A Fixed-Point Iteration Method With Quadratic Convergence

The fixed-point iteration algorithm is turned into a quadratically convergent scheme for a system of nonlinear equations. Most of the usual methods for obtaining the roots of a system of nonlinear equations rely on expanding the equation system about the roots in a Taylor series, and neglecting the higher order terms. Rearrangement of the resulting truncated system then results in the usual Newton-Raphson and Halley type approximations. In this paper the introduction of unit root functions avoids the direct expansion of the nonlinear system about the root, and relies, instead, on approximations which enable the unit root functions to considerably widen the radius of convergence of the iteration method. Methods for obtaining higher order rates of convergence and larger radii of convergence are discussed.

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Geometrically Constructed Family of the Simple Fixed Point Iteration Method

Mathematics ◽

10.3390/math9060694 ◽

2021 ◽

Vol 9 (6) ◽

pp. 694

Author(s):

Vinay Kanwar ◽

Puneet Sharma ◽

Ioannis K. Argyros ◽

Ramandeep Behl ◽

Christopher Argyros ◽

...

Keyword(s):

Fixed Point ◽

Fixed Points ◽

Nonlinear Equations ◽

Iteration Method ◽

Parameter Family ◽

Arbitrary Parameter ◽

Fixed Point Iteration ◽

Iterative Schemes ◽

Straight Line ◽

Predictor Corrector

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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Structural Analysis Methods for Differential Algebraic Equations via Fixed-Point Iteration

Journal of Computational and Theoretical Nanoscience ◽

10.1166/jctn.2016.4463 ◽

2016 ◽

Vol 13 (10) ◽

pp. 7705-7711 ◽

Cited By ~ 2

Author(s):

Juan Tang ◽

Wenyuan Wu ◽

Xiaolin Qin ◽

Yong Feng

Keyword(s):

Fixed Point ◽

Structural Analysis ◽

Iteration Method ◽

Large Scale ◽

Complexity Analysis ◽

Differential Algebraic Equations ◽

Iteration Algorithm ◽

Algebraic Equations ◽

Fixed Point Iteration ◽

Dae Systems

Motivated by Pryce’s structural analysis method for differential algebraic equations (DAEs), we show the complexity of the fixed-point iteration algorithm (FPIA) and propose a fixed-point iteration method with parameters. It leads to a block fixed-point iteration method (BFPIM) which can be applied to immediately calculate the crucial canonical offsets for large-scale (coupled) DAE systems with block-triangular structure, and its complexity analysis is also given in this paper. Moreover, preliminary numerical experiments show that the time complexity of BFPIM can be reduced by at least O(l) compared to the FPIA.

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A new fixed point iteration method for nonlinear third-order BVPs

International Journal of Computer Mathematics ◽

10.1080/00207160.2021.1883594 ◽

2021 ◽

pp. 1-13

Author(s):

S. A. Khuri ◽

I. Louhichi

Keyword(s):

Fixed Point ◽

Iteration Method ◽

Third Order ◽

Fixed Point Iteration

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Metode Iterasi Tiga Langkah untuk Menyelesaikan Persamaan NonLinear dengan Menggunakan Matlab

JISKA (Jurnal Informatika Sunan Kalijaga) ◽

10.14421/jiska.2019.42-05 ◽

2019 ◽

Vol 4 (2) ◽

pp. 34

Author(s):

Deasy Wahyuni ◽

Elisawati Elisawati

Keyword(s):

Numerical Simulations ◽

Nonlinear Equations ◽

Newton Method ◽

Iteration Method ◽

Newton's Method ◽

Order Of Convergence ◽

Higher Order ◽

Convergence Order ◽

Matlab Program ◽

Analytical Results

Newton method is one of the most frequently used methods to find solutions to the roots of nonlinear equations. Along with the development of science, Newton's method has undergone various modifications. One of them is the hasanov method and the newton method variant (vmn), with a higher order of convergence. In this journal focuses on the three-step iteration method in which the order of convergence is higher than the three methods. To find the convergence order of the three-step iteration method requires a program that can support the analytical results of both methods. One of them using the help of the matlab program. Which will then be compared with numerical simulations also using the matlab program. Keywords : newton method, newton method variant, Hasanov Method and order of convergence

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TENSILE PROPERTY AND CRACK PROPAGATION BEHAVIOR OF TUNGSTEN ALLOYS

International Journal of Modern Physics B ◽

10.1142/s0217979211100114 ◽

2011 ◽

Vol 25 (11) ◽

pp. 1475-1492 ◽

Cited By ~ 2

Author(s):

WEIDONG SONG ◽

HAIYAN LIU ◽

JIANGUO NING

Keyword(s):

Fixed Point ◽

Iteration Method ◽

Experimental System ◽

Unit Cell ◽

Cell Models ◽

Fixed Point Iteration ◽

Tungsten Alloys ◽

Propagation Behavior ◽

Numerical Predictions ◽

Microstructure Parameters

In situ SEM experimental system is employed to investigate the mechanical characteristics and the fracture behavior of 91W–6.3Ni–2.7Fe tungsten alloys. The crack initiation and propagation of tungsten alloys under tensile loadings are examined. Multi-particle unit cell models containing the microstructure characteristics of tungsten alloys are established. Fixed-point iteration method is firstly used for the multi-particle unit cell's boundary condition. By adopting the method, real displacement constrained conditions are applied on the multi-particle unit cell models. The mechanical and fracture behaviors of tungsten alloys under tensile loading are simulated. The effects of tungsten content, particle shape, particle size, and interface strength on the mechanical properties of tungsten alloys are analyzed. The relationship between the mechanical behaviors and the microstructure parameters is studied. A good agreement is obtained between the experimental results and the numerical predictions, verifying the rationality of the FE models using the fixed-point iteration method.

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Regenerasi Fungsi x2-9x-99 Dalam Pembangkit Bilangan Acak Berbasis CSRPNG Chaos

AITI ◽

10.24246/aiti.v16i2.125-134 ◽

2020 ◽

Vol 16 (2) ◽

pp. 125-134

Author(s):

David Lihananto ◽

Alz Danny Wowor

Keyword(s):

Fixed Point ◽

Iteration Method ◽

Quadratic Function ◽

Random Numbers ◽

Fixed Point Iteration ◽

Iteration Function ◽

Run Test ◽

Randomization Testing ◽

Correlation Level

This study examines whether the function f(x)=x2-9x-99 can be used as a key generator in cryptography. The quadratic function is regenerated using the fixed point iteration method into an iteration function. The distribution of digits to the output of iterative function to generate a number of chaos. Randomization testing uses run test and monobit testing. Followed by cryptographic testing to get the correlation between ciphertext and key which will be used as a decision whether the resulting key is random or not. Based on research that has been done iteration function xi = (xi-12-9xi-1-99)/9 can generate CSRPNG Chaos random numbers with the correlation level closest to the value of 0.

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Bounded Fixed Point Iteration

DAIMI Report Series ◽

10.7146/dpb.v20i359.6589 ◽

1991 ◽

Vol 20 (359) ◽

Author(s):

Hanne Riis Nielson ◽

Flemming Nielson

Keyword(s):

Fixed Point ◽

Abstract Interpretation ◽

Higher Order ◽

Monotone Functions ◽

Additive Functions ◽

Fixed Point Iteration ◽

Exponential Bound ◽

Strictness Analysis ◽

Long Chains

In the context of abstract interpretation for languages without higher-order features we study the number of times a functional need to be unfolded in order to give the least fixed point. For the cases of total or monotone functions we obtain an exponential bound and in the case of strict and additive (or distributive) functions we obtain a quadratic bound. These bounds are shown to be tight in that sufficiently long chains of functions can be shown to exist. Specializing the case of strict and additive functions to functionals of a form that would correspond to iterative programs we show that a linear bound is tight. This is related to several analyses studied in the literature (including strictness analysis).

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