Iteration Functions re-visited

We present an efficient method from Computational geometry, a branch of computer science devoted to the study of algorithms, for mathematical visualization of a third order root solver. For many decades the quality of iterative methods for solving nonlinear equations were analyzed only by using numerical experiments. The disadvantage of this approach is the inconvenient fact that convergence behavior strictly depends on the choice of initial approximations and the structure of functions whose zeros are sought, which often makes the convergence analysis very hard and incomplete. For this reason in this paper we apply dynamic study of iterative processes relied on basins of attraction, a new and powerful methodology developed at the beginning of the 21th century. This approach provides graphic visualization of the behavior of convergent sequences and, consequently, offers considerably better insight into the quality of applied root solvers, especially into the domain of convergence. For demonstration, we present dynamic study of one parameter family of Halley's type introduced in the first part of the paper. Characteristics of this family are discussed by basins of attractions for various values of the involved parameter. Special attention is devoted to clusters of polynomial roots, one of the most difficult problems in the topic. The analysis of the methods and presentation of basins of attractions are performed by the computer algebra system Mathematica.

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“…The expression of H (x) is very lengthy so that it is omitted. Setting x = α in (2), (4) and (5), we obtain…”

Section: A Family Of Third Order One Parameter Root-finding Methodsmentioning

confidence: 99%

“…The request 1) is tightly connected to the request 2); the best initial approximations provide the most accurate approximations to the zeros. Both problems were considered in details in the dissertation [4], the paper [5] and references cited there. For this reason we will not discuss the tasks 1) and 2).…”

Section: Dynamic Study Of the Third Order Familymentioning

confidence: 99%

Computational geometry as a tool for studying root-finding methods

Petković

Rancic

2019

Filomat

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“…This difficult task was considered only in few references, see, e.g. [32], [35], [43], [44]. To provide a better insight into global convergence characteristics of the proposed methods, we have employed an empirical methodology based on computer visualization of the flow of iterative process.…”

Section: Analysis Of Global Convergence Characteristics By Aberth-like Trajectoriesmentioning

confidence: 99%

Computer tools for the construction and analysis of some efficient root-finding simultaneous methods

Petković

Herceg

2020

Filomat

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Using the tools provided by computer algebra system Mathematica, we consider two iterative methods of high efficiency for the simultaneous approximation of simple or multiple (real or complex) zeros of algebraic polynomials. The proposed methods are based on the fourth-order Schr?der-like methods of the first and second kind. We prove that the order of convergence of both basic total-step simultaneous methods is equal to five. Using corrective approximations produced by methods of order two, three and four for finding a single multiple zero, the convergence order is increased from five to six, seven, and eight, respectively. The increased convergence speed is attained with negligible number of additional arithmetic operations, which significantly increases the computational efficiency of the accelerated methods. Convergence properties of the proposed methods are demonstrated by numerical examples and graphics visualization by plotting trajectories of zero approximations. Flows of iterative processes, presented by these trajectories, point to the stability and robustness of the proposed methods.

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On a high-order one-parameter family for the simultaneous determination of polynomial roots

Petković

2017

Applied Mathematics Letters

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Iteration Functions re-visited

Cited by 5 publications

References 13 publications

Computational geometry as a tool for studying root-finding methods

Computational geometry as a tool for studying root-finding methods

Computer tools for the construction and analysis of some efficient root-finding simultaneous methods

On a high-order one-parameter family for the simultaneous determination of polynomial roots

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