A NEW UNDERSTANDING OF THE QR METHOD

CHOHONG MIN 한국산업응용수학회 2010 Journal of the Korean Society for Industrial and A Vol.14 No.1

The QR method is one of the most common methods for calculating the eigenvalues of a square matrix, however its understanding would require complicated and sophisticated mathematical logics. In this article, we present a simple way to understand QR method only with a minimal mathematical knowledge. A deflation technique is introduced, and its combination with the power iteration leads to extracting all the eigenvectors. The orthogonal iteration is then shown to be compatible with the combination of deflation and power iteration. The connection of QR method to orthogonal iteration is then briefly reviewed. Our presentation is unique and easy to understand among many accounts for the QR method by introducing the orthogonal iteration in terms of deflation and power iteration.

Accuracy improvement of the most probable point‐based dimension reduction method using the hessian matrix

Kang, Seong Bin,Park, Jeong Woo,Lee, Ikjin John Wiley Sons, Ltd 2017 International journal for numerical methods in eng Vol.111 No.3

<P><B>Summary</B></P><P>This paper proposes a most probable point (MPP)‐based dimension reduction method (DRM) using the Hessian matrix called HeDRM to improve accuracy of reliability analysis in existing MPP‐based DRM methods. Conventional MPP‐based DRMs contain two types of errors: (1) error due to eliminating cross‐terms of a performance function by using the univariate DRM; (2) error because of dependency of an axis direction after a rotational transformation. The proposed method minimizes the aforementioned errors by utilizing the Hessian matrix of a performance function. By performing an orthogonal transformation using the eigenvectors of the Hessian matrix, the cross‐term effect of the performance function is minimized and the axis direction that results in the most accurate calculation is obtained because the Gaussian quadrature points for numerical integration are arranged along the eigenvector directions. In this way, the error incurred by exiting MPP‐based DRMs can be reduced that leads to more accurate probability of failure estimation. In addition, this paper proposes to allocate the Gaussian quadrature points using the magnitude of the eigenvalues of the Hessian matrix. This allocation makes it possible to predetermine the number of function evaluations required to estimate the probability of failure accurately and efficiently. Copyright © 2016 John Wiley & Sons, Ltd.</P>

Investigation on efficiency and applicability of subspace iteration method with accelerated starting vectors for calculating natural modes of structures

Kim, B.W.,Jung, H.J.,Hong, S.Y. Techno-Press 2011 Structural Engineering and Mechanics, An Int'l Jou Vol.37 No.5

For efficient calculation of natural modes of structures, a numerical scheme which accelerates convergence of the subspace iteration method by employing accelerated starting Lanczos vectors was proposed in 2005. This paper is an extension of the study. The previous study simply showed feasibility of the proposed method by analyzing structures with smaller degrees of freedom. While, the present study verifies efficiency of the proposed method more rigorously by comparing closeness of conventional and accelerated starting vectors to genuine eigenvectors. This study also analyzes an example structure with larger degrees of freedom and more complex constraints in order to investigate applicability of the proposed method.

Investigation on efficiency and applicability of subspace iteration method with accelerated starting vectors for calculating natural modes of structures

B.W. Kim,정형조,S.Y. Hong 국제구조공학회 2011 Structural Engineering and Mechanics, An Int'l Jou Vol.37 No.5

For efficient calculation of natural modes of structures, a numerical scheme which accelerates convergence of the subspace iteration method by employing accelerated starting Lanczos vectors was proposed in 2005. This paper is an extension of the study. The previous study simply showed feasibility of the proposed method by analyzing structures with smaller degrees of freedom. While, the present study verifies efficiency of the proposed method more rigorously by comparing closeness of conventional and accelerated starting vectors to genuine eigenvectors. This study also analyzes an example structure with larger degrees of freedom and more complex constraints in order to investigate applicability of the proposed method.

An Integrated Approach to Measuring Supply Chain Performance

Adisak Theeranuphattana,John C.S. Tang,Do Ba Khang 대한산업공학회 2012 Industrial Engineeering & Management Systems Vol.11 No.1

Chan and Qi (SCM 8/3 (2003) 209) developed an innovative measurement method that aggregates performance measures in a supply chain into an overall performance index. The method is useful and makes a significant contribution to supply chain management. Nevertheless, it can be cumbersome in computation due to its highly complex algorithmic fuzzy model. In aggregating the performance information, weights used by Chan and Qi-which aim to address the imprecision of human judgments-are incompatible with weights in additive models. Furthermore, the default assumption of linearity of its scoring procedure could lead to an inaccurate assessment of the overall performance. This paper addresses these limitations by developing an alternative measurement that takes care of the above. This research integrates three different approaches to multiple criteria decision analysis (MCDA)-the multiattribute value theory (MAVT), the swing weighting method and the eigenvector procedure-to develop a comprehensive assessment of supply chain performance. One case study is presented to demonstrate the measurement of the proposed method. The performance model used in the case study relies on the Supply Chain Operations Reference (SCOR) model level 1. With this measurement method, supply chain managers can easily benchmark the performance of the whole system, and then analyze the effectiveness and efficiency of the supply chain.

An Integrated Approach to Measuring Supply Chain Performance

Theeranuphattana, Adisak,Tang, John C.S.,Khang, Do Ba Korean Institute of Industrial Engineers 2012 Industrial Engineeering & Management Systems Vol.11 No.1

Chan and Qi (SCM 8/3 (2003) 209) developed an innovative measurement method that aggregates performance measures in a supply chain into an overall performance index. The method is useful and makes a significant contribution to supply chain management. Nevertheless, it can be cumbersome in computation due to its highly complex algorithmic fuzzy model. In aggregating the performance information, weights used by Chan and Qi-which aim to address the imprecision of human judgments-are incompatible with weights in additive models. Furthermore, the default assumption of linearity of its scoring procedure could lead to an inaccurate assessment of the overall performance. This paper addresses these limitations by developing an alternative measurement that takes care of the above. This research integrates three different approaches to multiple criteria decision analysis (MCDA)-the multiattribute value theory (MAVT), the swing weighting method and the eigenvector procedure-to develop a comprehensive assessment of supply chain performance. One case study is presented to demonstrate the measurement of the proposed method. The performance model used in the case study relies on the Supply Chain Operations Reference (SCOR) model level 1. With this measurement method, supply chain managers can easily benchmark the performance of the whole system, and then analyze the effectiveness and efficiency of the supply chain.

역거듭제곱방법의 비교

이규봉 배재대학교 자연과학연구소 2005 自然科學論文集 Vol.16 No.1

행렬의 고유치를 계산하는 방법을 가르칠 때 역거듭제곱방법을 소개한다. 이 방법의 알고리즘을 이론적인 면에서 이해하기 쉬운 알고리즘과 계산적인 면에서 효과적인 알고리즘을 비교한다. When teaching on the computing of eigenvalues of a matrix, we introduced the inverse power method. We compared the algorithm which is good for theoretical point of view with that is useful for computational point of view.

퍼지이론을 이용한 조직구성원의 업무수준결정

허식(Sik Heo),황승국(Seung-Gook Hwang) 한국지능시스템학회 2007 한국지능시스템학회논문지 Vol.17 No.2

본 논문에서는 고유벡터법과 평가기준의 관련성 평가에 의한 퍼지종속관계를 이용하여 농협지점의 조직구성원의 업무수준을 결정하는 모델을 제시하고자 한다. 업무수준평가를 위한 평가기준은 두 그룹, 즉 농협에서 이루어지는 업무그룹과 이업무를 하기 위해 필요로 하는 업무요구사항 그룹으로 나뉘어진다. 연구방법으로서는 업무그룹에 대한 가중치, 이행정도 및 신뢰도, 업무요구사항에 대한 항목별 가중치, 이행정도, 신뢰도, 업무그룹과 업무요구사항간의 관련성을 이용하여 현재의 농협지점의 조직구성원의 업무수준을 평가하므로서 농협지점 전체의 업무수준을 평가할 수 있도록 하였다. 이것은 각지점의 업무수준을 동일하게 보고 각 지점을 평가하여 순위를 내고 있는 현재의 평가방법에 개선의 여지가 있음을 보여주는 것으로서 농협 및 이와 유사한 업체의 조직구성원, 부서 및 지점의 평가시 많이 활용될 수 있을 것으로 기대된다. In this paper, we suggest the model how to evaluate the job level of the member of Nong-Hyup branch, using fuzzy subordination relation by estimating the relationship of criteria and eigenevector method. The criteria for the evaluation of job levels are divided into two groups, that is, the job group to do in Nong-Hyup and the job demanding details group that is needed to do this job. The study method used adding weight on the job group and the present level, the itemized weight about job demanding details and the present level, the relationship the job group and the job demanding details. This paper shows that there is room for improvement in the present evaluuation method, which regards the job level of each branch as equal, evaluates each branch and ranks. Therefore we will expect to utilize it a lot when the Nong-Hyup and the branchs and places of like this company are estimated.

Perturbation theory-based performance analysis of TLS-Prony method for natural frequency extraction

Lee, Kyu-Ho,Choi, Gyu-Hyeon,Lee, Joon-Ho Elsevier 2019 Digital signal processing Vol.85 No.-

<P><B>Abstract</B></P> <P>The total least squares Prony method is used for estimating <I>s</I>-plane natural frequencies from noisy late time response. It consists of three steps: In the first step, the minimum eigenvector of a matrix, whose entries are defined from the noisy late time response, should be computed. In the second step, the roots of a polynomial, whose coefficients are the entries of the minimum eigenvector defined in the first step, are estimated. The roots in the second step are defined as <I>z</I>-plane natural frequencies. In the third step, <I>s</I>-plane natural frequencies are obtained from <I>z</I>-plane natural frequencies. In this paper, a rigorous derivation of the mean square error of the estimator in each step due to an additive noise in the late time response is presented: In the first step, it is shown how the minimum eigenvector of a matrix is perturbed due to an additive noise in the late time response. In the second step, it is derived how the roots of a polynomial are perturbed due to perturbation in the coefficients of the polynomial. In the third step, it is derived how the <I>s</I>-plane natural frequencies are perturbed due to perturbation in <I>z</I>-plane natural frequencies. The mean square error for the first step, the second step and the third step are presented in , , and , respectively, and the validity of these expressions is illustrated in the results using simulated response and experimentally measured data. In conclusion, the mean square errors of the minimum eigenvector, the <I>z</I> plane natural frequencies, and the <I>s</I> plane natural frequencies for the total least squares Prony method can be obtained analytically from , and without a computationally intensive Monte-Carlo simulation.</P>

Discussion: Time-threshold maps: Using information from wavelet reconstructions with all threshold values simultaneously

오희석,임예지 한국통계학회 2012 Journal of the Korean Statistical Society Vol.41 No.2

In this study, we consider various methods to estimate the weights of a pairwise comparison matrix in the Analytic Hierarchy Process widely applied in various decision-making fields. This paper uses a data dependent simulation to evaluate the statistical accuracy, minimum violation and minimum norm of the obtaining weight methods from a reciprocal symmetric matrix. No method dominates others in all criteria. Least squares methods perform best in point of mean squared errors; however, the eigenvectors method has an advantage in the minimum norm.