A new fourth-order family of simultaneous methods for finding polynomial zeros

“…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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“…In numerical analysis, research on iterative approaches in finding the roots of different types of equations is highly relevant. It is mostly conducted and applied in the field of computer science [7,9,[18][19][20][21][22][23][24][25][26][27][28]. Thus, most of the iterative approaches developed have been meant for encoding into computer algorithms, and not for solving by hand.…”

Section: Introductionmentioning

confidence: 99%

“…Other studies focused on imaginary roots but the iteration formulas involved are too complex or impractical to use when calculations are to be done by hand since computations are tedious when imaginary numbers are being used in the iteration process. Though it has been established in previous studies [3,4,7,8,13,20,26] that determining a real root becomes a prior step in finding non-real roots, there has yet to be an iterative approach that assumes to find the non-real roots without the aid of information coming from the estimated real roots. This is particularly true for cubic polynomials.…”

Section: Introductionmentioning

confidence: 99%

Iteration functions for approximating complex roots of cubic polynomials

Yambao¹,

Decena²

2017

actamanil

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Iterative methods provide an alternative to finding the solutions of equations where analytical methods are inconvenient or even impossible to use. This study which focuses on cubic polynomial equation f (t) = at 3 + bt 2 + ct + d = 0, a > 0 with real coefficients and having an imaginary root, found that the fixed-point iterationwill always converge to the real part x of the imaginary root of f (t) = 0 whenever b 2 -3ac < 0. The only real root of g(t) = ½ f (t) f (t) -af (t) = 0 was found to be the real part x of the imaginary root of f (t) = 0 and is always outside the interval formed by the critical numbers of the function f.

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A new fourth-order family of simultaneous methods for finding polynomial zeros

Cited by 5 publications

References 12 publications

A family of root-finding methods with accelerated convergence

A family of root-finding methods with accelerated convergence

An efficient higher order family of root finders

Iteration functions for approximating complex roots of cubic polynomials

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