An optimal family of eighth-order simple-root finders with weight functions dependent on function-to-function ratios and their dynamics underlying extraneous fixed points

A generic family of optimal sixteenth-order multiple-root finders are theoretically developed from general settings of weight functions under the known multiplicity. Special cases of rational weight functions are considered and relevant coefficient relations are derived in such a way that all the extraneous fixed points are purely imaginary. A number of schemes are constructed based on the selection of desired free parameters among the coefficient relations. Numerical and dynamical aspects on the convergence of such schemes are explored with tabulated computational results and illustrated attractor basins. Overall conclusion is drawn along with future work on a different family of optimal root-finders.

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Asymptotics of the spectrum of even-order differential operators with discontinuos weight functions

Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva ◽

10.15507/2079-6900.22.202001.48-70 ◽

2020 ◽

Vol 22 (1) ◽

pp. 48-70

Author(s):

Sergey I. Mitrokhin

Keyword(s):

Asymptotic Behavior ◽

Boundary Value Problem ◽

Boundary Conditions ◽

Differential Operators ◽

Boundary Value ◽

Weight Functions ◽

Order Differential Operator ◽

Eighth Order ◽

Entire Family ◽

Indicator Diagram

The boundary-value problem for an eighth-order differential operator whose potential is a piecewise continuous function on the segment of the operator definition is studied. The weight function is piecewise constant. At the discontinuity points of the operator coefficients, the conditions of "conjugation" must be satislied which follow from physical considerations. The boundary conditions of the studied boundary value problem are separated and depend on several parameters. Thus, we simultaneously study the spectral properties of entire family of differential operators with discontinuous coefficients. The asymptotic behavior of the solutions of differential equations defining the operator is obtained for large values of the spectral parameter. Using these asymptotic expansions, the conditions of "conjugation" are investigated; as a result, the boundary conditions are studied. The equation on eigenvalues of the investigated boundary value problem is obtained. It is shown that the eigenvalues are the roots of some entire function. The indicator diagram of the eigenvalue equation is investigated. The asymptotic behavior of the eigenvalues in various sectors of the indicator diagram is found.

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A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points

Applied Mathematics and Computation ◽

10.1016/j.amc.2016.02.029 ◽

2016 ◽

Vol 283 ◽

pp. 120-140 ◽

Cited By ~ 20

Author(s):

Young Hee Geum ◽

Young Ik Kim ◽

Beny Neta

Keyword(s):

Fixed Points ◽

Multiple Root ◽

Sixth Order ◽

Extraneous Fixed Points

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HOW IS THE DYNAMICS OF KÖNIG ITERATION FUNCTIONS AFFECTED BY THEIR ADDITIONAL FIXED POINTS?

Fractals ◽

10.1142/s0218348x99000323 ◽

1999 ◽

Vol 07 (03) ◽

pp. 327-334 ◽

Cited By ~ 6

Author(s):

V. DRAKOPOULOS

Keyword(s):

Fixed Points ◽

Rational Functions ◽

Parameter Family ◽

Quadratic Polynomials ◽

Iteration Functions ◽

Extraneous Fixed Points ◽

Roots Of Equations ◽

The One ◽

Cubic Polynomials ◽

Raphson Method

König iteration functions are a generalization of Newton–Raphson method to determine roots of equations. These higher-degree rational functions possess additional fixed points, which are generally different from the desired roots. We first prove two new results: firstly, about these extraneous fixed points and, secondly, about the Julia sets of the König functions associated with the one-parameter family of quadratic polynomials. Then, after finding all the critical points of the König functions as applied to a one-parameter family of cubic polynomials, we examine the orbits of the ones available for convergence to an attracting periodic cycle, should such a cycle exist.

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