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The epsilon algorithm and related topics

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2000年第1-2期122卷 51-80页

作者： Graves-Morris, PR Roberts, DE Salam, A Univ Bradford Sch Comp & Math Bradford BD7 1DP W Yorkshire England Napier Univ Dept Math Edinburgh EH14 1DJ Midlothian Scotland Univ Littoral Lab Math Pures & Appl F-62228 Calais France

The epsilon algorithm is recommended as the best all-purpose acceleration method for slowly converging sequences. It exploits the numerical precision of the data to extrapolate the sequence to its limit. We explain its connections with Pade approximation and continued fractions which underpin its theoretical base. Then we review the most recent extensions of these principles to treat application of the epsilon algorithm to vector-valued sequences, and some related topics. In this paper, we consider the class of methods based on using generalised inverses of vectors, and the formulation specifically includes the complex case wherever possible. (C) 2000 Elsevier Science B.V. All rights reserved.

关键词： epsilon algorithm qd algorithm Pade vector-valued approximant Wynn cross rule star identity compass identity designant

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A topological optimization method for flexible multi-body dynamic system using epsilon algorithm

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STRUCTURAL ENGINEERING AND MECHANICS 2011年第5期37卷 475-487页

作者： Yang, Zhi-Jun Chen, Xin Kelly, Robert Guangdong Univ Technol Fac ElectroMech Engn Guangzhou 510006 Guangdong Peoples R China Columbia Univ Dept Mech Engn New York NY 10027 USA

In a flexible multi-body dynamic system the typical topological optimization method for structures cannot be directly applied, as the stiffness varies with position. In this paper, the topological optimization of the flexible multi-body dynamic system is converted into structural optimization using the equivalent static load method. First, the actual boundary conditions of the control system and the approximate stiffness curve of the mechanism are obtained from a flexible multi-body dynamical simulation. Second, the finite element models are built using the absolute nodal coordination for different positions according to the stiffness curve. For efficiency, the static reanalysis method is utilized to solve these finite element equilibrium equations. Specifically, the finite element equilibrium equations of key points in the stiffness curve are fully solved as the initial solution, and the following equilibrium equations are solved using a reanalysis method with an error controlled epsilon algorithm. In order to identify the efficiency of the elements, a non-dimensional measurement is introduced. Finally, an improved evolutional structural optimization (ESO) method is used to solve the optimization problem. The presented method is applied to the optimal design of a die bonder. The numerical results show that the presented method is practical and efficient when optimizing the design of the mechanism.

关键词： topological optimization the flexible multi-body dynamic system die bonder epsilon algorithm

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Acceleration of the EM algorithm: P-EM versus epsilon algorithm

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COMPUTATIONAL STATISTICS & DATA ANALYSIS 2012年第12期56卷 4122-4137页

作者： Berlinet, A. F. Roland, Ch Univ Montpellier 2 Equipe Probabilites & Stat UMR CNRS 5149 Inst Math & Modelisat Montpellier F-34095 Montpellier 5 France

Among recent methods designed for accelerating the EM algorithm without any modification in the structure of EM or in the statistical model, the parabolic acceleration (P-EM) has proved its efficiency. It does not involve any computation of gradient or hessian matrix and can be used as an additional software component of any fixed point algorithm maximizing some objective function. The vector epsilon algorithm was introduced to reach the same goals. Through geometric considerations, the relationships between the outputs of an improved version of P-EM and those of the vector epsilon algorithm are established. This sheds some light on their different behaviours and explains why the parabolic acceleration of EM outperforms its competitor in most numerical experiments. A detailed analysis of its trajectories in a variety of real or simulated data shows the ability of P-EM to choose the most efficient paths to the global maximum of the likelihood. (C) 2012 Elsevier B.V. All rights reserved.

关键词： EM algorithm Fixed point iteration Acceleration Convergence epsilon algorithm

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Multi-objective Control Method in Structural Strain Interval Reliability Based on epsilon algorithm

Multi-objective Control Method in Structural Strain Interval...

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3rd International Conference on Civil Engineering and Transportation (ICCET 2013)

作者： Sun Zuozhen Meng Guangwei Ma Kai Jilin Univ Coll Mech Sci & Engn Changchun 130022 Peoples R China

ISBN: (纸本)9783038350057

In this article, a kind of structural strain controlled reliability method was presented. The interval extension of multi-objective control algorithm was applied to control the structural strain reliability. The method has realized the goal of controlling the multiple static interval reliability indexes by the control of the structural interval parameters. In order to accelerate the speed of the structural reanalysis, the epsilon algorithm was used in the process of the structural reanalysis when a wide range of modification happened in the interval parameters. This method can both get a satisfactory accuracy, and improve the speed of the reanalysis. Numerical examples show that the method is effective and feasible.

关键词： epsilon algorithm Interval reliability Multi objective control Neumann series Strain control Static Analysis

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Multi-objective Control Method in Structural Stress Interval Reliability Based on epsilon algorithm

Multi-objective Control Method in Structural Stress Interval...

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3rd International Conference on Mechanical Engineering, Materials and Energy (ICMEME 2013)

作者： Sun Zuozhen Meng Guangwei Li Feng Ma Kai Jilin Univ Coll Mech Sci & Engn Changchun 130022 Peoples R China

ISBN: (纸本)9783037859834

A kind of structural stress reliability control method was presented. In this article, the interval extension of multi-objective control algorithm was applied to control the structural stress reliability. By the control of the structural interval parameters, this method has realized the goal of controlling the multiple static interval reliability indexes. To insure the accuracy of the structural reanalysis, the epsilon algorithm was used in the process of the structural reanalysis when a wide range of modification happened in the interval parameters. This method can both get a satisfactory accuracy, and improve the speed of the reanalysis. The results of the example further proved that the proposed method can be effectively applied in the multi-objective control of the structural stress interval reliability.

关键词： epsilon algorithm Interval reliability Multi objective control Neumann series

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A kind of Static Sensitivity Reanalysis of Structure Based on the Improved epsilon algorithm

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Procedia Computer Science 2022年 208卷 650-657页

作者： Kai Ma Bin Wu Jianhang Sun Sengkang Yan School of Mechanical and Aerospace Engineering Jilin University Changchun 13025 P.R. China

This study present a method to calculate first and second order displacement sensitivity based on epsilon algorithm and improved Neumann series. In this paper, analytical expressions of first and second-order sensitivity are derived. They are then decomposed into vectors using the improved Newman series method. Finally, epsilon is used to calculate the result. The method which is suitable for complex structure has the character of fast calculation and high efficiency. The high precision and fast convergence of the proposed method are proved by two numerical examples. The study show that the proposed method has engineering application value.

关键词： Structural reanalysis static sensitivity epsilon algorithm improved neumann series

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An Accelerating Method of Training Neural Networks Based on Vector epsilon algorithm

An Accelerating Method of Training Neural Networks Based on ...

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Third International Conference on Information and Computing(第三届信息与计算科学国际会议 ICIC 2010)

作者： Lian Lian Jianliang Li Yong Jiang School of Science Nanjing University of Science & Technology Nanjing 210094 P.R. China College of Math & Physics Nanjing University of Information Science & Technology Nanjing 210044P.R.

ISBN: (纸本)9781424470815;9780769540474

This paper studied the accelerating convergence of the vector sequences generated by BP algorithm with vector epsilon algorithm, and presented the conclusion that the algorithms is not only convergent but also accelerated. Finally, we tested them for three classical artificial neural network problems. By numerical experiments, results shown that can reduce CPU time for computation and improve the learning efficiency.

关键词： artificial neural networks BP algorithm epsilon algorithm accelerated convergence numerical analysis

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Construction of new generalizations of Wynn's epsilon and rho algorithm by solving finite difference equations in the transformation order

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NUMERICAL algorithmS 2020年第2期83卷 593-627页

作者： Chang, Xiang-Ke He, Yi Hu, Xing-Biao Sun, Jian-Qing Weniger, Ernst Joachim Chinese Acad Sci Acad Math & Syst Sci ICMSEC LSEC POB 2719 Beijing 100190 Peoples R China Univ Chinese Acad Sci Sch Math Sci Beijing 100049 Peoples R China Chinese Acad Sci Wuhan Inst Phys & Math Wuhan 430071 Peoples R China Ocean Univ China Sch Math Sci Qingdao 266100 Peoples R China Univ Regensburg Inst Phys & Theoret Chem D-93040 Regensburg Germany

We construct new sequence transformations based on Wynn's epsilon and rho algorithms. The recursions of the new algorithms include the recursions of Wynn's epsilon and rho algorithm and of Osada's generalized rho algorithm as special cases. We demonstrate the performance of our algorithms numerically by applying them to some linearly and logarithmically convergent sequences as well as some divergent series.

关键词： Convergence acceleration algorithm Sequence transformation epsilon algorithm Rho algorithm

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AUTOMATIC NUMERICAL INVERSION OF THE LAPLACE TRANSFORM [D5]

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ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE 1984年第3期10卷 348-353页

作者： PIESSENS, R HUYSMANS, R University of Leuven

A program for numerical inversion is presented with the following features: The user inserts (i) a subroutine for evaluating the complex function F (p) of the complex variable p, (ii) the abscissa of convergence of F(p), (iii) a value of t, and (iv) the requested absolute or relative accuracy epsilon . The program then returns an approximate value of f(t) in the hope of satisfying the accuracy requirement, or it exists with a statement that the accuracy cannot be achieved.

关键词： complex inversion formula epsilon algorithm integral transform Laplace transform numerical inversion

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A new method for accelerating convergence of alternating series

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APPLIED MATHEMATICS AND COMPUTATION 2007年第2期187卷 1502-1510页

作者： Thukral, R. Pade Res Ctr Leeds LS17 5JS W Yorkshire England

We describe a new method for accelerating the convergence of scalar sequence. We express the new method as a rational fraction, namely the rational approximant. The effectiveness of the new method is compared with the well established methods namely, the Lubkin transformation, the iterated Aitken Delta(2) algorithm, the Levin transformation, the epsilon algorithm and the Brezinski theta algorithm for approximating the partial sum of a given alternating series. Estimates of the partial sum produced by the new rational approximant method are found to be substantially more accurate than the classical methods. (c) 2006 Elsevier Inc. All rights reserved.

关键词： rational approximant iterated Aitken Delta(2) algorithm Lubkin transformation Levin transformation epsilon algorithm Brezinski theta algorithm convergence acceleration

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