixed point is normal to the velocity vector, Jacobian matrix is derived. Using the vector method, two different types of singularities of the SenSation can be identified and we discuss the physical significance of each of the three types of singularities. We will compare the kinematic performances(force manipulability ellipsoid, kinematic isotropy) of the SenSation with those of five-var parallel mechanism. By specifying that the potential energy be fixed, the conditions for the static balancing of the SenSation is derived. The static balancing is accomplished by changing the center of mass of the links.

• Computational Structural Engineering
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• v.11 no.1
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• pp.205-216
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• 1998

A solution method is presented to solve the eigenvalue problem arising in the dynamic analysis of nonclassicary damped structural systems with multiple eigenvalues. The proposed method is obtained by applying the modified Newton-Raphson technique and the orthonormal condition of the eigenvectors to the linear eigenproblem through matrix augmentation of the quadratic eigenvalue problem. In the iteration methods such as the inverse iteration method and the subspace iteration method, singularity may be occurred during the factorizing process when the shift value is close to an eigenvalue of the system. However, even though the shift value is an eigenvalue of the system, the proposed method provides nonsingularity, and that is analytically proved. Since the modified Newton-Raphson technique is adopted to the proposed method, initial values are need. Because the Lanczos method effectively produces better initial values than other methods, the results of the Lanczos method are taken as the initial values of the proposed method. Two numerical examples are presented to demonstrate the effectiveness of the proposed method and the results are compared with those of the well-known subspace iteration method and the Lanczos method.