Some modifications of King's family with optimal eighth order of convergence

F. Soleymani; S. Karimi Vanani; M. Khan; M. Sharifi

doi:10.1016/j.mcm.2011.10.016

Some modifications of King's family with optimal eighth order of convergence

F. Soleymani
, S. Karimi Vanani
, M. Khan
, M. Sharifi

Research output: Contribution to journal › Article › peer-review

33 Scopus citations

Abstract

Kung and Traub (1974)[4] conjectured that an iteration method without memory based on n+1 evaluations could achieve optimal convergence order 2 ⁿ. Hence, based on this conjecture, we derive two optimal three-step eighth-order classes of methods in which there are only four evaluations per full cycle. Analytical proofs of the presented derivative-involved classes are provided. Finally, a number of numerical examples are also proposed to illustrate the accuracy of the contributed methods by comparing with the new existing optimal eighth-order methods without memory in the literature.

Original language	English
Pages (from-to)	1373-1380
Number of pages	8
Journal	Mathematical and Computer Modelling
Volume	55
Issue number	3-4
DOIs	https://doi.org/10.1016/j.mcm.2011.10.016
State	Published - Feb 2012
Externally published	Yes

Keywords

Eighth-order convergence
Iterative methods
Kung-Traub conjecture
Multi-point iterations
Simple root

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10.1016/j.mcm.2011.10.016

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abstract = "Kung and Traub (1974)[4] conjectured that an iteration method without memory based on n+1 evaluations could achieve optimal convergence order 2 n. Hence, based on this conjecture, we derive two optimal three-step eighth-order classes of methods in which there are only four evaluations per full cycle. Analytical proofs of the presented derivative-involved classes are provided. Finally, a number of numerical examples are also proposed to illustrate the accuracy of the contributed methods by comparing with the new existing optimal eighth-order methods without memory in the literature.",

keywords = "Eighth-order convergence, Iterative methods, Kung-Traub conjecture, Multi-point iterations, Simple root",

author = "F. Soleymani and Vanani, \{S. Karimi\} and M. Khan and M. Sharifi",

year = "2012",

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T1 - Some modifications of King's family with optimal eighth order of convergence

AU - Soleymani, F.

AU - Vanani, S. Karimi

AU - Khan, M.

AU - Sharifi, M.

PY - 2012/2

Y1 - 2012/2

N2 - Kung and Traub (1974)[4] conjectured that an iteration method without memory based on n+1 evaluations could achieve optimal convergence order 2 n. Hence, based on this conjecture, we derive two optimal three-step eighth-order classes of methods in which there are only four evaluations per full cycle. Analytical proofs of the presented derivative-involved classes are provided. Finally, a number of numerical examples are also proposed to illustrate the accuracy of the contributed methods by comparing with the new existing optimal eighth-order methods without memory in the literature.

AB - Kung and Traub (1974)[4] conjectured that an iteration method without memory based on n+1 evaluations could achieve optimal convergence order 2 n. Hence, based on this conjecture, we derive two optimal three-step eighth-order classes of methods in which there are only four evaluations per full cycle. Analytical proofs of the presented derivative-involved classes are provided. Finally, a number of numerical examples are also proposed to illustrate the accuracy of the contributed methods by comparing with the new existing optimal eighth-order methods without memory in the literature.

KW - Eighth-order convergence

KW - Iterative methods

KW - Kung-Traub conjecture

KW - Multi-point iterations

KW - Simple root

UR - https://www.scopus.com/pages/publications/82255166574

U2 - 10.1016/j.mcm.2011.10.016

DO - 10.1016/j.mcm.2011.10.016

M3 - Article

AN - SCOPUS:82255166574

SN - 0895-7177

VL - 55

SP - 1373

EP - 1380

JO - Mathematical and Computer Modelling

JF - Mathematical and Computer Modelling

IS - 3-4

ER -

Some modifications of King's family with optimal eighth order of convergence

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Keywords

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