• East Asian mathematical journal
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• 제24권3호
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• pp.289-293
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• 2008

We recently showed in [5] a semilocal convergence theorem that guarantees convergence of Newton's method to a locally unique solution of a nonlinear equation under hypotheses weaker than those of the Newton-Kantorovich theorem [7]. Here, we first weaken Miranda's theorem [1], [9], [10], which is a generalization of the intermediate value theorem. Then, we show that operators satisfying the weakened Newton-Kantorovich conditions satisfy those of the weakened Miranda’s theorem.

• East Asian mathematical journal
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• 제25권2호
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• pp.147-151
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• 2009

We use an improved Newton-Kantorovich theorem introduced in [2] to analyze interior point methods. Our approach requires less number of steps than before [5] to achieve a certain error tolerance for both Newton's and Modified Newton's methods.

• Structural Engineering and Mechanics
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• 제52권1호
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• pp.115-136
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• 2014

In this study, an extended Kantorovich method, employing multi-term displacement functions, is applied to analyze the vibration problem of symmetrically laminated plates with arbitrary boundary conditions. The vibration behaviors of laminated plates are determined based on the variational principle of total energy minimization and the iterative Kantorovich method. The out-of-plane displacement is represented in the form of a series of a sum of products of functions in x and y directions. With a known function in the x or y directions, the formulation for the variation of total potential energy is transformed to a set of governing equations and a set of boundary conditions. The equations and boundary conditions are then numerically solved for the natural frequency and vibration mode shape. The solutions are verified with available solutions from the literature and solutions from the Ritz and finite element analysis. In most cases, the natural frequencies compare very well with the reference solutions. The vibration mode shapes are also very well modeled using the multi-term assumed displacement function in the terms of a power series. With the method used in this study, it is possible to solve the angle-ply plate problem, where the Kantorovich method with single-term displacement function is ineffective.

• 한국해양공학회지
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• 제17권3호
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• pp.21-26
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• 2003

An iteractive Kantorovich method is presented for the vibration analysis of rectangular orthotropic thick plates. Mindlin plate characteristic functions are derived in general forms using the Kantorovich method. Initially, Timoshenko beam functions consistent with the boundary conditions of the plate were used. Through numerical calculations of natural fairs of appropriate models, it has been confirmed that the method presented is superior to the Rayleigh-Ritz analysis or the finite element analysis in both accuracy and computational efficiency.

• 대한수학회지
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• 제51권1호
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• pp.137-162
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• 2014

In this work we study the local and semilocal convergence of the relaxed Newton's method, that is Newton's method with a relaxation parameter 0 < ${\lambda}$ < 2. We give a Kantorovich-like theorem that can be applied for operators defined between two Banach spaces. In fact, we obtain the recurrent sequence that majorizes the one given by the method and we characterize its convergence by a result that involves the relaxation parameter ${\lambda}$. We use a new technique that allows us on the one hand to generalize and on the other hand to extend the applicability of the result given initially by Kantorovich for ${\lambda}=1$.

• Korean Journal of Mathematics
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• 제27권4호
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• pp.1005-1025
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• 2019

The paper is devoted to study local convergence of a continuation method under the assumption of majorant conditions. The method is used to approximate a zero of an operator in Banach space and is of third order. It is seen that the famous Kantorovich-type and Smale-type conditions are special cases of our majorant conditions. This infers that our result is a generalized one in comparison to results based on Kantorovich-type and Smale-type conditions. Finally a number of numerical examples have been computed to show applicability of the convergence analysis.

• 대한조선학회논문집
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• 제33권2호
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• pp.85-95
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• 1996

직사각형 Mindlin판 유추계의 고유진동 및 동적응답의 해석에 대하여 종래의 방법들 보다 정확도 및 계산효율을 향상시킬 수 있는 방법을 추구하였다. Mindlin 판이론에 입각하여 탄성구속경계조건에 대해 정식화 하고, Timoshenko 보함수로부터 시작하는 Kantorovich 방법을 원용하여 Mindlin 판특성함수를 도출했다. 이 Mindlin판 특성함수에 기초하여 등방성후판 및 직교이방성후판에 대해서 반복적 Kantorovich 방법에 의해 고유해를 구하는 방법을 제시했다. 이 방법은 정확도 및 계산효율면에서 종래의 다른 근사해법보다 그 우월성이 인정되며, 특히 중복고유치를 갖는 경우 명확한 고유치 및 고유모드를 얻을 수 있다. 또 모드해석방법에 의한 동적응답계산에 있어서도 보다 더 정확한 결과를 얻을 수 있다.

• Journal of applied mathematics & informatics
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• 제16권1_2호
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• pp.1-17
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• 2004

The famous Newton-Kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of Newton method to a solution of an equation in connection with the Lipschitz continuity of the Frechet-derivative of the operator involved. Using Lipschitz and center-Lipschitz conditions we show that the Newton-Kantorovich hypothesis is weakened. The error bounds obtained under our semilocal convergence result are finer and the information on the location of the solution more precise than the corresponding ones given by the dominating Newton-Kantorovich theorem, and under the same hypotheses/computational cost, since the evaluation of the Lipschitz also requires the evaluation of the center-Lipschitz constant. In the case of local convergence we obtain a larger convergence radius than before. This observation is important in computational mathematics and can be used in connection to projection methods and in the construction of optimum mesh independence refinement strategies.

• 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.

• Structural Engineering and Mechanics
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• 제52권3호
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• pp.595-601
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• 2014

For analysis of the plates and membranes by numerical or analytical methods, the question of choice of the system of functions satisfying the different boundary conditions remains a major challenge to address. It is to this issue that is dedicated this work based on an approach of choice of combinations of trigonometric functions, which are shape functions of a bended beam with the boundary conditions corresponding to the plate support mode. To do this, the shape functions of beam vibrations for strength analysis of the rectangular plates by Kantorovich-Vlasov's method is considered. Using the properties of quasi-orthogonality of those functions allowed assessing to differential equation for every member of the series. Therefore it's proposed some new forms of integration of the beam functions, in order to simplify the problem.