scholarly journals Impact on Stability by the Use of Memory in Traub-Type Schemes

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 274
Author(s):  
Francisco I. Chicharro ◽  
Alicia Cordero ◽  
Neus Garrido ◽  
Juan R. Torregrosa
Keyword(s):  
Order Of Convergence ◽  
Nonlinear Problems ◽  
Higher Order ◽  
Lower Number ◽  
Dynamical Analysis ◽  
The Real ◽  
Original Scheme ◽  
Interpolation Polynomials ◽  
With Memory ◽  
Number Of Iterations

In this work, two Traub-type methods with memory are introduced using accelerating parameters. To obtain schemes with memory, after the inclusion of these parameters in Traub’s method, they have been designed using linear approximations or the Newton’s interpolation polynomials. In both cases, the parameters use information from the current and the previous iterations, so they define a method with memory. Moreover, they achieve higher order of convergence than Traub’s scheme without any additional functional evaluations. The real dynamical analysis verifies that the proposed methods with memory not only converge faster, but they are also more stable than the original scheme. The methods selected by means of this analysis can be applied for solving nonlinear problems with a wider set of initial estimations than their original partners. This fact also involves a lower number of iterations in the process.

scholarly journals Memorizing Schröder’s Method as an Efficient Strategy for Estimating Roots of Unknown Multiplicity

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2570
Author(s):  
Alicia Cordero ◽  
Beny Neta ◽  
Juan R. Torregrosa
Keyword(s):  
Order Of Convergence ◽  
Iterative Scheme ◽  
Basins Of Attraction ◽  
Higher Order ◽  
Multiple Roots ◽  
Higher Order Methods ◽  
Efficient Strategy ◽  
Schröder’S Method ◽  
With Memory

In this paper, we propose, to the best of our knowledge, the first iterative scheme with memory for finding roots whose multiplicity is unknown existing in the literature. It improves the efficiency of a similar procedure without memory due to Schröder and can be considered as a seed to generate higher order methods with similar characteristics. Once its order of convergence is studied, its stability is analyzed showing its good properties, and it is compared numerically in terms of their basins of attraction with similar schemes without memory for finding multiple roots.

scholarly journals Iterative Methods with Memory for Solving Systems of Nonlinear Equations Using a Second Order Approximation

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1069 ◽  
Author(s):  
Alicia Cordero ◽  
Javier G. Maimó ◽  
Juan R. Torregrosa ◽  
María P. Vassileva
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Order Approximation ◽  
Multidimensional Case ◽  
Systems Of Nonlinear Equations ◽  
Simple Method ◽  
The Right ◽  
Interpolation Polynomials ◽  
With Memory

Iterative methods for solving nonlinear equations are said to have memory when the calculation of the next iterate requires the use of more than one previous iteration. Methods with memory usually have a very stable behavior in the sense of the wideness of the set of convergent initial estimations. With the right choice of parameters, iterative methods without memory can increase their order of convergence significantly, becoming schemes with memory. In this work, starting from a simple method without memory, we increase its order of convergence without adding new functional evaluations by approximating the accelerating parameter with Newton interpolation polynomials of degree one and two. Using this technique in the multidimensional case, we extend the proposed method to systems of nonlinear equations. Numerical tests are presented to verify the theoretical results and a study of the dynamics of the method is applied to different problems to show its stability.

scholarly journals On Some Iterative Methods with Memory and High Efficiency Index for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Tahereh Eftekhari
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
High Efficiency ◽  
Efficiency Index ◽  
Order Convergence ◽  
Eighth Order ◽  
Additional Function ◽  
Numerical Comparisons ◽  
With Memory

Based on iterative methods without memory of eighth-order convergence proposed by Thukral (2012), some iterative methods with memory and high efficiency index are presented. We show that the order of convergence is increased without any additional function evaluations. Numerical comparisons are made to show the performance of the presented methods.

scholarly journals Reversals in the large-scale αΩ-dynamo with memory

2015 ◽  
Vol 22 (4) ◽  
pp. 361-369 ◽  
Author(s):  
L. K. Feschenko ◽  
G. M. Vodinchar
Keyword(s):  
Magnetic Field ◽  
Power Law ◽  
Geomagnetic Field ◽  
Large Scale ◽  
Magnetic Polarity ◽  
Power Law Model ◽  
The Real ◽  
Geomagnetic Polarity ◽  
With Memory ◽  
The Magnetic Field

Abstract. Inversion of the magnetic field in a model of large-scale αΩ-dynamo with α-effect with stochastic memory is under investigation. The model allows us to reproduce the main features of the geomagnetic field reversals. It was established that the polarity intervals in the model are distributed according to the power law. Model magnetic polarity timescale is fractal. Its dimension is consistent with the dimension of the real geomagnetic polarity timescale.

scholarly journals On a Derivative-Free Variant of King’s Family with Memory

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
M. Sharifi ◽  
S. Karimi Vanani ◽  
F. Khaksar Haghani ◽  
M. Arab ◽  
S. Shateyi
Keyword(s):  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Derivative Free ◽  
With Memory

The aim of this paper is to construct a method with memory according to King’s family of methods without memory for nonlinear equations. It is proved that the proposed method possesses higherR-order of convergence using the same number of functional evaluations as King’s family. Numerical experiments are given to illustrate the performance of the constructed scheme.

scholarly journals Metode Iterasi Tiga Langkah untuk Menyelesaikan Persamaan NonLinear dengan Menggunakan Matlab

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

scholarly journals EFEKTIVITAS PEMBELAJARAN BENTUK MOLEKUL DENGAN PEMODELAN REAL BERBASIS PENEMUAN TERBIMBING UNTUK MELATIHKAN KETERAMPILAN BERPIKIR TINGKAT TINGGI SISWA

2017 ◽  
Vol 3 (2) ◽  
pp. 332
Author(s):  
Sutrisno Sutrisno ◽  
Sri Poedjiastoeti ◽  
I Gusti Made Sanjaya
Keyword(s):  
Higher Order Thinking ◽  
Thinking Skills ◽  
Higher Order ◽  
Lesson Plans ◽  
Average Score ◽  
Learning Materials ◽  
Guided Discovery ◽  
Gain Score ◽  
The Real ◽  
Learning And Teaching

This study aimed to describe the effectiveness of learning materials on shape of the molecule with the real modeling supported by PhET media-based on guided discovery to facilitate the students’ high-order-thinking skills at odd semester XI class of SMAN 10. This research is developmental research using 4D models. Thetest of the learning materials use one group pretest-posttest design. The results of validity syllabus (3.87), lesson plans (3.71), students’ book (3.35), work sheet (3.63), and test of products (3.58) are categorized very good and reliability syllabus (99%), lesson plans (100%), students’ book (89%), work sheet (100%), and test of products (100%) are categorized reliable. The Achievement test of higher-order thinking skills showed that the individuals completeness an average score of 82.79, the average sensitivity of items was 0.74 and the average individual gain score of 0.82. Students' response to the learning and teaching activities in average were well-categorized. Based on the findings of the study, it can be concluded that the shape of molecule with the real modeling supported by PhET media based on guided discovery–was effective to train the students' higher-order thinking skills.Penelitian ini bertujuan untukmendeskripsikan efektivitas perangkat pembelajaran bentuk molekul dengan pemodelan real ditunjang media PhET berbasis penemuan terbimbinguntuk melatihkan keterampilan berpikir tingkat tinggi siswa kelas XI semester ganjil SMAN 10 Samarinda pada materi bentuk molekul.Perangkat pembelajaran yang digunakan dikembangkan dengan model 4D. Perangkatpembelajaran di uji cobakan menggunakan one group pretest-posttest design. Validitas Silabus (3,87), RPP (3,71), BAS (3,35), LKS (3,63), dan LP Produk (3,58) berkategori sangat baik dan reliabilitas Silabus (99%), RPP (100%), BAS (89), LKS (100%), LP Produk (100%) berkategori reliabel. Tes hasil belajar keterampilan berpikir tingkat tinggi menunjukkan ketuntasan individual rata-rata 82,79, sensitivitas butir soal rata-rata 0,74 dan gain score individual rata-rata 0,82. Respon siswa terhadap perangkat pembelajaran dan kegiatan pembelajaran rata-rata baik.Berdasarkan temuan hasil penelitian dapat disimpulkan bahwa perangkat pembelajaran bentuk molekul dengan pemodelan real ditunjang media PhET berbasis model penemuan terbimbingefektif untuk melatihkan keterampilan berpikir tingkat tinggi siswa.

Forecasting Groundwater Level Based on Relevance Vector Machine

2010 ◽  
Vol 121-122 ◽  
pp. 43-47 ◽  
Author(s):  
Li Ying Wang ◽  
Wei Guo Zhao
Keyword(s):  
Groundwater Level ◽  
Kernel Method ◽  
Nonlinear Problems ◽  
Relevance Vector Machine ◽  
Kernel Functions ◽  
Experimental Results ◽  
The Real

Relevance Vector Machine (RVM) is a novel kernel method based on sparse Bayesian, which has many advantages such as its kernel functions without the restriction of Mercer’s conditions, and the relevance vectors are automatically determined and have fewer parameters. In this paper, the RVM model is applied to forecasting groundwater level. The experimental results show the final RVM model achieved is sparser, the prediction precision is higher and the prediction values are in better agreement with the real values. It can be concluded that this technique can be seen as a very promising option to solve nonlinear problems such as forecasting groundwater level.

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