scholarly journals Ball Convergence of a Parametric Efficient Family of Iterative Methods for Solving Nonlinear Equations

Foundations ◽  
2021 ◽  
Vol 1 (1) ◽  
pp. 23-31
Author(s):  
Samundra Regmi ◽  
Christopher I. Argyros ◽  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Linear Convergence ◽  
Error Tolerance ◽  
First Derivative ◽  
Uniqueness Of The Solution ◽  
Ball Convergence ◽  
Space Case ◽  
Local Results ◽  
Lipschitz Conditions ◽  
Theoretical Results ◽  
General Banach Space

The goal is to extend the applicability of Newton-Traub-like methods in cases not covered in earlier articles requiring the usage of derivatives up to order seven that do not appear in the methods. The price we pay by using conditions on the first derivative that actually appear in the method is that we show only linear convergence. To find the convergence order is not our intention, however, since this is already known in the case where the spaces coincide with the multidimensional Euclidean space. Note that the order is rediscovered by using ACOC or COC, which require only the first derivative. Moreover, in earlier studies using Taylor series, no computable error distances were available based on generalized Lipschitz conditions. Therefore, we do not know, for example, in advance, how many iterates are needed to achieve a predetermined error tolerance. Furthermore, no uniqueness of the solution results is available in the aforementioned studies, but we also provide such results. Our technique can be used to extend the applicability of other methods in an analogous way, since it is so general. Finally note that local results of this type are important, since they demonstrate the difficulty in choosing initial points. Our approach also extends the applicability of this family of methods from the multi-dimensional Euclidean to the more general Banach space case. Numerical examples complement the theoretical results.

scholarly journals Extended ball convergence for a seventh order derivative free class of algorithms for nonlinear equations

2021 ◽  
Vol 56 (1) ◽  
pp. 72-82
Author(s):  
I.K. Argyros ◽  
D. Sharma ◽  
C.I. Argyros ◽  
S.K. Parhi ◽  
S.K. Sunanda ◽  
...  
Keyword(s):  
Convergence Analysis ◽  
Nonlinear Equations ◽  
Complex Polynomial ◽  
Taylor Formula ◽  
Polynomial Equations ◽  
First Derivative ◽  
Derivative Free ◽  
Uniqueness Of The Solution ◽  
Seventh Order ◽  
Ball Convergence

In the earlier work, expensive Taylor formula and conditions on derivatives up to the eighthorder have been utilized to establish the convergence of a derivative free class of seventh orderiterative algorithms. Moreover, no error distances or results on uniqueness of the solution weregiven. In this study, extended ball convergence analysis is derived for this class by imposingconditions on the first derivative. Additionally, we offer error distances and convergence radiustogether with the region of uniqueness for the solution. Therefore, we enlarge the practicalutility of these algorithms. Also, convergence regions of a specific member of this class are displayedfor solving complex polynomial equations. At the end, standard numerical applicationsare provided to illustrate the efficacy of our theoretical findings.

scholarly journals Ball Comparison between Three Sixth Order Methods for Banach Space Valued Operators

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 667
Author(s):  
Ramandeep Behl ◽  
Ioannis K. Argyros ◽  
Jose Antonio Tenreiro Machado
Keyword(s):  
Banach Space ◽  
Sixth Order ◽  
Dimensional Euclidean Space ◽  
Convergence Criteria ◽  
Convergence Domain ◽  
First Derivative ◽  
Uniqueness Of The Solution ◽  
A New Technique ◽  
Derivatives Of ◽  
General Banach Space

Three methods of sixth order convergence are tackled for approximating the solution of an equation defined on the finitely dimensional Euclidean space. This convergence requires the existence of derivatives of, at least, order seven. However, only derivatives of order one are involved in such methods. Moreover, we have no estimates on the error distances, conclusions about the uniqueness of the solution in any domain, and the convergence domain is not sufficiently large. Hence, these methods have limited usage. This paper introduces a new technique on a general Banach space setting based only the first derivative and Lipschitz type conditions that allow the study of the convergence. In addition, we find usable error distances as well as uniqueness of the solution. A comparison between the convergence balls of three methods, not possible to drive with the previous approaches, is also given. The technique is possible to use with methods available in literature improving, consequently, their applicability. Several numerical examples compare these methods and illustrate the convergence criteria.

scholarly journals Unified Local Convergence for Newton’s Method and Uniqueness of the Solution of Equations under Generalized Conditions in a Banach Space

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 463 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Ángel Alberto Magreñán ◽  
Lara Orcos ◽  
Íñigo Sarría
Keyword(s):  
Banach Space ◽  
Newton’S Method ◽  
Newton's Method ◽  
Convergence Region ◽  
Precise Information ◽  
Solution Of Equations ◽  
Lipschitz Constants ◽  
Uniqueness Of The Solution ◽  
Generalized Newton ◽  
Theoretical Results

Under the hypotheses that a function and its Fréchet derivative satisfy some generalized Newton–Mysovskii conditions, precise estimates on the radii of the convergence balls of Newton’s method, and of the uniqueness ball for the solution of the equations, are given for Banach space-valued operators. Some of the existing results are improved with the advantages of larger convergence region, tighter error estimates on the distances involved, and at-least-as-precise information on the location of the solution. These advantages are obtained using the same functions and Lipschitz constants as in earlier studies. Numerical examples are used to test the theoretical results.

scholarly journals Study of Local Convergence and Dynamics of a King-Like Two-Step Method with Applications

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1062
Author(s):  
Ioannis K. Argyros ◽  
Ángel Alberto Magreñán ◽  
Alejandro Moysi ◽  
Íñigo Sarría ◽  
Juan Antonio Sicilia Montalvo
Keyword(s):  
Nonlinear Equations ◽  
Local Convergence ◽  
Step Method ◽  
First Derivative ◽  
Family Behavior ◽  
Local Results

In this paper, we present the local results of the convergence of the two-step King-like method to approximate the solution of nonlinear equations. In this study, we only apply conditions to the first derivative, because we only need this condition to guarantee convergence. As a result, the applicability of the method is expanded. We also use different convergence planes to show family behavior. Finally, the new results are used to solve some applications related to chemistry.

scholarly journals Ball Convergence of an Efficient Eighth Order Iterative Method Under Weak Conditions

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 260 ◽  
Author(s):  
Janak Sharma ◽  
Ioannis Argyros ◽  
Sunil Kumar
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Higher Order ◽  
High Order ◽  
Convergence Order ◽  
Eighth Order ◽  
First Derivative ◽  
Ball Convergence ◽  
Derivatives Of

The convergence order of numerous iterative methods is obtained using derivatives of a higher order, although these derivatives are not involved in the methods. Therefore, these methods cannot be used to solve equations with functions that do not have such high-order derivatives, since their convergence is not guaranteed. The convergence in this paper is shown, relying only on the first derivative. That is how we expand the applicability of some popular methods.

scholarly journals A Fast Alternating Minimization Algorithm for Nonlocal Vectorial Total Variational Multichannel Image Denoising

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Rubing Xi ◽  
Zhengming Wang ◽  
Xia Zhao ◽  
Meihua Xie
Keyword(s):  
Complexity Analysis ◽  
Iterative Algorithms ◽  
Linear Convergence ◽  
Convergence Properties ◽  
Theoretical Derivation ◽  
Variational Models ◽  
Fixed Point Algorithm ◽  
Alternating Minimization Algorithm ◽  
Uniqueness Of The Solution ◽  
Nonlocal Regularization

The variational models with nonlocal regularization offer superior image restoration quality over traditional method. But the processing speed remains a bottleneck due to the calculation quantity brought by the recent iterative algorithms. In this paper, a fast algorithm is proposed to restore the multichannel image in the presence of additive Gaussian noise by minimizing an energy function consisting of anl2-norm fidelity term and a nonlocal vectorial total variational regularization term. This algorithm is based on the variable splitting and penalty techniques in optimization. Following our previous work on the proof of the existence and the uniqueness of the solution of the model, we establish and prove the convergence properties of this algorithm, which are the finite convergence for some variables and theq-linear convergence for the rest. Experiments show that this model has a fabulous texture-preserving property in restoring color images. Both the theoretical derivation of the computation complexity analysis and the experimental results show that the proposed algorithm performs favorably in comparison to the widely used fixed point algorithm.

scholarly journals Accelerated Incremental Gradient Descent using Momentum Acceleration with Scaling Factor

2019 ◽  
Author(s):  
Yuanyuan Liu ◽  
Fanhua Shang ◽  
Licheng Jiao
Keyword(s):  
Gradient Descent ◽  
Linear Convergence ◽  
Strongly Convex ◽  
Convex Problems ◽  
Accelerated Methods ◽  
Gradient Descent Methods ◽  
Gradient Descent Algorithm ◽  
Oracle Complexity ◽  
Incremental Gradient Descent ◽  
Theoretical Results

Recently, research on variance reduced incremental gradient descent methods (e.g., SAGA) has made exciting progress (e.g., linear convergence for strongly convex (SC) problems). However, existing accelerated methods (e.g., point-SAGA) suffer from drawbacks such as inflexibility. In this paper, we design a novel and simple momentum to accelerate the classical SAGA algorithm, and propose a direct accelerated incremental gradient descent algorithm. In particular, our theoretical result shows that our algorithm attains a best known oracle complexity for strongly convex problems and an improved convergence rate for the case of n>=L/\mu. We also give experimental results justifying our theoretical results and showing the effectiveness of our algorithm.

scholarly journals Ball convergence theorem for a Steffensen-type third-order method

2017 ◽  
Vol 51 (1) ◽  
pp. 1-14
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Nonlinear Equation ◽  
Error Bounds ◽  
Order Method ◽  
Third Order ◽  
Numerical Examples ◽  
First Derivative ◽  
Fourth Derivative ◽  
Ball Convergence ◽  
Computable Error Bounds ◽  
Local Convergence Analysis

We present a local convergence analysis for a family of Steffensen-type third-order methods in order to approximate a solution of a nonlinear equation. We use hypothesis up to the first derivative in contrast to earlier studies such as [2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] using hypotheses up to the fourth derivative. This way the applicability of these methods is extended under weaker hypothesis. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.

scholarly journals Parameter-robust Uzawa-type iterative methods for double saddle point problems arising in Biot’s consolidation and multiple-network poroelasticity models

2020 ◽  
Vol 30 (13) ◽  
pp. 2523-2555
Author(s):  
Qingguo Hong ◽  
Johannes Kraus ◽  
Maria Lymbery ◽  
Fadi Philo
Keyword(s):  
Saddle Point ◽  
Linear Convergence ◽  
Iterative Solution ◽  
Displacement Fields ◽  
Type Method ◽  
Pressure Fields ◽  
Fluid Networks ◽  
Multiple Network ◽  
Physical Quantities ◽  
Theoretical Results

This work is concerned with the iterative solution of systems of quasi-static multiple-network poroelasticity equations describing flow in elastic porous media that is permeated by single or multiple fluid networks. Here, the focus is on a three-field formulation of the problem in which the displacement field of the elastic matrix and, additionally, one velocity field and one pressure field for each of the [Formula: see text] fluid networks are the unknown physical quantities. Generalizing Biot’s model of consolidation, which is obtained for [Formula: see text], the MPET equations for [Formula: see text] exhibit a double saddle point structure. The proposed approach is based on a framework of augmenting and splitting this three-by-three block system in such a way that the resulting block Gauss–Seidel preconditioner defines a fully decoupled iterative scheme for the flux-, pressure-, and displacement fields. In this manner, one obtains an augmented Lagrangian Uzawa-type method, the analysis of which is the main contribution of this work. The parameter-robust uniform linear convergence of this fixed-point iteration is proved by showing that its rate of contraction is strictly less than one independent of all physical and discretization parameters. The theoretical results are confirmed by a series of numerical tests that compare the new fully decoupled scheme to the very popular partially decoupled fixed-stress split iterative method, which decouples only flow — the flux and pressure fields remain coupled in this case — from the mechanics problem. We further test the performance of the block-triangular preconditioner defining the new scheme when used to accelerate the generalized minimal residual method (GMRES) algorithm.

COMPARISON THEOREMS FOR CAPUTO–HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS

Fractals ◽  
2019 ◽  
Vol 27 (03) ◽  
pp. 1950036 ◽  
Author(s):  
LI MA
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Fractional Derivatives ◽  
Comparison Theorems ◽  
Hadamard Fractional Differential Equations ◽  
Fractional Differential ◽  
The One ◽  
Lipschitz Conditions ◽  
Theoretical Results

The main purpose of this paper is to investigate the comparison theorems for fractional differential equations involving Caputo–Hadamard fractional derivatives. First, we indicate the continuous dependence on parameters of solutions for Caputo–Hadamard fractional differential equations (C-HFDEs). Then, the first and second comparison theorems for C-HFDEs are proposed and proved, respectively. In addition, we establish the generalized comparisons for C-HFDEs under the one-side Lipschitz conditions. At last, the corresponding examples are also provided to verify the theoretical results.

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