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Journal articles on the topic 'Iterative methods (Mathematics)'

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Published: 4 June 2021

Last updated: 1 August 2024

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1

Rafiq, Arif, Sifat Hussain, Farooq Ahmad, and Muhammad Awais. "New iterative methods." Applied Mathematics and Computation 189, no. 2 (June 2007): 1260–67. http://dx.doi.org/10.1016/j.amc.2006.12.042.

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2

Beauwens, Robert. "Iterative solution methods." Applied Numerical Mathematics 51, no. 4 (December 2004): 437–50. http://dx.doi.org/10.1016/j.apnum.2004.06.003.

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3

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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4

Geiser, Jürgen. "Computing Exponential for Iterative Splitting Methods: Algorithms and Applications." Journal of Applied Mathematics 2011 (2011): 1–27. http://dx.doi.org/10.1155/2011/193781.

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Iterative splitting methods have a huge amount to compute matrix exponential. Here, the acceleration and recovering of higher-order schemes can be achieved. From a theoretical point of view, iterative splitting methods are at least alternating Picards fix-point iteration schemes. For practical applications, it is important to compute very fast matrix exponentials. In this paper, we concentrate on developing fast algorithms to solve the iterative splitting scheme. First, we reformulate the iterative splitting scheme into an integral notation of matrix exponential. In this notation, we consider fast approximation schemes to the integral formulations, also known as -functions. Second, the error analysis is explained and applied to the integral formulations. The novelty is to compute cheaply the decoupled exp-matrices and apply only cheap matrix-vector multiplications for the higher-order terms. In general, we discuss an elegant way of embedding recently survey on methods for computing matrix exponential with respect to iterative splitting schemes. We present numerical benchmark examples, that compared standard splitting schemes with the higher-order iterative schemes. A real-life application in contaminant transport as a two phase model is discussed and the fast computations of the operator splitting method is explained.

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5

Khattri, S. K., and R. P. Agarwal. "Derivative-Free Optimal Iterative Methods." Computational Methods in Applied Mathematics 10, no. 4 (2010): 368–75. http://dx.doi.org/10.2478/cmam-2010-0022.

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AbstractIn this study, we develop an optimal family of derivative-free iterative methods. Convergence analysis shows that the methods are fourth order convergent, which is also verified numerically. The methods require three functional evaluations during each iteration. Though the methods are independent of derivatives, computa- tional results demonstrate that the family of methods are efficient and demonstrate equal or better performance as compared with many well-known methods and the clas- sical Newton method. Through optimization we derive an optimal value for the free parameter and implement it adaptively, which enhances the convergence order without increasing functional evaluations.

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6

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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7

Vabishchevich, Petr N. "Iterative Methods for Solving Convection-diffusion Problem." Computational Methods in Applied Mathematics 2, no. 4 (2002): 410–44. http://dx.doi.org/10.2478/cmam-2002-0023.

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AbstractTo obtain an approximate solution of the steady-state convectiondiffusion problem, it is necessary to solve the corresponding system of linear algebraic equations. The basic peculiarity of these LA systems is connected with the fact that they have non-symmetric matrices. We discuss the questions of approximate solution of 2D convection-diffusion problems on the basis of two- and three-level iterative methods. The general theory of iterative methods of solving grid equations is used to present the material of the paper. The basic problems of constructing grid approximations for steady-state convection-diffusion problems are considered. We start with the consideration of the Dirichlet problem for the differential equation with a convective term in the divergent, nondivergent, and skew-symmetric forms. Next, the corresponding grid problems are constructed. And, finally, iterative methods are used to solve approximately the above grid problems. Primary consideration is given to the study of the dependence of the number of iteration on the Peclet number, which is the ratio of the convective transport to the diffusive one.

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8

Bai, Zhong-Zhi. "Regularized HSS iteration methods for stabilized saddle-point problems." IMA Journal of Numerical Analysis 39, no. 4 (July 31, 2018): 1888–923. http://dx.doi.org/10.1093/imanum/dry046.

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Abstract We extend the regularized Hermitian and skew-Hermitian splitting (RHSS) iteration methods for standard saddle-point problems to stabilized saddle-point problems and establish the corresponding unconditional convergence theory for the resulting methods. Besides being used as stationary iterative solvers, this class of RHSS methods can also be used as preconditioners for Krylov subspace methods. It is shown that the eigenvalues of the corresponding preconditioned matrix are clustered at a small number of points in the interval $(0, \, 2)$ when the iteration parameter is close to $0$ and, furthermore, they can be clustered near $0$ and $2$ when the regularization matrix is appropriately chosen. Numerical results on stabilized saddle-point problems arising from finite element discretizations of an optimal boundary control problem and of a Cahn–Hilliard image inpainting problem, as well as from the Gauss–Newton linearization of a nonlinear image restoration problem, show that the RHSS iteration method significantly outperforms the Hermitian and skew-Hermitian splitting iteration method in iteration counts and computing times when they are used either as linear iterative solvers or as matrix splitting preconditioners for Krylov subspace methods, and optimal convergence behavior can be achieved when using inexact variants of the proposed RHSS preconditioners.

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9

Javidi, M. "Iterative methods to nonlinear equations." Applied Mathematics and Computation 193, no. 2 (November 2007): 360–65. http://dx.doi.org/10.1016/j.amc.2007.03.068.

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10

Huhtanen, Marko, and Olavi Nevanlinna. "Minimal decompositions and iterative methods." Numerische Mathematik 86, no. 2 (August 2000): 257–81. http://dx.doi.org/10.1007/pl00005406.

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11

Frommer, Andreas, and Daniel B. Szyld. "Asynchronous two-stage iterative methods." Numerische Mathematik 69, no. 2 (December 1994): 141–53. http://dx.doi.org/10.1007/s002110050085.

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12

Ho, Andy C., and Michael K. Ng. "Iterative methods for Robbins problems." Applied Mathematics and Computation 165, no. 1 (June 2005): 103–25. http://dx.doi.org/10.1016/j.amc.2004.04.025.

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13

Regmi, Samundra, Ioannis K. Argyros, Jinny Ann John, and Jayakumar Jayaraman. "Extended Convergence of Two Multi-Step Iterative Methods." Foundations 3, no. 1 (March 13, 2023): 140–53. http://dx.doi.org/10.3390/foundations3010013.

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Iterative methods which have high convergence order are crucial in computational mathematics since the iterates produce sequences converging to the root of a non-linear equation. A plethora of applications in chemistry and physics require the solution of non-linear equations in abstract spaces iteratively. The derivation of the order of the iterative methods requires expansions using Taylor series formula and higher-order derivatives not present in the method. Thus, these results cannot prove the convergence of the iterative method in these cases when such higher-order derivatives are non-existent. However, these methods may still converge. Our motivation originates from the need to handle these problems. No error estimates are given that are controlled by constants. The process introduced in this paper discusses both the local and the semi-local convergence analysis of two step fifth and multi-step 5+3r order iterative methods obtained using only information from the operators on these methods. Finally, the novelty of our process relates to the fact that the convergence conditions depend only on the functions and operators which are present in the methods. Thus, the applicability is extended to these methods. Numerical applications complement the theory.

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14

Khattri, Sanjay Kumar. "Optimal Eighth Order Iterative Methods." Mathematics in Computer Science 5, no. 2 (June 2011): 237–43. http://dx.doi.org/10.1007/s11786-011-0064-7.

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15

Hemeda, A. A., and E. E. Eladdad. "New Iterative Methods for Solving Fokker-Planck Equation." Mathematical Problems in Engineering 2018 (November 28, 2018): 1–9. http://dx.doi.org/10.1155/2018/6462174.

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In this article, we propose the new iterative method and introduce the integral iterative method to solve linear and nonlinear Fokker-Planck equations and some similar equations. The results obtained by the two methods are compared with those obtained by both Adomian decomposition and variational iteration methods. Comparison shows that the two methods are more effective and convenient to use and overcome the difficulties arising in calculating Adomian polynomials and Lagrange multipliers, which means that the considered methods can simply and successfully be applied to a large class of problems.

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16

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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17

Rhee, Noah H. "Homotopy Understanding of Iterative Methods." Missouri Journal of Mathematical Sciences 6, no. 2 (May 1994): 78–90. http://dx.doi.org/10.35834/1994/0602078.

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18

Hanke, Martin. "Iterative Solution Methods (Owe Axeisson)." SIAM Review 37, no. 3 (September 1995): 466–67. http://dx.doi.org/10.1137/1037104.

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19

Chen, Huijuan, and Xintao Zheng. "Improved Newton Iterative Algorithm for Fractal Art Graphic Design." Complexity 2020 (November 27, 2020): 1–11. http://dx.doi.org/10.1155/2020/6623049.

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Fractal art graphics are the product of the fusion of mathematics and art, relying on the computing power of a computer to iteratively calculate mathematical formulas and present the results in a graphical rendering. The selection of the initial value of the first iteration has a greater impact on the final calculation result. If the initial value of the iteration is not selected properly, the iteration will not converge or will converge to the wrong result, which will affect the accuracy of the fractal art graphic design. Aiming at this problem, this paper proposes an improved optimization method for selecting the initial value of the Gauss-Newton iteration method. Through the area division method of the system composed of the sensor array, the effective initial value of iterative calculation is selected in the corresponding area for subsequent iterative calculation. Using the special skeleton structure of Newton’s iterative graphics, such as infinitely finely inlaid chain-like, scattered-point-like composition, combined with the use of graphic secondary design methods, we conduct fractal art graphics design research with special texture effects. On this basis, the Newton iterative graphics are processed by dithering and MATLAB-based mathematical morphology to obtain graphics and then processed with the help of weaving CAD to directly form fractal art graphics with special texture effects. Design experiments with the help of electronic Jacquard machines proved that it is feasible to transform special texture effects based on Newton's iterative graphic design into Jacquard fractal art graphics.

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20

Gubar, I. G. "Iterative methods of solving Theodorsen's equation." Researches in Mathematics, no. 1 (July 10, 2021): 49. http://dx.doi.org/10.15421/246708.

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21

Zhou, Lu, and Homer F. Walker. "Residual Smoothing Techniques for Iterative Methods." SIAM Journal on Scientific Computing 15, no. 2 (March 1994): 297–312. http://dx.doi.org/10.1137/0915021.

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22

Vogel, C. R., and M. E. Oman. "Iterative Methods for Total Variation Denoising." SIAM Journal on Scientific Computing 17, no. 1 (January 1996): 227–38. http://dx.doi.org/10.1137/0917016.

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23

Ching, Wai Ki, and Anthony W. Loh. "Iterative methods for flexible manufacturing systems." Applied Mathematics and Computation 141, no. 2-3 (September 2003): 553–64. http://dx.doi.org/10.1016/s0096-3003(02)00275-8.

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24

Noor, Muhammad Aslam, Khalida Inayat Noor, Waseem Asghar Khan, and Faizan Ahmad. "On iterative methods for nonlinear equations." Applied Mathematics and Computation 183, no. 1 (December 2006): 128–33. http://dx.doi.org/10.1016/j.amc.2006.05.054.

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25

Tian, Zhaolu, and Chuanqing Gu. "The iterative methods for centrosymmetric matrices." Applied Mathematics and Computation 187, no. 2 (April 2007): 902–11. http://dx.doi.org/10.1016/j.amc.2006.09.030.

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26

Djordjević, Dragan S. "Iterative methods for computing generalized inverses." Applied Mathematics and Computation 189, no. 1 (June 2007): 101–4. http://dx.doi.org/10.1016/j.amc.2006.11.063.

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27

Meerbergen, K. "A survey of preconditioned iterative methods." Journal of Computational and Applied Mathematics 66, no. 1-2 (January 1996): N4—N5. http://dx.doi.org/10.1016/0377-0427(96)80472-9.

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28

Ali, M. S. S. "Descent methods for convex optimization problems in Banach spaces." International Journal of Mathematics and Mathematical Sciences 2005, no. 15 (2005): 2347–57. http://dx.doi.org/10.1155/ijmms.2005.2347.

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We consider optimization problems in Banach spaces, whose cost functions are convex and smooth, but do not possess strengthened convexity properties. We propose a general class of iterative methods, which are based on combining descent and regularization approaches and provide strong convergence of iteration sequences to a solution of the initial problem.

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29

Djordjević, Dragan S., and Predrag S. Stanimirović. "Iterative methods for computing generalized inverses related with optimization methods." Journal of the Australian Mathematical Society 78, no. 2 (April 2005): 257–72. http://dx.doi.org/10.1017/s1446788700008077.

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AbstractWe develop several iterative methods for computing generalized inverses using both first and second order optimization methods in C*-algebras. Known steepest descent iterative methods are generalized in C*-algebras. We introduce second order methods based on the minimization of the norms ‖Ax − b‖2 and ‖x‖2 by means of the known second order unconstrained minimization methods. We give several examples which illustrate our theory.

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30

Ramos, J. I. "Iterative and non-iterative methods for non-linear Volterra integro-differential equations." Applied Mathematics and Computation 214, no. 1 (August 2009): 287–96. http://dx.doi.org/10.1016/j.amc.2009.03.067.

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31

Mitin, Alexander V., and Gerhard Hirsch. "Linear extrapolation in iterative methods." Journal of Mathematical Chemistry 15, no. 1 (December 1994): 109–13. http://dx.doi.org/10.1007/bf01277552.

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32

Krukier, L. A., and B. L. Krukier. "Convergence of skew-symmetric iterative methods." Russian Mathematics 55, no. 6 (May 25, 2011): 64–67. http://dx.doi.org/10.3103/s1066369x11060090.

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33

Kellogg, R. B. "Iterative methods for convection dominated flow." Rendiconti del Seminario Matematico e Fisico di Milano 60, no. 1 (December 1990): 167–76. http://dx.doi.org/10.1007/bf02925084.

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34

Ito, Kazufumi, and Jari Toivanen. "Preconditioned iterative methods on sparse subspaces." Applied Mathematics Letters 19, no. 11 (November 2006): 1191–97. http://dx.doi.org/10.1016/j.aml.2005.11.027.

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35

Konovalov, A. N. "Variational optimization of iterative split methods." Siberian Mathematical Journal 38, no. 2 (April 1997): 267–80. http://dx.doi.org/10.1007/bf02674625.

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36

Vrahatis, M. N., G. D. Magoulas, and V. P. Plagianakos. "From linear to nonlinear iterative methods." Applied Numerical Mathematics 45, no. 1 (April 2003): 59–77. http://dx.doi.org/10.1016/s0168-9274(02)00235-0.

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37

Su, Haiyan, Pengzhan Huang, Juan Wen, and Xinlong Feng. "Three Iterative Finite Element Methods for the Stationary Smagorinsky Model." East Asian Journal on Applied Mathematics 4, no. 2 (May 2014): 132–51. http://dx.doi.org/10.4208/eajam.230913.120314a.

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AbstractThree iterative stabilised finite element methods based on local Gauss integration are proposed in order to solve the steady two-dimensional Smagorinsky model numerically. The Stokes iterative scheme, the Newton iterative scheme and the Oseen iterative scheme are adopted successively to deal with the nonlinear terms involved. Numerical experiments are carried out to demonstrate their effectiveness. Furthermore, the effect of the parameters Re (the Reynolds number) and δ (the spatial filter radius) on the performance of the iterative numerical results is discussed.

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38

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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39

Arnal, J., V. Migallon, J. Penades, and D. B. Szyld. "Newton additive and multiplicative Schwarz iterative methods." IMA Journal of Numerical Analysis 28, no. 1 (March 16, 2007): 143–61. http://dx.doi.org/10.1093/imanum/drm015.

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40

Liu, Chengzhi, Xuli Han, and Li Zhang. "Unconditional convergence of iterative approximation methods." Engineering Analysis with Boundary Elements 126 (May 2021): 161–68. http://dx.doi.org/10.1016/j.enganabound.2021.03.001.

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41

Climent, Joan-Josep, Carmen Perea, Leandro Tortosa, and Antonio Zamora. "Convergence theorems for parallel alternating iterative methods." Applied Mathematics and Computation 148, no. 2 (January 2004): 497–517. http://dx.doi.org/10.1016/s0096-3003(02)00916-5.

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42

Chan, R. H. "Iterative methods for overflow queueing models I." Numerische Mathematik 51, no. 2 (March 1987): 143–80. http://dx.doi.org/10.1007/bf01396747.

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43

Wang, Ke, and Bing Zheng. "Block iterative methods for fuzzy linear systems." Journal of Applied Mathematics and Computing 25, no. 1-2 (September 2007): 119–36. http://dx.doi.org/10.1007/bf02832342.

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44

Chan, R. H. "Iterative methods for overflow queuing models II." Numerische Mathematik 54, no. 1 (January 1988): 57–78. http://dx.doi.org/10.1007/bf01403891.

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45

Prévost, Marc, Michela Redivo-Zaglia, and Franck Wielonsky. "International conference on approximation and iterative methods." Journal of Computational and Applied Mathematics 219, no. 2 (October 2008): 327–28. http://dx.doi.org/10.1016/j.cam.2007.11.012.

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46

Wang, Li, and Yongzhong Song. "Preconditioned AOR iterative methods for M-matrices." Journal of Computational and Applied Mathematics 226, no. 1 (April 2009): 114–24. http://dx.doi.org/10.1016/j.cam.2008.05.022.

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47

Noor, Muhammad Aslam, Faizan Ahmad, and Shumaila Javeed. "Two-step iterative methods for nonlinear equations." Applied Mathematics and Computation 181, no. 2 (October 2006): 1068–75. http://dx.doi.org/10.1016/j.amc.2006.01.065.

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48

Aslam Noor, Muhammad, and Khalida Inayat Noor. "Three-step iterative methods for nonlinear equations." Applied Mathematics and Computation 183, no. 1 (December 2006): 322–27. http://dx.doi.org/10.1016/j.amc.2006.05.055.

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49

Aslam Noor, Muhammad, and Khalida Inayat Noor. "Improved iterative methods for solving nonlinear equations." Applied Mathematics and Computation 184, no. 2 (January 2007): 270–75. http://dx.doi.org/10.1016/j.amc.2006.05.165.

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50

Loizou, Nicolas, and Peter Richtárik. "Convergence Analysis of Inexact Randomized Iterative Methods." SIAM Journal on Scientific Computing 42, no. 6 (January 2020): A3979—A4016. http://dx.doi.org/10.1137/19m125248x.

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