• Bulletin of the Korean Mathematical Society
• /
• v.48 no.5
• /
• pp.969-990
• /
• 2011

Assume that X, partitioned into $2{\times}2$ block form, is a solution of the system of quaternion matrix equations $A_1XB_1$ = $C_1,A_2XB_2=C_2$. We in this paper give the maximal and minimal ranks of the submatrices in X, and establish necessary and sufficient conditions for the submatrices to be zero, unique as well as independent. As applications, we consider the common inner inverse G, partitioned into $2{\times}2$ block form, of two quaternion matrices M and N. We present the formulas of the maximal and minimal ranks of the submatrices of G, and describe the properties of the submatrices of G as well. The findings of this paper generalize some known results in the literature.

• Journal of the Korean Society of Surveying, Geodesy, Photogrammetry and Cartography
• /
• v.3 no.2
• /
• pp.48-62
• /
• 1985

A method for the free network adjustment is proposed, based on the application of generalized inverse matrix (g-inverse). If the network adjustment is executed according to the solution with parameters, especially when all coordinates are considered as parameters to keep unity strength, the matrix of normal equation will be singular. This paper discusses the problem of singular matrix and the analysis of accuracy between conventional method and the free network adjustment of trilateration. In case of the adjustment, the RMS errors of adjusted X, Y coordinates are increased to 35.6% in a polygon, central-point figure, and 50.5% in a quadrilateral. In the elements of error ellipse, the RMS errors are decreased by $\pm$24.5% (a) and $\pm$5.0 % (b) in the polygon, $\pm$42.6% (a) and $\pm$49.2% (b) in the quadrilateral. Introduction of free network adjustment, therefore, could be applied to improvement of relative accuracy in the horizontal positioning.

• Journal of KSNVE
• /
• v.9 no.1
• /
• pp.103-112
• /
• 1999

A new modeling and analysis technique for one-dimensional distributed parameter systems is presented. First. discretized equations of motion in Laplace domain are derived by applying discretization methods for partial differential equations of a one-dimensional structure with respect to spatial coordinate. Secondly. the z and inverse z transformations are applied to the discretized equations of motion for obtaining a dynamic matrix for a uniform element. Four different discretization methods are tested with an example. Finally, taking infinite on the number of step for a uniform element leads to an exact dynamic matrix for the uniform element. A generalized modal analysis procedure for eigenvalue analysis and modal expansion is also presented. The resulting element dynamic matrix is tested with a numerical example. Another application example is provided to demonstrate the applicability of the proposed method.

• Computational Structural Engineering
• /
• v.11 no.1
• /
• pp.217-226
• /
• 1998

The purpose of this study is to develop a technique for the shape analysis of plane truss structures under prescribed displacement modes. The shape analysis is performed based on the existence theorem of the solution and the Moore-Penrose generalized inverse matrix. In this paper, the homologous deformation of structures was proposed as prescribed displacement modes, the shape of the structure is determined from these various modes and applied loads. In general, the shape analysis is a kind of inverse problem different from stress analysis, and the governing equation becomes nonlinear. In this regard, Newton-Raphson method was used to solve the nonlinear equation. Three different shape models are investigated as numerical examples to show the accuracy and the effectiveness of the proposed method.

• Architectural research
• /
• v.8 no.2
• /
• pp.37-42
• /
• 2006

Starting from the quadratic optimal control algorithm, this study obtains the relation of the performance index for constrained systems and Gauss's principle. And minimizing a function of the variation in kinetic energy at constrained and unconstrained states with respect to the velocity variation, the dynamic equation is derived and it is shown that the result compares with the generalized inverse method proposed by Udwadia and Kalaba. It is investigated that the responses of a 10-story building are constrained by the installation of a two-bar structure as an application to utilize the derived equations. The structural responses are affected by various factors like the length of each bar, damping, stiffness of the bar structure, and the junction positions of two structures. Under an assumption that the bars have the same mass density, this study determines the junction positions to minimize the total dynamic responses of the structure.

• Journal of Korean Association for Spatial Structures
• /
• v.3 no.2 s.8
• /
• pp.101-107
• /
• 2003

There exists a structural problem for link structures in the unstable state. The primary characteristics of this problem are in the existence of rigid body displacements without strain, and in the possibility of the introduction of prestressing to change an unstable state into a stable state. When we make local linearized incremental equations in order to obtain knowledge about these unstable structures, the determinant of the coefficient matrices is zero, so that we face a numerically unstable situation. This is similar to the situation in the stability problem. To avoid such a difficult situation, in this paper a simple and straightforward method was presented by means of the generalized inverse for the numerical analysis of stability problem.

• Proceedings of the KIEE Conference
• /
• 1995.07b
• /
• pp.729-732
• /
• 1995

This paper presents a Generalized Iteration (GI) which includes power method, inverse power method, shifted inverse power method, and Rayleigh quotient iteration (RQI), and modified RQI (MRQI). Furthermore, we propose a GI-based algorithm to find arbitrary eigenpairs for Hermitian matrices. The proposed algorithm appears to be much faster and more accurate than the valuable generalized MRQI of Hu (GMRQI-Hu). The idea of GI is also employed to speed up the GMRQI-Hu and we propose a modified version of Hu's GMRQI (GMRQI-Hu-mod) which is improved in the convergence rate. Some numerical simulation results are presented to confirm our contributions

• Journal of the Korean Mathematical Society
• /
• v.47 no.4
• /
• pp.831-843
• /
• 2010

Let $\cal{H}$ and $\cal{K}$ be Hilbert spaces and let T, $\tilde{T}$ = T + ${\delta}T$ be bounded operators from $\cal{H}$ into $\cal{K}$. In this article, two facts related to the perturbation bounds are studied. The first one is to find the upper bound of $\parallel\tilde{T}^+\;-\;T^+\parallel$ which extends the results obtained by the second author and enriches the perturbation theory for the Moore-Penrose inverse. The other one is to develop explicit representations of projectors $\parallel\tilde{T}\tilde{T}^+\;-\;TT^+\parallel$ and $\parallel\tilde{T}^+\tilde{T}\;-\;T^+T\parallel$. In addition, some spectral cases related to these results are analyzed.

• Journal of applied mathematics & informatics
• /
• v.32 no.1_2
• /
• pp.61-74
• /
• 2014

This paper describes an iterative method for orthogonal projections $AA^+$ and $A^+A$ of an arbitrary matrix A, where $A^+$ represents the Moore-Penrose inverse. Convergence analysis along with the first and second order error estimates of the method are investigated. Three numerical examples are worked out to show the efficacy of our work. The first example is on a full rank matrix, whereas the other two are on full rank and rank deficient randomly generated matrices. The results obtained by the method are compared with those obtained by another iterative method. The performance measures in terms of mean CPU time (MCT) and the error bounds for computing orthogonal projections are listed in tables. If $Z_k$, k = 0,1,2,... represents the k-th iterate obtained by our method then the sequence of the traces {trace($Z_k$)} is a monotonically increasing sequence converging to the rank of (A). Also, the sequence of traces {trace($I-Z_k$)} is a monotonically decreasing sequence converging to the nullity of $A^*$.

• Journal of Mechanical Science and Technology
• /
• v.18 no.11
• /
• pp.1891-1899
• /
• 2004

The characteristics of dynamic systems subjected to multiple linear constraints are determined by considering the constrained effects. Although there have been many researches to investigate the dynamic characteristics of constrained systems, most of them depend on numerical analysis like Lagrange multipliers method. In 1992, Udwadia and Kalaba presented an explicit form to describe the motion for constrained discrete systems. Starting from the method, this study determines the dynamic characteristics of the systems to have positive semidefinite mass matrix and the continuous systems. And this study presents a closed form to calculate frequency response matrix for constrained systems subjected to harmonic forces. The proposed methods that do not depend on any numerical schemes take more generalized forms than other research results.