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

A biparametric family of derivative-free optimal iterative methods of order four, for solving nonlinear equations, is presented. From the error equation of this class, different families of iterative schemes with memory can be designed increasing the order of convergence up to six. The real stability analysis of the biparametric family without memory is made on quadratic polynomials, finding areas in the parametric plane with good performance. Moreover, in order to study the real behavior of the parametric class with memory, we associate it with a discrete multidimensional dynamical system. By analyzing the fixed and critical points of its vectorial rational function, we can select those methods with best stability properties.

/pdf/king-type-derivative-free-iterative-families-real-and-memory-3qx2bow7h4.pdf

King-Type Derivative-Free Iterative Families: Real and Memory Dynamics

In this paper, we present a new parametric family of three-step iterative for solving nonlinear equations. First, we design a fourth-order triparametric family that, by holding only one of its parameters, we get to accelerate its convergence and finally obtain a sixth-order uniparametric family. With this last family, we study its convergence, its complex dynamics (stability), and its numerical behavior. The parameter spaces and dynamical planes are presented showing the complexity of the family. From the parameter spaces, we have been able to determine different members of the family that have bad convergence properties, as attracting periodic orbits and attracting strange fixed points appear in their dynamical planes. Moreover, this same study has allowed us to detect family members with especially stable behavior and suitable for solving practical problems. Several numerical tests are performed to illustrate the efficiency and stability of the presented family.

https://www.mdpi.com/1999-4893/14/4/101/pdf

Chaos and Stability in a New Iterative Family for Solving Nonlinear Equations

In this paper, a two-step class of fourth-order iterative methods for solving systems of nonlinear equations is presented. We further extend the two-step class to establish a new sixth-order family which requires only one additional functional evaluation. The convergence analysis of the proposed classes is provided under several mild conditions. A complete dynamical analysis is made, by using real multidimensional discrete dynamics, in order to select the most stable elements of both families of fourth- and sixth-order of convergence. To get this aim, a novel tool based on the existence of critical points has been used, the parameter line. The analytical discussion of the work is upheld by performing numerical experiments on some application-oriented problems. We provide an implementation of the proposed scheme on nonlinear optimization problem and zero-residual nonlinear least-squares problems taken from the constrained and unconstrained testing environment test set. Finally, based on numerical results, it has been concluded that our methods are comparable with the existing ones of similar nature in terms of order, efficiency, and computational time and also that the stability results provide the most efficient member of each class of iterative schemes.

New fourth- and sixth-order classes of iterative methods for solving systems of nonlinear equations and their stability analysis

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.

/pdf/iterative-methods-with-memory-for-solving-systems-of-1kzkxkgfxf.pdf

Iterative methods with memory for solving systems of nonlinear equations using a second order approximation

The parameter space associated to the parametric family of Chebyshev-Halley on quadratic polynomials shows a dynamical richness worthy of study. This analysis has been initiated by the authors in previous works. Every value of the parameter belonging to the same connected component of the parameter space gives rise to similar dynamical behavior. In this paper, we focus on the search of regions in the parameter space that gives rise to the appearance of attractive orbits of period two.

/pdf/bulbs-of-period-two-in-the-family-of-chebyshev-halley-18fg959olr.pdf

Alicia Cordero

Papers

King-Type Derivative-Free Iterative Families: Real and Memory Dynamics

Chaos and Stability in a New Iterative Family for Solving Nonlinear Equations

New fourth- and sixth-order classes of iterative methods for solving systems of nonlinear equations and their stability analysis

Iterative methods with memory for solving systems of nonlinear equations using a second order approximation

Bulbs of Period Two in the Family of Chebyshev-Halley Iterative Methods on Quadratic Polynomials