Family of simultaneous methods of Hansen–Patrick's type

Petković, Marko D.; Sakurai, Tetsuya; Rančić, L.

doi:10.1016/j.apnum.2004.01.004

Cited by 8 publications

(19 citation statements)

References 22 publications

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“…A(−h) means A × 10 −h . of the family, see [31]. From tables 1 and 2 we observe that even the methods from the family stated for α = 2 and α = −3 produce worse results compared with the methods stated for |α| 1.…”

Section: Numerical Examplesmentioning

confidence: 75%

“…Such a family can be usefully applied for finding zeros of a wide class of analytic functions, see the recent paper [31]. However, the family (2.6) cannot be implemented to analytic functions since then the calculation of some functions involved in the fixed point relation (2.4) uses an approximate numerical integration which blight the inclusion property, necessary for the construction of any interval method.…”

Section: Remarkmentioning

confidence: 98%

“…We hope to address the problem concerning clusters in future works. Since the study of clusters is very actual at present time, we refer the papers [2,11,16,19,21,23,31,35] for more details.…”

Section: Numerical Examplesmentioning

confidence: 99%

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A higher order family for the simultaneous inclusion of multiple zeros of polynomials

Petković

Milošević

2005

Numer Algor

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Starting from a suitable fixed point relation, a new family of iterative methods for the simultaneous inclusion of multiple complex zeros in circular complex arithmetic is constructed. The order of convergence of the basic family is four. Using Newton's and Halley's corrections, we obtain families with improved convergence. Faster convergence of accelerated methods is attained with only few additional numerical operations, which provides a high computational efficiency of these methods. Convergence analysis of the presented methods and numerical results are given.

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Section: Numerical Examplesmentioning

confidence: 75%

Section: Remarkmentioning

confidence: 98%

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A higher order family for the simultaneous inclusion of multiple zeros of polynomials

Petković

Milošević

2005

Numer Algor

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“…The rational function W * i (z) was applied in [26,30] for constructing two simultaneous methods of the fourth order for finding simple zeros of polynomials.…”

Section: Family Of Iterative Methods For Simple Zerosmentioning

confidence: 99%

“…The presented family is more general and contains all those methods as special cases. Other families of simultaneous methods were recently constructed in [26] (of the order four) and [24] (of the order four, five and six). The proposed family contains the first of them as a special case, and it is more efficient than the second one.…”

Section: Introductionmentioning

confidence: 99%

An efficient higher order family of root finders

Petković¹,

Rančić²,

Petković³

2008

Journal of Computational and Applied Mathematics

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A one parameter family of iterative methods for the simultaneous approximation of simple complex zeros of a polynomial, based on a cubically convergent Hansen-Patrick's family, is studied. We show that the convergence of the basic family of the fourth order can be increased to five and six using Newton's and Halley's corrections, respectively. Since these corrections use the already calculated values, the computational efficiency of the accelerated methods is significantly increased. Further acceleration is achieved by applying the Gauss-Seidel approach (single-step mode). One of the most important problems in solving nonlinear equations, the construction of initial conditions which provide both the guaranteed and fast convergence, is considered for the proposed accelerated family. These conditions are computationally verifiable; they depend only on the polynomial coefficients, its degree and initial approximations, which is of practical importance. Some modifications of the considered family, providing the computation of multiple zeros of polynomials and simple zeros of a wide class of analytic functions, are also studied. Numerical examples demonstrate the convergence properties of the presented family of root-finding methods.

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Polynomials orthogonal on the semicircle

Reichel

2013

Walter Gautschi, Volume 2

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Family of simultaneous methods of Hansen–Patrick's type

Cited by 8 publications

References 22 publications

A higher order family for the simultaneous inclusion of multiple zeros of polynomials

A higher order family for the simultaneous inclusion of multiple zeros of polynomials

An efficient higher order family of root finders

Polynomials orthogonal on the semicircle

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