On the convergence radius of the modified Newton method for multiple roots under the center–Hölder condition

2013 ◽  
Vol 65 (2) ◽  
pp. 221-232 ◽  
Author(s):  
Xiaojian Zhou ◽  
Xin Chen ◽  
Yongzhong Song
Keyword(s):  
Newton Method ◽  
Multiple Roots ◽  
Hölder Condition ◽  
Convergence Radius ◽  
Modified Newton Method

scholarly journals New Families of Third-Order Iterative Methods for Finding Multiple Roots

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
R. F. Lin ◽  
H. M. Ren ◽  
Z. Šmarda ◽  
Q. B. Wu ◽  
Y. Khan ◽  
...  
Keyword(s):  
Iterative Methods ◽  
Newton Method ◽  
Multiple Roots ◽  
Third Order ◽  
Cubic Convergence ◽  
Numerical Examples ◽  
Mild Conditions ◽  
First Derivative ◽  
Modified Newton Method ◽  
Iteration Methods

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.

scholarly journals On the local convergence of the Modified Newton method

2019 ◽  
Vol 57 (1) ◽  
pp. 13-22
Author(s):  
Ştefan Măruşter
Keyword(s):  
Newton Method ◽  
Local Convergence ◽  
Convergence Order ◽  
Convergence Radius ◽  
First Derivative ◽  
Modified Newton Method

Abstract The aim of this paper is to investigate the local convergence of the Modified Newton method, i.e. the classical Newton method in which the first derivative is re-evaluated periodically after m steps. The convergence order is shown to be m + 1. A new algorithm is proposed for the estimation the convergence radius of the method. We propose also a threshold for the number of steps after which is recommended to re-evaluate the first derivative in the Modified Newton method.

scholarly journals A Modified Newton Method for Multilinear PageRank

2018 ◽  
Vol 22 (5) ◽  
pp. 1161-1171
Author(s):  
Pei-Chang Guo ◽  
Shi-Chen Gao ◽  
Xiao-Xia Guo
Keyword(s):  
Newton Method ◽  
Modified Newton Method

scholarly journals A Fast Sparse Recovery Algorithm for Compressed Sensing Using Approximate l0 Norm and Modified Newton Method

Materials ◽  
2019 ◽  
Vol 12 (8) ◽  
pp. 1227 ◽  
Author(s):  
Dingfei Jin ◽  
Yue Yang ◽  
Tao Ge ◽  
Daole Wu
Keyword(s):  
Compressed Sensing ◽  
Newton Method ◽  
Simulation Experiment ◽  
Sparse Recovery ◽  
Signal Recovery ◽  
Recovery Efficiency ◽  
Recovery Algorithm ◽  
Modified Newton Method ◽  
Computer Simulation Experiment ◽  
Recovery Algorithms

In this paper, we propose a fast sparse recovery algorithm based on the approximate l0 norm (FAL0), which is helpful in improving the practicability of the compressed sensing theory. We adopt a simple function that is continuous and differentiable to approximate the l0 norm. With the aim of minimizing the l0 norm, we derive a sparse recovery algorithm using the modified Newton method. In addition, we neglect the zero elements in the process of computing, which greatly reduces the amount of computation. In a computer simulation experiment, we test the image denoising and signal recovery performance of the different sparse recovery algorithms. The results show that the convergence rate of this method is faster, and it achieves nearly the same accuracy as other algorithms, improving the signal recovery efficiency under the same conditions.

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