Journal articles: 'Newton-type method'

1

Fischer, A. "A special newton-type optimization method." Optimization 24, no. 3-4 (January 1992): 269–84. http://dx.doi.org/10.1080/02331939208843795.

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Păvăloiu, Ion, and Emil Cătinaş. "On an Aitken–Newton type method." Numerical Algorithms 62, no. 2 (May 6, 2012): 253–60. http://dx.doi.org/10.1007/s11075-012-9577-7.

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Păvăloiu, I., and E. Cătinaş. "On a Newton–Steffensen type method." Applied Mathematics Letters 26, no. 6 (June 2013): 659–63. http://dx.doi.org/10.1016/j.aml.2013.01.003.

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4

Ramm, A. G. "On the DSM Newton-type method." Journal of Applied Mathematics and Computing 38, no. 1-2 (June 9, 2011): 523–33. http://dx.doi.org/10.1007/s12190-011-0494-z.

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Jnawali, Jivandhar. "New Modified Newton Type Iterative Methods." Nepal Journal of Mathematical Sciences 2, no. 1 (April 30, 2021): 17–24. http://dx.doi.org/10.3126/njmathsci.v2i1.36559.

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In this work, we present two Newton type iterative methods for finding the solution of nonlinear equations of single variable. One is obtained as variant of McDougall and Wotherspoon method, and another is obtained by amalgamation of Potra and Pta’k method and our newly introduced method. The order of convergence of these methods are 1 + √2 and 3.5615. Some numerical examples are given to compare the performance of these methods with some similar existing methods.

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PAVALOIU, ION. "On an Aitken-Steffensen-Newton type method." Carpathian Journal of Mathematics 34, no. 1 (2018): 85–92. http://dx.doi.org/10.37193/cjm.2018.01.09.

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We consider an Aitken-Steffensen type method in which the nodes are controlled by Newton and two-step Newton iterations. We prove a local convergence result showing the q-convergence order 7 of the iterations. Under certain supplementary conditions, we obtain monotone convergence of the iterations, providing an alternative to the usual ball attraction theorems. Numerical examples show that this method may, in some cases, have larger (possibly sided) convergence domains than other methods with similar convergence orders.

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Vijesh, V. Antony, and P. V. Subrahmanyam. "A Newton-type method and its application." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–9. http://dx.doi.org/10.1155/ijmms/2006/23674.

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We prove an existence and uniqueness theorem for solving the operator equationF(x)+G(x)=0, whereFis a continuous and Gâteaux differentiable operator and the operatorGsatisfies Lipschitz condition on an open convex subset of a Banach space. As corollaries, a recent theorem of Argyros (2003) and the classical convergence theorem for modified Newton iterates are deduced. We further obtain an existence theorem for a class of nonlinear functional integral equations involving the Urysohn operator.

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Bayrak, Mine Aylin, Ali Demir, and Ebru Ozbilge. "On Fractional Newton-Type Method for Nonlinear Problems." Journal of Mathematics 2022 (November 21, 2022): 1–10. http://dx.doi.org/10.1155/2022/7070253.

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The current manuscript is concerned with the development of the Newton–Raphson method, playing a significant role in mathematics and various other disciplines such as optimization, by using fractional derivatives and fractional Taylor series expansion. The development and modification of the Newton–Raphson method allow us to establish two new methods, which are called first- and second-order fractional Newton–Raphson (FNR) methods. We provide convergence analysis of first- and second-order fractional methods and give a general condition for the convergence of higher-order FNR. Finally, some illustrative examples are considered to confirm the accuracy and effectiveness of both methods.

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Jnawali, Jivandhar, and Chet Raj Bhatta. "Iterative Methods for Solving Nonlinear Equations with Fourth-Order Convergence." Tribhuvan University Journal 30, no. 2 (December 1, 2016): 65–72. http://dx.doi.org/10.3126/tuj.v30i2.25548.

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In this paper, we obtain fourth order iterative method for solving nonlinear equations by combining arithmetic mean Newton method, harmonic mean Newton method and midpoint Newton method uniquely. Also, some variant of Newton type methods based on inverse function have been developed. These methods are free from second order derivatives.

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10

Zhao, Wan-Chen, and Xin-Hui Shao. "New matrix splitting iteration method for generalized absolute value equations." AIMS Mathematics 8, no. 5 (2023): 10558–78. http://dx.doi.org/10.3934/math.2023536.

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<abstract><p>In this paper, a relaxed Newton-type matrix splitting (RNMS) iteration method is proposed for solving the generalized absolute value equations, which includes the Picard method, the modified Newton-type (MN) iteration method, the shift splitting modified Newton-type (SSMN) iteration method and the Newton-based matrix splitting (NMS) iteration method. We analyze the sufficient convergence conditions of the RNMS method. Lastly, the efficiency of the RNMS method is analyzed by numerical examples involving symmetric and non-symmetric matrices.</p></abstract>

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Jnawali, Jivandhar. "Higher Order Convergent Newton Type Iterative Methods." Journal of Nepal Mathematical Society 1, no. 2 (August 5, 2018): 32–39. http://dx.doi.org/10.3126/jnms.v1i2.41488.

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Newton method is one of the most widely used numerical methods for solving nonlinear equations. McDougall and Wotherspoon [Appl. Math. Lett., 29 (2014), 20-25] modified this method in predictor-corrector form and get an order of convergence 1+√2. More on the PDF

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12

Gugat, Matin. "Semi-infinite terminal problems: a newton type method." Optimization 44, no. 1 (January 1998): 25–48. http://dx.doi.org/10.1080/02331939808844398.

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13

Amat, S., S. Busquier, A. Escudero, and F. Manzano. "A wavelet adaptive two-step Newton type method." Journal of the Franklin Institute 348, no. 5 (June 2011): 823–31. http://dx.doi.org/10.1016/j.jfranklin.2011.02.006.

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Amat, S., Á. A. Magreñán, and N. Romero. "On a two-step relaxed Newton-type method." Applied Mathematics and Computation 219, no. 24 (August 2013): 11341–47. http://dx.doi.org/10.1016/j.amc.2013.04.061.

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15

Santos, P. J. S., P. S. M. Santos, and S. Scheimberg. "A proximal Newton-type method for equilibrium problems." Optimization Letters 12, no. 5 (October 10, 2017): 997–1009. http://dx.doi.org/10.1007/s11590-017-1204-z.

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16

Fliege, Jörg, Andrey Tin, and Alain Zemkoho. "Gauss–Newton-type methods for bilevel optimization." Computational Optimization and Applications 78, no. 3 (January 10, 2021): 793–824. http://dx.doi.org/10.1007/s10589-020-00254-3.

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AbstractThis article studies Gauss–Newton-type methods for over-determined systems to find solutions to bilevel programming problems. To proceed, we use the lower-level value function reformulation of bilevel programs and consider necessary optimality conditions under appropriate assumptions. First, under strict complementarity for upper- and lower-level feasibility constraints, we prove the convergence of a Gauss–Newton-type method in computing points satisfying these optimality conditions under additional tractable qualification conditions. Potential approaches to address the shortcomings of the method are then proposed, leading to alternatives such as the pseudo or smoothing Gauss–Newton-type methods for bilevel optimization. Our numerical experiments conducted on 124 examples from the recently released Bilevel Optimization LIBrary (BOLIB) compare the performance of our method under different scenarios and show that it is a tractable approach to solve bilevel optimization problems with continuous variables.

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17

Candelario, Giro, Alicia Cordero, and Juan R. Torregrosa. "Multipoint Fractional Iterative Methods with (2α + 1)th-Order of Convergence for Solving Nonlinear Problems." Mathematics 8, no. 3 (March 20, 2020): 452. http://dx.doi.org/10.3390/math8030452.

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In the recent literature, some fractional one-point Newton-type methods have been proposed in order to find roots of nonlinear equations using fractional derivatives. In this paper, we introduce a new fractional Newton-type method with order of convergence α + 1 and compare it with the existing fractional Newton method with order 2 α . Moreover, we also introduce a multipoint fractional Traub-type method with order 2 α + 1 and compare its performance with that of its first step. Some numerical tests and analysis of the dependence on the initial estimations are made for each case, including a comparison with classical Newton ( α = 1 of the first step of the class) and classical Traub’s scheme ( α = 1 of fractional proposed multipoint method). In this comparison, some cases are found where classical Newton and Traub’s methods do not converge and the proposed methods do, among other advantages.

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18

Zhou, Wen, and Jisheng Kou. "Third-Order Newton-Type Methods Combined with Vector Extrapolation for Solving Nonlinear Systems." Abstract and Applied Analysis 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/601745.

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We present a third-order method for solving the systems of nonlinear equations. This method is a Newton-type scheme with the vector extrapolation. We establish the local and semilocal convergence of this method. Numerical results show that the composite method is more robust and efficient than a number of Newton-type methods with the other vector extrapolations.

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19

Bisheh-Niasar, Morteza, and Abbas Saadatmandi. "Some novel Newton-type methods for solving nonlinear equations." Boletim da Sociedade Paranaense de Matemática 38, no. 3 (February 18, 2019): 111–23. http://dx.doi.org/10.5269/bspm.v38i3.37351.

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The aim of this paper is to present a new nonstandard Newton iterative method for solving nonlinear equations. The convergence of the proposed method is proved and it is shown that the new method has cubic convergence. Furthermore, two new multi-point methods with sixth-order convergence, based on the introduced method, are presented. Also, we describe the basins of attraction for these methods. Finally, some numerical examples are given to show the performance of our methods by comparing with some other methods available in the literature

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20

Rashid, MH, and A. Sarder. "Convergence of the Newton-Type Method for Generalized Equations." GANIT: Journal of Bangladesh Mathematical Society 35 (June 28, 2016): 27–40. http://dx.doi.org/10.3329/ganit.v35i0.28565.

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Let X and Y be real or complex Banach spaces. Suppose that f: X->Y is a Frechet differentiable function and F: X => 2Yis a set-valued mapping with closed graph. In the present paper, we study the Newton-type method for solving generalized equation 0 ? f(x) + F(x). We prove the existence of the sequence generated by the Newton-type method and establish local convergence of the sequence generated by this method for generalized equation.GANIT J. Bangladesh Math. Soc.Vol. 35 (2015) 27-40

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21

Guo, Miao, and Qingbiao Wu. "Two effective inexact iteration methods for solving the generalized absolute value equations." AIMS Mathematics 7, no. 10 (2022): 18675–89. http://dx.doi.org/10.3934/math.20221027.

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<abstract><p>Modified Newton-type methods are efficient for addressing the generalized absolute value equations. In this paper, to further speed up the modified Newton-type methods, two new inexact modified Newton-type iteration methods are proposed. The sufficient conditions for the convergence of the two proposed inexact iteration methods are given. Moreover, to demonstrate the efficacy of the new method, several numerical examples are provided.</p></abstract>

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22

J. Wolters, Hans. "A Newton-type method for computing best segment approximations." Communications on Pure & Applied Analysis 3, no. 1 (2004): 133–48. http://dx.doi.org/10.3934/cpaa.2004.3.133.

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23

Jnawali, Jivandhar. "A Newton Type Iterative Method with Fourth-order Convergence." Journal of the Institute of Engineering 12, no. 1 (March 6, 2017): 87–95. http://dx.doi.org/10.3126/jie.v12i1.16729.

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The aim of this paper is to propose a fourth-order Newton type iterative method for solving nonlinear equations in a single variable. We obtained this method by combining the iterations of contra harmonic Newton’s method with secant method. The proposed method is free from second order derivative. Some numerical examples are given to illustrate the performance and to show this method’s advantage over other compared methods.Journal of the Institute of Engineering, 2016, 12 (1): 87-95

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24

Lee, Joon-Ho, Sung-Woo Cho, and Hyung Seok Kim. "Newton-type method in spectrum estimaion-based AOA estimation." IEICE Electronics Express 9, no. 12 (2012): 1036–43. http://dx.doi.org/10.1587/elex.9.1036.

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25

Amat, S., M. J. Légaz, and P. Pedregal. "On a Newton-Type Method for Differential-Algebraic Equations." Journal of Applied Mathematics 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/718608.

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This paper deals with the approximation of systems of differential-algebraic equations based on a certain error functional naturally associated with the system. In seeking to minimize the error, by using standard descent schemes, the procedure can never get stuck in local minima but will always and steadily decrease the error until getting to the solution sought. Starting with an initial approximation to the solution, we improve it by adding the solution of some associated linear problems, in such a way that the error is significantly decreased. Some numerical examples are presented to illustrate the main theoretical conclusions. We should mention that we have already explored, in some previous papers (Amat et al., in press, Amat and Pedregal, 2009, and Pedregal, 2010), this point of view for regular problems. However, the main hypotheses in these papers ask for some requirements that essentially rule out the application to singular problems. We are also preparing a much more ambitious perspective for the theoretical analysis of nonlinear DAEs based on this same approach.

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26

Basu, Dhiman. "Composite Fourth Order Newton Type Method for Simple Root." International Journal for Computational Methods in Engineering Science and Mechanics 9, no. 4 (May 30, 2008): 201–10. http://dx.doi.org/10.1080/15502280802069889.

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27

Tao, Yuxi, and Xiaofeng Wan. "A Newton Type Iterative Method with Seventh-Order Convergence." Journal of Physics: Conference Series 1648 (October 2020): 042123. http://dx.doi.org/10.1088/1742-6596/1648/4/042123.

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28

Azé, D., and C. C. Chou. "On a Newton Type Iterative Method for Solving Inclusions." Mathematics of Operations Research 20, no. 4 (November 1995): 790–800. http://dx.doi.org/10.1287/moor.20.4.790.

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29

Zhou, Guanglu, and Liqun Qi. "On the convergence of an inexact Newton-type method." Operations Research Letters 34, no. 6 (November 2006): 647–52. http://dx.doi.org/10.1016/j.orl.2005.11.001.

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30

Fang, Liang, Guoping He, and Zhongyong Hu. "A cubically convergent Newton-type method under weak conditions." Journal of Computational and Applied Mathematics 220, no. 1-2 (October 2008): 409–12. http://dx.doi.org/10.1016/j.cam.2007.08.013.

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31

Amat, S., S. Busquier, and S. Plaza. "Chaotic dynamics of a third-order Newton-type method." Journal of Mathematical Analysis and Applications 366, no. 1 (June 2010): 24–32. http://dx.doi.org/10.1016/j.jmaa.2010.01.047.

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32

Cárdenas, Elkin, Rodrigo Castro, and Willy Sierra. "A Newton-type midpoint method with high efficiency index." Journal of Mathematical Analysis and Applications 491, no. 2 (November 2020): 124381. http://dx.doi.org/10.1016/j.jmaa.2020.124381.

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33

Amat, S., S. Busquier, and Á. A. Magreñán. "Reducing Chaos and Bifurcations in Newton-Type Methods." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/726701.

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We study the dynamics of some Newton-type iterative methods when they are applied of polynomials degrees two and three. The methods are free of high-order derivatives which are the main limitation of the classical high-order iterative schemes. The iterative schemes consist of several steps of damped Newton's method with the same derivative. We introduce a damping factor in order to reduce thebadzones of convergence. The conclusion is that the damped schemes become real alternative to the classical Newton-type method since both chaos and bifurcations of the original schemes are reduced. Therefore, the new schemes can be utilized to obtain good starting points for the original schemes.

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34

Shen, Jie, Li-Ping Pang, and Dan Li. "An Approximate Quasi-Newton Bundle-Type Method for Nonsmooth Optimization." Abstract and Applied Analysis 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/697474.

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An implementable algorithm for solving a nonsmooth convex optimization problem is proposed by combining Moreau-Yosida regularization and bundle and quasi-Newton ideas. In contrast with quasi-Newton bundle methods of Mifflin et al. (1998), we only assume that the values of the objective function and its subgradients are evaluated approximately, which makes the method easier to implement. Under some reasonable assumptions, the proposed method is shown to have a Q-superlinear rate of convergence.

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35

Guo, Pei-Chang. "A Fast Newton-Shamanskii Iteration for a Matrix Equation Arising from M/G/1-Type Markov Chains." Mathematical Problems in Engineering 2017 (2017): 1–8. http://dx.doi.org/10.1155/2017/4018239.

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For the nonlinear matrix equations arising in the analysis of M/G/1-type and GI/M/1-type Markov chains, the minimal nonnegative solution G or R can be found by Newton-like methods. We prove monotone convergence results for the Newton-Shamanskii iteration for this class of equations. Starting with zero initial guess or some other suitable initial guess, the Newton-Shamanskii iteration provides a monotonically increasing sequence of nonnegative matrices converging to the minimal nonnegative solution. A Schur decomposition method is used to accelerate the Newton-Shamanskii iteration. Numerical examples illustrate the effectiveness of the Newton-Shamanskii iteration.

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36

Liang, Ke, Mostafa M. Abdalla, and Qin Sun. "A modified Newton-type Koiter-Newton method for tracing the geometrically nonlinear response of structures." International Journal for Numerical Methods in Engineering 113, no. 10 (November 22, 2017): 1541–60. http://dx.doi.org/10.1002/nme.5709.

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37

Khaksar Haghani, F. "A Third-Order Newton-Type Method for Finding Polar Decomposition." Advances in Numerical Analysis 2014 (September 30, 2014): 1–5. http://dx.doi.org/10.1155/2014/576325.

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It is attempted to present an iteration method for finding polar decomposition. The approach is categorized in the scope of Newton-type methods. Error analysis and rate of convergence are studied. Some illustrations are also given to disclose the numerical behavior of the proposed method.

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38

Kumar, Manoj. "On an new Open type variant of Newton´s method." Journal of Applied Mathematics, Statistics and Informatics 10, no. 2 (December 1, 2014): 21–31. http://dx.doi.org/10.2478/jamsi-2014-0010.

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Abstract The aim of the present paper is to introduce and investigate a new Open type variant of Newton's method for solving nonlinear equations. The order of convergence of the proposed method is three. In addition to numerical tests verifying the theory, a comparison of the results for the proposed method and some of the existing ones have also been given.

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39

Ohtsuka, Toshiyuki. "Quasi-Newton-Type Continuation Method for Nonlinear Receding Horizon Control." Journal of Guidance, Control, and Dynamics 25, no. 4 (July 2002): 685–92. http://dx.doi.org/10.2514/2.4935.

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40

Pieraccini, S. "Hybrid Newton-Type Method for a Class of Semismooth Equations." Journal of Optimization Theory and Applications 112, no. 2 (February 2002): 381–402. http://dx.doi.org/10.1023/a:1013610108041.

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41

Bai, Zheng-Jian, and Shufang Xu. "An inexact Newton-type method for inverse singular value problems." Linear Algebra and its Applications 429, no. 2-3 (July 2008): 527–47. http://dx.doi.org/10.1016/j.laa.2008.03.008.

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42

H�u�ler, W. M. "A Kantorovich-type convergence analysis for the Gauss-Newton-Method." Numerische Mathematik 48, no. 1 (January 1986): 119–25. http://dx.doi.org/10.1007/bf01389446.

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43

Zhang, Xinzhen, Hefeng Jiang, and Yiju Wang. "A smoothing Newton-type method for generalized nonlinear complementarity problem." Journal of Computational and Applied Mathematics 212, no. 1 (February 2008): 75–85. http://dx.doi.org/10.1016/j.cam.2006.03.042.

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44

Hoppe, R. H. W., S. I. Petrova, and V. Schulz. "Primal-Dual Newton-Type Interior-Point Method for Topology Optimization." Journal of Optimization Theory and Applications 114, no. 3 (September 2002): 545–71. http://dx.doi.org/10.1023/a:1016070928600.

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45

Thukral, R. "Introduction to a Newton-type method for solving nonlinear equations." Applied Mathematics and Computation 195, no. 2 (February 2008): 663–68. http://dx.doi.org/10.1016/j.amc.2007.05.013.

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46

Deng, Haoyang, and Toshiyuki Ohtsuka. "A parallel Newton-type method for nonlinear model predictive control." Automatica 109 (November 2019): 108560. http://dx.doi.org/10.1016/j.automatica.2019.108560.

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47

He, Guo-qiang, and Ze-hong Meng. "A Newton type iterative method for heat-conduction inverse problems." Applied Mathematics and Mechanics 28, no. 4 (April 2007): 531–39. http://dx.doi.org/10.1007/s10483-007-0414-y.

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48

Fischer, A. "A Newton-type method for positive-semidefinite linear complementarity problems." Journal of Optimization Theory and Applications 86, no. 3 (September 1995): 585–608. http://dx.doi.org/10.1007/bf02192160.

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49

Argyros, Ioannis K., and Saïd Hilout. "An improved local convergence analysis for Newton–Steffensen-type method." Journal of Applied Mathematics and Computing 32, no. 1 (January 27, 2009): 111–18. http://dx.doi.org/10.1007/s12190-009-0236-7.

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50

Khaton, M., M. Rashid, and M. Hossain. "Convergence Properties of Extended Newton-type Iteration Method for Generalized Equations." Journal of Mathematics Research 10, no. 4 (April 19, 2018): 1. http://dx.doi.org/10.5539/jmr.v10n4p1.

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In this paper, we introduce and study the extended Newton-type method for solving generalized equation $0\in f(x)+g(x)+\mathcal F(x)$, where $f:\Omega\subseteq\mathcal X\to \mathcal Y$ is Fr\'{e}chet differentiable in a neighborhood $\Omega$ of a point $\bar{x}$ in $\mathcal X$, $g:\Omega\subseteq \mathcal X\to \mathcal Y$ is linear and differentiable at a point $\bar{x}$, and $\mathcal F$ is a set-valued mapping with closed graph acting in Banach spaces $\mathcal X$ and $\mathcal Y$. Semilocal and local convergence of the extended Newton-type method are analyzed.

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Journal articles on the topic 'Newton-type method'

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