Newton's method and complex dynamical systems

DOI10.1007/BF00047501zbMath0671.30023OpenAlexW4248072974MaRDI QIDQ1119771

Heinz-Otto Peitgen, Fritz von Haeseler

Publication date: 1988

Published in: Acta Applicandae Mathematicae (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/bf00047501

zbMATH Keywords

Newton methodMandelbrot setiteration of rational functionspolynomial-like mappings

Mathematics Subject Classification ID

Topological dynamics (37B99) Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable (30D05) Numerical computation of solutions to single equations (65H05) Length, area, volume, other geometric measure theory (28A75)

Related Items

The Julia set of Newton's method for multiple root, Iteration of meromorphic functions, GENERALIZATIONS OF NEWTON'S METHOD, Geometry and Dynamics of Quadratic Rational Maps, FINITENESS OF THE AREA OF BASINS OF ATTRACTION OF RELAXED NEWTON METHOD FOR CERTAIN HOLOMORPHIC FUNCTIONS, Quadratic and cubic Newton maps of rational functions, Global aspects of the continuous and discrete Newton method: A case study, Dynamics and limiting behavior of Julia sets of König's method for multiple roots, CONJECTURES ABOUT SIMPLE DYNAMICS FOR SOME REAL NEWTON MAPS ON ℝ2, JULIA SETS OF GENERALIZED NEWTON'S METHOD, Berkovich dynamics of Newton maps, Attracting cycles for the relaxed Newton's method, On solving systems of autonomous ordinary differential equations by reduction to a variable of an algebra, Contrasts in the basins of attraction of structurally identical iterative root finding methods, Julia sets for the super-Newton method, Cauchy’s method, and Halley’s method, The complex dynamics of Newton's method for a double root, On infinite area for complex exponential function, Convergence and dynamics of structurally identical root finding methods, The complex dynamic of conjugate gradient method, Dynamics of the Newton maps of rational functions, Conjugacies of the Newton maps of rational functions, Newton's method and a class of meromorphic functions without wandering domains

Cites Work

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• On the iteration of a rational function: Computer experiments with Newton's method
• The Plancherel formula for complex algebraic unimodular groups
• Cayley's problem and Julia sets
• Are there critical points on the boundaries of singular domains?
• The Bieberbach conjecture
• Quasiconformal homeomorphisms and dynamics. I: Solution of the Fatou- Julia problem on wandering domains
• On the existence of generally convergent algorithms
• Extraneous fixed points, basin boundaries and chaotic dynamics for Schröder and König rational iteration functions
• Families of rational maps and iterative root-finding algorithms
• The iteration of cubic polynomials. I: The global topology of parameter space
• Discrete versus continuous Newton's method: A case study
• Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. (On smooth conjugacy of diffeomorphisms of the circle with rotations)
• Über die Einzugsbereiche der Nullstellen von Polynomen beim Newton-Verfahren
• Computing the Laurent series of map \(\Psi:C-\overline D\to C-M\)
• Quantitative universality for a class of nonlinear transformations
• Invariant sets under iteration of rational functions
• Über die Iteration der ganzen transzendenten Funktionen, insbesondere von \(\sin z\) und \(\cos z\)
• Iteration of analytic functions
• On the dynamics of rational maps
• Newton's Method and Symbolic Dynamics
• Complex analytic dynamics on the Riemann sphere
• Newton's Method, Circle Maps, and Chaotic Motion
• Some entire functions with multiply-connected wandering domains
• On conformal mapping and iteration of rational functions
• Multiple attractors in Newton's method
• On the dynamics of polynomial-like mappings
• Newton’s Algorithm and Chaotic Dynamical Systems
• On the efficiency of algorithms of analysis
• Julia Sets and Mandelbrot-Like Sets Associated With Higher Order Schroder Rational Iteration Functions: A Computer Assisted Study
• On the quasiconformal surgery of rational functions
• On Sullivan's Proof of the Finiteness Theorem and the Eventual Periodicity Theorem
• On iterates of e^z
• Period Three Implies Chaos
• Exemples de fractions rationnelles ayant une orbite dense sur la sphère de Riemann