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Journal articles on the topic 'High order convergence'

Author: Grafiati

Published: 4 June 2021

Last updated: 17 February 2022

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1

Vaarmann, Otu. "HIGH ORDER ITERATIVE METHODS FOR DECOMPOSITION‐COORDINATION PROBLEMS." Technological and Economic Development of Economy 12, no. 1 (March 31, 2006): 56–61. http://dx.doi.org/10.3846/13928619.2006.9637723.

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Many real‐life optimization problems are of the multiobjective type and highdimensional. Possibilities for solving large scale optimization problems on a computer network or multiprocessor computer using a multi‐level approach are studied. The paper treats numerical methods in which procedural and rounding errors are unavoidable, for example, those arising in mathematical modelling and simulation. For the solution of involving decomposition‐coordination problems some rapidly convergent interative methods are developed based on the classical cubically convergent method of tangent hyperbolas (Chebyshev‐Halley method) and the method of tangent parabolas (Euler‐Chebyshev method). A family of iterative methods having the convergence order equal to four is also considered. Convergence properties and computational aspects of the methods under consideration are examined. The problems of their global implementation and polyalgorithmic strategy are discussed as well.

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2

Albizuri, F. Xabier, Alicia d'Anjou, Manuel Graña, and J. Antonio Lozano. "Convergence Properties of High-order Boltzmann Machines." Neural Networks 9, no. 9 (December 1996): 1561–67. http://dx.doi.org/10.1016/s0893-6080(96)00026-3.

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3

Bokanowski, Olivier, Athena Picarelli, and Christoph Reisinger. "High-order filtered schemes for time-dependent second order HJB equations." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 1 (January 2018): 69–97. http://dx.doi.org/10.1051/m2an/2017039.

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In this paper, we present and analyse a class of “filtered” numerical schemes for second order Hamilton–Jacobi–Bellman (HJB) equations. Our approach follows the ideas recently introduced in B.D. Froese and A.M. Oberman, Convergent filtered schemes for the Monge-Ampère partial differential equation, SIAM J. Numer. Anal. 51 (2013) 423–444, and more recently applied by other authors to stationary or time-dependent first order Hamilton–Jacobi equations. For high order approximation schemes (where “high” stands for greater than one), the inevitable loss of monotonicity prevents the use of the classical theoretical results for convergence to viscosity solutions. The work introduces a suitable local modification of these schemes by “filtering” them with a monotone scheme, such that they can be proven convergent and still show an overall high order behaviour for smooth enough solutions. We give theoretical proofs of these claims and illustrate the behaviour with numerical tests from mathematical finance, focussing also on the use of backward differencing formulae for constructing the high order schemes.

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4

Hemker, Pieter W., Grigorii I. Shishkin, and Lidia P. Shishkina. "High-order Time-accurate Schemes for Singularly Perturbed Parabolic Convection-diffusion Problems with Robin Boundary Conditions." Computational Methods in Applied Mathematics 2, no. 1 (2002): 3–25. http://dx.doi.org/10.2478/cmam-2002-0001.

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AbstractThe boundary-value problem for a singularly perturbed parabolic PDE with convection is considered on an interval in the case of the singularly perturbed Robin boundary condition; the highest space derivatives in the equation and in the boundary condition are multiplied by the perturbation parameter ε. The order of convergence for the known ε-uniformly convergent schemes does not exceed 1. In this paper, using a defect correction technique, we construct ε-uniformly convergent schemes of highorder time-accuracy. The efficiency of the new defect-correction schemes is confirmed by numerical experiments. A new original technigue for experimental studying of convergence orders is developed for the cases where the orders of convergence in the x-direction and in the t-direction can be substantially different.

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5

Behl, Ramandeep, Ioannis K. Argyros, and Fouad Othman Mallawi. "Some High-Order Convergent Iterative Procedures for Nonlinear Systems with Local Convergence." Mathematics 9, no. 12 (June 14, 2021): 1375. http://dx.doi.org/10.3390/math9121375.

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In this study, we suggested the local convergence of three iterative schemes that works for systems of nonlinear equations. In earlier results, such as from Amiri et al. (see also the works by Behl et al., Argryos et al., Chicharro et al., Cordero et al., Geum et al., Guitiérrez, Sharma, Weerakoon and Fernando, Awadeh), authors have used hypotheses on high order derivatives not appearing on these iterative procedures. Therefore, these methods have a restricted area of applicability. The main difference of our study to earlier studies is that we adopt only the first order derivative in the convergence order (which only appears on the proposed iterative procedure). No work has been proposed on computable error distances and uniqueness in the aforementioned studies given on Rk. We also address these problems too. Moreover, by using Banach space, the applicability of iterative procedures is extended even further. We have examined the convergence criteria on several real life problems along with a counter problem that completes this study.

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6

Artidiello, Santiago, Alicia Cordero, Juan R. Torregrosa, and María P. Vassileva. "Design of High-Order Iterative Methods for Nonlinear Systems by Using Weight Function Procedure." Abstract and Applied Analysis 2015 (2015): 1–12. http://dx.doi.org/10.1155/2015/289029.

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We present two classes of iterative methods whose orders of convergence are four and five, respectively, for solving systems of nonlinear equations, by using the technique of weight functions in each step. Moreover, we show an extension to higher order, adding only one functional evaluation of the vectorial nonlinear function. We perform numerical tests to compare the proposed methods with other schemes in the literature and test their effectiveness on specific nonlinear problems. Moreover, some real basins of attraction are analyzed in order to check the relation between the order of convergence and the set of convergent starting points.

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7

Kiran, Quanita, and Tayyab Kamran. "Nadler’s type principle with high order of convergence." Nonlinear Analysis: Theory, Methods & Applications 69, no. 11 (December 2008): 4106–20. http://dx.doi.org/10.1016/j.na.2007.10.041.

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8

Wang, Wen Kai, and Huan Xin Peng. "High-Order Distributed Consensus with One-Bit Adaptive Quantization." Advanced Materials Research 591-593 (November 2012): 1299–302. http://dx.doi.org/10.4028/www.scientific.net/amr.591-593.1299.

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The convergence performance of distributed consensus algorithm with adaptive quantization communication depends on the convergence rate of the distributed consensus algorithm. In order to improve the convergence performance of distributed consensus under adaptive quantization communication, based on the one-bit adaptive quantization scheme, we propose the high-order distributed consensus to update the state of every node. We analyze the convergence performance and calculate the mean square error of the high-order distributed consensus algorithm with one-bit adaptive quantization. The high-order distributed consensus with one-bit adaptive quantization achieves a consensus in a mean square sense. Simultaneously, Simulations are done about the high-order distributed consensus based on one-bit adaptive quantization. Results show that the high-order distributed consensus algorithm based on one-bit adaptive quantization can reach an average consensus, and its convergence rate is higher than those of the first-order adaptive quantized distributed consensus algorithm and second-order adaptive quantized distributed consensus algorithm.

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9

Bin Jebreen, Haifa. "Constructing a High-Order Globally Convergent Iterative Method for Calculating the Matrix Sign Function." Mathematical Problems in Engineering 2018 (June 21, 2018): 1–9. http://dx.doi.org/10.1155/2018/8973867.

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This work is concerned with the construction of a new matrix iteration in the form of an iterative method which is globally convergent for finding the sign of a square matrix having no eigenvalues on the axis of imaginary. Toward this goal, a new method is built via an application of a new four-step nonlinear equation solver on a particulate matrix equation. It is discussed that the proposed scheme has global convergence with eighth order of convergence. To illustrate the effectiveness of the theoretical results, several computational experiments are worked out.

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10

Proinov, Petko D., and Maria T. Vasileva. "A New Family of High-Order Ehrlich-Type Iterative Methods." Mathematics 9, no. 16 (August 5, 2021): 1855. http://dx.doi.org/10.3390/math9161855.

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One of the famous third-order iterative methods for finding simultaneously all the zeros of a polynomial was introduced by Ehrlich in 1967. In this paper, we construct a new family of high-order iterative methods as a combination of Ehrlich’s iteration function and an arbitrary iteration function. We call these methods Ehrlich’s methods with correction. The paper provides a detailed local convergence analysis of presented iterative methods for a large class of iteration functions. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with computer verifiable initial condition). As special cases of the main results, we study the convergence of several particular iterative methods. The paper ends with some experiments that show the applicability of our semilocal convergence theorems.

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11

Argyros, Ioannis K., and Santhosh George. "Local convergence for some high convergence order Newton-like methods with frozen derivatives." SeMA Journal 70, no. 1 (July 24, 2015): 47–59. http://dx.doi.org/10.1007/s40324-015-0039-8.

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12

Du, Rui, Zhao-peng Hao, and Zhi-zhong Sun. "Lubich Second-Order Methods for Distributed-Order Time-Fractional Differential Equations with Smooth Solutions." East Asian Journal on Applied Mathematics 6, no. 2 (May 2016): 131–51. http://dx.doi.org/10.4208/eajam.020615.030216a.

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AbstractThis article is devoted to the study of some high-order difference schemes for the distributed-order time-fractional equations in both one and two space dimensions. Based on the composite Simpson formula and Lubich second-order operator, a difference scheme is constructed with convergence in the L1(L∞)-norm for the one-dimensional case, where τ,h and σ are the respective step sizes in time, space and distributed-order. Unconditional stability and convergence are proven. An ADI difference scheme is also derived for the two-dimensional case, and proven to be unconditionally stable and convergent in the L1(L∞)-norm, where h1 and h2 are the spatial step sizes. Some numerical examples are also given to demonstrate our theoretical results.

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13

JENSEN, M. S. "HIGH CONVERGENCE ORDER FINITE ELEMENTS WITH LUMPED MASS MATRIX." International Journal for Numerical Methods in Engineering 39, no. 11 (June 15, 1996): 1879–88. http://dx.doi.org/10.1002/(sici)1097-0207(19960615)39:11<1879::aid-nme933>3.0.co;2-2.

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14

Ezquerro, J. A., M. A. Hernández, and M. A. Salanova. "Construction of iterative processes with high order of convergence." International Journal of Computer Mathematics 69, no. 1-2 (January 1998): 191–201. http://dx.doi.org/10.1080/00207169808804717.

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15

Rossi, Louis F. "Achieving High-Order Convergence Rates with Deforming Basis Functions." SIAM Journal on Scientific Computing 26, no. 3 (January 2005): 885–906. http://dx.doi.org/10.1137/s1064827503425286.

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16

Westernacher-Schneider, John Ryan. "Extremely high-order convergence in simulations of relativistic stars." Classical and Quantum Gravity 38, no. 14 (June 15, 2021): 145003. http://dx.doi.org/10.1088/1361-6382/ac0234.

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17

Liu, Xiaoji, and Zemeng Zuo. "A High-Order Iterate Method for ComputingAT,S(2)." Journal of Applied Mathematics 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/741368.

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We investigate a new higher order iterative method for computing the generalized inverseAT,S(2)for a given matrixA. We also discuss how the new method could be applied for finding approximate inverses of nonsingular square matrices. Analysis of convergence is included to show that the proposed scheme has at least fifteenth-order convergence. Some tests are also presented to show the superiority of the new method.

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18

Ding, Hengfei, and Changpin Li. "High-order algorithms for riesz derivative and their applications (IV)." Fractional Calculus and Applied Analysis 22, no. 6 (December 18, 2019): 1537–60. http://dx.doi.org/10.1515/fca-2019-0080.

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Abstract The main goal of this article is to establish a new 4th-order numerical differential formula to approximate Riesz derivatives which is obtained by means of a newly established generating function. Then the derived formula is used to solve the Riesz space fractional advection-dispersion equation. Meanwhile, by the energy method, it is proved that the difference scheme is unconditionally stable and convergent with order 𝓞(τ2 + h4). Finally, several numerical examples are given to show that the numerical convergence orders of the numerical differential formulas and the finite difference scheme are in line with the theoretical analysis.

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19

Abel, Ulrich. "High order algorithms for calculating roots." Mathematical Gazette 100, no. 549 (October 17, 2016): 420–28. http://dx.doi.org/10.1017/mag.2016.106.

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In a recent Note [1] Michael D. Hirschhorn presented high order algorithms for calculating numerically square roots and cube roots. In particular, he obtained the method(1)with , where the convergence is of tenth order:We recall his idea in the case of an arbitrary square root with a > 0. Let p ⩾ 2 be a fixed integer. Our starting point is the relationChoose

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20

Graça, Mario Meireles, and Pedro Miguel Lima. "On High Order Barycentric Root-Finding Methods." TEMA (São Carlos) 17, no. 3 (December 20, 2016): 321. http://dx.doi.org/10.5540/tema.2016.017.03.0321.

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To approximate a simple root of a real function f we construct a family of iterative maps, which we call Newton-barycentric functions, and analyse their convergence order. The performance of the resulting methods is illustrated by means of numerical examples.

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21

Argyros, Ioannis K., Santhosh George, and Á. Alberto Magreñán. "Local convergence for multi-point-parametric Chebyshev–Halley-type methods of high convergence order." Journal of Computational and Applied Mathematics 282 (July 2015): 215–24. http://dx.doi.org/10.1016/j.cam.2014.12.023.

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22

Davis and Warnick. "High-order convergence with a low-order discretization of the 2-D MFIE." IEEE Antennas and Wireless Propagation Letters 3 (2004): 355–58. http://dx.doi.org/10.1109/lawp.2004.840254.

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23

Amorós, Cristina, Ioannis Argyros, Ruben González, Á. Magreñán, Lara Orcos, and Íñigo Sarría. "Study of a High Order Family: Local Convergence and Dynamics." Mathematics 7, no. 3 (February 28, 2019): 225. http://dx.doi.org/10.3390/math7030225.

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The study of the dynamics and the analysis of local convergence of an iterative method, when approximating a locally unique solution of a nonlinear equation, is presented in this article. We obtain convergence using a center-Lipschitz condition where the ball radii are greater than previous studies. We investigate the dynamics of the method. To validate the theoretical results obtained, a real-world application related to chemistry is provided.

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24

Haghani, F. Khaksar, and F. Soleymani. "A New High-Order Stable Numerical Method for Matrix Inversion." Scientific World Journal 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/830564.

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A stable numerical method is proposed for matrix inversion. The new method is accompanied by theoretical proof to illustrate twelfth-order convergence. A discussion of how to achieve the convergence using an appropriate initial value is presented. The application of the new scheme for finding Moore-Penrose inverse will also be pointed out analytically. The efficiency of the contributed iterative method is clarified on solving some numerical examples.

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25

Behl, Ramandeep, Alicia Cordero, Juan R. Torregrosa, and Sonia Bhalla. "A New High-Order Jacobian-Free Iterative Method with Memory for Solving Nonlinear Systems." Mathematics 9, no. 17 (September 1, 2021): 2122. http://dx.doi.org/10.3390/math9172122.

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We used a Kurchatov-type accelerator to construct an iterative method with memory for solving nonlinear systems, with sixth-order convergence. It was developed from an initial scheme without memory, with order of convergence four. There exist few multidimensional schemes using more than one previous iterate in the very recent literature, mostly with low orders of convergence. The proposed scheme showed its efficiency and robustness in several numerical tests, where it was also compared with the existing procedures with high orders of convergence. These numerical tests included large nonlinear systems. In addition, we show that the proposed scheme has very stable qualitative behavior, by means of the analysis of an associated multidimensional, real rational function and also by means of a comparison of its basin of attraction with those of comparison methods.

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26

Liu, Xiaoji, and Naping Cai. "High-Order Iterative Methods for the DMP Inverse." Journal of Mathematics 2018 (May 7, 2018): 1–6. http://dx.doi.org/10.1155/2018/8175935.

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We investigate two iterative methods for computing the DMP inverse. The necessary and sufficient conditions for convergence of our schemes are considered and the error estimate is also derived. Numerical examples are given to test the accuracy and effectiveness of our methods.

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27

Argyros, Ioannis K., Ramandeep Behl, Daniel González, and S. S. Motsa. "Local convergence for multistep high order methods under weak conditions." Applicationes Mathematicae 47, no. 2 (2020): 293–304. http://dx.doi.org/10.4064/am2374-1-2019.

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28

Georgiev, Pando, Panos Pardalos, and Andrzej Cichocki. "Algorithms with high order convergence speed for blind source extraction." Computational Optimization and Applications 38, no. 1 (May 18, 2007): 123–31. http://dx.doi.org/10.1007/s10589-007-9031-2.

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29

Moore, Kevin L., and YangQuan Chen. "ON MONOTONIC CONVERGENCE OF HIGH ORDER ITERATIVE LEARNING UPDATE LAWS." IFAC Proceedings Volumes 35, no. 1 (2002): 19–24. http://dx.doi.org/10.3182/20020721-6-es-1901.00989.

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30

Li, Bin, and Nan Wu. "Convergence Analysis of Gaussian SPAWN Under High-Order Graphical Models." IEEE Signal Processing Letters 27 (2020): 1725–29. http://dx.doi.org/10.1109/lsp.2020.3025066.

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31

Bonito, Andrea, J. Manuel Cascón, Khamron Mekchay, Pedro Morin, and Ricardo H. Nochetto. "High-Order AFEM for the Laplace–Beltrami Operator: Convergence Rates." Foundations of Computational Mathematics 16, no. 6 (November 23, 2016): 1473–539. http://dx.doi.org/10.1007/s10208-016-9335-7.

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32

Beirão da Veiga, L., K. Lipnikov, and G. Manzini. "Convergence analysis of the high-order mimetic finite difference method." Numerische Mathematik 113, no. 3 (May 28, 2009): 325–56. http://dx.doi.org/10.1007/s00211-009-0234-6.

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33

Liu, Don, Weijia Kuang, and Andrew Tangborn. "High-Order Compact Implicit Difference Methods For Parabolic Equations in Geodynamo Simulation." Advances in Mathematical Physics 2009 (2009): 1–23. http://dx.doi.org/10.1155/2009/568296.

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A series of compact implicit schemes of fourth and sixth orders are developed for solving differential equations involved in geodynamics simulations. Three illustrative examples are described to demonstrate that high-order convergence rates are achieved while good efficiency in terms of fewer grid points is maintained. This study shows that high-order compact implicit difference methods provide high flexibility and good convergence in solving some special differential equations on nonuniform grids.

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34

Song, Chunlin, Changzhu Wei, Feng Yang, and Naigang Cui. "High-Order Sliding Mode-Based Fixed-Time Active Disturbance Rejection Control for Quadrotor Attitude System." Electronics 7, no. 12 (November 26, 2018): 357. http://dx.doi.org/10.3390/electronics7120357.

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This article presents a fixed-time active disturbance rejection control approach for the attitude control problem of quadrotor unmanned aerial vehicle in the presence of dynamic wind, mass eccentricity and an actuator fault. The control scheme applies the feedback linearization technique and enhances the performance of the traditional active disturbance rejection control (ADRC) based on the fixed-time high-order sliding mode method. A switching-type uniformly convergent differentiator is used to improve the extended state observer for estimating and attenuating the lumped disturbance more accurately. A multivariable high-order sliding mode feedback law is derived to achieve fixed time convergence. The timely convergence of the designed extended state observer and the feedback law is proved theoretically. Mathematical simulations with detailed actuator models and real time experiments are performed to demonstrate the robustness and practicability of the proposed control scheme.

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35

Boutin, B., T. H. T. Nguyen, A. Sylla, S. Tran-Tien, and J. F. Coulombel. "High order numerical schemes for transport equations on bounded domains." ESAIM: Proceedings and Surveys 70 (2021): 84–106. http://dx.doi.org/10.1051/proc/202107006.

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This article is an account of the NABUCO project achieved during the summer camp CEMRACS 2019 devoted to geophysical fluids and gravity flows. The goal is to construct finite difference approximations of the transport equation with nonzero incoming boundary data that achieve the best possible convergence rate in the maximum norm. We construct, implement and analyze the so-called inverse Lax-Wendroff procedure at the incoming boundary. Optimal convergence rates are obtained by combining sharp stability estimates for extrapolation boundary conditions with numerical boundary layer expansions. We illustrate the results with the Lax-Wendroff and O3 schemes.

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36

Behl, Ramandeep, and Eulalia Martínez. "A New High-Order and Efficient Family of Iterative Techniques for Nonlinear Models." Complexity 2020 (January 30, 2020): 1–11. http://dx.doi.org/10.1155/2020/1706841.

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In this paper, we want to construct a new high-order and efficient iterative technique for solving a system of nonlinear equations. For this purpose, we extend the earlier scalar scheme [16] to a system of nonlinear equations preserving the same convergence order. Moreover, by adding one more additional step, we obtain minimum 5th-order convergence for every value of a free parameter, θ∈ℝ, and for θ=−1, the method reaches maximum 6-order convergence. We present an extensive convergence analysis of our scheme. The analytical discussion of the work is upheld by performing numerical experiments on some applied science problems and a large system of nonlinear equations. Furthermore, numerical results demonstrate the validity and reliability of the suggested methods.

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37

Artidiello, S., A. Cordero, Juan R. Torregrosa, and M. P. Vassileva. "Optimal High-Order Methods for Solving Nonlinear Equations." Journal of Applied Mathematics 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/591638.

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A class of optimal iterative methods for solving nonlinear equations is extended up to sixteenth-order of convergence. We design them by using the weight function technique, with functions of three variables. Some numerical tests are made in order to confirm the theoretical results and to compare the new methods with other known ones.

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38

Zhou, Zhongguo, and Lin Li. "The high accuracy conserved splitting domain decomposition scheme for solving the parabolic equations." Applied Mathematics and Nonlinear Sciences 3, no. 2 (December 31, 2018): 583–92. http://dx.doi.org/10.2478/amns.2018.2.00045.

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AbstractIn this paper, the high accuracy mass-conserved splitting domain decomposition method for solving the parabolic equations is proposed. In our scheme, the time extrapolation and local multi-point weighted average schemes are used to approximate the interface fluxes on interfaces of sub-domains, while the interior solutions are computed by one dimension high-order implicit schemes in sub-domains. The important feature is that the developed scheme keeps mass conservation and are of second-order convergent in time and fourth-order convergent in space. Numerical experiments confirm the convergence.

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39

Behl, Ramandeep, and Ioannis K. Argyros. "Local Convergence for Multi-Step High Order Solvers under Weak Conditions." Mathematics 8, no. 2 (February 2, 2020): 179. http://dx.doi.org/10.3390/math8020179.

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Our aim in this article is to suggest an extended local convergence study for a class of multi-step solvers for nonlinear equations valued in a Banach space. In comparison to previous studies, where they adopt hypotheses up to 7th Fŕechet-derivative, we restrict the hypotheses to only first-order derivative of considered operators and Lipschitz constants. Hence, we enlarge the suitability region of these solvers along with computable radii of convergence. In the end of this study, we choose a variety of numerical problems which illustrate that our works are applicable but not earlier to solve nonlinear problems.

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40

Argyros, Ioannis, Ramandeep Behl, and S. S. Motsa. "Local Convergence of an Efﬁcient High Convergence Order Method Using Hypothesis Only on the First Derivative." Algorithms 8, no. 4 (November 20, 2015): 1076–87. http://dx.doi.org/10.3390/a8041076.

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41

Antona, Rubén, Renato Vacondio, Diego Avesani, Maurizio Righetti, and Massimiliano Renzi. "Towards a High Order Convergent ALE-SPH Scheme with Efficient WENO Spatial Reconstruction." Water 13, no. 17 (September 4, 2021): 2432. http://dx.doi.org/10.3390/w13172432.

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This paper studies the convergence properties of an arbitrary Lagrangian–Eulerian (ALE) Riemann-based SPH algorithm in conjunction with a Weighted Essentially Non-Oscillatory (WENO) high-order spatial reconstruction, in the framework of the DualSPHysics open-source code. A convergence analysis is carried out for Lagrangian and Eulerian simulations and the numerical results demonstrate that, in absence of particle disorder, the overall convergence of the scheme is close to the one guaranteed by the WENO spatial reconstruction. Moreover, an alternative method for the WENO spatial reconstruction is introduced which guarantees a speed-up of 3.5, in comparison with the classical Moving Least-Squares (MLS) approach.

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42

He, Fangchao. "Optimal convergence rates of high order Parzen windows with unbounded sampling." Statistics & Probability Letters 92 (September 2014): 26–32. http://dx.doi.org/10.1016/j.spl.2014.04.023.

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43

Xu, Meng, Linze Song, Kai Song, and Qiang Shi. "Convergence of high order perturbative expansions in open system quantum dynamics." Journal of Chemical Physics 146, no. 6 (February 14, 2017): 064102. http://dx.doi.org/10.1063/1.4974926.

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44

Hayakawa, Kantaro, and Yuusuke Iso. "High-order uniform convergence estimation of boundary solutions for Laplace's equation." Publications of the Research Institute for Mathematical Sciences 27, no. 2 (1991): 333–45. http://dx.doi.org/10.2977/prims/1195169841.

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45

Wang, Xia, and Liping Liu. "Modified Ostrowski’s method with eighth-order convergence and high efficiency index." Applied Mathematics Letters 23, no. 5 (May 2010): 549–54. http://dx.doi.org/10.1016/j.aml.2010.01.009.

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46

Fernández-Torres, Gustavo. "Derivative free iterative methods with memory of arbitrary high convergence order." Numerical Algorithms 67, no. 3 (December 4, 2013): 565–80. http://dx.doi.org/10.1007/s11075-013-9808-6.

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47

Shishkin, G. I. "ROBUST NOVEL HIGH-ORDER ACCURATE NUMERICAL METHODS FOR SINGULARLY PERTURBED CONVECTION‐DIFFUSION PROBLEMS." Mathematical Modelling and Analysis 10, no. 4 (December 31, 2005): 393–412. http://dx.doi.org/10.3846/13926292.2005.9637296.

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For singularly perturbed boundary value problems, numerical methods convergent ϵ‐uniformly have the low accuracy. So, for parabolic convection‐diffusion problem the order of convergence does not exceed one even if the problem data are sufficiently smooth. However, already for piecewise smooth initial data this order is not higher than 1/2. For problems of such type, using newly developed methods such as the method based on the asymptotic expansion technique and the method of the additive splitting of singularities, we construct ϵ‐uniformly convergent schemes with improved order of accuracy. Straipsnyje nagrinejami nedidelio tikslumo ϵ‐tolygiai konvertuojantys skaitmeniniai metodai, singuliariai sutrikdytiems kraštiniams uždaviniams. Paraboliniam konvekcijos‐difuzijos uždaviniui konvergavimo eile neviršija vienos antrosios, jeigu uždavinio duomenys yra pakankamai glodūs. Tačiau trūkiems pradiniams duomenims eile yra ne aukštesne už 2−1. Šio tipo uždaviniams, naudojant naujai išvestus metodus, darbe sukonstruotos ϵ‐tolygiai konvertuojančios schemos aukštesniu tikslumu.

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48

Lei, Siu-Long, Xu Chen, and Xinhe Zhang. "Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations." East Asian Journal on Applied Mathematics 6, no. 2 (May 2016): 109–30. http://dx.doi.org/10.4208/eajam.060815.180116a.

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AbstractHigh-dimensional two-sided space fractional diffusion equations with variable diffusion coefficients are discussed. The problems can be solved by an implicit finite difference scheme that is proven to be uniquely solvable, unconditionally stable and first-order convergent in the infinity norm. A nonsingular multilevel circulant pre-conditoner is proposed to accelerate the convergence rate of the Krylov subspace linear system solver efficiently. The preconditoned matrix for fast convergence is a sum of the identity matrix, a matrix with small norm, and a matrix with low rank under certain conditions. Moreover, the preconditioner is practical, with an O(NlogN) operation cost and O(N) memory requirement. Illustrative numerical examples are also presented.

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49

Chicharro, Francisco I., Alicia Cordero, Neus Garrido, and Juan R. Torregrosa. "Generalized High-Order Classes for Solving Nonlinear Systems and Their Applications." Mathematics 7, no. 12 (December 5, 2019): 1194. http://dx.doi.org/10.3390/math7121194.

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A generalized high-order class for approximating the solution of nonlinear systems of equations is introduced. First, from a fourth-order iterative family for solving nonlinear equations, we propose an extension to nonlinear systems of equations holding the same order of convergence but replacing the Jacobian by a divided difference in the weight functions for systems. The proposed GH family of methods is designed from this fourth-order family using both the composition and the weight functions technique. The resulting family has order of convergence 9. The performance of a particular iterative method of both families is analyzed for solving different test systems and also for the Fisher’s problem, showing the good performance of the new methods.

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50

Sun, Zhi-Zhong. "A High-Order Difference Scheme for a Nonlocal Boundary-Value Problem for the Heat Equation." Computational Methods in Applied Mathematics 1, no. 4 (2001): 398–414. http://dx.doi.org/10.2478/cmam-2001-0026.

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Abstract This paper is concerned with a high order difference scheme for a non- local boundary-value problem of parabolic equation. The integrals in the boundary equations are approximated by the composite Simpson rule. The unconditional solv- ability and L_∞ convergence of the difference scheme is proved by the energy method. The convergence rate of the difference scheme is second order in time and fourth order in space. Some numerical examples are provided to illustrate the convergence.

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