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Journal articles on the topic 'Iteration method'

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Published: 4 June 2021
Last updated: 21 February 2023

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1

Maibed, Zena Hussein, and Ali Qasem Thajil. "Zenali Iteration Method For Approximating Fixed Point of A δZA - Quasi Contractive mappings." Ibn AL- Haitham Journal For Pure and Applied Sciences 34, no. 4 (October 20, 2021): 78–92. http://dx.doi.org/10.30526/34.4.2705.

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This article will introduce a new iteration method called the zenali iteration method for the approximation of fixed points. We show that our iteration process is faster than the current leading iterations like Mann, Ishikawa, oor, D- iterations, and *- iteration for new contraction mappings called quasi contraction mappings. And we proved that all these iterations (Mann, Ishikawa, oor, D- iterations and *- iteration) equivalent to approximate fixed points of quasi contraction. We support our analytic proof by a numerical example, data dependence result for contraction mappings type by employing zenali iteration also discussed.
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2

Zhong, Deyun, Liguan Wang, Jinmiao Wang, and Mingtao Jia. "An Efficient Mine Ventilation Solution Method Based on Minimum Independent Closed Loops." Energies 13, no. 22 (November 10, 2020): 5862. http://dx.doi.org/10.3390/en13225862.

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In this paper, according to the analysis of optimum circuits, we present an efficient ventilation network solution based on minimum independent closed loops. Our main contribution is optimizing the circuit dividing strategy to improve the iteration convergence and the efficiency of a single iteration. In contrast to a traditional circuit, a minimum closed loop may contain one or more co-tree branches but fewer high-resistance branches and fan branches. It is helpful in solving the problem of divergence or slow convergence for complex ventilation networks. Moreover, we analyze the dividing rules of closed loops and improve the dividing algorithm of minimum independent closed loops. Compared with the traditional Hardy Cross iteration method, the improved solution method has better iteration convergence and computation efficiency. The experimental results of real-world mine ventilation networks show that the improved solution method converges rapidly within a small number of iterations. We also investigate the influence of network complexity, iterative precision, and initial airflow on the iteration convergence.
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3

Sun, Zhen, and Zilong Zou. "Towards an efficient method of predicting vehicle-induced response of bridge." Engineering Computations 33, no. 7 (October 3, 2016): 2067–89. http://dx.doi.org/10.1108/ec-02-2015-0034.

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Purpose The purpose of this paper is to present a practical and efficient iterative method for predicting vehicle-induced response of bridge. Design/methodology/approach The vehicle-bridge interaction (VBI) problem is generalized mathematically and a computational algorithm for VBI is proposed. This method rests on an iterative procedure, which utilizes the whole interaction process for iteration. By this means, vehicle and bridge become totally uncoupled and are only linked by the contact force history. This method provides flexibility to choose simplified or refined vehicle and bridge models for the VBI problem, as well as open options for different commercial FEM software without specialized codes. Findings The method is verified through two numerical examples. The first example uses a simple 1D beam bridge model, which illustrates the procedure of this method and demonstrates its fast convergence in several iterations. The second example employs a realistic full 3D finite element bridge model, which shows that the method easily connects complex FEM bridge models in ABAQUS with a calibrated vehicle model in Matlab. The dynamic response of the bridge is reliably calculated within only a few iterations. Originality/value The proposed iterative method separates vehicle and bridge into independent subsystems in the computational process, thus providing more flexibility to utilize commercial FEM softwares. Its efficiency is realized through choosing the whole interaction force process for iteration, which considerably reduces the iteration steps.
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4

Zhao, Duo, and Yong Yang. "An Iterative Learning Control Design Method for Nonlinear Discrete-Time Systems with Unknown Iteration-Varying Parameters and Control Direction." Mathematical Problems in Engineering 2016 (2016): 1–7. http://dx.doi.org/10.1155/2016/8971407.

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An iterative learning control (ILC) scheme is designed for a class of nonlinear discrete-time dynamical systems with unknown iteration-varying parameters and control direction. The iteration-varying parameters are described by a high-order internal model (HOIM) such that the unknown parameters in the current iteration are a linear combination of the counterparts in the previous certain iterations. Under the framework of ILC, the learning convergence condition is derived through rigorous analysis. It is shown that the adaptive ILC law can achieve perfect tracking of system state in presence of iteration-varying parameters and unknown control direction. The effectiveness of the proposed control scheme is verified by simulations.
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Rehman, Habib ur, Poom Kumam, Ioannis K. Argyros, Nasser Aedh Alreshidi, Wiyada Kumam, and Wachirapong Jirakitpuwapat. "A Self-Adaptive Extra-Gradient Methods for a Family of Pseudomonotone Equilibrium Programming with Application in Different Classes of Variational Inequality Problems." Symmetry 12, no. 4 (April 2, 2020): 523. http://dx.doi.org/10.3390/sym12040523.

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The main objective of this article is to propose a new method that would extend Popov’s extragradient method by changing two natural projections with two convex optimization problems. We also show the weak convergence of our designed method by taking mild assumptions on a cost bifunction. The method is evaluating only one value of the bifunction per iteration and it is uses an explicit formula for identifying the appropriate stepsize parameter for each iteration. The variable stepsize is going to be effective for enhancing iterative algorithm performance. The variable stepsize is updating for each iteration based on the previous iterations. After numerical examples, we conclude that the effect of the inertial term and variable stepsize has a significant improvement over the processing time and number of iterations.
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6

Liu, Yi Di. "Research on Iterative Method in Solving Linear Equations on the Hadoop Platform." Applied Mechanics and Materials 347-350 (August 2013): 2763–68. http://dx.doi.org/10.4028/www.scientific.net/amm.347-350.2763.

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Solving linear equations is ubiquitous in many engineering problems, and iterative method is an efficient way to solve this question. In this paper, we propose a general iteration method for solving linear equations. Our general iteration method doesnt contain denominators in its iterative formula, and this relaxes the limits that traditional iteration methods require the coefficient aii to be non-zero. Moreover, as there is no division operation, this method is more efficient. We implement this method on the Hadoop platform, and compare it with the Jacobi iteration, the Guass-Seidel iteration and the SOR iteration. Experiments show that our proposed general iteration method is not only more efficient, but also has a good scalability.
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7

Tian, Zhaolu, Xiaoyan Liu, Yudong Wang, and P. H. Wen. "The modified matrix splitting iteration method for computing PageRank problem." Filomat 33, no. 3 (2019): 725–40. http://dx.doi.org/10.2298/fil1903725t.

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In this paper, based on the iteration methods [3,10], we propose a modified multi-step power-inner-outer (MMPIO) iteration method for solving the PageRank problem. In the MMPIO iteration method, we use the multi-step matrix splitting iterations instead of the power method, and combine with the inner-outer iteration [24]. The convergence of the MMPIO iteration method is analyzed in detail, and some comparison results are also given. Several numerical examples are presented to illustrate the effectiveness of the proposed algorithm.
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8

Cao, Jing. "Inner Sequential Single Solid Method for Layout Optimization of Multi-Materials." Journal of Physics: Conference Series 2235, no. 1 (May 1, 2022): 012091. http://dx.doi.org/10.1088/1742-6596/2235/1/012091.

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Abstract In multiple materials layout optimization, the optimal solution is difficult to achieve due to the number of design variables being too large and the material interpolation scheme becoming complicated when using density-like method. To bypass this problem, an inner sequential single solid optimization (ISSSO) method is presented in this study. There are two types of iterations in the optimization process of this method: the inner and outer iterations. In the inner iteration, the original layout optimization of many materials is replaced with a series of single solid sub-optimizations. In each sub-optimization, only one solid is chosen to update and the rest keeps unchanged. The inner iteration stops when all the solids are updated consequently. The next loop of inner iteration starts when the current solution does not meet the requirement of convergence. The outer iteration starts only if the current inner iteration comes to an end. In the outer iteration, the new inner iterations for update of all the solid materials are carried out. The effectiveness of ISSSO method is verified by comparing the calculation results with other methods.
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9

Li, Xu, Yu-Jiang Wu, Ai-Li Yang, and Jin-Yun Yuan. "A Generalized HSS Iteration Method for Continuous Sylvester Equations." Journal of Applied Mathematics 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/578102.

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Based on the Hermitian and skew-Hermitian splitting (HSS) iteration technique, we establish a generalized HSS (GHSS) iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and positive definite/semidefinite matrices. The GHSS method is essentially a four-parameter iteration which not only covers the standard HSS iteration but also enables us to optimize the iterative process. An exact parameter region of convergence for the method is strictly proved and a minimum value for the upper bound of the iterative spectrum is derived. Moreover, to reduce the computational cost, we establish an inexact variant of the GHSS (IGHSS) iteration method whose convergence property is discussed. Numerical experiments illustrate the efficiency and robustness of the GHSS iteration method and its inexact variant.
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Al-shameri, Wadia Faid Hassan, and Mohamed El Sayed. "Fractals Generated via Numerical Iteration Method." Fractal and Fractional 6, no. 4 (March 31, 2022): 196. http://dx.doi.org/10.3390/fractalfract6040196.

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In this research article, a modified algorithm for the generation of a fractal pattern resulting from the iteration of an algebraic function using the numerical iteration method is presented. This fractal pattern shows the dynamical behavior of the numerical iterations. A nonstandard convergence test of the displayable fractal pattern was applied.
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11

Walker, W. Thomas, Thomas F. Rossi, and Nazrul Islam. "Method of Successive Averages Versus Evans Algorithm: Iterating a Regional Travel Simulation Model to the User Equilibrium Solution." Transportation Research Record: Journal of the Transportation Research Board 1645, no. 1 (January 1998): 32–40. http://dx.doi.org/10.3141/1645-05.

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The results of comparative tests of two methods for iterating a regional travel demand model system are presented. Model iteration is necessary to ensure consistency between model input and output speeds, as required by current federal legislation. Two methods were tested: the Evans algorithm and the method of successive averages. A series of tests using alternative assignment techniques was conducted for each method. Criteria for evaluating the iteration methods included convergence error, average highway speeds compared with observations, highway vehicle miles traveled compared with Highway Performance Monitoring System estimates, transit boardings compared with observations, and computer running time. It was concluded that the Evans algorithm performed the best, primarily on the basis of superior computational efficiency, although good results were obtained by using the method of successive averages. Use of the Evans algorithm is recommended, embedded within a formal assignment restart, for iterating the model system. Multiple iterations of highway assignment should be used in the initial model loop and all-or-nothing assignments in subsequent iterations of the modeling chain.
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12

Jiang, Kai, Jianghao Su, and Juan Zhang. "A Data-Driven Parameter Prediction Method for HSS-Type Methods." Mathematics 10, no. 20 (October 14, 2022): 3789. http://dx.doi.org/10.3390/math10203789.

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Some matrix-splitting iterative methods for solving systems of linear equations contain parameters that need to be specified in advance, and the choice of these parameters directly affects the efficiency of the corresponding iterative methods. This paper uses a Bayesian inference-based Gaussian process regression (GPR) method to predict the relatively optimal parameters of some HSS-type iteration methods and provide extensive numerical experiments to compare the prediction performance of the GPR method with other existing methods. Numerical results show that using GPR to predict the parameters of the matrix-splitting iterative methods has the advantage of smaller computational effort, predicting more optimal parameters and universality compared to the currently available methods for finding the parameters of the HSS-type iteration methods.
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13

Al-Mahdawi, Hassan K. Ibrahim, Hussein Alkattan, Mostafa Abotaleb, Ammar Kadi, and El-Sayed M. El-kenawy. "Updating the Landweber Iteration Method for Solving Inverse Problems." Mathematics 10, no. 15 (August 7, 2022): 2798. http://dx.doi.org/10.3390/math10152798.

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The Landweber iteration method is one of the most popular methods for the solution of linear discrete ill-posed problems. The diversity of physical problems and the diversity of operators that result from them leads us to think about updating the main methods and algorithms to achieve the best results. We considered in this work the linear operator equation and the use of a new version of the Landweber iterative method as an iterative solver. The main goal of updating the Landweber iteration method is to make the iteration process fast and more accurate. We used a polar decomposition to achieve a symmetric positive definite operator instead of an identity operator in the classical Landweber method. We carried out the convergence and other necessary analyses to prove the usability of the new iteration method. The residual method was used as an analysis method to rate the convergence of the iteration. The modified iterative method was compared to the classical Landweber method. A numerical experiment illustrates the effectiveness of this method by applying it to solve the inverse boundary value problem of the heat equation (IBVP).
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14

Kobelkov, George M., and Eckart Schnack. "Connection between the existence of a priori estimate for a flux and the convergence of iterative methods for diffusion equation with highly varying coefficients." Russian Journal of Numerical Analysis and Mathematical Modelling 37, no. 3 (June 1, 2022): 143–47. http://dx.doi.org/10.1515/rnam-2022-0012.

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Abstract An iterative method with the number of iterations independent of the coefficient jumps is proposed for the boundary value problem for a diffusion equation with highly varying coefficient. The method applies one solution of the Poisson equation at each step of iteration. In the present paper we extend the class of domains the iterative method is justified for.
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15

Hong, Baojian, and Dianchen Lu. "Modified Fractional Variational Iteration Method for Solving the Generalized Time-Space Fractional Schrödinger Equation." Scientific World Journal 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/964643.

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Based on He’s variational iteration method idea, we modified the fractional variational iteration method and applied it to construct some approximate solutions of the generalized time-space fractional Schrödinger equation (GFNLS). The fractional derivatives are described in the sense of Caputo. With the help of symbolic computation, some approximate solutions and their iterative structure of the GFNLS are investigated. Furthermore, the approximate iterative series and numerical results show that the modified fractional variational iteration method is powerful, reliable, and effective when compared with some classic traditional methods such as homotopy analysis method, homotopy perturbation method, adomian decomposition method, and variational iteration method in searching for approximate solutions of the Schrödinger equations.
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16

Li, Yazhe, Kai Zhou, and Zhen Zhang. "The flow-difference feedback iteration method for aerostatic thrust bearings and its convergence characteristics." Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 233, no. 11 (April 29, 2019): 1743–52. http://dx.doi.org/10.1177/1350650119846230.

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The flow-difference feedback iteration method for aerostatic thrust bearings is proposed to the flow balance-based iteration, and two modification methods are further provided to improve the adaptability. The bearing capability calculated by the proposed method is validated by the experimental data. Moreover, the influence of convergence rate factors, iterative initial values, and mesh grids on the iteration ratio is investigated. Compared with the conventional iteration methods, the proposed method with appropriate convergence rate factors provides a higher convergence efficiency. In addition, good convergence behavior under different iterative initial values and the mesh grid size is shown, and the convergence rate is insensitive to the finite difference method parameters. A series of calculations are conducted to investigate the generality of the proposed methods.
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17

Mat Ali, Nur Afza, Jumat Sulaiman, Azali Saudi, and Nor Syahida Mohamad. "Performance of similarity explicit group iteration for solving 2D unsteady convection-diffusion equation." Indonesian Journal of Electrical Engineering and Computer Science 23, no. 1 (July 1, 2021): 471. http://dx.doi.org/10.11591/ijeecs.v23.i1.pp471-478.

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In this paper, a similarity finite difference (SFD) solution is addressed for thetwo-dimensional (2D) parabolic partial differential equation (PDE), specifically on the unsteady convection-diffusion problem. Structuring the similarity transformation using wave variables, we reduce the parabolic PDE into elliptic PDE. The numerical solution of the corresponding similarity equation is obtained using a second-order central SFD discretization schemeto get the second-order SFD approximation equation. We propose a four-point similarity explicit group (4-point SEG) iterative methodasa numericalsolution of the large-scale and sparse linear systems derived from SFD discretization of 2D unsteady convection-diffusion equation (CDE). To showthe 4-point SEG iteration efficiency, two iterative methods, such as Jacobiand Gauss-Seidel (GS) iterations, are also considered. The numerical experiments are carried out using three different problems to illustrate our proposed iterative method's performance. Finally, the numerical results showed that our proposed iterative method is more efficient than the Jacobiand GS iterations in terms of iteration number and execution time.
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18

Sablin, M. "An iterative method for solving difference problems of gas dynamics in the mixed Euler-Lagrangian variables." Journal of Physics: Conference Series 2099, no. 1 (November 1, 2021): 012013. http://dx.doi.org/10.1088/1742-6596/2099/1/012013.

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Abstract The method proposed is intended to solve implicit conservative operator difference schemes for a grid initial-boundary value problems on a simplex grid for a system of equations of gas dynamics in the mixed Euler-Lagrangian variables. To find a solution to such a scheme at a time step, it is represented as a single equation for a nonlinear function of two arguments from space – the direct product of the grid spaces of gas-dynamic quantities. To solve such an equation, a combination of the generalized Gauss-Seidel iterative method (external iterations) and an implicit two-layer iteration scheme (internal iterations at each external iteration) is used. The feature of the method is that, the equation, which is solved by internal iterations, is obtained from the equation of the difference scheme using symmetrization – such a non-degenerate linear transformation that the function in this equation has a self-adjoint positive Frechet derivative.
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19

Hernández, A., V. Petuya, and E. Amezua. "A method for the solution of the forward position problems of planar mechanisms with prismatic and revolute joints." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 216, no. 4 (April 1, 2001): 395–407. http://dx.doi.org/10.1243/0954406021525197.

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In this paper, a method to solve the forward position problems of planar linkages with prismatic and revolute joints is presented. These linkages can have any number of degrees of freedom. This method has been named the geometrical iterative method and is based on geometrical concepts. An iteration sequence that corresponds to the system of non-linear equations describing closure of the mechanism loops is defined. This sequence is applied in successive iterations to obtain the position of the mechanism. In order to achieve convergence, the iteration sequence must fulfil two fundamental conditions. A searching algorithm has been developed to obtain a useful iteration sequence. It is based on the use of hierarchical rules and criteria. The method has been implemented in a simulation program developed by the authors. Several illustrative examples are presented using representative linkages.
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20

Nasikun, Ahmad, and Klaus Hildebrandt. "The Hierarchical Subspace Iteration Method for Laplace–Beltrami Eigenproblems." ACM Transactions on Graphics 41, no. 2 (April 30, 2022): 1–14. http://dx.doi.org/10.1145/3495208.

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Sparse eigenproblems are important for various applications in computer graphics. The spectrum and eigenfunctions of the Laplace–Beltrami operator, for example, are fundamental for methods in shape analysis and mesh processing. The Subspace Iteration Method is a robust solver for these problems. In practice, however, Lanczos schemes are often faster. In this article, we introduce the Hierarchical Subspace Iteration Method (HSIM) , a novel solver for sparse eigenproblems that operates on a hierarchy of nested vector spaces. The hierarchy is constructed such that on the coarsest space all eigenpairs can be computed with a dense eigensolver. HSIM uses these eigenpairs as initialization and iterates from coarse to fine over the hierarchy. On each level, subspace iterations, initialized with the solution from the previous level, are used to approximate the eigenpairs. This approach substantially reduces the number of iterations needed on the finest grid compared to the non-hierarchical Subspace Iteration Method. Our experiments show that HSIM can solve Laplace–Beltrami eigenproblems on meshes faster than state-of-the-art methods based on Lanczos iterations, preconditioned conjugate gradients, and subspace iterations.
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21

Wang, Shou Jun, and Ming Wei Wei. "Application of Accelerated Iterative Method in Calculating Wave Length in Harbor Engineering." Applied Mechanics and Materials 130-134 (October 2011): 3481–84. http://dx.doi.org/10.4028/www.scientific.net/amm.130-134.3481.

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In this paper, a specific application accelerated iterative method is presented for calculating wave length in harbor engineering, which includes calculation method of wave length and specific implement in Excel. Different wavelengths into the iteration formula to calculate the same result can be obtained, but the calculation speed of different methods have significant differences to arrive at the fastest method . Calculated by accelerating the iteration method can significantly increase the computing speed and calculation steps. After the derivation of several methods and calculations show that Newton iteration is the fastest way to convergence speed, in the practical range of about 10 steps through the iterative convergence results can be obtained.
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Svirakova, Eva. "Dynamic Iteration Method: New Approach to Cultural Events Management." International Journal of Systems Applications, Engineering & Development 15 (November 26, 2021): 116–23. http://dx.doi.org/10.46300/91015.2021.15.16.

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This article provides project managers, the cultural events organizers, a new approach to plan preparation and to the monitoring of events realization. The Dynamic Iteration Method introduced in this article is based on the system dynamic modelling and on the principles of project iterative development. The plan model and the reality model are structurally similar; they differ in values of exogenous variables. The new approach enables to easily monitor the real project course in close connection with the plan and to take timely controlling steps. The effects resulting from the manager´s decisionmaking process are compared with the plan in regular iterations. The method thus reminds of a traveller whose route is adjusted by a GPS navigation system.
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23

Faraz, Naeem, Yasir Khan, and Francis Austin. "An Alternative Approach to Differential-Difference Equations Using the Variational Iteration Method." Zeitschrift für Naturforschung A 65, no. 12 (December 1, 2010): 1055–59. http://dx.doi.org/10.1515/zna-2010-1206.

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Although a variational iteration algorithm was proposed by Yildirim (Math. Prob. Eng. 2008 (2008), Article ID 869614) that successfully solves differential-difference equations, the method involves some repeated and unnecessary iterations in each step. An alternative iteration algorithm (variational iteration algorithm-II) is constructed in this paper that overcomes this shortcoming and promises to provide a universal mathematical tool for many differential-difference equations.
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BAKIR, YASEMİN, OYA MERT, and ÖZLEM ORHAN. "ON COMPARISON OF SOLUTION OF ORDINARY DIFFERENTIAL EQUATION WITH HAAR WAVELET METHOD AND THE MODIFIED ISHIKAWA ITERATION METHOD." Journal of Science and Arts 22, no. 2 (June 30, 2022): 389–94. http://dx.doi.org/10.46939/j.sci.arts-22.2-a12.

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In this study, we have used a newly modified Ishikawa iteration method and the Haar wavelet method to solve an ordinary linear differential equation with initial conditions. Using the modified Ishikawa iteration approach, we derive approximate solutions to the issue as well as the related iterative schemes. For this problem, the Ishikawa Iteration Method is applied for different lambda and gamma values and approximation solutions for these values are compared with the approximate solution of Haar wavelet collocation and its exact solution. Finally, the error tables are written and the graphs are shown.
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Xu, Kai, and Zhi Xiong. "Nonparametric Tensor Completion Based on Gradient Descent and Nonconvex Penalty." Symmetry 11, no. 12 (December 12, 2019): 1512. http://dx.doi.org/10.3390/sym11121512.

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Existing tensor completion methods all require some hyperparameters. However, these hyperparameters determine the performance of each method, and it is difficult to tune them. In this paper, we propose a novel nonparametric tensor completion method, which formulates tensor completion as an unconstrained optimization problem and designs an efficient iterative method to solve it. In each iteration, we not only calculate the missing entries by the aid of data correlation, but consider the low-rank of tensor and the convergence speed of iteration. Our iteration is based on the gradient descent method, and approximates the gradient descent direction with tensor matricization and singular value decomposition. Considering the symmetry of every dimension of a tensor, the optimal unfolding direction in each iteration may be different. So we select the optimal unfolding direction by scaled latent nuclear norm in each iteration. Moreover, we design formula for the iteration step-size based on the nonconvex penalty. During the iterative process, we store the tensor in sparsity and adopt the power method to compute the maximum singular value quickly. The experiments of image inpainting and link prediction show that our method is competitive with six state-of-the-art methods.
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Zhang, Yanmei, Xia Cui, and Guangwei Yuan. "Nonlinear iteration acceleration solution for equilibrium radiation diffusion equation." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 5 (June 26, 2020): 1465–90. http://dx.doi.org/10.1051/m2an/2019095.

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This paper discusses accelerating iterative methods for solving the fully implicit (FI) scheme of equilibrium radiation diffusion problem. Together with the FI Picard factorization (PF) iteration method, three new nonlinear iterative methods, namely, the FI Picard-Newton factorization (PNF), FI Picard-Newton (PN) and derivative free Picard-Newton factorization (DFPNF) iteration methods are studied, in which the resulting linear equations can preserve the parabolic feature of the original PDE. By using the induction reasoning technique to deal with the strong nonlinearity of the problem, rigorous theoretical analysis is performed on the fundamental properties of the four iteration methods. It shows that they all have first-order time and second-order space convergence, and moreover, can preserve the positivity of solutions. It is also proved that the iterative sequences of the PF iteration method and the three Newton-type iteration methods converge to the solution of the FI scheme with a linear and a quadratic speed respectively. Numerical tests are presented to confirm the theoretical results and highlight the high performance of these Newton acceleration methods.
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27

He, Y., and A. Bilgic. "Iterative least squares method for global positioning system." Advances in Radio Science 9 (August 1, 2011): 203–8. http://dx.doi.org/10.5194/ars-9-203-2011.

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Abstract. The efficient implementation of positioning algorithms is investigated for Global Positioning System (GPS). In order to do the positioning, the pseudoranges between the receiver and the satellites are required. The most commonly used algorithm for position computation from pseudoranges is non-linear Least Squares (LS) method. Linearization is done to convert the non-linear system of equations into an iterative procedure, which requires the solution of a linear system of equations in each iteration, i.e. linear LS method is applied iteratively. CORDIC-based approximate rotations are used while computing the QR decomposition for solving the LS problem in each iteration. By choosing accuracy of the approximation, e.g. with a chosen number of optimal CORDIC angles per rotation, the LS computation can be simplified. The accuracy of the positioning results is compared for various numbers of required iterations and various approximation accuracies using real GPS data. The results show that very coarse approximations are sufficient for reasonable positioning accuracy. Therefore, the presented method reduces the computational complexity significantly and is highly suited for hardware implementation.
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Delgado, Paul M., V. M. Krushnarao Kotteda, and Vinod Kumar. "Hybrid Fixed-Point Fixed-Stress Splitting Method for Linear Poroelasticity." Geosciences 9, no. 1 (January 8, 2019): 29. http://dx.doi.org/10.3390/geosciences9010029.

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Efficient and accurate poroelasticity models are critical in modeling geophysical problems such as oil exploration, gas-hydrate detection, and hydrogeology. We propose an efficient operator splitting method for Biot’s model of linear poroelasticity based on fixed-point iteration and constrained stress. In this method, we eliminate the constraint on time step via combining our fixed-point approach with a physics-based restraint between iterations. Three different cases are considered to demonstrate the stability and consistency of the method for constant and variable parameters. The results are validated against the results from the fully coupled approach. In case I, a single iteration is used for continuous coefficients. The relative error decreases with an increase in time. In case II, material coefficients are assumed to be linear. In the single iteration approach, the relative error grows significantly to 40% before rapidly decaying to zero. This is an artifact of the approximate solutions approaching the asymptotic solution. The error in the multiple iterations oscillates within 10 − 6 before decaying to the asymptotic solution. Nine iterations per time step are enough to achieve the relative error close to 10 − 7 . In the last case, the hybrid method with multiple iterations requires approximately 16 iterations to make the relative error 5 × 10 − 6 .
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29

Douglas, Jim, Jeffrey L. Hensley, and Jean E. Roberts. "An alternating-direction iteration method for Helmholtz problems." Applications of Mathematics 38, no. 4 (1993): 289–300. http://dx.doi.org/10.21136/am.1993.104557.

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30

Wang, Fengpeng, Feifan Fan, Deren Yuan, and Xinghua Wang. "In-Line and Off-Axis Hybrid Digital Holographic Imaging Method Based on Off-Axis Hologram Constraints." International Journal of Optics 2022 (November 25, 2022): 1–9. http://dx.doi.org/10.1155/2022/6577057.

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We put forward a novel hybrid iterative algorithm to improve the imaging quality of digital holography. An off-axis hologram is added to the iteration process via interference and inverse interference process and becomes part of the constraints. A frequency domain filter varying with the number of iterations is used to improve the competitive advantage of low frequency information in the early iterations, while retaining the high frequency information. In practical applications, an additional iterative process is used after averaging filtering to suppress the influence of the imperfect consistency between the reconstructed reference wave and the actual reference wave. Numerical simulations and experiments show that image reconstruction may be significantly improved compared to the conventional method.
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31

Pour, H. Noormohammadi, and H. Sadeghi Goughery. "Generalized Accelerated Hermitian and Skew-Hermitian Splitting Methods for Saddle-Point Problems." Numerical Mathematics: Theory, Methods and Applications 10, no. 1 (February 2017): 167–85. http://dx.doi.org/10.4208/nmtma.2017.m1524.

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AbstractWe generalize the accelerated Hermitian and skew-Hermitian splitting (AHSS) iteration methods for large sparse saddle-point problems. These methods involve four iteration parameters whose special choices can recover the preconditioned HSS and accelerated HSS iteration methods. Also a new efficient case is introduced and we theoretically prove that this new method converges to the unique solution of the saddle-point problem. Numerical experiments are used to further examine the effectiveness and robustness of iterations.
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32

Huang, Dejian, Yanqing Li, and Donghe Pei. "Determining An Unknown Boundary Condition by An Iteration Method." Symmetry 10, no. 9 (September 18, 2018): 409. http://dx.doi.org/10.3390/sym10090409.

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This paper investigates the boundary value in the heat conduction problem by a variational iteration method. Applying the iteration method, a sequence of convergent functions is constructed, the limit approximates the exact solution of the heat conduction equation in a few iterations using only the initial condition. This method does not require discretization of the variables. Numerical results show that this method is quite simple and straightforward for models that are currently under research.
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33

Castaño, F., L. Laín, M. N. Sanchez, and A. Torre. "A general iterative time-independent perturbation theory." Canadian Journal of Physics 63, no. 9 (September 1, 1985): 1157–61. http://dx.doi.org/10.1139/p85-189.

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An iterative method for time-independent perturbation theory is presented. Lennard-Jones–Brillouin–Wigner (LBW) and Rayleigh–Schrödinger (RS) power series are shown to be particular cases of the iteration and the combined expansion–iteration. Improvements in convergence of the power series are suggested and analyzed.The iterative method gives considerable insight into the nature and relative convergence of the currently used time-independent perturbation methods.
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34

Zhou, Jituan, Ruirui Wang, and Qiang Niu. "A Preconditioned Iteration Method for Solving Sylvester Equations." Journal of Applied Mathematics 2012 (2012): 1–12. http://dx.doi.org/10.1155/2012/401059.

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A preconditioned gradient-based iterative method is derived by judicious selection of two auxil- iary matrices. The strategy is based on the Newton’s iteration method and can be regarded as a generalization of the splitting iterative method for system of linear equations. We analyze the convergence of the method and illustrate that the approach is able to considerably accelerate the convergence of the gradient-based iterative method.
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35

RAMADHANI UTAMI, NANDA NINGTYAS, I. NYOMAN WIDANA, and NI MADE ASIH. "PERBANDINGAN SOLUSI SISTEM PERSAMAAN NONLINEAR MENGGUNAKAN METODE NEWTON-RAPHSON DAN METODE JACOBIAN." E-Jurnal Matematika 2, no. 2 (May 31, 2013): 11. http://dx.doi.org/10.24843/mtk.2013.v02.i02.p032.

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System of nonlinear equations is a collection of some nonlinear equations. The Newton-Raphson method and Jacobian method are methods used for solving systems of nonlinear equations. The Newton-Raphson methods uses first and second derivatives and indeed does perform better than the steepest descent method if the initial point is close to the minimizer. Jacobian method is a method of resolving equations through iteration process using simultaneous equations. If the Newton-Raphson methods and Jacobian methods are compared with the exact value, the Jacobian method is the closest to exact value but has more iterations. In this study the Newton-Raphson method gets the results faster than the Jacobian method (Newton-Raphson iteration method is 5 and 58 in the Jacobian iteration method). In this case, the Jacobian method gets results closer to the exact value.
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36

Cheng, Jun-Bo, Yueling Jia, Song Jiang, Eleuterio F. Toro, and Ming Yu. "A Second-Order Cell-Centered Lagrangian Method for Two-Dimensional Elastic-Plastic Flows." Communications in Computational Physics 22, no. 5 (October 31, 2017): 1224–57. http://dx.doi.org/10.4208/cicp.oa-2016-0173.

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AbstractFor 2D elastic-plastic flows with the hypo-elastic constitutive model and von Mises’ yielding condition, the non-conservative character of the hypo-elastic constitutive model and the von Mises’ yielding condition make the construction of the solution to the Riemann problem a challenging task. In this paper, we first analyze the wave structure of the Riemann problem and develop accordingly aFour-Rarefaction wave approximateRiemannSolver withElastic waves (FRRSE). In the construction of FRRSE one needs to use an iterative method. A direct iteration procedure for four variables is complex and computationally expensive. In order to simplify the solution procedure we develop an iteration based on two nested iterations upon two variables, and our iteration method is simple in implementation and efficient. Based on FRRSE as a building block, we propose a 2nd-order cell-centered Lagrangian numerical scheme. Numerical results with smooth solutions show that the scheme is of second-order accuracy. Moreover, a number of numerical experiments with shock and rarefaction waves demonstrate the scheme is essentially non-oscillatory and appears to be convergent. For shock waves the present scheme has comparable accuracy to that of the scheme developed by Maire et al., while it is more accurate in resolving rarefaction waves.
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37

Lin, R. F., H. M. Ren, Z. Šmarda, Q. B. Wu, Y. Khan, and J. L. Hu. "New Families of Third-Order Iterative Methods for Finding Multiple Roots." Journal of Applied Mathematics 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/812072.

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Two families of third-order iterative methods for finding multiple roots of nonlinear equations are developed in this paper. Mild conditions are given to assure the cubic convergence of two iteration schemes (I) and (II). The presented families include many third-order methods for finding multiple roots, such as the known Dong's methods and Neta's method. Some new concrete iterative methods are provided. Each member of the two families requires two evaluations of the function and one of its first derivative per iteration. All these methods require the knowledge of the multiplicity. The obtained methods are also compared in their performance with various other iteration methods via numerical examples, and it is observed that these have better performance than the modified Newton method, and demonstrate at least equal performance to iterative methods of the same order.
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Thorbecke, Jan, Evert Slob, Joeri Brackenhoff, Joost van der Neut, and Kees Wapenaar. "Implementation of the Marchenko method." GEOPHYSICS 82, no. 6 (November 1, 2017): WB29—WB45. http://dx.doi.org/10.1190/geo2017-0108.1.

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The Marchenko method makes it possible to compute subsurface-to-surface Green’s functions from reflection measurements at the surface. Applications of the Marchenko method have already been discussed in many papers, but its implementation aspects have not yet been discussed in detail. Solving the Marchenko equation is an inverse problem. The Marchenko method computes a solution of the Marchenko equation by an (adaptive) iterative scheme or by a direct inversion. We have evaluated the iterative implementation based on a Neumann series, which is considered to be the conventional scheme. At each iteration of this scheme, a convolution in time and an integration in space are performed between a so-called focusing (update) function and the reflection response. In addition, by applying a time window, one obtains an update, which becomes the input for the next iteration. In each iteration, upgoing and downgoing focusing functions are updated with these terms. After convergence of the scheme, the resulting upgoing and downgoing focusing functions are used to compute the upgoing and downgoing Green’s functions with a virtual-source position in the subsurface and receivers at the surface. We have evaluated this algorithm in detail and developed an implementation that reproduces our examples. The software fits into the Seismic Unix software suite of the Colorado School of Mines.
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39

Proinov, Petko D. "Two Classes of Iteration Functions and Q-Convergence of Two Iterative Methods for Polynomial Zeros." Symmetry 13, no. 3 (February 25, 2021): 371. http://dx.doi.org/10.3390/sym13030371.

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In this work, two broad classes of iteration functions in n-dimensional vector spaces are introduced. They are called iteration functions of the first and second kind at a fixed point of the corresponding iteration function. Two general local convergence theorems are presented for Picard-type iterative methods with high Q-order of convergence. In particular, it is shown that if an iterative method is generated by an iteration function of first or second kind, then it is Q-convergent under each initial approximation that is sufficiently close to the fixed point. As an application, a detailed local convergence analysis of two fourth-order iterative methods is provided for finding all zeros of a polynomial simultaneously. The new results improve the previous ones for these methods in several directions.
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40

Ben-Romdhane, Mohamed, and Helmi Temimi. "An Iterative Numerical Method for Solving the Lane–Emden Initial and Boundary Value Problems." International Journal of Computational Methods 15, no. 04 (May 24, 2018): 1850020. http://dx.doi.org/10.1142/s0219876218500202.

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In this paper, we propose fast iterative methods based on the Newton–Raphson–Kantorovich approximation in function space [Bellman and Kalaba, (1965)] to solve three kinds of the Lane–Emden type problems. First, a reformulation of the problem is performed using a quasilinearization technique which leads to an iterative scheme. Such scheme consists in an ordinary differential equation that uses the approximate solution from the previous iteration to yield the unknown solution of the current iteration. At every iteration, a further discretization of the problem is achieved which provides the numerical solution with low computational cost. Numerical simulation shows the accuracy as well as the efficiency of the method.
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41

Xia, Yu, Wenzheng Kong, Yingye Yu, Yiying Hu, and Jinyou Li. "Improved Response Surface Method Based on Linear Gradient Iterative Criterion." Advances in Civil Engineering 2023 (February 8, 2023): 1–9. http://dx.doi.org/10.1155/2023/6360796.

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Aiming at the problem of big calculation error in solving reliability index by the traditional response surface method, an improved response surface method based on the linear gradient iteration criterion is proposed. First, the linear gradient iteration criterion is proposed to reduce the iteration step size with the increase of iteration times. It will improve the fitting accuracy of response surface and lead to a better convergence while approaching the limit state surface. Then, the reduction coefficient of the linear gradient iterative criterion is studied. The optimal value of coefficient is 0.2. The improved response surface method will get a more accurate reliability index quickly. Examples show that the proposed method has obvious advantages of high accuracy and efficiency. The application of this method can also be expanded in other similar engineering structure.
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42

Kanwar, Vinay, Puneet Sharma, Ioannis K. Argyros, Ramandeep Behl, Christopher Argyros, Ali Ahmadian, and Mehdi Salimi. "Geometrically Constructed Family of the Simple Fixed Point Iteration Method." Mathematics 9, no. 6 (March 23, 2021): 694. http://dx.doi.org/10.3390/math9060694.

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This study presents a new one-parameter family of the well-known fixed point iteration method for solving nonlinear equations numerically. The proposed family is derived by implementing approximation through a straight line. The presence of an arbitrary parameter in the proposed family improves convergence characteristic of the simple fixed point iteration as it has a wider domain of convergence. Furthermore, we propose many two-step predictor–corrector iterative schemes for finding fixed points, which inherit the advantages of the proposed fixed point iterative schemes. Finally, several examples are given to further illustrate their efficiency.
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43

Liao, Li Dan, and Guo Feng Zhang. "Efficient Preconditioner and Iterative Method for Large Complex Symmetric Linear Algebraic Systems." East Asian Journal on Applied Mathematics 7, no. 3 (August 2017): 530–47. http://dx.doi.org/10.4208/eajam.240316.290417a.

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AbstractWe discuss an efficient preconditioner and iterative numerical method to solve large complex linear algebraic systems of the form (W + iT)u = c, where W and T are symmetric matrices, and at least one of them is nonsingular. When the real part W is dominantly stronger or weaker than the imaginary part T, we propose a block multiplicative (BM) preconditioner or its variant (VBM), respectively. The BM and VBM preconditioned iteration methods are shown to be parameter-free, in terms of eigenvalue distributions of the preconditioned matrix. Furthermore, when the relationship between W and T is obscure, we propose a new preconditioned BM method (PBM) to overcome this difficulty. Both convergent properties of these new iteration methods and spectral properties of the corresponding preconditioned matrices are discussed. The optimal value of iteration parameter for the PBM method is determined. Numerical experiments involving the Helmholtz equation and some other applications show the effectiveness and robustness of the proposed preconditioners and corresponding iterative methods.
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Zhang, N. L., and W. Zhang. "Speeding Up the Convergence of Value Iteration in Partially Observable Markov Decision Processes." Journal of Artificial Intelligence Research 14 (February 1, 2001): 29–51. http://dx.doi.org/10.1613/jair.761.

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Partially observable Markov decision processes (POMDPs) have recently become popular among many AI researchers because they serve as a natural model for planning under uncertainty. Value iteration is a well-known algorithm for finding optimal policies for POMDPs. It typically takes a large number of iterations to converge. This paper proposes a method for accelerating the convergence of value iteration. The method has been evaluated on an array of benchmark problems and was found to be very effective: It enabled value iteration to converge after only a few iterations on all the test problems.
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45

Sasaki, Nopparut, and Pattrawut Chansangiam. "Modified Jacobi-Gradient Iterative Method for Generalized Sylvester Matrix Equation." Symmetry 12, no. 11 (November 5, 2020): 1831. http://dx.doi.org/10.3390/sym12111831.

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We propose a new iterative method for solving a generalized Sylvester matrix equation A1XA2+A3XA4=E with given square matrices A1,A2,A3,A4 and an unknown rectangular matrix X. The method aims to construct a sequence of approximated solutions converging to the exact solution, no matter the initial value is. We decompose the coefficient matrices to be the sum of its diagonal part and others. The recursive formula for the iteration is derived from the gradients of quadratic norm-error functions, together with the hierarchical identification principle. We find equivalent conditions on a convergent factor, relied on eigenvalues of the associated iteration matrix, so that the method is applicable as desired. The convergence rate and error estimation of the method are governed by the spectral norm of the related iteration matrix. Furthermore, we illustrate numerical examples of the proposed method to show its capability and efficacy, compared to recent gradient-based iterative methods.
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46

DOGAN, KADRI, and VATAN KARAKAYA. "A study in the fixed point theory for a new iterative scheme and a class of generalized mappings." Creative Mathematics and Informatics 27, no. 2 (2018): 151–60. http://dx.doi.org/10.37193/cmi.2018.02.07.

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In this study, we introduce a new iteration scheme and prove the strong convergence result for this iteration method. We also compare the rate of convergence with the iterative scheme and the fixed point iteration scheme known as Picard-S due to Gursoy. Then we prove that this new iteration method is equivalent to convergence of the iteration schemes given in the introduction section of the manuscript. Moreover, we show the result of its data dependency.
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47

Wang, Qian, Xiaoyan Chen, Di Wang, Zichen Wang, Xinyu Zhang, Na Xie, and Lili Liu. "Regularization Solver Guided FISTA for Electrical Impedance Tomography." Sensors 23, no. 4 (February 16, 2023): 2233. http://dx.doi.org/10.3390/s23042233.

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Electrical impedance tomography (EIT) is non-destructive monitoring technology that can visualize the conductivity distribution in the observed area. The inverse problem for imaging is characterized by a serious nonlinear and ill-posed nature, which leads to the low spatial resolution of the reconstructions. The iterative algorithm is an effective method to deal with the imaging inverse problem. However, the existing iterative imaging methods have some drawbacks, such as random and subjective initial parameter setting, very time consuming in vast iterations and shape blurring with less high-order information, etc. To solve these problems, this paper proposes a novel fast convergent iteration method for solving the inverse problem and designs an initial guess method based on an adaptive regularization parameter adjustment. This method is named the Regularization Solver Guided Fast Iterative Shrinkage Threshold Algorithm (RS-FISTA). The iterative solution process under the L1-norm regular constraint is derived in the LASSO problem. Meanwhile, the Nesterov accelerator is introduced to accelerate the gradient optimization race in the ISTA method. In order to make the initial guess contain more prior information and be independent of subjective factors such as human experience, a new adaptive regularization weight coefficient selection method is introduced into the initial conjecture of the FISTA iteration as it contains more accurate prior information of the conductivity distribution. The RS-FISTA method is compared with the methods of Landweber, CG, NOSER, Newton—Raphson, ISTA and FISTA, six different distributions with their optimal parameters. The SSIM, RMSE and PSNR of RS-FISTA methods are 0.7253, 3.44 and 37.55, respectively. In the performance test of convergence, the evaluation metrics of this method are relatively stable at 30 iterations. This shows that the proposed method not only has better visualization, but also has fast convergence. It is verified that the RS-FISTA algorithm is the better algorithm for EIT reconstruction from both simulation and physical experiments.
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48

Gander, Walter. "On Halley's Iteration Method." American Mathematical Monthly 92, no. 2 (February 1985): 131. http://dx.doi.org/10.2307/2322644.

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49

Acan, Omer, Omer Firat, Yildiray Keskin, and Galip Oturanc. "Conformable variational iteration method." New Trends in Mathematical Science 1, no. 5 (February 11, 2017): 172–78. http://dx.doi.org/10.20852/ntmsci.2017.135.

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50

Tari, Hafez. "Modified variational iteration method." Physics Letters A 369, no. 4 (September 2007): 290–93. http://dx.doi.org/10.1016/j.physleta.2007.04.090.

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