Newton’s Algorithm and Chaotic Dynamical Systems

Hurley, Mike; Martin, Christoph

doi:10.1137/0515020

Cited by 36 publications

(28 citation statements)

References 15 publications

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“…[17,27]), the global convergence behaviour of (1.3) can be very wild and unsurveyable, even in the case of simple mappings F. In order to get a better insight into the global aspects of Newton's iteration method (1.3), it is not, therefore, unreasonable to treat -as a first stepa continuous, desingularized version (such as (1.4)). A key fact is the following observation: multiplying both sides of (1.4) by DF(x(t)) yields, in view of (1.…”

Section: R N Outside the Set C We Have I)f(x) F(x) = Det Df(x) • Dfmentioning

confidence: 99%

The continuous, desingularized Newton method for meromorphic functions

1988

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Abstract. For any (nonconstant) meromorphic function, we present a real analytic dynamical system, which may be interpreted as an infinitesimal version of Newton's method for finding its zeros. A fairly complete description of the local and global features of the phase portrait of such a system is obtained (especially, if the function behaves not too bizarre at infinity). Moreover, in the case of rational functions, structural stability aspects are studied. For a generic class of rational functions, we give a complete graph-theoretical characterization, resp. classification, of these systems. Finally, we present some results on the asymptotic behaviour of meromorphic functions.

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Section: R N Outside the Set C We Have I)f(x) F(x) = Det Df(x) • Dfmentioning

confidence: 99%

The continuous, desingularized Newton method for meromorphic functions

1988

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“…It may be relevant to compare Rayleigh quotient iteration with Newton's method. Newton's method may succeed or fail on a polynomial with real coefficients (see [15,9,10] Finally, one can define an analogous algorithm over C. For an n x n complex-valued matrix B, the Rayleigh quotient iteration formula induces a map of CP"~ . This map has been studied in [ 13], and a related map is studied in [8], Note that if B has real entries, then RP"~ is an invariant submanifold.…”

Section: Further Problemsmentioning

confidence: 99%

Rayleigh quotient iteration for nonsymmetric matrices

Batterson¹,

Smillie²

1990

Math. Comp.

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Abstract.Rayleigh quotient iteration is an iterative algorithm for the calculation of approximate eigenvectors of a matrix. Given a matrix, the algorithm supplies a function whose iteration of an initial vector, vQ , produces a sequence of vectors, vn . If the matrix is symmetric, then for almost any choice of v0 the sequence will converge to an eigenvector at an eventually cubic rate. In this paper we show that there exist open sets of real matrices, each of which possesses an open set of initial vectors for which the algorithm will not converge to an eigenspace. The proof employs techniques from dynamical systems and bifurcation theory.

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“…4 One particular scheme that has been successful in computing unstable periodic orbits is that of Recently, the global behavior of Newton iterations has come under investigation in terms of dynamical systems and chaos [11,12,13]. In this paper, we consider how NM globally changes the structure of the basins of attraction for a map, T, which is bistable; i.e., the map has two stable fixed points, and the basins of attraction are intertwined in a complicated manner.…”

Section: Introductionmentioning

confidence: 99%