scholarly journals Extending the Convergence Domain of Methods of Linear Interpolation for the Solution of Nonlinear Equations

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1093
Author(s):  
Ioannis K. Argyros ◽  
Stepan Shakhno ◽  
Halyna Yarmola
Keyword(s):  
Iterative Methods ◽  
Order Of Convergence ◽  
A Priori ◽  
Semilocal Convergence ◽  
Linear Interpolation ◽  
A Posteriori ◽  
Convergence Domain ◽  
Solving Equations ◽  
Abstract Spaces ◽  
Kurchatov’S Method

Solving equations in abstract spaces is important since many problems from diverse disciplines require it. The solutions of these equations cannot be obtained in a form closed. That difficulty forces us to develop ever improving iterative methods. In this paper we improve the applicability of such methods. Our technique is very general and can be used to expand the applicability of other methods. We use two methods of linear interpolation namely the Secant as well as the Kurchatov method. The investigation of Kurchatov’s method is done under rather strict conditions. In this work, using the majorant principle of Kantorovich and our new idea of the restricted convergence domain, we present an improved semilocal convergence of these methods. We determine the quadratical order of convergence of the Kurchatov method and order 1 + 5 2 for the Secant method. We find improved a priori and a posteriori estimations of the method’s error.

A Traub type result for one-point iterative methods with memory

2014 ◽  
Vol 12 (03) ◽  
pp. 323-340 ◽  
Author(s):  
J. A. Ezquerro ◽  
M. Grau-Sánchez ◽  
M. A. Hernández-Verón ◽  
M. Noguera
Keyword(s):  
Iterative Methods ◽  
Order Of Convergence ◽  
Semilocal Convergence ◽  
Type Result ◽  
Simple Modification ◽  
Kurchatov’S Method ◽  
With Memory

We present an extension of a well-known result of Traub to increase the R-order of convergence of one-point iterative methods by a simple modification of this type of methods. We consider the extension to one-point iterative methods with memory and present a particular case where Kurchatov's method is used. Moreover, we analyze the efficiency and the semilocal convergence of this method. Finally, two applications are presented, where differentiable and nondifferentiable equations are considered, that illustrate the above-mentioned.

scholarly journals Displacement uncertainties with multiview correlation schemes

2020 ◽  
Vol 55 (7-8) ◽  
pp. 199-211
Author(s):  
François Hild ◽  
Stéphane Roux
Keyword(s):  
Finite Element ◽  
A Priori Estimates ◽  
A Priori ◽  
Linear Interpolation ◽  
A Posteriori ◽  
Noise Floor ◽  
Registration Procedure ◽  
Priori Estimates ◽  
Nodal Displacements ◽  
Good Agreement

Multiview correlation is based on the use of explicitly parameterizing surfaces whose deformation is to be measured. In the present analyses, the surfaces are modeled as triangular facets (for local approaches), and with finite-element meshes (for global approaches) using a linear interpolation between nodal displacements. Displacement uncertainty due to the presence of noise in the images is considered. The noise floor levels of the latter are compared for both local and global cases with two different sets of pictures. A priori estimates based on the Hessian of the registration procedure are derived analytically, and a simplified expression is provided that is shown to be in very good agreement with their a posteriori evaluations. The uncertainty difference between local and global approaches is interpreted from their different spatial resolutions.

scholarly journals An Algorithm Derivative-Free to Improve the Steffensen-Type Methods

Symmetry ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 4
Author(s):  
Miguel A. Hernández-Verón ◽  
Sonia Yadav ◽  
Ángel Alberto Magreñán ◽  
Eulalia Martínez ◽  
Sukhjit Singh
Keyword(s):  
Iterative Methods ◽  
Iterative Process ◽  
Quadratic Convergence ◽  
Semilocal Convergence ◽  
Divided Differences ◽  
Derivative Free ◽  
Solving Equations ◽  
Predictor Corrector ◽  
Steffensen Method ◽  
Theoretical Results

Solving equations of the form H(x)=0 is one of the most faced problem in mathematics and in other science fields such as chemistry or physics. This kind of equations cannot be solved without the use of iterative methods. The Steffensen-type methods, defined using divided differences are derivative free, are usually considered to solve these problems when H is a non-differentiable operator due to its accuracy and efficiency. However, in general, the accessibility of these iterative methods is small. The main interest of this paper is to improve the accessibility of Steffensen-type methods, this is the set of starting points that converge to the roots applying those methods. So, by means of using a predictor–corrector iterative process we can improve this accessibility. For this, we use a predictor iterative process, using symmetric divided differences, with good accessibility and then, as corrector method, we consider the Center-Steffensen method with quadratic convergence. In addition, the dynamical studies presented show, in an experimental way, that this iterative process also improves the region of accessibility of Steffensen-type methods. Moreover, we analyze the semilocal convergence of the predictor–corrector iterative process proposed in two cases: when H is differentiable and H is non-differentiable. Summing up, we present an effective alternative for Newton’s method to non-differentiable operators, where this method cannot be applied. The theoretical results are illustrated with numerical experiments.

A priori/a posteriori

2017 ◽  
Author(s):  
Heinrich Schepers ◽  
Giorgio Tonelli ◽  
Rudolf Eisler
Keyword(s):  
A Priori ◽  
A Posteriori

A Critique of Modernist Synthesis in Islamic Thought

1994 ◽  
Vol 11 (4) ◽  
pp. 475-503
Author(s):  
Masudul Alum Choudhury
Keyword(s):  
Scientific Inquiry ◽  
A Priori ◽  
Scientific Investigation ◽  
Muslim World ◽  
A Posteriori ◽  
Islamic Thought ◽  
And Behavior ◽  
Muslim Scholars

Is it the realm of theoretical constructs or positive applications thatdefines the essence of scientific inquiry? Is there unison between thenormative and the positive, between the inductive and deductivecontents, between perception and reality, between the micro- andmacro-phenomena of reality as technically understood? In short, isthere a possibility for unification of knowledge in modernist epistemologicalcomprehension? Is knowledge perceived in conceptionand application as systemic dichotomy between the purely epistemic(in the metaphysically a priori sense) and the purely ontic (in thepurely positivistically a posteriori sense) at all a reflection of reality?Is knowledge possible in such a dichotomy or plurality?Answers to these foundational questions are primal in order tounderstand a critique of modernist synthesis in Islamic thought thathas been raging among Muslim scholars for some time now. Theconsequences emanating from the modernist approach underlie muchof the nature of development in methodology, thinking, institutions,and behavior in the Muslim world throughout its history. They arefound to pervade more intensively, I will argue here, as the consequenceof a taqlid of modernism among Islamic thinkers. I will thenargue that this debility has arisen not because of a comparativemodem scientific investigation, but due to a failure to fathom theuniqueness of a truly Qur'anic epistemological inquiry in the understandingof the nature of the Islamic socioscientific worldview ...

Qualifier la qualité des données localisées numériques des réseaux d’eau et d’assainissement

2019 ◽  
Vol 11 ◽  
pp. 51-64
Author(s):  
M. LE MOAL
Keyword(s):  
A Priori ◽  
A Posteriori ◽  
Systèmes D’Information Géographique ◽  
Systèmes D’Information

Les systèmes d’information géographique (SIG) sont devenus incontournables dans la gestion des réseaux d’eau et d’assainissement et leur efficacité repose en très grande partie sur la qualité des données exploitées. Parallèlement, les évolutions réglementaires et les pratiques des utilisateurs augmentant notamment les échanges d’informations renforcent le rôle central des données et de leur qualité. Si la plupart des solutions SIG du marché disposent de fonctions dédiées à la qualification de la qualité des données, elles procèdent de la traduction préalable de spécifications des données en règles informatiques avant de procéder aux tests qualitatifs. Cette approche chronophage requiert des compétences métier. Pour éviter ces contraintes, Axes Conseil a élaboré un procédé de contrôle des données SIG rapide et accessible à des acteurs métier de l’eau et de l’assainissement. Plutôt qu’une lourde approche de modélisation a priori, le principe est de générer un ensemble d’indicateurs explicites facilement exploitables a posteriori par les acteurs du métier. Cette approche offre une grande souplesse d’analyse et ne nécessite pas de compétences informatiques avancées.

Kant’s ‘Transcendental Deduction’

2018 ◽  
Author(s):  
Barry Stroud
Keyword(s):  
A Priori ◽  
Human Mind ◽  
Critical Assessment ◽  
Structural Description ◽  
A Posteriori ◽  
Human Knowledge ◽  
Transcendental Deduction ◽  
The World ◽  
Empirical Realism

This chapter presents a straightforward structural description of Immanuel Kant’s conception of what the transcendental deduction is supposed to do, and how it is supposed to do it. The ‘deduction’ Kant thinks is needed for understanding the human mind would establish and explain our ‘right’ or ‘entitlement’ to something we seem to possess and employ in ‘the highly complicated web of human knowledge’. This is: experience, concepts, and principles. The chapter explains the point and strategy of the ‘deduction’ as Kant understands it, as well as the demanding conditions of its success, without entering into complexities of interpretation or critical assessment of the degree of success actually achieved. It also analyses Kant’s arguments regarding a priori concepts as well as a posteriori knowledge of the world around us, along with his claim that our position in the world must be understood as ‘empirical realism’.

Positive Psychology’s A Priori Problem

2017 ◽  
Vol 58 (3) ◽  
pp. 313-342 ◽  
Author(s):  
Barbara S. Held
Keyword(s):  
Conceptual Framework ◽  
A Priori ◽  
A Posteriori ◽  
Negative Numbers ◽  
Psychological States ◽  
Psychological Reality ◽  
Negative Results ◽  
Optimal Functioning ◽  
Conceptual Muddles ◽  
Negative Phenomena

The positive/negative distinction works well in many fields—for example, in mathematics negative numbers hold their own, and in medical pathology negative results are usually celebrated. But in positive psychology negativity should be replaced with positivity for flourishing/optimal functioning to occur. That the designation of the psychological states and processes deemed positive (good/desirable) and negative (bad/undesirable) is made a priori, independent of circumstantial particularity, both intrapersonal and interpersonal, does not seem to bother positive psychologists. But it should, as it results in conceptual muddles and dead ends that cannot be solved within their conceptual framework of positivity and negativity. Especially problematic is an ambiguity I find in positive psychologists’ a priori and a posteriori understandings of positivity and negativity, an ambiguity about constitutive and causal relations that pervades their science and the conclusions drawn from it. By eliminating their a priori dichotomy of positivity and negativity, positive psychologists might well find themselves in a better position to put back together the psychological reality that they have fractured in their ontologically dubious move of carving up psychological reality a priori into positive and negative phenomena. They then might find themselves better placed to “broaden and build” their own science of flourishing.

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