• Title, Summary, Keyword: Secant Method

### NEW CONVERGENCE CONDITIONS OF SECANT METHODS VIA ALPHA THEORY

• KIM, S.
• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.5 no.2
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• pp.101-115
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• 2001
• Recent theoretical analysis of numerical methods for solving nonlinear systems of equations is represented by alpha theory of Newton method developed Smale et al. The theory was extended to Secant method by providing convergence conditions by Yakoubsohn which the Secant method is treated as an operator defined for analytical functions. We use Secant methods as an iterative scheme with approximations, which results in new convergence conditions. We compare the two conditions and show that the new conditions represent the features of Secant method in a more precise way.

### ON THE ORDER AND RATE OF CONVERGENCE FOR PSEUDO-SECANT-NEWTON'S METHOD LOCATING A SIMPLE REAL ZERO

• Kim, Young Ik
• Journal of the Chungcheong Mathematical Society
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• v.19 no.2
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• pp.133-139
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• 2006
• By combining the classical Newton's method with the pseudo-secant method, pseudo-secant-Newton's method is constructed and its order and rate of convergence are investigated. Given a function $f:\mathbb{R}{\rightarrow}\mathbb{R}$ that has a simple real zero ${\alpha}$ and is sufficiently smooth in a small neighborhood of ${\alpha}$, the convergence behavior is analyzed near ${\alpha}$ for pseudo-secant-Newton's method. The order of convergence is shown to be cubic and the rate of convergence is proven to be $$\frac{f^{{\prime}{\prime}}(\alpha)}{2f^{\prime}(\alpha)}$^2$. Numerical experiments show the validity of the theory presented here and are confirmed via high-precision programming in Mathematica.

### Moment Redistribution for Moment-Resisting Frames using Secant Stiffness Analysis Method (할선강성해석법을 이용한 모멘트저항골조의 모멘트 재분배)

• Park, Hong-Gun;Kim, Chang-Soo;Eom, Tae-Sung
• Proceedings of the Korea Concrete Institute Conference
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• pp.221-224
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• 2008
• A secant stiffness linear analysis method was developed for moment redistribution of moment-resisting frames. In the proposed method, rotational spring models are used for plastic hinges of the members whose flexural moments are needed to be redistributed. At the plastic hinges, secant stiffness is used to address the effect of the flexural stiffness reduced by inelastic deformation. Linear analysis is repeated with adjusted secant stiffness until the flexural equilibrium is satisfied in the structure and members. By using the secant stiffness analysis, the effect of the inelastic deformation on the moment redistribution can be considered. Further, the safety of plastic hinges can be evaluated by comparing the inelastic rotation resulting from the secant stiffness analysis with the rotational capacity of the plastic hinges. For verification, the proposed method was applied to a continuous beam tested in previous study. A application example for a multiple story moment-resisting frame was presented.

### Face Recognition by Using FP-ICA Based on Secant Method

• Cho, Yong-Hyun
• International Journal of Fuzzy Logic and Intelligent Systems
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• v.5 no.2
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• pp.131-135
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• 2005
• This paper proposes an efficient face recognition using independent component analysis(ICA) derived from the fixed point(FP) algorithm based on secant method. The secant method can exclude the complex computation of differential process from the FP based on Newton method. The proposed ICA has been applied to recognize the 20 Yale face images of $324\times324$ pixels. The experimental results show that the proposed ICA is superior to PCA not only in the restoration performance of basis images but also in the recognition performance of the trained images and the test images. Then negative angle as similarity measures has better recognition ratio than city-block and Euclidean.

### EXPANDING THE APPLICABILITY OF SECANT METHOD WITH APPLICATIONS

• Magrenan, A. Alberto;Argyros, Ioannis K.
• Bulletin of the Korean Mathematical Society
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• v.52 no.3
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• pp.865-880
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• 2015
• We present new sufficient convergence criteria for the convergence of the secant-method to a locally unique solution of a nonlinear equation in a Banach space. Our idea uses Lipschitz and center-Lipschitz instead of just Lipschitz conditions in the convergence analysis. The new convergence criteria can always be weaker than the corresponding ones in earlier studies. Numerical examples are also provided in this study to solve equations in cases not possible before.

### Multi Area Economic Dispatch using Secant Method

• Sudhakar, A.V.V.;Chandram, Karri;Jayalaxmi, A.
• Journal of Electrical Engineering and Technology
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• v.8 no.4
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• pp.744-751
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• 2013
• In this paper, Secant method is proposed to solve multi area economic dispatch (MAED) problem. Generator limits of all generators in each area are calculated at given area power demands plus export (or import) using secant method and the generator limits of all generators are modified as modified generator limits. Central economic dispatch (CED) problem is used to determine the output powers of all generators. Here, Secant method is applied to solve the CED problem. The proposed approach has been tested on two-area (two generators in each area) system and four-area (four generators in each area) system. It is observed from various cases that the proposed approach provides optimally best solution in terms of cost with tie line loss with less computational burden.

### A UNIFIED CONVERGENCE ANALYSIS FOR SECANT-TYPE METHODS

• Argyros, Ioannis Konstantinos;Magrenan, Angel Alberto
• Journal of the Korean Mathematical Society
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• v.51 no.6
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• pp.1155-1175
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• 2014
• We present a unified local and semilocal convergence analysis for secant-type methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis includes the computation of the bounds on the limit points of the majorizing sequences involved. Under the same computational cost our semilocal convergence criteria can be weaker; the error bounds more precise and in the local case the convergence balls can be larger and the error bounds tighter than in earlier studies such as [1-3,7-14,16,20,21] at least for the cases of Newton's method and the secant method. Numerical examples are also presented to illustrate the theoretical results obtained in this study.

### AN IMPROVED LOCAL CONVERGENCE ANALYSIS FOR SECANT-LIKE METHOD

• Argyros, Ioannis K.;Hilout, Said
• East Asian mathematical journal
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• v.23 no.2
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• pp.261-270
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• 2007
• We provide a local convergence analysis for Secant-like algorithm for solving nonsmooth variational inclusions in Banach spaces. An existence-convergence theorem and an improvement of the ratio of convergence of this algorithm are given under center-conditioned divided difference and Aubin's continuity concept. Our result compare favorably with related obtained in .

### Secant Method for Economic Dispatch with Generator Constraints and Transmission Losses

• Chandram, K.;Subrahmanyam, N.;Sydulu, M.
• Journal of Electrical Engineering and Technology
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• v.3 no.1
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• pp.52-59
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• 2008
• This paper describes the secant method for solving the economic dispatch (ED) problem with generator constraints and transmission losses. The ED problem is an important optimization problem in the economic operation of a power system. The proposed algorithm involves selection of minimum and maximum incremental costs (lambda values) and then the evaluation of optimal lambda at required power demand is done by secant method. The proposed algorithm has been tested on a power system having 6, 15, and 40 generating units. Studies have been made on the proposed method to solve the ED problem by taking 120 and 200 units with generator constraints. Simulation results of the proposed approach were compared in terms of solution quality, convergence characteristics, and computation efficiency with conventional methods such as lambda iterative method, heuristic methods such as genetic algorithm, and meta-heuristic methods like particle swarm optimization. It is observed from different case studies that the proposed method provides qualitative solutions with less computational time compared to various methods available in the literature.

### Direct Inelastic Slab Design (직접비탄성 슬래브 설계법의 개발)

• Jung Won-Hee;Park Hong-Gun
• Proceedings of the Korea Concrete Institute Conference
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• pp.498-501
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• 2004
• A new slab design using secant stiffness, Direct Inelastic Slab Design, was developed. Since basically the proposed design method uses linear analysis, it is convenient and stable in numerical analysis. At the same time, the proposed design method can accurately estimate the inelastic strength and ductility demands of slab because it can analyzes the inelastic behavior of structure using iterative calculations for secant stiffness. In the present study, the procedure of the proposed design method was established, and a computer program incorporating the proposed method was developed. Design examples using the proposed method were presented, and compared with traditional nonlinear analysis, and experiments. The Direct Inelastic Slab Design, as an integrated analysis/design method, can directly address the design strategy intended by the engineer, such as moment strength and ductility limit. As a result, economical and safe design can be achieved.