Dynamics of Newton-like root finding methods
B. Campos, Jordi Canela, P. Vindel
Abstract
Abstract When exploring the literature, it can be observed that the operator obtained when applying Newton-like root finding algorithms to the quadratic polynomials z 2 − c has the same form regardless of which algorithm has been used. In this paper, we justify why this expression is obtained. This is done by studying the symmetries of the operators obtained after applying Newton-like algorithms to a family of degree d polynomials p ( z ) = z d − c . Moreover, we provide an iterative procedure to obtain the expression of new Newton-like algorithms. We also carry out a dynamical study of the given generic operator and provide general conclusions of this type of methods.
Topics & Concepts
MathematicsOperator (biology)Newton's methodTheory of computationQuadratic equationRoot (linguistics)Homogeneous spaceExpression (computer science)Root-finding algorithmCarry (investment)Dynamics (music)Applied mathematicsDegree (music)Algebra over a fieldAlgorithmPure mathematicsComputer scienceGeometryTranscription factorEconomicsGeneChemistryBiochemistryQuantum mechanicsFinanceNonlinear systemProgramming languagePhilosophyPhysicsAcousticsLinguisticsRepressorIterative Methods for Nonlinear EquationsAdvanced Numerical Analysis TechniquesNumerical Methods and Algorithms