高等学校計算数学学報 = Numerical mathematics

Marine Carbon BiogeochemistryBiogeochemical Cycles and ClimateGlobal Biogeochemical Cycles in the Climate SystemOcean Dynamics and the Carbon CycleLive Long and EvolveBiogeochemistry of Marine Dissolved Organic MatterOcean Biogeochemical DynamicsMarine Ecosystems and Global ChangeBiological OceanographyNitrogen in the SeaBiogeochemistry of EstuariesInteractions of C, N, P and S Biogeochemical Cycles and Global ChangeGlobal EnvironmentComputational Science — ICCS 2004The Oceans and ClimateMarine Microbiome and Biogeochemical Cycles in Marine Productive AreasMetal Ions in Biological Systems, Volume 43 Biogeochemical Cycles of ElementsNitrogen in the Marine EnvironmentIntroduction to Marine BiogeochemistryOcean Circulation and ClimatePrimary Productivity and Biogeochemical Cycles in the SeaThe Biogeochemical Cycle of Silicon in the OceanChemical OceanographyTowards a Model of Ocean Biogeochemical ProcessesOcean Dynamics and the Carbon CycleOcean BiogeochemistryParticle Analysis in OceanographyBiogeochemical CyclesBiogeochemical Dynamics at Major River-Coastal InterfacesThe Ocean Carbon Cycle and ClimateSensitivity of global ocean biogeochemical dynamics to ecosystem structure in a future climateModeling Methods for Marine ScienceAn Introduction to the Chemistry of the SeaEncyclopedia of Ocean SciencesCO2 in Seawater: Equilibrium, Kinetics, IsotopesThe Biogeochemical Cycle of Silicon in the OceanKuroshio CurrentThe Mediterranean Sea in the Era of Global Change 1Ocean Biogeochemical DynamicsOcean Mixing

Ocean Biogeochemical Dynamics

Variants of Newton’s Method using fifth-order quadrature formulas☆

Different anomalies in a Jarratt family of iterative root-finding methods☆

A reduced composition technique has been used on Newton and Jarratt’s methods in order to obtain an optimal relation between convergence order, functional evaluations and number of operations. Following this aim, a family of methods is obtained whose efficiency indices are proved to be better for systems of nonlinear equations.

A modified Newton-Jarratt’s composition

In this paper, we present a new third-order family of iterative methods in order to compute the multiple roots of nonlinear equations when the multiplicity (m≥1) is known in advance. There is a plethora of third-order point-to-point methods, available in the literature; but our methods are based on geometric derivation and converge to the required zero even though derivative becomes zero or close to zero in vicinity of the required zero. We use the exponential fitted curve and tangency conditions for the development of our schemes. Well-known Chebyshev, Halley, super-Halley and Chebyshev–Halley are the special members of our schemes for m=1. Complex dynamics techniques allows us to see the relation between the element of the family of iterative schemes and the wideness of the basins of attraction of the simple and multiple roots, on quadratic polynomials. Several applied problems are considered in order to demonstrate the performance of our methods and for comparison with the existing ones. Based on the numerical outcomes, we deduce that our methods illustrate better performance over the earlier methods even though in the case of multiple roots of high multiplicity.

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A New Third-Order Family of Multiple Root-Findings Based on Exponential Fitted Curve

Research interest in iterative multipoint schemes to solve nonlinear problems has increased recently because of the drawbacks of point-to-point methods, which need high-order derivatives to increase the order of convergence. However, this order is not the only key element to classify the iterative schemes. We aim to design new multipoint fixed point classes without memory, that improve or bring together the existing ones in different areas such as computational efficiency, stability and also convergence order. In this manuscript, we present a family of parametric iterative methods, whose order of convergence is four, that has been designed by using composition and weight function techniques. A qualitative analysis is made, based on complex discrete dynamics, to select those elements of the class with best stability properties on low-degree polynomials. This stable behavior is directly related with the simplicity of the fractals defined by the basins of attraction. In the opposite, particular methods with unstable performance present high-complexity in the fractals of their basins. The stable members are demonstrated also be the best ones in terms of numerical performance of non-polynomial functions, with special emphasis on Colebrook-White equation, with wide applications in Engineering.

/pdf/parametric-family-of-root-finding-iterative-methods-fractals-1mpxalg3.pdf

Parametric Family of Root-Finding Iterative Methods: Fractals of the Basins of Attraction

In this work, we use the technique of recurrence relations to prove the semilocal convergence in Banach spaces of the multidimensional extension of Chun’s iterative method. This is an iterative method of fourth order, that can be transferred to the multivariable case by using the divided difference operator. We obtain the domain of existence and uniqueness by taking a suitable starting point and imposing a Lipschitz condition to the first Frechet derivative in the whole domain. Moreover, we apply the theoretical results obtained to a nonlinear integral equation of Hammerstein type, showing the applicability of our results.

/pdf/semilocal-convergence-of-the-extension-of-chun-s-method-3rzeoorqy8.pdf

Semilocal Convergence of the Extension of Chun’s Method

In this work, we design a family of new iterative methods with memory, using some known schemes without memory keeping or increasing its order of convergence. As starting point we use the Ostrowski-Chum bi-parametric family of methods without memory, to design a new bi-parametric family of methods with memory, increasing the original order of convergence without adding new functional evaluations.

Bi-parametric Family of Methods with Memory Based of Ostrowski-Chun Method

/pdf/iterative-methods-for-solving-nonlinear-equations-and-1ygsam6cmf.pdf

Alicia Cordero

Papers

A New Third-Order Family of Multiple Root-Findings Based on Exponential Fitted Curve

Parametric Family of Root-Finding Iterative Methods: Fractals of the Basins of Attraction

Semilocal Convergence of the Extension of Chun’s Method

Bi-parametric Family of Methods with Memory Based of Ostrowski-Chun Method

Iterative Methods for Solving Nonlinear Equations and Systems