• Journal of applied mathematics & informatics
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• 제33권5_6호
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• pp.657-670
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• 2015

In this paper, we provide numerical analysis of so-called Legendre Gauss-Radau and Legendre-Gauss collocation methods for ordinary differential equations. After recasting these collocation methods as Runge-Kutta methods, we prove that the Legendre-Gauss collocation method is equivalent to the well-known Gauss method, while the Legendre-Gauss-Radau collocation method does not belong to the classes of Radau IA or Radau IIA methods in the Runge-Kutta literature. Making use of the well-established theory of Runge-Kutta methods, we study stability and accuracy of the Legendre-Gauss-Radau collocation method. Numerical experiments are conducted to confirm our theoretical results on the accuracy and numerical stability of the Legendre-Gauss-Radau collocation method, and compare Legendre-Gauss collocation method with the Gauss method.

• Journal of Information Processing Systems
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• 제19권5호
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• pp.563-575
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• 2023

Trajectory planning is a key technology for unmanned aerial vehicles (UAVs) to achieve complex flight missions. In this paper, a terminal constraints conversion-based Gauss pseudospectral trajectory planning optimization method is proposed. Firstly, the UAV trajectory planning mathematical model is established with considering the boundary conditions and dynamic constraints of UAV. Then, a terminal constraint handling strategy is presented to tackle terminal constraints by introducing new penalty parameters so as to improve the performance index. Combined with Gauss-Legendre collocation discretization, the improved Gauss pseudospectral method is given in detail. Finally, simulation tests are carried out on a four-quadrotor UAV model with different terminal constraints to verify the performance of the proposed method. Test studies indicate that the proposed method performances well in handling complex terminal constraints and the improvements are efficient to obtain better performance indexes when compared with the traditional Gauss pseudospectral method.

• 대한수학회지
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• 제33권4호
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• pp.865-878
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• 1996

We consider numerical methods for finding a solution to a nonlinear system of algebraic equations $$ (1) f(x) = 0, $$ where the function $f : R^n \to R^n$ is ain $x \in R^n$. In [10], a quasi-Gauss-Newton method is proposed and shown the computational efficiency over SQRT algorithm by numerical experiments. The convergence rate of the method has not been proved theoretically. In this paper, we show theoretically that the iterate $x_k$ obtained from the quasi-Gauss-Newton method for the problem (1) actually converges to a root by Kantorovich-type convergence analysis. We also show the rate of convergence of the method is superlinear.

• 대한수학회보
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• 제48권2호
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• pp.303-314
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• 2011

For Ax = b, it has recently been reported that the convergence of the preconditioned Gauss-Seidel iterative method which uses a matrix of the type P = I + S (${\alpha}$) to perform certain elementary row operations on is faster than the basic Gauss-Seidel method. In this paper, we discuss the adaptive Gauss-Seidel iterative method which uses P = I + S (${\alpha}$) + $\bar{K}({\beta})$ as a preconditioner. We present some comparison theorems, which show the rate of convergence of the new method is faster than the basic method and the method in [7] theoretically. Numerical examples show the effectiveness of our algorithm.

• 대한기계학회논문집B
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• 제41권2호
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• pp.117-124
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• 2017

본 연구에서는 Coloring 기법을 적용한 Gauss-Seidel 해법의 연산 성능을 분석하기 위해 2차원과 3차원 전도 열전달 문제를 다양한 격자 크기에서 해석하였다. 지배방정식의 이산화는 유한차분법과 유한요소법을 사용하였다. CPU의 경우에는 상대적으로 작은 격자계에서 연산 성능이 좋으며, 계산에 사용되는 메모리의 크기가 캐시메모리보다 크게 되면 연산 성능이 급격히 떨어진다. 반면에, GPU는 메모리 지연시간 숨김 특성으로 인하여 격자의 수가 충분히 많을 때 연산 성능이 좋다. GPU에 기반한 Colored Gauss-Seidel 해법은 단일 CPU를 이용한 연산에 비해서 각각 최대 7배의 속도 향상을 보인다. 또한, GPU 기반에서 Colored Gauss-Seidel 해법은 Jacobi 보다 약 2배 빠름을 확인하였다.

• 한국전자파학회지:전자파기술
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• 제6권4호
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• pp.20-27
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• 1995

본 논문에서는 직선 배열 응원에 의한 초음파 트랜스듀서의 목적 지향성에 대한 Beam forming을 Gauss 소거법을 이용하여 수치 계산하였다. 하나의 System으로 여러 가지의 조건에 대한 지향성 합성의 실현을 목적으로 하였으며, 목적 지향성으로는 계산에 의해 합성된 선음원에 의한 지향성과 임의로 설정한 Beam 폭의 변화와 방사 방향의 회전에 대한 준이상 Beam을 선택하여 지향성 합성을 시융레이션 하였다. 수치 계산에는 PC(CPU: 80486DX2, RAM 16Mbyte)를 이용하였으며, 수치 계산 결과, 반복법(LMS볍, DFP볍)에 비해 훨씬 빠른 지향성 합성이 가능하고, 또 System 안정도 면에서도 매우 양호함을 확인했다.

• Ocean Systems Engineering
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• 제7권4호
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• pp.399-412
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• 2017

The Gauss-Legendre integral method is applied to numerically evaluate the Green function and its derivatives in finite water depth. In this method, the singular point of the function in the traditional integral equation can be avoided. Moreover, based on the improved Gauss-Laguerre integral method proposed in the previous research, a new methodology is developed through the Gauss-Legendre integral. Using this new methodology, the Green function with the field and source points near the water surface can be obtained, which is less mentioned in the previous research. The accuracy and efficiency of this new method is investigated. The numerical results using a Gauss-Legendre integral method show good agreements with other numerical results of direct calculations and series form in the far field. Furthermore, the cases with the field and source points near the water surface are also considered. Considering the computational efficiency, the method using the Gauss-Legendre integral proposed in this paper could obtain the accurate numerical results of the Green function and its derivatives in finite water depth and can be adopted in the near field.

• 대한수학회논문집
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• 제28권3호
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• pp.603-613
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• 2013

In 2009, Zheng and Miao [B. Zheng and S.-X. Miao, Two new modified Gauss-Seidel methods for linear system with M-matrices, J. Comput. Appl. Math. 233 (2009), 922-930] considered the modified Gauss-Seidel method for solving M-matrix linear system with the preconditioner $P_{max}$. In this paper, we consider the modified Gauss-Seidel method for solving the linear system with the generalized preconditioner $P_{max}({\alpha})$, and study its convergent properties when the coefficient matrix is an H-matrix. Numerical experiments are performed with different examples, and the numerical results verify our theoretical analysis.

• 대한수학회지
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• 제56권2호
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• pp.311-327
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• 2019

The aim of this paper is to extend the applicability of Gauss-Newton method for solving underdetermined nonlinear least squares problems in cases not covered before. The novelty of the paper is the introduction of a restricted convergence domain. We find a more precise location where the Gauss-Newton iterates lie than in earlier studies. Consequently the Lipschitz constants are at least as small as the ones used before. This way and under the same computational cost, we extend the local as well the semilocal convergence of Gauss-Newton method. The new developmentes are obtained under the same computational cost as in earlier studies, since the new Lipschitz constants are special cases of the constants used before. Numerical examples further justify the theoretical results.

• 대한전자공학회논문지TE
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• 제39권3호
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• pp.15-20
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• 2002

본 논문에서 훼손된 신호를 복원하는 방법에 대해서 연구하였다. 기존의 처리방법은 특이점이나 악조건일 경우 수렴속도가 늦어진다는 점과 처리시간이 많이 소요되는 단점이 있다. 이러한 단점을 보완하기 위해 Gauss-Seidel 방법으로 처리하는 방법이 있으나 이러한 경우 신호를 반복해서 처리해야 하므로 처리시간이 많이 소요된다. 이러한 단점(수렴속도, 전체처리시간)을 개선하기 위하여 본 논문에서는 기존의 신호처리(Gauss-Seidel)와 제안된 알고리즘을 적용시켜 비교하여 봄으로써 특이점 혹은 악조건일 경우에도 수렴속도를 고속화 하여 기존의 Gauss-Seidel 신호처리방법보다 처리시간을 단축할 수 있는 신호 처리 방법을 제시하였다.